Which Question Is Your Attention Metric Answering? Attention Rows as Compositional Data

# Which Question Is Your Attention Metric Answering?Attention Rows as Compositional Data
Source: [https://arxiv.org/html/2608.14712](https://arxiv.org/html/2608.14712)
## Which Question Is Your Attention Metric Answering? Attention Rows as Compositional DataThanks:Department of Statistics and Data Science, The Wharton School, University of Pennsylvania, 3733 Spruce Street, Philadelphia, PA 19104\-6340,ruanej@wharton\.upenn\.edu

Marios PapamichalisThanks:Human Nature Lab, Yale University, New Haven, CT 06511,marios\.papamichalis@yale\.eduAffiliation:Human Nature Lab, Yale UniversityAffiliation:andAffiliation:Regina RuaneAffiliation:Department of Statistics and Data Science, The Wharton School,Affiliation:University of Pennsylvania

###### Abstract

Each row of a transformer’s attention matrix is a probability distribution over tokens, and in trained models most of that probability lands on a single*sink*token, usually the first\. Standard tools for comparing attention rows \(cosine similarity, Jensen–Shannon divergence, Shannon entropy\) therefore hinge on a choice papers rarely report: keep the sink, or drop it and renormalize\. This choice can reverse conclusions\. On ten pretrained models from five families, 17–47% of verdicts about which of two heads is more similar flip with the convention, and the most prominent structure in a standard BERT head\-clustering pipeline is an artifact of it\. The reason is that one\-number summaries mix two questions: how much attention the sink takes, and how the rest is divided among the content tokens\. Treating rows as compositional data separates them exactly: the Aitchison distance splits orthogonally into a sink term and a content term, entropy splits by an exact identity, and the content distance is characterized by invariances the transformer itself possesses\. The separation matters in practice: most measured entropy collapse during training is the sink growing, not attention sharpening \(30% of the drop at 70M parameters, 95% at 1B, 79% at 1\.4B\), and pruning heads with the wrong channel can inflate perplexity more than a hundredfold\. We map where each convention is safe, test a frozen out\-of\-sample predictor \(one confirmation, one abstention, one failure\), and release code regenerating every number\.

## 1Introduction

Figure 1:The reversal in one construction \(Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)\)\.Three attention rows over\{sink,a,b\}\\\{\\text\{sink\},a,b\\\}:p=\(0\.90,0\.08,0\.02\)p=\(0\.90,0\.08,0\.02\),q=\(0\.70,0\.24,0\.06\)q=\(0\.70,0\.24,0\.06\),r=\(0\.90,0\.02,0\.08\)r=\(0\.90,0\.02,0\.08\)\. \(a\) With the sink column,ppandrrshare the same sink mass whileqqdiffers\. \(b\) Dropping the sink and renormalizing, a routine, usually undocumented preprocessing choice, makesp′p^\{\\prime\}andq′q^\{\\prime\}*identical*\(both allocate content4:14\{:\}1toa:ba\{:\}b\) whiler′r^\{\\prime\}mirrors them\. \(c\) Cosine, JS, and Euclidean distance all sayrris the closer head with the sink andqqwithout it: the scientific conclusion is an artifact of the convention\. The Aitchison distance returns the same verdict in both views, and its content partdA⟂d\_\{A\}^\{\\perp\}is numerically identical\.Softmax attention produces, for every query position, a probability distribution over keys, a point on the simplex\. This is precisely a*composition*in the sense of[1](https://arxiv.org/html/2608.14712#bib.bib1): a vector of positive parts carrying only relative information\. The interpretability and training\-stability literatures routinely compare and summarize these rows\.[4](https://arxiv.org/html/2608.14712#bib.bib10)cluster BERT heads by the JS divergence between attention rows; cosine similarity between attention maps is pervasive;[26](https://arxiv.org/html/2608.14712#bib.bib14)track the Shannon entropy of attention rows to diagnose*entropy collapse*; head\-role taxonomies and pruning scores\([23](https://arxiv.org/html/2608.14712#bib.bib13);[16](https://arxiv.org/html/2608.14712#bib.bib12);[18](https://arxiv.org/html/2608.14712#bib.bib21)\)summarize populations of attention rows with Euclidean averages\. These statistics are valid for their intended targets, but full\-row comparisons need not preserve conclusions after a coordinate is removed and the row is renormalized, and Shannon entropy combines marginal sink mass with conditional content entropy\. Compositional data analysis \(CoDA\) identifies the properties at stake: perturbation invariance and subcompositional coherence\([1](https://arxiv.org/html/2608.14712#bib.bib1);[2](https://arxiv.org/html/2608.14712#bib.bib2);[7](https://arxiv.org/html/2608.14712#bib.bib3)\)\.

CoDA critiques of Euclidean statistics date back four decades\. What makes them consequential here is a second, recent empirical fact: attention mass concentrates overwhelmingly on a few*sink*coordinates, chiefly the first or⟨bos⟩\\langle\\mathrm\{bos\}\\rangletoken\([24](https://arxiv.org/html/2608.14712#bib.bib16);[22](https://arxiv.org/html/2608.14712#bib.bib18);[10](https://arxiv.org/html/2608.14712#bib.bib19)\)\. In a typical Llama\-3\.1\-405B prompt roughly80%80\\%of the attention mass sits on the⟨bos⟩\\langle\\mathrm\{bos\}\\rangletoken\([3](https://arxiv.org/html/2608.14712#bib.bib20)\), and the fraction of sink heads grows with model scale\([10](https://arxiv.org/html/2608.14712#bib.bib19)\)\. A sink is exactly the kind of dominant shared part whose inclusion or exclusion CoDA warns changes Euclidean and divergence\-based comparisons\. Whether an analysis keeps the sink column, or masks it and renormalizes, is today an undocumented analyst choice, and this choice can determine the answer: two heads that divide their non\-sink attention identically can be scored as distant, and a head with the opposite content preference ranked their nearest neighbor, purely because all three share a large sink \(Figure[1](https://arxiv.org/html/2608.14712#S1.F1)\)\.

Two estimands\.Full\-row statistics answer questions about total allocation, sink included; sink\-dropped statistics answer questions about allocation conditional on ignoring the sink\. Both are legitimate; the failure mode is mixing them under an unstated convention, which is what one\-number summaries do when a shared sink holds most of the mass, and these summaries feed head taxonomies, pruning decisions, and training\-stability monitors\. Our claim is not that the classical metrics are wrong but that they answer a convention\-dependent question; we separate the channels exactly, characterize the content channel’s canonical metric from the transformer’s own symmetries, and measure what the separation changes\. Concurrent 2026 work arrives at the same geometry for different questions \(§[2](https://arxiv.org/html/2608.14712#S2)\); the measurement problem, the sink/content separation, and the convention audit are, to our knowledge, new\.

Contributions\.

- •Diagnosis\.We identify the keep\-versus\-drop sink convention as an unreported analyst choice and prove it reverses cosine, JS, and Euclidean closer\-head verdicts on a positive\-measure set \(Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)\)\.
- •Toolkit\.An exact orthogonal split of the Aitchison distance into sink and content terms, an exact entropy identityH=Hb​\(s\)\+\(1−s\)​H​\(π\)H=H\_\{b\}\(s\)\+\(1\-s\)H\(\\pi\), invariance statements for the transformer’s own nuisance transformations, and a five\-axiom characterization with a quotient version for the content distance \(Lemmas[1](https://arxiv.org/html/2608.14712#Thmlemma1),[2](https://arxiv.org/html/2608.14712#Thmlemma2); Prop\.[2](https://arxiv.org/html/2608.14712#Thmproposition2); Thm\.[2](https://arxiv.org/html/2608.14712#Thmtheorem2)\)\.
- •Failure geometry\.In the sink\-dominated regime all classical pairwise dissimilarities collapse into vanishing bands while the content diagnostics are unaffected \(Theorem[3](https://arxiv.org/html/2608.14712#Thmtheorem3); Prop\.[3](https://arxiv.org/html/2608.14712#Thmproposition3)\)\.
- •Measurement\.1717–47%47\\%of closer\-head verdicts flip on ten pretrained models, tracking curves computed before any model was run; prospectively specified re\-analyses on taxonomies \(including Clark et al\.’s pipeline on its full support\), scale, and training checkpoints are scored against their predictions \(§[3](https://arxiv.org/html/2608.14712#S5.F3)–[5\.3](https://arxiv.org/html/2608.14712#S5.SS3)\)\.
- •Function\.As pruning criteria every metric fails somewhere; we map the regimes across ten models, show each criterion is stable along a different axis, and test a frozen regime predictor out of sample, reporting one confirmation, one abstention, and one failure \(§[3](https://arxiv.org/html/2608.14712#S5.F3)\)\.

## 2Related work

Attention\-analysis metrics\.[4](https://arxiv.org/html/2608.14712#bib.bib10)compute head–head distances as summed JS divergences between attention rows and cluster the result;[13](https://arxiv.org/html/2608.14712#bib.bib11)taxonomize attention patterns and study redundancy; cosine similarity between flattened attention maps is a default in many analyses\.Entropy collapse and stability\.[26](https://arxiv.org/html/2608.14712#bib.bib14)define attention entropy row\-wise and tie pathologically low entropy to training instability;[6](https://arxiv.org/html/2608.14712#bib.bib15)study rank collapse of pure attention\.Attention sinks and registers\.StreamingLLM identifies the first\-token sink and its role in windowed inference\([24](https://arxiv.org/html/2608.14712#bib.bib16)\);[5](https://arxiv.org/html/2608.14712#bib.bib17)find high\-norm register tokens in ViTs;[22](https://arxiv.org/html/2608.14712#bib.bib18)trace sinks to massive activations acting as implicit attention biases;[10](https://arxiv.org/html/2608.14712#bib.bib19)measure how the fraction of sink heads scales;[3](https://arxiv.org/html/2608.14712#bib.bib20)explain why models attend to the first token and quantify the mass involved\.Head roles and pruning\.[23](https://arxiv.org/html/2608.14712#bib.bib13)taxonomize heads and prune withL0L\_\{0\}gates;[16](https://arxiv.org/html/2608.14712#bib.bib12)prune most heads at test time;[18](https://arxiv.org/html/2608.14712#bib.bib21)characterize induction heads\.Compositional data analysis\.The log\-ratio approach originates with[1](https://arxiv.org/html/2608.14712#bib.bib1);[2](https://arxiv.org/html/2608.14712#bib.bib2); ILR coordinates and balances come from[7](https://arxiv.org/html/2608.14712#bib.bib3);[20](https://arxiv.org/html/2608.14712#bib.bib4)is the standard treatment; zero handling follows[15](https://arxiv.org/html/2608.14712#bib.bib5);[9](https://arxiv.org/html/2608.14712#bib.bib7);[8](https://arxiv.org/html/2608.14712#bib.bib8)quantify and reappraise subcompositional incoherence\.[21](https://arxiv.org/html/2608.14712#bib.bib22)bring log\-contrasts to deep models for compositional*inputs*\.Concurrent compositional views of attention\.Three concurrent works arrive at this geometry from other directions\.[11](https://arxiv.org/html/2608.14712#bib.bib24)introduce the Aitchison distance to quantify within\-row token distinguishability in long contexts and derive a linear relation between temperature scaling and Aitchison distance, independently corroborating our powering axiom\.[14](https://arxiv.org/html/2608.14712#bib.bib25)identifies the row\-centered attention logit with the CLR transform and studies spectral invariants of the resulting field\.[25](https://arxiv.org/html/2608.14712#bib.bib26)construct tree\-aligned orthonormal bases of the Aitchison simplex and note the isomorphism between shift\-equivalent logits and the CLR hyperplane\. None studies the measurement problem addressed here: cross\-head similarity, entropy diagnostics, taxonomies, and pruning under sink conventions \([17](https://arxiv.org/html/2608.14712#bib.bib9)embed graph nodes as compositions for learned representations\)\. Information geometry \(Fisher–Rao\) equips the simplex with a different metric with different invariances; it is not perturbation\-invariant and does not decompose sink from content, which are the properties the sink problem demands \(§[6](https://arxiv.org/html/2608.14712#S6)\)\.

To our knowledge, this is the first work to bring log\-ratio geometry to the measurement methodology of attention analysis, and the first to separate the sink and content channels within it\.

## 3Aitchison geometry of attention rows

Compositions\.Let𝒮D=\{p∈ℝ\>0D:∑ipi=1\}\\mathcal\{S\}^\{D\}=\\\{p\\in\\mathbb\{R\}^\{D\}\_\{\>0\}:\\sum\_\{i\}p\_\{i\}=1\\\}denote the \(interior of the\) simplex withDDparts, and𝒞⁡\(x\)=x/∑ixi\\mathcal\{C\}\(x\)=x/\\sum\_\{i\}x\_\{i\}the closure operation\. A softmax attention row overDDkeys is a point of𝒮D\\mathcal\{S\}^\{D\}\(structural zeros from masking are handled below\)\. The simplex is a\(D−1\)\(D\{\-\}1\)\-dimensional real vector space under*perturbation*p⊕q=𝒞⁡\(p1​q1,…,pD​qD\)p\\oplus q=\\mathcal\{C\}\(p\_\{1\}q\_\{1\},\\dots,p\_\{D\}q\_\{D\}\)and*powering*α⊙p=𝒞⁡\(p1α,…,pDα\)\\alpha\\odot p=\\mathcal\{C\}\(p\_\{1\}^\{\\alpha\},\\dots,p\_\{D\}^\{\\alpha\}\), with identity the uniform compositioneeand inverse⊖p=𝒞⁡\(1/p1,…,1/pD\)\\ominus p=\\mathcal\{C\}\(1/p\_\{1\},\\dots,1/p\_\{D\}\)\([1](https://arxiv.org/html/2608.14712#bib.bib1);[20](https://arxiv.org/html/2608.14712#bib.bib4)\)\. Writingg⁡\(p\)g\(p\)for the geometric mean of the parts, the*centered log\-ratio*isclr⁡\(p\)=\(log⁡pig⁡\(p\)\)i\\clr\(p\)=\\big\(\\log\\tfrac\{p\_\{i\}\}\{g\(p\)\}\\big\)\_\{i\}, mapping𝒮D\\mathcal\{S\}^\{D\}isomorphically onto the hyperplaneℋ=\{x∈ℝD:∑ixi=0\}\\mathcal\{H\}=\\\{x\\in\\mathbb\{R\}^\{D\}:\\sum\_\{i\}x\_\{i\}=0\\\}, and an*isometric log\-ratio*\(ILR\) map isilr⁡\(p\)=V⊤​clr⁡\(p\)\\ilr\(p\)=V^\{\\top\}\\clr\(p\)for any orthonormal contrast basisV∈ℝD×\(D−1\)V\\in\\mathbb\{R\}^\{D\\times\(D\-1\)\},V⊤​V=IV^\{\\top\}V=I,V⊤​𝟏=0V^\{\\top\}\\mathbf\{1\}=0\([7](https://arxiv.org/html/2608.14712#bib.bib3)\)\. The*Aitchison inner product, norm, and distance*are the Euclidean ones pulled back throughclr\\clr\(equivalently anyilr\\ilr\):

dA​\(p,q\)=∥clr⁡\(p\)−clr⁡\(q\)∥2=∥ilr⁡\(p\)−ilr⁡\(q\)∥2\.d\_\{A\}\(p,q\)\\;=\\;\\lVert\\clr\(p\)\-\\clr\(q\)\\rVert\_\{2\}\\;=\\;\\lVert\\ilr\(p\)\-\\ilr\(q\)\\rVert\_\{2\}\.\(1\)
Subcompositions and coherence\.ForS⊆\{1,…,D\}S\\subseteq\\\{1,\\dots,D\\\}the*subcomposition*p\(S\)=𝒞⁡\(\(pi\)i∈S\)p^\{\(S\)\}=\\mathcal\{C\}\\big\(\(p\_\{i\}\)\_\{i\\in S\}\\big\)discards the other parts and renormalizes, exactly what an analyst does when masking the sink column\.*Subcompositional coherence*demands that conclusions about the parts inSSnot depend on whether the analysis was run onppor onp\(S\)p^\{\(S\)\}; a dissimilarity is*subcompositionally dominant*ifδ⁡\(p\(S\),q\(S\)\)≤δ⁡\(p,q\)\\delta\\big\(p^\{\(S\)\},q^\{\(S\)\}\\big\)\\leq\\delta\(p,q\)\([2](https://arxiv.org/html/2608.14712#bib.bib2);[9](https://arxiv.org/html/2608.14712#bib.bib7)\)\. In ILR coordinates, taking a subcomposition is an orthogonal projection\([17](https://arxiv.org/html/2608.14712#bib.bib9), Lemma 3\.1\), sodAd\_\{A\}is dominant and log\-ratios among the parts ofSSare untouched by the operation\. Cosine, JS, and Euclidean distance are neither subcompositionally coherent nor dominant, and none is stable under sink removal, as §[4\.1](https://arxiv.org/html/2608.14712#S4.SS1)makes constructive\.

Sink/content notation\.We index the sink coordinate as00and the remaining*content*coordinates as1,…,D−11,\\dots,D\{\-\}1; any row factors uniquely asp=\(s,\(1−s\)​π\)p=\(s,\(1\-s\)\\pi\)with sink masss=p0s=p\_\{0\}and content compositionπ∈𝒮D−1\\pi\\in\\mathcal\{S\}^\{D\-1\}\. We writep′=p\(\{1,…,D−1\}\)=πp^\{\\prime\}=p^\{\(\\\{1,\\dots,D\-1\\\}\)\}=\\pifor the sink\-dropped, renormalized row\. \(Multiple sinks group identically; Appendix[5](https://arxiv.org/html/2608.14712#A5.T5)\.\)

Structural zeros and masking\.Causal masking creates structural zeros, which log\-ratios forbid; the principled route is comparison on the*common unmasked support*, a subcomposition, hence an orthogonal projection, and our protocol additionally fixes the key support across aggregated query positions \(Appendix[5](https://arxiv.org/html/2608.14712#A5.T5)\)\. Residual numerical zeros are handled by multiplicative replacement withε=10−6\\varepsilon=10^\{\-6\}\([15](https://arxiv.org/html/2608.14712#bib.bib5);[19](https://arxiv.org/html/2608.14712#bib.bib6)\), and we report sensitivity toε∈\{10−5,10−6,10−7\}\\varepsilon\\in\\\{10^\{\-5\},10^\{\-6\},10^\{\-7\}\\\}\.

## 4Theory

### 4\.1The standard toolkit is subcompositionally incoherent

Writec⁡\(p,q\)c\(p,q\)for cosine similarity,JS⁡\(p,q\)\\mathrm\{JS\}\(p,q\)for the Jensen–Shannon divergence \(natural log\),∥p−q∥\\lVert p\-q\\rVertfor Euclidean distance, andH\(p\)=−∑ipilogpiH\(p\)=\-\\sum\_\{i\}p\_\{i\}\\log p\_\{i\}for Shannon entropy\.

###### Theorem 1\(Sink\-driven ranking reversal\)\.

FixD≥3D\\geq 3, index rowsx∈𝒮Dx\\in\\mathcal\{S\}^\{D\}by0,…,D−10,\\dots,D\-1with coordinate00the sink, and writeR0​\(x\):=\(x1,…,xD−1\)/\(1−x0\)∈𝒮D−1R\_\{0\}\(x\):=\(x\_\{1\},\\dots,x\_\{D\-1\}\)/\(1\-x\_\{0\}\)\\in\\mathcal\{S\}^\{D\-1\}\. Fixs′∈\(0,1\)s^\{\\prime\}\\in\(0,1\)andπ∈𝒮D−1\\pi\\in\\mathcal\{S\}^\{D\-1\}, for each permutationτ\\tauof the content coordinates withτ​π≠π\\tau\\pi\\neq\\piands∈\(s′,1\)s\\in\(s^\{\\prime\},1\)setps:=\(s,\(1−s\)​π\)p\_\{s\}:=\(s,\(1\-s\)\\pi\),q:=\(s′,\(1−s′\)​π\)q:=\(s^\{\\prime\},\(1\-s^\{\\prime\}\)\\pi\), andrs,τ:=\(s,\(1−s\)​τ​π\)r\_\{s,\\tau\}:=\(s,\(1\-s\)\\tau\\pi\); fix one suchσ\\sigmaand writeπ~:=σ​π\\widetilde\{\\pi\}:=\\sigma\\piandrs:=rs,σr\_\{s\}:=r\_\{s,\\sigma\}\. Then:*\(i\)*R0​\(ps\)=R0​\(q\)=πR\_\{0\}\(p\_\{s\}\)=R\_\{0\}\(q\)=\\piandR0​\(rs\)=π~R\_\{0\}\(r\_\{s\}\)=\\widetilde\{\\pi\}, so after sink removalqqis strictly closer topsp\_\{s\}thanrsr\_\{s\}under cosine,JS\\mathrm\{JS\}, Euclidean distance, and*every*metric on𝒮D−1\\mathcal\{S\}^\{D\-1\}\.*\(ii\)*There is a thresholds∗=s∗​\(π,s′\)∈\(s′,1\)s\_\{\\ast\}=s\_\{\\ast\}\(\\pi,s^\{\\prime\}\)\\in\(s^\{\\prime\},1\), uniform over all permutationsτ\\tauwithτ​π≠π\\tau\\pi\\neq\\pi, such that for everys∈\(s∗,1\)s\\in\(s\_\{\\ast\},1\)and every suchτ\\tauthe with\-sink verdict is simultaneously reversed for all three classical measures:c⁡\(ps,rs,τ\)\>c⁡\(ps,q\)c\(p\_\{s\},r\_\{s,\\tau\}\)\>c\(p\_\{s\},q\),JS⁡\(ps,rs,τ\)<JS⁡\(ps,q\)\\mathrm\{JS\}\(p\_\{s\},r\_\{s,\\tau\}\)<\\mathrm\{JS\}\(p\_\{s\},q\), and∥ps−rs,τ∥<∥ps−q∥\\lVert p\_\{s\}\-r\_\{s,\\tau\}\\rVert<\\lVert p\_\{s\}\-q\\rVert\.*\(iii\)*The set of triples in\(𝒮D\)3\(\\mathcal\{S\}^\{D\}\)^\{3\}on which all three classical verdicts flip between the two conventions is open and has positive3​\(D−1\)3\(D\-1\)\-dimensional Lebesgue measure in the affine hull\.*\(iv\)*dA⟂​\(x,y\):=dA​\(R0​\(x\),R0​\(y\)\)d\_\{A\}^\{\\perp\}\(x,y\):=d\_\{A\}\(R\_\{0\}\(x\),R\_\{0\}\(y\)\)is a pseudometric on𝒮D\\mathcal\{S\}^\{D\}\(a metric on𝒮D/∼0\\mathcal\{S\}^\{D\}/\{\\sim\_\{0\}\},x∼0yx\\sim\_\{0\}yiffR0​\(x\)=R0​\(y\)R\_\{0\}\(x\)=R\_\{0\}\(y\)\); on the family,dA⟂​\(ps,q\)=0d\_\{A\}^\{\\perp\}\(p\_\{s\},q\)=0anddA⟂​\(ps,rs\)=dA​\(π,π~\)=:B\>0d\_\{A\}^\{\\perp\}\(p\_\{s\},r\_\{s\}\)=d\_\{A\}\(\\pi,\\widetilde\{\\pi\}\)=:B\>0under*both*conventions\. FulldAd\_\{A\}satisfiesdA​\(ps,q\)=αD​log⁡s⁡\(1−s′\)s′​\(1−s\)d\_\{A\}\(p\_\{s\},q\)=\\alpha\_\{D\}\\log\\tfrac\{s\(1\-s^\{\\prime\}\)\}\{s^\{\\prime\}\(1\-s\)\}\(αD=\(D−1\)/D\\alpha\_\{D\}=\\sqrt\{\(D\-1\)/D\}\) anddA​\(ps,rs\)=Bd\_\{A\}\(p\_\{s\},r\_\{s\}\)=B, so it ranksqqcloser exactly whens<sAs<s\_\{\\mathrm\{A\}\}for an explicit cutoffsA∈\(s′,1\)s\_\{\\mathrm\{A\}\}\\in\(s^\{\\prime\},1\); the Table[1](https://arxiv.org/html/2608.14712#S4.T1)witness satisfiess<sAs<s\_\{\\mathrm\{A\}\}, which is why fulldAd\_\{A\}agrees across conventions there\. \(Proof: Appendix[B](https://arxiv.org/html/2608.14712#A2)\.\)

Table 1:Exact witness for Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)withp=\(0\.90,0\.08,0\.02\)p=\(0\.90,0\.08,0\.02\),q=\(0\.70,0\.24,0\.06\)q=\(0\.70,0\.24,0\.06\),r=\(0\.90,0\.02,0\.08\)r=\(0\.90,0\.02,0\.08\):qqsharespp’s content ratio \(4:14\{:\}1\) andrrsharespp’s sink mass with mirrored content\. “Closer” means larger cosine similarity or smaller dissimilarity\. All three classical verdicts reverse; the JS and Euclidean values also violate subcompositional dominance \(JS⁡\(p′,r′\)\>JS⁡\(p,r\)\\mathrm\{JS\}\(p^\{\\prime\},r^\{\\prime\}\)\>\\mathrm\{JS\}\(p,r\),∥p′−r′∥\>∥p−r∥\\lVert p^\{\\prime\}\-r^\{\\prime\}\\rVert\>\\lVert p\-r\\rVert\)\.The witness entries are exact rationals; every displayed quantity is evaluated in closed form in Appendices[B](https://arxiv.org/html/2608.14712#A2)and[C](https://arxiv.org/html/2608.14712#A3)\.

###### Proposition 1\(Entropy inversion under sink removal\)\.

For everyD≥3D\\geq 3there existp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}withH⁡\(p\)<H⁡\(q\)H\(p\)<H\(q\)butH⁡\(p′\)\>H⁡\(q′\)H\(p^\{\\prime\}\)\>H\(q^\{\\prime\}\): entropy\-collapse comparisons, across heads, layers, or training steps, can invert with the analyst’s convention\. A witness atD=3D=3isP=\(1920,140,140\)P=\\big\(\\tfrac\{19\}\{20\},\\tfrac\{1\}\{40\},\\tfrac\{1\}\{40\}\\big\),Q=\(12,9992000,12000\)Q=\\big\(\\tfrac\{1\}\{2\},\\tfrac\{999\}\{2000\},\\tfrac\{1\}\{2000\}\\big\):H⁡\(P\)≈0\.233<0\.697≈H⁡\(Q\)H\(P\)\\approx 0\.233<0\.697\\approx H\(Q\), yetH⁡\(P′\)=log⁡2\>H⁡\(Q′\)≈0\.0079H\(P^\{\\prime\}\)=\\log 2\>H\(Q^\{\\prime\}\)\\approx 0\.0079\. The convention\-invariant content statistics agree with the conditional view:H⁡\(P′\)=log⁡2H\(P^\{\\prime\}\)=\\log 2is maximal andCA⟂​\(P\)=0C\_\{A\}^\{\\perp\}\(P\)=0\(content exactly uniform\), versusH⁡\(Q′\)≈0\.008H\(Q^\{\\prime\}\)\\approx 0\.008andCA⟂​\(Q\)≈4\.88C\_\{A\}^\{\\perp\}\(Q\)\\approx 4\.88\(content nearly degenerate\)\. \(Proof and sharpness: Appendix[B](https://arxiv.org/html/2608.14712#A2)\.\)

The mechanism is the chain ruleH⁡\(p\)=Hb​\(s\)\+\(1−s\)​H​\(π\)H\(p\)=H\_\{b\}\(s\)\+\(1\-s\)H\(\\pi\)\(Appendix[B](https://arxiv.org/html/2608.14712#A2)\): a growing sink drivesHHdown regardless of what the content distribution does, soHHconflates two unrelated quantities, sink strength and content spread, that the Aitchison geometry separates exactly \(Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\)\. The constructive remedy is to report the pair\(s,H⁡\(π\)\)\(s,H\(\\pi\)\)in place ofH⁡\(p\)H\(p\);CA⟂C\_\{A\}^\{\\perp\}is a complementary log\-ratio dispersion of the same content channel, not an entropy substitute \(Remark[1](https://arxiv.org/html/2608.14712#Thmremark1)\)\.

### 4\.2A coherent toolkit, and why it is essentially unique

###### Definition 1\(Coherent attention diagnostics\)\.

For rowsp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}with sink part00:*\(a\)*the*head signature*of a set of rows\{pt\}\\\{p\_\{t\}\\\}is their Aitchison meancen⁡\(\{pt\}\)=𝒞⁡\(exp⁡\(1T​∑tlog⁡pt\)\)\\cen\(\\\{p\_\{t\}\\\}\)=\\mathcal\{C\}\\big\(\\exp\\big\(\\tfrac\{1\}\{T\}\\sum\_\{t\}\\log p\_\{t\}\\big\)\\big\), the Fréchet mean underdAd\_\{A\};*\(b\)*head dissimilarity isdAd\_\{A\}of \([1](https://arxiv.org/html/2608.14712#S3.E1)\), and the direction similarity is the ILR\-cosineρA​\(p,q\)=⟨clr⁡p,clr⁡q⟩∥clr⁡p∥​∥clr⁡q∥\\rho\_\{A\}\(p,q\)=\\tfrac\{\\langle\\clr p,\\clr q\\rangle\}\{\\lVert\\clr p\\rVert\\lVert\\clr q\\rVert\};*\(c\)*the*sink balance*isb⁡\(p\)=D−1D​log⁡p0g⁡\(p1,…,pD−1\)b\(p\)=\\sqrt\{\\tfrac\{D\-1\}\{D\}\}\\,\\log\\tfrac\{p\_\{0\}\}\{g\(p\_\{1\},\\dots,p\_\{D\-1\}\)\};*\(d\)*the*content distance*isdA⟂​\(p,q\)=dA​\(p′,q′\)d\_\{A\}^\{\\perp\}\(p,q\)=d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\)and the*content log\-ratio dispersion*isCA⟂​\(p\)=∥clr⁡\(p′\)∥2=dA​\(p′,e\)C\_\{A\}^\{\\perp\}\(p\)=\\lVert\\clr\(p^\{\\prime\}\)\\rVert\_\{2\}=d\_\{A\}\(p^\{\\prime\},e\), the Aitchison distance of the content to uniform \(it orders by log\-ratio dispersion, not by majorization; Remark[1](https://arxiv.org/html/2608.14712#Thmremark1)\);*\(e\)*dispersion of a set of rows is the total Aitchison variance1n​∑idA​\(pi,cen\)2\\tfrac\{1\}\{n\}\\sum\_\{i\}d\_\{A\}\\big\(p\_\{i\},\\cen\\big\)^\{2\}\.

###### Lemma 1\(Pythagorean sink decomposition\)\.

LetD≥2D\\geq 2,p,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}with designated sink00, andb⁡\(⋅\)b\(\\cdot\)the sink balance of Definition[1](https://arxiv.org/html/2608.14712#Thmdefinition1)\(c\)\. In CLR space the exact orthogonal identityclr⁡\(p\)=b⁡\(p\)​ub\+J​clr⁡\(p′\)\\clr\(p\)=b\(p\)\\,u\_\{b\}\+J\\clr\(p^\{\\prime\}\)holds, whereubu\_\{b\}is the unit balance direction andJJis the isometric zero\-extension of the content CLR space; hence

dA​\(p,q\)2=dA⟂​\(p,q\)2\+\(b⁡\(p\)−b⁡\(q\)\)2\.d\_\{A\}\(p,q\)^\{2\}=d\_\{A\}^\{\\perp\}\(p,q\)^\{2\}\+\\big\(b\(p\)\-b\(q\)\\big\)^\{2\}\.There is a sink\-adapted ILR basis whose first coordinate isb⁡\(⋅\)b\(\\cdot\)and whose remainingD−2D\-2coordinates are ILR coordinates of the sink\-dropped composition\. ConsequentlydA⟂d\_\{A\}^\{\\perp\}andCA⟂C\_\{A\}^\{\\perp\}depend only on the content composition: varying the sink mass with the content held fixed changes neither, and both take identical values whether computed in the balance\-orthogonal coordinates of the full rows or after dropping the sink and re\-closing\. \(Proof: Appendix[C](https://arxiv.org/html/2608.14712#A3)\.\)

###### Proposition 2\(Invariances and equivariances\)\.

With softmax rowsp=𝒞⁡\(eℓ\)p=\\mathcal\{C\}\(e^\{\\ell\}\),q=𝒞⁡\(eℓ′\)q=\\mathcal\{C\}\(e^\{\\ell^\{\\prime\}\}\):*\(i\)*a shared finite logit biasuuacts as perturbation bycu=𝒞⁡\(eu\)c\_\{u\}=\\mathcal\{C\}\(e^\{u\}\), anddAd\_\{A\}anddA⟂d\_\{A\}^\{\\perp\}are invariant under it;CA⟂C\_\{A\}^\{\\perp\}\(forD≥3D\\geq 3\), Euclidean distance,JS\\mathrm\{JS\}, cosine, andρA\\rho\_\{A\}are not\.*\(ii\)*a temperatureτ\\tauacts as powering by1/τ1/\\tau:dAd\_\{A\},dA⟂d\_\{A\}^\{\\perp\}, andCA⟂C\_\{A\}^\{\\perp\}scale by1/τ1/\\tau\(so one common temperature preserves all rankings and ties\), total Aitchison variance scales by1/τ21/\\tau^\{2\}, andρA\\rho\_\{A\}is invariant to*separate*positive per\-head temperatures applied uniformly within each head \(row\-dependent temperatures can change it\)\.*\(iii\)*sink coherence:dA⟂​\(p,q\)d\_\{A\}^\{\\perp\}\(p,q\)andCA⟂​\(p\)C\_\{A\}^\{\\perp\}\(p\)are independent of the sink masses; fulldAd\_\{A\}is not sink\-invariant but decomposes exactly \(Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\)\.*\(iv\)*a subcomposition onSS\(\|S\|≥2\|S\|\\geq 2\) is an orthogonal projection in CLR space, sodA​\(p\(S\),q\(S\)\)≤dA​\(p,q\)d\_\{A\}\(p^\{\(S\)\},q^\{\(S\)\}\)\\leq d\_\{A\}\(p,q\); hard masks require re\-closure on a common positive support; and*\(v\)*forD≥3D\\geq 3, Euclidean distance,JS\\mathrm\{JS\}, and1−cos1\-\\cosare*not*subcompositionally dominant\. \(Full statements with all qualifications, proofs, and scope: Appendix[C](https://arxiv.org/html/2608.14712#A3)\.\)

The axioms are the transformer’s own knobs\.Given the operational meaning the transformer itself assigns to⊕\\oplus\(shared logit biases\) and⊙\\odot\(temperature\), four interpretable axioms force the Aitchison distance, and the invariance axioms among them are transformations the architecture performs\. A quotient version characterizes the content distance itself \(Lemma[2](https://arxiv.org/html/2608.14712#Thmlemma2)\)\.

###### Theorem 2\(Characterization of the Aitchison distance\)\.

Letδ:𝒮D×𝒮D→ℝ≥0\\delta:\\mathcal\{S\}^\{D\}\\times\\mathcal\{S\}^\{D\}\\to\\mathbb\{R\}\_\{\\geq 0\}satisfy:*\(A1\)*δ\\deltais a metric;*\(A2\)*perturbation invariance:δ⁡\(c⊕p,c⊕q\)=δ⁡\(p,q\)\\delta\(c\\oplus p,\\,c\\oplus q\)=\\delta\(p,q\)for allcc;*\(A3\)*powering homogeneity:δ⁡\(α⊙p,α⊙q\)=\|α\|​δ​\(p,q\)\\delta\(\\alpha\\odot p,\\,\\alpha\\odot q\)=\|\\alpha\|\\,\\delta\(p,q\)for allα∈ℝ\\alpha\\in\\mathbb\{R\};*\(A4\)*the parallelogram law: writingN⁡\(v\)=δ⁡\(v,e\)N\(v\)=\\delta\(v,e\),N​\(v⊕w\)2\+N​\(v⊖w\)2=2​N​\(v\)2\+2​N​\(w\)2\\;N\(v\\oplus w\)^\{2\}\+N\(v\\ominus w\)^\{2\}=2N\(v\)^\{2\}\+2N\(w\)^\{2\};*\(A5\)*permutation invariance:δ⁡\(σ​p,σ​q\)=δ⁡\(p,q\)\\delta\(\\sigma p,\\sigma q\)=\\delta\(p,q\)for allσ∈SD\\sigma\\in S\_\{D\}\. Thenδ=c​dA\\delta=c\\,d\_\{A\}for some constantc\>0c\>0\.

The proof \(Appendix[C](https://arxiv.org/html/2608.14712#A3)\) reducesδ\\deltato a norm via A2, upgrades it to an inner\-product norm via A4 by[12](https://arxiv.org/html/2608.14712#bib.bib23), and pins the inner product to a multiple of the CLR one via A5 and irreducibility of the permutation representation on the sum\-zero hyperplane\. The axioms are tight:N1​\(v\)=∥clr⁡v∥1N\_\{1\}\(v\)=\\lVert\\clr v\\rVert\_\{1\}satisfies A1–A3 and A5 but not A4 \(Remark[C\.3](https://arxiv.org/html/2608.14712#A3.Thmaremark3)\)\. A2 and A3 are exactly invariance to shared logit biases and consistency under temperature, the two nuisance transformations the transformer parameterization itself supplies\. Cross\-dimensional comparisons need one common calibration constant, under which dominance, the sink decomposition, and the bias/temperature behavior follow rather than being assumed \(Corollary[2](https://arxiv.org/html/2608.14712#Thmcorollary2), Remark[C\.2](https://arxiv.org/html/2608.14712#A3.Thmaremark2)\)\.

### 4\.3Consequences: dynamic range, taxonomies, and training curves

Three consequences sharpen what §[5](https://arxiv.org/html/2608.14712#S5)measures\.

###### Theorem 3\(Dynamic\-range collapse in the sink\-dominated regime\)\.

FixD≥3D\\geq 3ands0∈\(12,1\)s\_\{0\}\\in\(\\tfrac\{1\}\{2\},1\)\.*\(i\)*Ifp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}have sink masses at leasts0s\_\{0\}, then, strictly,

1−c⁡\(p,q\)<\(1−s0\)2s02\+\(1−s0\)2,∥p−q∥<2​\(1−s0\),JS⁡\(p,q\)<\(1−s0\)​log⁡2,1\-c\(p,q\)<\\frac\{\(1\-s\_\{0\}\)^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\},\\qquad\\lVert p\-q\\rVert<\\sqrt\{2\}\\,\(1\-s\_\{0\}\),\\qquad\\mathrm\{JS\}\(p,q\)<\(1\-s\_\{0\}\)\\log 2,and each right\-hand side is the exact, nonattained supremum \(the attainable values fill\[0,⋅\)\[0,\\cdot\)\); the cosine band shrinks quadratically,∼\(1−s0\)2\\sim\(1\-s\_\{0\}\)^\{2\}\.*\(ii\)*The content\-preserving sink transformationsTt​r:=\(t,\(1−t\)​r′\)T\_\{t\}r:=\\big\(t,\(1\-t\)r^\{\\prime\}\\big\)leave the entiredA⟂d\_\{A\}^\{\\perp\}dissimilarity matrix and allCA⟂C\_\{A\}^\{\\perp\}values of any finite head family exactly unchanged; in particular, sinks can strengthen arbitrarily while the content geometry stays fixed\. \(Proof: Appendix[D](https://arxiv.org/html/2608.14712#A4)\.\)

Concretely, heads with sink mass≥0\.8\\geq 0\.8\(Llama\-3\.2\-1B’s mean is0\.840\.84\) have*all*pairwise cosine similarities above0\.940\.94, Euclidean distances below0\.290\.29, andJS\\mathrm\{JS\}below0\.140\.14nats: the classical dynamic range shrinks into a band comparable to estimation noise, so the orderings inside it, nearest neighbors, redundancy ranks, cluster merges, carry vanishing information \(a mechanism candidate for the pruning catastrophe of §[3](https://arxiv.org/html/2608.14712#S5.F3)\)\.

###### Proposition 3\(Taxonomies: stability and instability\)\.

*\(i\)*For any finite labelled head set, thedA⟂d\_\{A\}^\{\\perp\}dissimilarity matrix computed with the sink retained equals, entrywise, thedAd\_\{A\}matrix of the explicitly sink\-dropped heads\. Hence every clustering rule, deterministic or randomized, whose sole data\-dependent input is this labelled matrix returns identical outputs in the two analyses \(equally, in distribution and under a common coupling, when randomized\)\.*\(ii\)*There is a nonempty open set of head triples of positive3​\(D−1\)3\(D\-1\)\-dimensional measure on which, simultaneously for1−cos1\-\\cos,JS\\mathrm\{JS\}, and Euclidean distance, the unique closest pair is\(p,r\)\(p,r\)with the sink and\(p,q\)\(p,q\)after the drop; single\- and complete\-linkage clustering atK=2K\{=\}2therefore return\{p,r\}​\{q\}\\\{p,r\\\}\\\{q\\\}versus\{p,q\}​\{r\}\\\{p,q\\\}\\\{r\\\}\. The exact rational witness is Table[1](https://arxiv.org/html/2608.14712#S4.T1)\. \(Proof: Appendix[D](https://arxiv.org/html/2608.14712#A4)\.\)

###### Corollary 1\(Checkpoint entropy decomposition and sink\-only collapse\)\.

Along any checkpoint familypt=\(st,\(1−st\)​πt\)p\_\{t\}=\\big\(s\_\{t\},\(1\-s\_\{t\}\)\\pi\_\{t\}\\big\),H⁡\(pt\)=Hb​\(st\)\+\(1−st\)​H​\(πt\)H\(p\_\{t\}\)=H\_\{b\}\(s\_\{t\}\)\+\(1\-s\_\{t\}\)H\(\\pi\_\{t\}\)pointwise\. If the content is fixed,πt≡π\\pi\_\{t\}\\equiv\\pi, thenCA⟂​\(pt\)C\_\{A\}^\{\\perp\}\(p\_\{t\}\)is constant whiles↦H⁡\(s,\(1−s\)​π\)s\\mapsto H\\big\(s,\(1\-s\)\\pi\\big\)is strictly decreasing on\[12,1\)\[\\tfrac\{1\}\{2\},1\)\(its unique maximizer isθπ=\(1\+eH⁡\(π\)\)−1<12\\theta\_\{\\pi\}=\\big\(1\+e^\{H\(\\pi\)\}\\big\)^\{\-1\}<\\tfrac\{1\}\{2\}\); with uniform content the apparent collapse can approachlog⁡D\\log DwithCA⟂≡0C\_\{A\}^\{\\perp\}\\equiv 0\. This is a possibility result, not a claim that sink mass grows along any empirical trajectory\. \(Proof: Appendix[D](https://arxiv.org/html/2608.14712#A4)\.\)

§[5\.3](https://arxiv.org/html/2608.14712#S5.SS3)turns all three into prospectively specified measurements\.

## 5Experiments

All numbers in this section are exact evaluations, Monte Carlo estimates, or single\-seed model measurements produced by the released code with fixed seeds; §[5\.1](https://arxiv.org/html/2608.14712#S5.SS1)isolates the mechanism in silico and §[3](https://arxiv.org/html/2608.14712#S5.F3)measures it on pretrained models\.

### 5\.1Exact constructions and calibrated simulations

Exact witnesses\.Table[1](https://arxiv.org/html/2608.14712#S4.T1)and Figure[1](https://arxiv.org/html/2608.14712#S1.F1)evaluate the Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)witness; Proposition[1](https://arxiv.org/html/2608.14712#Thmproposition1)is instantiated in its statement; every invariance claim of Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2)is verified to machine precision in the released tests\.

How often do verdicts flip?Figure[3](https://arxiv.org/html/2608.14712#S5.F3)samples head triples with sink structure calibrated to reported statistics\([3](https://arxiv.org/html/2608.14712#bib.bib20);[10](https://arxiv.org/html/2608.14712#bib.bib19)\)and asks the basic analyst’s question,*which ofq,rq,ris closer topp?*, under both sink conventions\. At mean sink masss¯=0\.5/0\.7/0\.9\\bar\{s\}=0\.5/0\.7/0\.9the conventions disagree on36/39/41%36/39/41\\%\(cosine\),23/31/40%23/31/40\\%\(JS\), and27/37/45%27/37/45\\%\(Euclidean\) of triples, growing with sink mass toward the50%50\\%chance level\. The content distancedA⟂d\_\{A\}^\{\\perp\}is identical in both pipelines by Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1), hence0%0\\%everywhere\. The positive\-measure region of Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)is not thin at realistic sink levels; stars overlay the §[3](https://arxiv.org/html/2608.14712#S5.F3)measurements at each model’s signature sink mass\.

Figure 2:Entropy conflates sink strength with content spread;CA⟂C\_\{A\}^\{\\perp\}separates them\.Synthetic 12\-layer models,4,0004\{,\}000rows per layer\. \(a\) Sink mass grows0\.06→0\.850\.06\\to 0\.85with content log\-ratio dispersion held fixed: mean Shannon entropy falls2\.71→0\.822\.71\\to 0\.82nats \(−70%\-70\\%, a textbook “entropy collapse”\) whileCA⟂C\_\{A\}^\{\\perp\}moves48\.1→48\.048\.1\\to 48\.0\(−0\.3%\-0\.3\\%\): the collapse is phantom\. \(b\) Content genuinely collapses while the sink recedes0\.70→0\.050\.70\\to 0\.05: entropy drifts1\.73→1\.271\.73\\to 1\.27\(−27%\-27\\%, reads as mild\) whileCA⟂C\_\{A\}^\{\\perp\}rises10\.0→70\.410\.0\\to 70\.4\(7×7\\times\): the real collapse is masked\.Collapse diagnostics\.Figure[2](https://arxiv.org/html/2608.14712#S5.F2)constructs two 12\-layer scenarios\. In Scenario \(a\), only the sink grows; Shannon entropy reports a70%70\\%collapse while content log\-ratio dispersion is flat, exactly the failure Proposition[1](https://arxiv.org/html/2608.14712#Thmproposition1)predicts, and the mode to which entropy\-collapse monitoring\([26](https://arxiv.org/html/2608.14712#bib.bib14)\)is exposed when sinks strengthen\([10](https://arxiv.org/html/2608.14712#bib.bib19)\)\. In Scenario \(b\), the content genuinely collapses while the sink recedes; entropy barely moves\.CA⟂C\_\{A\}^\{\\perp\}responds in both cases because it is a statistic of the content subcomposition only\.

### 5\.2Pretrained models across five model families

Figure 3:Calibrated reversal curves and ten measured models\.Curves: Monte Carlo closer\-head disagreement rates between sink conventions for cosine, JS, and Euclidean distance, computed before any pretrained model was run \(solidα=0\.1\\alpha\{=\}0\.1, dashedα=1\\alpha\{=\}1\)\. Stars: the ten models, placed at the sink mass of the signatures the distances are computed on; mean absolute deviation from the solid curves is4\.84\.8points \(max1313\)\. ThedA⟂d\_\{A\}^\{\\perp\}rates are zero identically \(Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\), plotted as a code check\.Protocol\(released; two consumer GPUs, a few hours, no training; ten models, GPT\-2, Pythia 70M–1\.4B, Qwen2\.5\-1\.5B, Llama\-3\.2\-1B/3B, ViT\-B/16; three resampling seeds each, varying data order and triple sampling for one fixed public checkpoint per model\)\.Extraction:forward hooks on the attention softmax; for LMs, query positionst∈\[w,T\]t\\in\[w,T\]restricted to the firstW=64W=64keys \(a fixed common support; no zero imputation needed\), with key00\(the⟨bos⟩\\langle\\mathrm\{bos\}\\rangle/first token\) as the sink column; for ViT\-B/16, the CLS column as sink\.Aggregation:per\-head Aitchison mean over query positions and inputs \(Definition[1](https://arxiv.org/html/2608.14712#Thmdefinition1)\(a\)\)\.Measurements:\(1\) closer\-head ranking\-reversal rates over50,00050\{,\}000head triples between the with\-sink and sink\-dropped conventions, for cosine/JS/Euclidean versusdA⟂d\_\{A\}^\{\\perp\}; \(2\) layer\-wise meanHH,H⁡\(π\)H\(\\pi\), and sink mass; \(3\) redundancy pruning at matched sparsity, prune them∈\{10,20,30\}%m\\in\\\{10,20,30\\\}\\%most redundant heads \(nearest\-neighbor distance\) under JS, under the content distancedA⟂d\_\{A\}^\{\\perp\}, and under the fulldAd\_\{A\}\(sink balance included\), comparing perplexity \(LMs\) and top\-1 agreement with the unpruned model \(ViT, so no labels are needed\); \(4\) anε\\varepsilonsweep over\{10−5,10−6,10−7\}\\\{10^\{\-5\},10^\{\-6\},10^\{\-7\}\\\}\. Data:2,0002\{,\}000WikiText\-103 validation sequences \(LMs\),2,0482\{,\}048COCO val2017 images \(ViT\); details in Appendix[5](https://arxiv.org/html/2608.14712#A5.T5)\.

Table 2:Measured convention\-dependence on ten pretrained models\(mean±\\pmsd over three seeds\)\. Reversal==% of50,00050\{,\}000head triples whose closer\-head verdict flips between sink conventions \(bit\-identical across theε\\varepsilonsweep\)\. Rates rise with signature sink mass \(Figure[3](https://arxiv.org/html/2608.14712#S5.F3)\)\. Hell\./FR: Hellinger reversal; Fisher–Rao is identical since both are strictly monotone in the Bhattacharyya coefficient, so every ranking verdict coincides\. Collapse disagr\.: fraction of adjacent\-layer transitions on whichHHand the content entropyH⁡\(π\)H\(\\pi\)move in opposite directions\. Sink \(row/sig\.\): arithmetic row\-level mean and the sink mass of the Aitchison\-mean signatures the distances use\. The content\-distance column is omitted: its rate is zero by construction \(Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\)\.Table 3:Redundancy pruning at20%20\\%sparsity, all ten models\(mean±\\pmsd, three resampling seeds;10/20/30%10/20/30\\%sweep in Table[10](https://arxiv.org/html/2608.14712#A5.T10); random and total\-dAd\_\{A\}baselines in Table[11](https://arxiv.org/html/2608.14712#A5.T11)\)\. LMs: perplexity, lower is better; ViT: top\-1 agreement with the unpruned model, a stability rather than accuracy metric, higher is better\. JS′is JS on the sink\-dropped rows; Hellinger on the content rows selects identical prune sets \(Appendix[E\.1](https://arxiv.org/html/2608.14712#A5.SS1)\)\. “\-L1” restricts neighbors to the same layer and defers a head whose nearest neighbor is already selected \(one representative of each mutual pair is retained\)\. Bold: best of the five criteria; the regime structure is described in the text\.Figure 4:Measured layer\-wise collapse decomposed\.Mean Shannon entropyHH\(red\) versus content entropyH⁡\(π\)H\(\\pi\)\(green\), sink mass dotted \(seed 0\); shaded bands mark adjacent\-layer transitions on which the two disagree in sign\. Where the sink strengthens with depth,HHreports deepening collapse that the content channel does not show: Proposition[1](https://arxiv.org/html/2608.14712#Thmproposition1)’s conflation observed in pretrained models\.Results: measurement\.Table[3](https://arxiv.org/html/2608.14712#S5.T3)confirms the theory on ten pretrained models across five families\. Row\-level sink masses on the fixed6464\-key support span0\.040\.04\(ViT CLS\) to0\.860\.86, corroborating[3](https://arxiv.org/html/2608.14712#bib.bib20); within each family sink mass grows with scale\. The signatures the distances are computed on concentrate further, geometric averaging amplifying a shared sink \(row/signature masses in Table[3](https://arxiv.org/html/2608.14712#S5.T3); Pythia\-70M0\.08→0\.430\.08\\to 0\.43\)\. Indexed by the signature sink, reversal rates rise from1717–23%23\\%on ViT to4444–47%47\\%, approaching the50%50\\%chance level, on the Llamas, and track the calibrated curves of Figure[3](https://arxiv.org/html/2608.14712#S5.F3)with mean absolute deviation4\.84\.8points \(max1313\); reversal underdA⟂d\_\{A\}^\{\\perp\}is zero by construction \(verified to the bit\); classical rates areε\\varepsilon\-invariant with resampling sd≤0\.4\\leq 0\.4points\. The support choice is not decisive: the window retains4848–71%71\\%of row mass, reversal moves at most22points acrossW∈\{32,64,128\}W\\in\\\{32,64,128\\\}, anddA⟂d\_\{A\}^\{\\perp\}distances correlate atρ≥0\.87\\rho\\geq 0\.87across windows \(Appendix[E\.3](https://arxiv.org/html/2608.14712#A5.SS3)\)\. Entropy and the content entropyH⁡\(π\)H\(\\pi\)disagree in sign on99–47%47\\%of layer transitions across models \(Figure[4](https://arxiv.org/html/2608.14712#S5.F4)shows three\)\. \(A measured Llama triple realizes Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)verbatim: Fig\.[7](https://arxiv.org/html/2608.14712#A1.F7)\.\) Appendix[E\.1](https://arxiv.org/html/2608.14712#A5.SS1)audits the estimator \(tail coordinates carry≤4\.6%\\leq 4\.6\\%of squareddA⟂d\_\{A\}^\{\\perp\}; dropped\-row JS agrees on coarse ranking but not on nearest neighbors or taxonomies\)\.

Figure 5:Prospectively specified downstream outcomes\.*\(a\)*reversal tracks sink mass, which strengthens with scale within each family\.*\(b\)*at Pythia\-410MHHcollapses whileH⁡\(π\)H\(\\pi\)barely moves; the share1−Δ​H​\(π\)/Δ​H1\-\\Delta H\(\\pi\)/\\Delta His30/53/70/95/79%30/53/70/95/79\\%across 70M–1\.4B, rising to95%95\\%at 1B \(79%79\\%at 1\.4B\)\.Results: pruning\.Table[3](https://arxiv.org/html/2608.14712#S5.T3)is a regime map\. Under the global protocol, conditioning out the sink is the decisive step and the choice of geometry within the content channel is secondary: on Llama\-3\.2\-1B, with\-sink JS collapses \(15\.3→2094±14015\.3\\to 2094\{\\pm\}140\) while dropped\-row JS \(70\.5±9\.670\.5\{\\pm\}9\.6\) anddA⟂d\_\{A\}^\{\\perp\}\(63\.6±4\.163\.6\{\\pm\}4\.1\) both avoid the failure, and dropped\-row JS wins outright on Llama\-3\.2\-3B \(25\.325\.3versus40\.240\.2\); the fulldAd\_\{A\}, re\-mixing the sink channel, fails like JS \(2480±2112480\{\\pm\}211\), and random orderings beat globaldA⟂d\_\{A\}^\{\\perp\}on1515of3030sweep cells\. The separation appears under a protocol closer to practice \(same\-layer neighbors, one representative kept per redundant pair\):dA⟂d\_\{A\}^\{\\perp\}\-L1 is the best of all nine criteria on every strong\-sink model \(45\.045\.0,26\.926\.9,23\.423\.4on Llama\-1B/Qwen/Llama\-3B;0\.5810\.581on ViT\), whereas each classical criterion fails severely on at least one of them \(JS′\-L1 reaches26422642on Llama\-3\.2\-1B; JS\-L1 reaches17331733on Llama\-3\.2\-3B\)\.dA⟂d\_\{A\}^\{\\perp\}\-L1 in turn fails on GPT\-2 \(449449\), and with\-sink JS\(\-L1\) wins wherever sinks are weak\. Across resampling seeds thedA⟂d\_\{A\}^\{\\perp\}\-L1 sets are the less stable \(Jaccard0\.360\.36–0\.770\.77versus0\.920\.92–1\.001\.00for JS\-L1\) even where their outcomes are stable: each criterion is stable along a different axis\. Sink mass alone does not predict the regime \(GPT\-2 and Qwen2\.5 shares=0\.68s=0\.68with opposite verdicts\); the cross\-head dispersion of the sink share separates all nine LMs \(coefficient of variation, CV,≤0\.36\\leq 0\.36in the content regime,≥0\.42\\geq 0\.42in the JS regime\); a frozen out\-of\-sample test on three unseen models returned one confirmation, one boundary abstention, and one failure, so the rule is presented as descriptive, not validated \(Appendix[E\.2](https://arxiv.org/html/2608.14712#A5.SS2)\)\. The experiments support two claims: measurements feeding any criterion are convention\-dependent under classical metrics \(Table[3](https://arxiv.org/html/2608.14712#S5.T3)\); which channel carries the functional signal is model\-dependent, a question the decomposition makes precise\. Two stress tests separate the Aitchison choice from simply dropping the sink: rescaling the content logits by an unobservable temperatureτ∈\[0\.5,2\]\\tau\\in\[0\.5,2\]leaves everydA⟂d\_\{A\}^\{\\perp\}verdict and prune set bit\-identical \(Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2)\) while dropped\-row JS changes up to38%38\\%of its top\-20%20\\%prune\-set membership, and across sink definitions \(first\-1 versus first\-4\)dA⟂d\_\{A\}^\{\\perp\}is the less sensitive on every model, with nested\-projection dominance verified \(Appendix[E\.2](https://arxiv.org/html/2608.14712#A5.SS2)\)\.

### 5\.3Prospectively specified downstream tests

Are*conclusions*built on these measurements convention\-dependent too? Three predictions were stated before the runs \(in the released repository; no external registry\); we report each outcome\.

*\(1\) Taxonomies\.*Clark\-style hierarchical head clustering \(agglomerative,K∈\{4,6,8\}K\\in\\\{4,6,8\\\}\) on BERT\-base, GPT\-2, and Llama\-3\.2\-1B\([4](https://arxiv.org/html/2608.14712#bib.bib10)\)\. Prediction \(Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)\): classical between\-convention ARIs sit materially below11, decreasing with sink mass; thedA⟂d\_\{A\}^\{\\perp\}partitions are identical by construction\.*Outcome*: at6464keys, ARIs sit at or below chance on all three \(Table[5](https://arxiv.org/html/2608.14712#A5.T5)\)\. On the full128128\-token support with[4](https://arxiv.org/html/2608.14712#bib.bib10)’s own pipeline \(JS, average linkage\), cross\-convention ARI is0\.010\.01–0\.050\.05, and the SEP\-head block \(6868/144144heads\) drops from Jaccard0\.870\.87–0\.920\.92to≈0\.48\\approx 0\.48when the sink is dropped \(Appendix[E\.3](https://arxiv.org/html/2608.14712#A5.SS3)\): its most prominent structure is the convention\.

*\(2\) Scale\.*Sink mass and reversal rate across Pythia 70M–1\.4B; prediction \(Theorem[3](https://arxiv.org/html/2608.14712#Thmtheorem3);[10](https://arxiv.org/html/2608.14712#bib.bib19)\): both increase with parameters\.*Outcome*: sink mass is monotone within each family and reversal tracks it near the pre\-computed curves \(Fig\.[5](https://arxiv.org/html/2608.14712#S5.F5)a\)\.*\(3\) Training curves\.*HHversusCA⟂C\_\{A\}^\{\\perp\}across Pythia checkpoints; prediction \(Corollary[1](https://arxiv.org/html/2608.14712#Thmcorollary1)\): during sink emergenceHHfalls withCA⟂C\_\{A\}^\{\\perp\}approximately flat\.*Outcome on the frozen endpoint: not confirmed\.*CA⟂C\_\{A\}^\{\\perp\}changes by57/72/85%57/72/85\\%at 70M/160M/410M \(16%16\\%at 1B,42%42\\%at 1\.4B\), failing at the three smaller sizes and approximately holding at 1B\.*Exploratory analysis*via the exact identityH=Hb​\(s\)\+\(1−s\)​H​\(π\)H=H\_\{b\}\(s\)\+\(1\-s\)H\(\\pi\)\(Fig\.[5](https://arxiv.org/html/2608.14712#S5.F5)b\): at 410MHHcollapses3\.63→1\.523\.63\\to 1\.52whileH⁡\(π\)H\(\\pi\)moves0\.60\.6nats, and at 1BH⁡\(π\)H\(\\pi\)moves0\.070\.07nats through a1\.61\.6\-nat collapse; the share1−Δ​H​\(π\)/Δ​H1\-\\Delta H\(\\pi\)/\\Delta H\(the fraction of the drop not attributable to a fall inH⁡\(π\)H\(\\pi\)\) is30/53/70/95/79%30/53/70/95/79\\%across 70M–1\.4B, rising to95%95\\%at 1B and remaining high \(79%79\\%\) at 1\.4B \(signed decomposition: Appendix[E\.2](https://arxiv.org/html/2608.14712#A5.SS2)\)\.

## 6Conclusion

Attention rows are compositions, and the field’s standard summaries mix a sink question with a content question under an unreported convention\. We separated the channels exactly, showed the separation changes published\-style conclusions \(verdict flips near chance, a clustering whose main structure is the convention, a collapse that is mostly sink\), and mapped when each channel matters functionally\.What the theory does and does not claim\.Theorems[1](https://arxiv.org/html/2608.14712#Thmtheorem1)–[2](https://arxiv.org/html/2608.14712#Thmtheorem2)are statements about*measurement*: for relative allocation among content tokens, the classical toolkit answers a convention\-dependent question, the Aitchison toolkit a well\-posed one\. We make no claim that computation “uses” log\-ratios\.Limitations\.The scale law rests on one family; the frozen regime test returned one confirmation, one abstention, one failure \(OPT\-125M: uniformly high sink, benign pruning, a third behavior the map lacks\);dA⟂d\_\{A\}^\{\\perp\}\-L1 selects unstable sets across resampling; audits are signature\-level rather than per\-row\.Future work:fittingdλ2=\(dA⟂\)2\+λ​\(Δ​b\)2d\_\{\\lambda\}^\{2\}=\(d\_\{A\}^\{\\perp\}\)^\{2\}\+\\lambda\(\\Delta b\)^\{2\}from held\-out functional evidence, per\-row audits, and value\-weighted extensions\.

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## Appendix AWorked example and the geometry of the reversal

The witness in full\.The triple of Figure[1](https://arxiv.org/html/2608.14712#S1.F1)and Table[1](https://arxiv.org/html/2608.14712#S4.T1)isp=\(0\.90,0\.08,0\.02\)p=\(0\.90,0\.08,0\.02\),q=\(0\.70,0\.24,0\.06\)q=\(0\.70,0\.24,0\.06\),r=\(0\.90,0\.02,0\.08\)r=\(0\.90,0\.02,0\.08\)over parts\(sink,a,b\)\(\\text\{sink\},a,b\)\. In the parameterization of Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1):π=\(0\.8,0\.2\)\\pi=\(0\.8,0\.2\),π~=\(0\.2,0\.8\)\\tilde\{\\pi\}=\(0\.2,0\.8\)\(the transposition of the two content parts\),s=0\.90s=0\.90,s′=0\.70s^\{\\prime\}=0\.70\. All values in Table[1](https://arxiv.org/html/2608.14712#S4.T1)are exact evaluations \(code:experiments/exact\_examples\.py\)\. The sink balances areb⁡\(p\)=b⁡\(r\)=2\.5422b\(p\)=b\(r\)=2\.5422andb⁡\(q\)=1\.4400b\(q\)=1\.4400, and the Pythagorean decomposition of Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)verifies to machine precision:

dA​\(p,q\)2=1\.214868=0\+\(2\.5422−1\.4400\)2,dA​\(p,r\)2=3\.843624=1\.96052\+0\.d\_\{A\}\(p,q\)^\{2\}=1\.214868=0\+\(2\.5422\-1\.4400\)^\{2\},\\qquad d\_\{A\}\(p,r\)^\{2\}=3\.843624=1\.9605^\{2\}\+0\.The\(p,q\)\(p,q\)discrepancy is entirely sink balance; the\(p,r\)\(p,r\)discrepancy is entirely content\. Cosine, JS, and Euclidean distance mix the two components in a renormalization\-dependent way, which is precisely what the reversal exploits\.

![Refer to caption](https://arxiv.org/html/2608.14712v1/fig_wedge.png)Figure 6:The reversal region is not thin\.Fixingppandrrof the witness and sweeping the third rowqqover the interior of the simplex \(20,00020\{,\}000Dirichlet\(1,1,1\)\(1,1,1\)samples with all parts\>0\.004\>0\.004\), the cosine verdict about which ofq,rq,ris closer toppdiffers between the with\-sink and sink\-dropped conventions on77\.9%77\.9\\%of positions \(red\)\.How large is the flip region for one anchor pair?Figure[6](https://arxiv.org/html/2608.14712#A1.F6)fixes\(p,r\)\(p,r\)and sweepsqq: the cosine verdict is convention\-dependent on77\.9%77\.9\\%of the simplex\. This complements the positive\-measure statement of Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)\(iii\): near strong sinks, convention\-dependence is the typical case, not the exception\.

Transparency note: totaldAd\_\{A\}versus contentdA⟂d\_\{A\}^\{\\perp\}\.In the Monte Carlo of Figure[3](https://arxiv.org/html/2608.14712#S5.F3), the*total*Aitchison distance and the*content*distancedA⟂d\_\{A\}^\{\\perp\}disagree about the closer head on0\.1%0\.1\\%of triples, at every sink level\. The two are different, explicitly declared estimands; whenever they disagree the discrepancy is exactly the sink\-balance term of Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1), in contrast to the undisclosed convention\-dependence of cosine/JS/Euclidean\. An analyst who wants sink\-inclusive similarity reportsdAd\_\{A\}; one who wants content similarity reportsdA⟂d\_\{A\}^\{\\perp\}; both are stable under the keep\-vs\-drop choice\.

Figure 7:A measured Llama\-3\.2\-1B head triple realizing Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)\.Found automatically byfind\_real\_triple\.pyfrom the saved seed\-0 signatures: with the sink, cosine, JS, and Euclidean distance all rankrrcloser topp; after dropping the sink and re\-closing, all three rankqqcloser; thedA⟂d\_\{A\}^\{\\perp\}verdict is identical in both pipelines\.
## Appendix BProofs for Section[4\.1](https://arxiv.org/html/2608.14712#S4.SS1)

#### Standing convention\.

For every integerm≥2m\\geq 2, let𝒮m:=\{x∈\(0,∞\)m:𝟏⊤​x=1\}\\mathcal\{S\}^\{m\}:=\\\{x\\in\(0,\\infty\)^\{m\}:\\mathbf\{1\}^\{\\top\}x=1\\\}\. Forx∈𝒮mx\\in\\mathcal\{S\}^\{m\}, letlog⁡x\\log xdenote the componentwise logarithm and write

clrm⁡\(x\):=log⁡x−1m​\(𝟏⊤​log⁡x\)​𝟏\.\\clr\_\{m\}\(x\):=\\log x\-\\frac\{1\}\{m\}\\bigl\(\\mathbf\{1\}^\{\\top\}\\log x\\bigr\)\\mathbf\{1\}\.WheneverdAd\_\{A\}is applied to twomm\-part compositions, it denotes the unrescaled CLR distancedA​\(x,y\):=∥clrm⁡\(x\)−clrm⁡\(y\)∥2d\_\{A\}\(x,y\):=\\lVert\\clr\_\{m\}\(x\)\-\\clr\_\{m\}\(y\)\\rVert\_\{2\}; the dimension of the CLR map is determined by the arguments ofdAd\_\{A\}, so full\-row and sink\-removed distances useclrD\\clr\_\{D\}andclrD−1\\clr\_\{D\-1\}, respectively\. This is a metric on𝒮m\\mathcal\{S\}^\{m\}: ifclrm⁡\(x\)=clrm⁡\(y\)\\clr\_\{m\}\(x\)=\\clr\_\{m\}\(y\), thenlog⁡x−log⁡y=c​𝟏\\log x\-\\log y=c\\mathbf\{1\}for somec∈ℝc\\in\\mathbb\{R\}, sox=ec​yx=e^\{c\}y; since both vectors sum to one,c=0c=0andx=yx=y\. All logarithms are natural, andH\(x\):=−∑jxjlogxjH\(x\):=\-\\sum\_\{j\}x\_\{j\}\\log x\_\{j\}with0​log⁡0:=00\\log 0:=0\.

###### Proof of Theorem[1](https://arxiv.org/html/2608.14712#Thmtheorem1)\.

Put

t:=1−s,u:=1−s′,a:=∥π∥22,β:=⟨π,π~⟩,Δ:=a−β\.t:=1\-s,\\qquad u:=1\-s^\{\\prime\},\\qquad a:=\\lVert\\pi\\rVert\_\{2\}^\{2\},\\qquad\\beta:=\\langle\\pi,\\widetilde\{\\pi\}\\rangle,\\qquad\\Delta:=a\-\\beta\.Becauseπ~\\widetilde\{\\pi\}is a permutation ofπ\\pi,∥π~∥22=a\\lVert\\widetilde\{\\pi\}\\rVert\_\{2\}^\{2\}=a\. Cauchy–Schwarz givesβ≤a\\beta\\leq a, and equality would forceπ~\\widetilde\{\\pi\}to be a positive scalar multiple ofπ\\pi\. Since both vectors sum to one, equality would implyπ~=π\\widetilde\{\\pi\}=\\pi, contrary to assumption\. Thus

0<a<1,0<β<a,0<Δ<a,∥π−π~∥22=2​Δ\.0<a<1,\\qquad 0<\\beta<a,\\qquad 0<\\Delta<a,\\qquad\\lVert\\pi\-\\widetilde\{\\pi\}\\rVert\_\{2\}^\{2\}=2\\Delta\.Here

1−a=2​∑i<jπi​πj\>0,1\-a=2\\sum\_\{i<j\}\\pi\_\{i\}\\pi\_\{j\}\>0,becauseD−1≥2D\-1\\geq 2and every coordinate ofπ\\piis positive\.

*Proof of\(i\)\.*The identities forR0​\(ps\),R0​\(q\),R0​\(rs\)R\_\{0\}\(p\_\{s\}\),R\_\{0\}\(q\),R\_\{0\}\(r\_\{s\}\)follow directly from the definition\. HenceR0​\(ps\)=R0​\(q\)R\_\{0\}\(p\_\{s\}\)=R\_\{0\}\(q\)andR0​\(ps\)≠R0​\(rs\)R\_\{0\}\(p\_\{s\}\)\\neq R\_\{0\}\(r\_\{s\}\)\. Metric definiteness proves the Euclidean and arbitrary\-metric assertions\. Jensen–Shannon divergence is zero if and only if its arguments agree\. Finally,

c⁡\(R0​\(ps\),R0​\(q\)\)=1,c⁡\(R0​\(ps\),R0​\(rs\)\)=βa<1\.c\\bigl\(R\_\{0\}\(p\_\{s\}\),R\_\{0\}\(q\)\\bigr\)=1,\\qquad c\\bigl\(R\_\{0\}\(p\_\{s\}\),R\_\{0\}\(r\_\{s\}\)\\bigr\)=\\frac\{\\beta\}\{a\}<1\.
*Proof of\(ii\)\.*Direct calculation gives

∥ps−q∥22\\displaystyle\\lVert p\_\{s\}\-q\\rVert\_\{2\}^\{2\}=\(s−s′\)2​\(1\+a\)=\(u−t\)2​\(1\+a\),\\displaystyle=\(s\-s^\{\\prime\}\)^\{2\}\(1\+a\)=\(u\-t\)^\{2\}\(1\+a\),\(2\)∥ps−rs∥22\\displaystyle\\lVert p\_\{s\}\-r\_\{s\}\\rVert\_\{2\}^\{2\}=2​Δ​t2,\\displaystyle=2\\Delta t^\{2\},\(3\)c⁡\(ps,rs\)\\displaystyle c\(p\_\{s\},r\_\{s\}\)=\(1−t\)2\+β​t2\(1−t\)2\+a​t2=1−Δ​t2\(1−t\)2\+a​t2\.\\displaystyle=\\frac\{\(1\-t\)^\{2\}\+\\beta t^\{2\}\}\{\(1\-t\)^\{2\}\+at^\{2\}\}=1\-\\frac\{\\Delta t^\{2\}\}\{\(1\-t\)^\{2\}\+at^\{2\}\}\.\(4\)Also,

c⁡\(ps,q\)=\(1−t\)​s′\+t​u​a\(1−t\)2\+a​t2​\(s′\)2\+a​u2\.c\(p\_\{s\},q\)=\\frac\{\(1\-t\)s^\{\\prime\}\+tua\}\{\\sqrt\{\(1\-t\)^\{2\}\+at^\{2\}\}\\sqrt\{\(s^\{\\prime\}\)^\{2\}\+au^\{2\}\}\}\.\(5\)
For Jensen–Shannon divergence,

mp​q:=ps\+q2=\(s\+s′2,t\+u2​π\)\.m\_\{pq\}:=\\frac\{p\_\{s\}\+q\}\{2\}=\\left\(\\frac\{s\+s^\{\\prime\}\}\{2\},\\frac\{t\+u\}\{2\}\\pi\\right\)\.Consequently,

KL\(ps∥mp​q\)\\displaystyle\\mathrm\{KL\}\(p\_\{s\}\\\|m\_\{pq\}\)=s​log⁡2​ss\+s′\+t​log⁡2​tt\+u,\\displaystyle=s\\log\\frac\{2s\}\{s\+s^\{\\prime\}\}\+t\\log\\frac\{2t\}\{t\+u\},KL\(q∥mp​q\)\\displaystyle\\mathrm\{KL\}\(q\\\|m\_\{pq\}\)=s′​log⁡2​s′s\+s′\+u​log⁡2​ut\+u\.\\displaystyle=s^\{\\prime\}\\log\\frac\{2s^\{\\prime\}\}\{s\+s^\{\\prime\}\}\+u\\log\\frac\{2u\}\{t\+u\}\.Therefore

JS⁡\(ps,q\)=JSBer​\(s,s′\),\\mathrm\{JS\}\(p\_\{s\},q\)=\\mathrm\{JS\}\_\{\\rm Ber\}\(s,s^\{\\prime\}\),\(6\)where

JSBer​\(v,w\):=JS⁡\(\(v,1−v\),\(w,1−w\)\)\.\\mathrm\{JS\}\_\{\\rm Ber\}\(v,w\):=\\mathrm\{JS\}\\bigl\(\(v,1\-v\),\(w,1\-w\)\\bigr\)\.Similarly,

mp​r:=ps\+rs2=\(s,t​π\+π~2\),m\_\{pr\}:=\\frac\{p\_\{s\}\+r\_\{s\}\}\{2\}=\\left\(s,t\\frac\{\\pi\+\\widetilde\{\\pi\}\}\{2\}\\right\),and direct substitution gives

JS⁡\(ps,rs\)=t​JS​\(π,π~\)\.\\mathrm\{JS\}\(p\_\{s\},r\_\{s\}\)=t\\,\\mathrm\{JS\}\(\\pi,\\widetilde\{\\pi\}\)\.\(7\)
Set

γ:=s′\(s′\)2\+a​\(1−s′\)2∈\(0,1\),η:=1−γ\>0,\\gamma:=\\frac\{s^\{\\prime\}\}\{\\sqrt\{\(s^\{\\prime\}\)^\{2\}\+a\(1\-s^\{\\prime\}\)^\{2\}\}\}\\in\(0,1\),\\qquad\\eta:=1\-\\gamma\>0,and define

ε∗:=min⁡\{u2,u28​log⁡2,η4​1\+a,η8​a\},s∗:=1−ε∗\.\\varepsilon\_\{\\ast\}:=\\min\\left\\\{\\frac\{u\}\{2\},\\frac\{u^\{2\}\}\{8\\log 2\},\\frac\{\\eta\}\{4\\sqrt\{1\+a\}\},\\sqrt\{\\frac\{\\eta\}\{8a\}\}\\right\\\},\\qquad s\_\{\\ast\}:=1\-\\varepsilon\_\{\\ast\}\.Every entry in the minimum is strictly positive\. Moreover,ε∗≤u/2<u\\varepsilon\_\{\\ast\}\\leq u/2<u, sos∗∈\(s′,1\)s\_\{\\ast\}\\in\(s^\{\\prime\},1\)\. This threshold depends ons′s^\{\\prime\}and onπ\\pionly througha=∥π∥22a=\\lVert\\pi\\rVert\_\{2\}^\{2\}; it is therefore uniform over all admissible permutations\.

Supposes\>s∗s\>s\_\{\\ast\}, so0<t<ε∗0<t<\\varepsilon\_\{\\ast\}\. Use

TV⁡\(P,Q\):=12​∥P−Q∥1\.\\operatorname\{TV\}\(P,Q\):=\\tfrac\{1\}\{2\}\\lVert P\-Q\\rVert\_\{1\}\.IfP,QP,Qare Bernoulli laws with parameterss,s′s,s^\{\\prime\}andM=\(P\+Q\)/2M=\(P\+Q\)/2, then

TV⁡\(P,M\)=TV⁡\(Q,M\)=\|s−s′\|2\.\\operatorname\{TV\}\(P,M\)=\\operatorname\{TV\}\(Q,M\)=\\frac\{\|s\-s^\{\\prime\}\|\}\{2\}\.Pinsker’s inequality applied to both terms gives

JSBer​\(s,s′\)≥\(s−s′\)22=\(u−t\)22\.\\mathrm\{JS\}\_\{\\rm Ber\}\(s,s^\{\\prime\}\)\\geq\\frac\{\(s\-s^\{\\prime\}\)^\{2\}\}\{2\}=\\frac\{\(u\-t\)^\{2\}\}\{2\}\.Furthermore,JS⁡\(π,π~\)≤log⁡2\\mathrm\{JS\}\(\\pi,\\widetilde\{\\pi\}\)\\leq\\log 2\. Indeed, ifm=\(π\+π~\)/2m=\(\\pi\+\\widetilde\{\\pi\}\)/2, thenmi≥πi/2m\_\{i\}\\geq\\pi\_\{i\}/2andmi≥π~i/2m\_\{i\}\\geq\\widetilde\{\\pi\}\_\{i\}/2, so each KL divergence in the definition ofJS\\mathrm\{JS\}is at mostlog⁡2\\log 2\. Consequently,

JS⁡\(ps,q\)≥\(u−t\)22\>u28\>t​log⁡2≥JS⁡\(ps,rs\)\.\\mathrm\{JS\}\(p\_\{s\},q\)\\geq\\frac\{\(u\-t\)^\{2\}\}\{2\}\>\\frac\{u^\{2\}\}\{8\}\>t\\log 2\\geq\\mathrm\{JS\}\(p\_\{s\},r\_\{s\}\)\.
The Euclidean identities, together withΔ<a<1\\Delta<a<1andt<u/2t<u/2, give

∥ps−rs∥2\\displaystyle\\lVert p\_\{s\}\-r\_\{s\}\\rVert\_\{2\}=t​2​Δ<t​2​a<u2​2​a<u2​1\+a\\displaystyle=t\\sqrt\{2\\Delta\}<t\\sqrt\{2a\}<\\frac\{u\}\{2\}\\sqrt\{2a\}<\\frac\{u\}\{2\}\\sqrt\{1\+a\}<\(u−t\)​1\+a=∥ps−q∥2\.\\displaystyle<\(u\-t\)\\sqrt\{1\+a\}=\\lVert p\_\{s\}\-q\\rVert\_\{2\}\.
It remains to compare cosine similarities\. Lete0=\(1,0,…,0\)e\_\{0\}=\(1,0,\\ldots,0\)andx^=x/∥x∥2\\widehat\{x\}=x/\\lVert x\\rVert\_\{2\}forx≠0x\\neq 0\. Sincet<u/2<1/2t<u/2<1/2,

\(1−t\)2\+a​t2\>14,\(1\-t\)^\{2\}\+at^\{2\}\>\\frac\{1\}\{4\},and equation \([4](https://arxiv.org/html/2608.14712#A2.E4)\) gives

c⁡\(ps,rs\)\>1−4​a​t2\>1−η2\.c\(p\_\{s\},r\_\{s\}\)\>1\-4at^\{2\}\>1\-\\frac\{\\eta\}\{2\}\.Moreover,γ=⟨e0,q^⟩\\gamma=\\langle e\_\{0\},\\widehat\{q\}\\rangleand

\|c⁡\(ps,q\)−γ\|\\displaystyle\\bigl\|c\(p\_\{s\},q\)\-\\gamma\\bigr\|=\|⟨p^s−e0,q^⟩\|≤∥p^s−e0∥2\\displaystyle=\\bigl\|\\langle\\widehat\{p\}\_\{s\}\-e\_\{0\},\\widehat\{q\}\\rangle\\bigr\|\\leq\\lVert\\widehat\{p\}\_\{s\}\-e\_\{0\}\\rVert\_\{2\}≤2​∥ps−e0∥2=2​t​1\+a<η2\.\\displaystyle\\leq 2\\lVert p\_\{s\}\-e\_\{0\}\\rVert\_\{2\}=2t\\sqrt\{1\+a\}<\\frac\{\\eta\}\{2\}\.Here

∥p^s−e0∥2\\displaystyle\\lVert\\widehat\{p\}\_\{s\}\-e\_\{0\}\\rVert\_\{2\}≤∥p^s−ps∥2\+∥ps−e0∥2\\displaystyle\\leq\\lVert\\widehat\{p\}\_\{s\}\-p\_\{s\}\\rVert\_\{2\}\+\\lVert p\_\{s\}\-e\_\{0\}\\rVert\_\{2\}=\|1−∥ps∥2\|\+∥ps−e0∥2\\displaystyle=\|1\-\\lVert p\_\{s\}\\rVert\_\{2\}\|\+\\lVert p\_\{s\}\-e\_\{0\}\\rVert\_\{2\}≤2​∥ps−e0∥2,\\displaystyle\\leq 2\\lVert p\_\{s\}\-e\_\{0\}\\rVert\_\{2\},where the last inequality is the reverse triangle inequality\. Since1−η/2=γ\+η/21\-\\eta/2=\\gamma\+\\eta/2,

c⁡\(ps,rs\)\>1−η2=γ\+η2\>c⁡\(ps,q\)\.c\(p\_\{s\},r\_\{s\}\)\>1\-\\frac\{\\eta\}\{2\}=\\gamma\+\\frac\{\\eta\}\{2\}\>c\(p\_\{s\},q\)\.This proves all three with\-sink inequalities\. Part \(i\) then gives the simultaneous reversals\.

The argument above applies verbatim to every permutationτ\\tauwithτ​π≠π\\tau\\pi\\neq\\pi\(replaceπ~\\widetilde\{\\pi\}byτ​π\\tau\\pi\); since there are finitely many suchτ\\tau, takings∗s\_\{\\ast\}to be the maximum of the per\-τ\\tauthresholds yields the threshold of \(ii\), uniform overτ\\tau\.

*Proof of\(iii\)\.*Forz=\(x,y,w\)∈\(𝒮D\)3z=\(x,y,w\)\\in\(\\mathcal\{S\}^\{D\}\)^\{3\}, define

F1​\(z\)\\displaystyle F\_\{1\}\(z\):=c⁡\(x,w\)−c⁡\(x,y\),\\displaystyle:=c\(x,w\)\-c\(x,y\),F2​\(z\)\\displaystyle F\_\{2\}\(z\):=c⁡\(R0​\(x\),R0​\(y\)\)−c⁡\(R0​\(x\),R0​\(w\)\),\\displaystyle:=c\\bigl\(R\_\{0\}\(x\),R\_\{0\}\(y\)\\bigr\)\-c\\bigl\(R\_\{0\}\(x\),R\_\{0\}\(w\)\\bigr\),F3​\(z\)\\displaystyle F\_\{3\}\(z\):=JS⁡\(x,y\)−JS⁡\(x,w\),\\displaystyle:=\\mathrm\{JS\}\(x,y\)\-\\mathrm\{JS\}\(x,w\),F4​\(z\)\\displaystyle F\_\{4\}\(z\):=JS⁡\(R0​\(x\),R0​\(w\)\)−JS⁡\(R0​\(x\),R0​\(y\)\),\\displaystyle:=\\mathrm\{JS\}\\bigl\(R\_\{0\}\(x\),R\_\{0\}\(w\)\\bigr\)\-\\mathrm\{JS\}\\bigl\(R\_\{0\}\(x\),R\_\{0\}\(y\)\\bigr\),F5​\(z\)\\displaystyle F\_\{5\}\(z\):=∥x−y∥22−∥x−w∥22,\\displaystyle:=\\lVert x\-y\\rVert\_\{2\}^\{2\}\-\\lVert x\-w\\rVert\_\{2\}^\{2\},F6​\(z\)\\displaystyle F\_\{6\}\(z\):=∥R0​\(x\)−R0​\(w\)∥22−∥R0​\(x\)−R0​\(y\)∥22\.\\displaystyle:=\\lVert R\_\{0\}\(x\)\-R\_\{0\}\(w\)\\rVert\_\{2\}^\{2\}\-\\lVert R\_\{0\}\(x\)\-R\_\{0\}\(y\)\\rVert\_\{2\}^\{2\}\.Then

ℛD=⋂j=16Fj−1​\(\(0,∞\)\)\.\\mathcal\{R\}\_\{D\}=\\bigcap\_\{j=1\}^\{6\}F\_\{j\}^\{\-1\}\\bigl\(\(0,\\infty\)\\bigr\)\.Squaring a nonnegative distance does not change a strict comparison\. Moreover,

1−x0=∑i=1D−1xi\>01\-x\_\{0\}=\\sum\_\{i=1\}^\{D\-1\}x\_\{i\}\>0on𝒮D\\mathcal\{S\}^\{D\}, soR0R\_\{0\}is smooth there\. Cosine similarity, Jensen–Shannon divergence, and squared Euclidean distance are continuous on the relevant positive simplices\. Thus everyFjF\_\{j\}is continuous\. Choosing anys∈\(s∗,1\)s\\in\(s\_\{\\ast\},1\), parts \(i\)–\(ii\) give

\(ps,q,rs\)∈ℛD,\(p\_\{s\},q,r\_\{s\}\)\\in\\mathcal\{R\}\_\{D\},soℛD\\mathcal\{R\}\_\{D\}is nonempty\.

Let

𝒜D:=\{x∈ℝD:∑i=0D−1xi=1\}\.\\mathcal\{A\}\_\{D\}:=\\left\\\{x\\in\\mathbb\{R\}^\{D\}:\\sum\_\{i=0\}^\{D\-1\}x\_\{i\}=1\\right\\\}\.Because

𝒮D=𝒜D∩\(0,∞\)D,\\mathcal\{S\}^\{D\}=\\mathcal\{A\}\_\{D\}\\cap\(0,\\infty\)^\{D\},the set\(𝒮D\)3\(\\mathcal\{S\}^\{D\}\)^\{3\}is open in the3​\(D−1\)3\(D\-1\)\-dimensional affine space𝒜D3\\mathcal\{A\}\_\{D\}^\{3\}\. HenceℛD\\mathcal\{R\}\_\{D\}is open in𝒜D3\\mathcal\{A\}\_\{D\}^\{3\}\. Every nonempty open subset of that affine space contains a nondegenerate relative Euclidean ball and therefore has positive intrinsic3​\(D−1\)3\(D\-1\)\-dimensional Lebesgue measure\. This proves part \(iii\)\.

*Proof of\(iv\)\.*BecausedAd\_\{A\}is a metric andR0R\_\{0\}is a map, its pullbackdA⟂d\_\{A\}^\{\\perp\}is nonnegative and symmetric and satisfies the triangle inequality\. Moreover,

dA⟂\(x,y\)=0⟺R0\(x\)=R0\(y\)⟺x∼0y\.d\_\{A\}^\{\\perp\}\(x,y\)=0\\quad\\Longleftrightarrow\\quad R\_\{0\}\(x\)=R\_\{0\}\(y\)\\quad\\Longleftrightarrow\\quad x\\sim\_\{0\}y\.ThereforedA⟂d\_\{A\}^\{\\perp\}is a pseudometric on𝒮D\\mathcal\{S\}^\{D\}, and

dA,0​\(\[x\],\[y\]\):=dA⟂​\(x,y\)d\_\{A,0\}\(\[x\],\[y\]\):=d\_\{A\}^\{\\perp\}\(x,y\)defines a genuine metric on𝒮D/∼0\\mathcal\{S\}^\{D\}/\{\\sim\_\{0\}\}\. It is well defined becausex∼0x′x\\sim\_\{0\}x^\{\\prime\}andy∼0y′y\\sim\_\{0\}y^\{\\prime\}implyR0​\(x\)=R0​\(x′\)R\_\{0\}\(x\)=R\_\{0\}\(x^\{\\prime\}\)andR0​\(y\)=R0​\(y′\)R\_\{0\}\(y\)=R\_\{0\}\(y^\{\\prime\}\)\.

The identities fordA⟂d\_\{A\}^\{\\perp\}follow from its definition and part \(i\)\. Moreover,B\>0B\>0becausedAd\_\{A\}is a metric andπ≠π~\\pi\\neq\\widetilde\{\\pi\}\.

Put

Ls:=log⁡\(s⁡\(1−s′\)s′​\(1−s\)\)\>0\.L\_\{s\}:=\\log\\\!\\left\(\\frac\{s\(1\-s^\{\\prime\}\)\}\{s^\{\\prime\}\(1\-s\)\}\\right\)\>0\.Direct centering of the coordinatewise log\-ratio gives

clrD⁡\(ps\)−clrD⁡\(q\)=LsD​\(D−1,−1,…,−1\)\.\\clr\_\{D\}\(p\_\{s\}\)\-\\clr\_\{D\}\(q\)=\\frac\{L\_\{s\}\}\{D\}\(D\-1,\-1,\\ldots,\-1\)\.Therefore

dA​\(ps,q\)=‖clrD⁡\(ps\)−clrD⁡\(q\)‖2=αD​Ls=αD​\(logit⁡\(s\)−logit⁡\(s′\)\)\.d\_\{A\}\(p\_\{s\},q\)=\\left\\lVert\\clr\_\{D\}\(p\_\{s\}\)\-\\clr\_\{D\}\(q\)\\right\\rVert\_\{2\}=\\alpha\_\{D\}L\_\{s\}=\\alpha\_\{D\}\\bigl\(\\operatorname\{logit\}\(s\)\-\\operatorname\{logit\}\(s^\{\\prime\}\)\\bigr\)\.
Next set

hi:=log⁡πiπ~i\.h\_\{i\}:=\\log\\frac\{\\pi\_\{i\}\}\{\\widetilde\{\\pi\}\_\{i\}\}\.Becauseπ~\\widetilde\{\\pi\}is a permutation ofπ\\pi,

∑i=1D−1hi=log⁡\(∏i=1D−1πi∏i=1D−1π~i\)=0\.\\sum\_\{i=1\}^\{D\-1\}h\_\{i\}=\\log\\\!\\left\(\\frac\{\\prod\_\{i=1\}^\{D\-1\}\\pi\_\{i\}\}\{\\prod\_\{i=1\}^\{D\-1\}\\widetilde\{\\pi\}\_\{i\}\}\\right\)=0\.Consequently,

clrD⁡\(ps\)−clrD⁡\(rs\)=\(0,h1,…,hD−1\),\\clr\_\{D\}\(p\_\{s\}\)\-\\clr\_\{D\}\(r\_\{s\}\)=\(0,h\_\{1\},\\ldots,h\_\{D\-1\}\),while

clrD−1⁡\(π\)−clrD−1⁡\(π~\)=\(h1,…,hD−1\)\.\\clr\_\{D\-1\}\(\\pi\)\-\\clr\_\{D\-1\}\(\\widetilde\{\\pi\}\)=\(h\_\{1\},\\ldots,h\_\{D\-1\}\)\.Thus

dA​\(ps,rs\)=dA​\(π,π~\)=B\.d\_\{A\}\(p\_\{s\},r\_\{s\}\)=d\_\{A\}\(\\pi,\\widetilde\{\\pi\}\)=B\.
Let

E:=exp⁡\(B/αD\)\>1\.E:=\\exp\(B/\\alpha\_\{D\}\)\>1\.Then

sA=s′​E1−s′\+s′​E,s\_\{\\mathrm\{A\}\}=\\frac\{s^\{\\prime\}E\}\{1\-s^\{\\prime\}\+s^\{\\prime\}E\},and

sA−s′=s′​\(1−s′\)​\(E−1\)1−s′\+s′​E\>0,1−sA=1−s′1−s′\+s′​E\>0\.s\_\{\\mathrm\{A\}\}\-s^\{\\prime\}=\\frac\{s^\{\\prime\}\(1\-s^\{\\prime\}\)\(E\-1\)\}\{1\-s^\{\\prime\}\+s^\{\\prime\}E\}\>0,\\qquad 1\-s\_\{\\mathrm\{A\}\}=\\frac\{1\-s^\{\\prime\}\}\{1\-s^\{\\prime\}\+s^\{\\prime\}E\}\>0\.HencesA∈\(s′,1\)s\_\{\\mathrm\{A\}\}\\in\(s^\{\\prime\},1\)\. Moreover,

LsA=log⁡E=BαD,dd​s​Ls=1s⁡\(1−s\)\>0\.L\_\{s\_\{\\mathrm\{A\}\}\}=\\log E=\\frac\{B\}\{\\alpha\_\{D\}\},\\qquad\\frac\{d\}\{ds\}L\_\{s\}=\\frac\{1\}\{s\(1\-s\)\}\>0\.ThereforedA​\(ps,q\)<Bd\_\{A\}\(p\_\{s\},q\)<Bexactly whens<sAs<s\_\{\\mathrm\{A\}\}, with equality ats=sAs=s\_\{\\mathrm\{A\}\}and the reverse inequality whens\>sAs\>s\_\{\\mathrm\{A\}\}\.

In the special caseD=3D=3, for the Table[1](https://arxiv.org/html/2608.14712#S4.T1)values,

\(h1,h2\)=\(log⁡4,−log⁡4\),Ls=log⁡\(9/10\)​\(3/10\)\(7/10\)​\(1/10\)=log⁡277\.\(h\_\{1\},h\_\{2\}\)=\(\\log 4,\-\\log 4\),\\qquad L\_\{s\}=\\log\\frac\{\(9/10\)\(3/10\)\}\{\(7/10\)\(1/10\)\}=\\log\\frac\{27\}\{7\}\.Hence

B=2​log⁡4,dA​\(ps,q\)=23​log⁡277\.B=\\sqrt\{2\}\\log 4,\\qquad d\_\{A\}\(p\_\{s\},q\)=\\sqrt\{\\frac\{2\}\{3\}\}\\log\\frac\{27\}\{7\}\.Finally,

23​log⁡277<23​log⁡4<2​log⁡4,\\sqrt\{\\frac\{2\}\{3\}\}\\log\\frac\{27\}\{7\}<\\sqrt\{\\frac\{2\}\{3\}\}\\log 4<\\sqrt\{2\}\\log 4,which completes the proof\. ∎

###### Proof of Proposition[1](https://arxiv.org/html/2608.14712#Thmproposition1)\.

We first establish the entropy identity for the sink–content factorization\. Let

r=\(s,\(1−s\)​π\),s∈\(0,1\),π∈𝒮D−1,r=\\bigl\(s,\(1\-s\)\\pi\\bigr\),\\qquad s\\in\(0,1\),\\quad\\pi\\in\\mathcal\{S\}^\{D\-1\},and define the binary entropy

Hb​\(s\):=−s​log⁡s−\(1−s\)​log⁡\(1−s\)\.H\_\{\\mathrm\{b\}\}\(s\):=\-s\\log s\-\(1\-s\)\\log\(1\-s\)\.Since∑j=1D−1πj=1\\sum\_\{j=1\}^\{D\-1\}\\pi\_\{j\}=1, direct expansion gives

H⁡\(r\)\\displaystyle H\(r\)=−s​log⁡s−∑j=1D−1\(1−s\)​πj​log⁡\(\(1−s\)​πj\)\\displaystyle=\-s\\log s\-\\sum\_\{j=1\}^\{D\-1\}\(1\-s\)\\pi\_\{j\}\\log\\\!\\bigl\(\(1\-s\)\\pi\_\{j\}\\bigr\)=−s​log⁡s−\(1−s\)​log⁡\(1−s\)−\(1−s\)​∑j=1D−1πj​log⁡πj\\displaystyle=\-s\\log s\-\(1\-s\)\\log\(1\-s\)\-\(1\-s\)\\sum\_\{j=1\}^\{D\-1\}\\pi\_\{j\}\\log\\pi\_\{j\}=Hb​\(s\)\+\(1−s\)​H​\(π\)\.\\displaystyle=H\_\{\\mathrm\{b\}\}\(s\)\+\(1\-s\)H\(\\pi\)\.\(8\)
FixD≥3D\\geq 3, and putm:=D−1≥2m:=D\-1\\geq 2\. Define

π\(p\):=\(1m,…,1m\)∈𝒮m,H⁡\(π\(p\)\)=log⁡m\.\\pi^\{\(p\)\}:=\\left\(\\frac\{1\}\{m\},\\ldots,\\frac\{1\}\{m\}\\right\)\\in\\mathcal\{S\}^\{m\},\\qquad H\\bigl\(\\pi^\{\(p\)\}\\bigr\)=\\log m\.For0<ε<1/m0<\\varepsilon<1/m, define

πε\(q\):=\(1−\(m−1\)​ε,ε,…,ε\)∈𝒮m\\pi^\{\(q\)\}\_\{\\varepsilon\}:=\\bigl\(1\-\(m\-1\)\\varepsilon,\\varepsilon,\\ldots,\\varepsilon\\bigr\)\\in\\mathcal\{S\}^\{m\}and setηε:=H⁡\(πε\(q\)\)\\eta\_\{\\varepsilon\}:=H\\bigl\(\\pi^\{\(q\)\}\_\{\\varepsilon\}\\bigr\)\. The vectorπε\(q\)\\pi^\{\(q\)\}\_\{\\varepsilon\}hasmmstrictly positive coordinates that sum to one, and

ηε=−\(1−\(m−1\)​ε\)​log⁡\(1−\(m−1\)​ε\)−\(m−1\)​ε​log⁡ε⟶0\\eta\_\{\\varepsilon\}=\-\\bigl\(1\-\(m\-1\)\\varepsilon\\bigr\)\\log\\bigl\(1\-\(m\-1\)\\varepsilon\\bigr\)\-\(m\-1\)\\varepsilon\\log\\varepsilon\\longrightarrow 0asε↓0\\varepsilon\\downarrow 0\. Indeed, the first term tends to zero by continuity of−x​log⁡x\-x\\log xatx=1x=1, while the second tends to zero byx​log⁡x→0x\\log x\\to 0asx↓0x\\downarrow 0\. Becausem≥2m\\geq 2and every coordinate ofπε\(q\)\\pi^\{\(q\)\}\_\{\\varepsilon\}lies in\(0,1\)\(0,1\), we also haveηε\>0\\eta\_\{\\varepsilon\}\>0\. We may therefore fixε∈\(0,1/m\)\\varepsilon\\in\(0,1/m\)such that0<ηε<log⁡20<\\eta\_\{\\varepsilon\}<\\log 2\.

Fors∈\(0,1\)s\\in\(0,1\), set

ps:=\(s,\(1−s\)​π\(p\)\),q:=\(12,12​πε\(q\)\)\.p\_\{s\}:=\\bigl\(s,\(1\-s\)\\pi^\{\(p\)\}\\bigr\),\\qquad q:=\\left\(\\frac\{1\}\{2\},\\frac\{1\}\{2\}\\pi^\{\(q\)\}\_\{\\varepsilon\}\\right\)\.Both vectors belong to𝒮D\\mathcal\{S\}^\{D\}\. By \([8](https://arxiv.org/html/2608.14712#A2.E8)\),

H⁡\(ps\)=Hb​\(s\)\+\(1−s\)​log⁡m⟶0\(s↑1\),H\(p\_\{s\}\)=H\_\{\\mathrm\{b\}\}\(s\)\+\(1\-s\)\\log m\\longrightarrow 0\\qquad\(s\\uparrow 1\),whereas

H⁡\(q\)=Hb​\(12\)\+12​ηε=log⁡2\+12​ηε\>0\.H\(q\)=H\_\{\\mathrm\{b\}\}\\\!\\left\(\\frac\{1\}\{2\}\\right\)\+\\frac\{1\}\{2\}\\eta\_\{\\varepsilon\}=\\log 2\+\\frac\{1\}\{2\}\\eta\_\{\\varepsilon\}\>0\.SinceH⁡\(q\)\>0H\(q\)\>0is fixed, there existss0∈\(0,1\)s\_\{0\}\\in\(0,1\), sufficiently close to one, such thatH⁡\(ps0\)<H⁡\(q\)H\(p\_\{s\_\{0\}\}\)<H\(q\)\. Setp:=ps0p:=p\_\{s\_\{0\}\}\. Removing the sink and re\-closing givesp′=π\(p\)p^\{\\prime\}=\\pi^\{\(p\)\}andq′=πε\(q\)q^\{\\prime\}=\\pi^\{\(q\)\}\_\{\\varepsilon\}, and hence

H⁡\(p′\)=log⁡m≥log⁡2\>ηε=H⁡\(q′\)\.H\(p^\{\\prime\}\)=\\log m\\geq\\log 2\>\\eta\_\{\\varepsilon\}=H\(q^\{\\prime\}\)\.This proves the asserted inversion for everyD≥3D\\geq 3\.

We now verify the displayedD=3D=3witness by exact inequalities\. Direct re\-closure gives

P′=\(12,12\),Q′=\(9991000,11000\)\.P^\{\\prime\}=\\left\(\\frac\{1\}\{2\},\\frac\{1\}\{2\}\\right\),\\qquad Q^\{\\prime\}=\\left\(\\frac\{999\}\{1000\},\\frac\{1\}\{1000\}\\right\)\.ForPP,

H⁡\(P\)=1920​log⁡2019\+120​log⁡40,H\(P\)=\\frac\{19\}\{20\}\\log\\frac\{20\}\{19\}\+\\frac\{1\}\{20\}\\log 40,so

20​H​\(P\)=log⁡\[40​\(2019\)19\]\.20H\(P\)=\\log\\\!\\left\[40\\left\(\\frac\{20\}\{19\}\\right\)^\{19\}\\right\]\.Since

\(1\+119\)19<e<3,\\left\(1\+\\frac\{1\}\{19\}\\right\)^\{19\}<e<3,we obtain

40​\(2019\)19<120<220\.40\\left\(\\frac\{20\}\{19\}\\right\)^\{19\}<120<2^\{20\}\.Taking logarithms and dividing by2020yieldsH⁡\(P\)<log⁡2H\(P\)<\\log 2\. Applying \([8](https://arxiv.org/html/2608.14712#A2.E8)\) toQQgives

H⁡\(Q\)=log⁡2\+12​Hb​\(11000\)\>log⁡2\.H\(Q\)=\\log 2\+\\frac\{1\}\{2\}H\_\{\\mathrm\{b\}\}\\\!\\left\(\\frac\{1\}\{1000\}\\right\)\>\\log 2\.ThusH⁡\(P\)<H⁡\(Q\)H\(P\)<H\(Q\)exactly\.

Next,

Hb′\(t\)=log1−tt,Hb′′\(t\)=−1t⁡\(1−t\)<0\(0<t<1\)\.H\_\{\\mathrm\{b\}\}^\{\\prime\}\(t\)=\\log\\frac\{1\-t\}\{t\},\\qquad H\_\{\\mathrm\{b\}\}^\{\\prime\\prime\}\(t\)=\-\\frac\{1\}\{t\(1\-t\)\}<0\\qquad\(0<t<1\)\.HenceHbH\_\{\\mathrm\{b\}\}is strictly concave and has its unique maximumlog⁡2\\log 2att=1/2t=1/2\. Since1/1000≠1/21/1000\\neq 1/2,

H⁡\(Q′\)=Hb​\(11000\)<log⁡2=H⁡\(P′\)\.H\(Q^\{\\prime\}\)=H\_\{\\mathrm\{b\}\}\\\!\\left\(\\frac\{1\}\{1000\}\\right\)<\\log 2=H\(P^\{\\prime\}\)\.
Finally, for a binary compositionu=\(a,b\)u=\(a,b\),

clr2⁡\(u\)=\(12​log⁡ab,−12​log⁡ab\)\.\\clr\_\{2\}\(u\)=\\left\(\\frac\{1\}\{2\}\\log\\frac\{a\}\{b\},\-\\frac\{1\}\{2\}\\log\\frac\{a\}\{b\}\\right\)\.It follows that

clr2⁡\(P′\)=\(0,0\),clr2⁡\(Q′\)=\(12​log​999,−12​log​999\),\\clr\_\{2\}\(P^\{\\prime\}\)=\(0,0\),\\qquad\\clr\_\{2\}\(Q^\{\\prime\}\)=\\left\(\\frac\{1\}\{2\}\\log 999,\-\\frac\{1\}\{2\}\\log 999\\right\),and therefore

CA⟂​\(P\)=0,CA⟂​\(Q\)=log⁡9992\.C\_\{A\}^\{\\perp\}\(P\)=0,\\qquad C\_\{A\}^\{\\perp\}\(Q\)=\\frac\{\\log 999\}\{\\sqrt\{2\}\}\.This completes the verification of the witness and the concentration values\. ∎

## Appendix CProofs for Section[4\.2](https://arxiv.org/html/2608.14712#S4.SS2)

Table 4:Invariance matrix\.Perm\.==permutation invariance; Perturb\.==invariance to a shared logit bias \(Prop\.[2](https://arxiv.org/html/2608.14712#Thmproposition2)\(i\)\); Temp\.==global\-temperature ranking preservation; Sink==coherent under keep\-vs\-drop of the sink column; Dom\.==subcompositional dominance\.†\\dagger:dAd\_\{A\}is1/τ1/\\tau\-equivariant, so rankings are exact;ρA\\rho\_\{A\}is invariant even to per\-head temperatures\.‡\\ddagger:dAd\_\{A\}decomposes exactly \(Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\);dA⟂,CA⟂d\_\{A\}^\{\\perp\},C\_\{A\}^\{\\perp\}are exactly sink\-invariant\. “n/a”: not applicable\.#### Standing notation\.

All logarithms are natural\. For every positive vectorx=\(x0,…,xm−1\)x=\(x\_\{0\},\\ldots,x\_\{m\-1\}\), write𝒞⁡\(x\):=x/∑ixi\\mathcal\{C\}\(x\):=x/\\sum\_\{i\}x\_\{i\}\. For every integerm≥1m\\geq 1, let𝒮m\\mathcal\{S\}^\{m\}be the openmm\-part simplex,em:=1m​𝟏me\_\{m\}:=\\frac\{1\}\{m\}\\mathbf\{1\}\_\{m\},

g⁡\(x1,…,xm\):=\(∏i=1mxi\)1/m,clrm⁡\(x\):=\(log⁡xi−m−1​∑jlog⁡xj\)i,g\(x\_\{1\},\\ldots,x\_\{m\}\):=\\Bigl\(\\prod\_\{i=1\}^\{m\}x\_\{i\}\\Bigr\)^\{1/m\},\\qquad\\clr\_\{m\}\(x\):=\\bigl\(\\log x\_\{i\}\-m^\{\-1\}\\textstyle\\sum\_\{j\}\\log x\_\{j\}\\bigr\)\_\{i\},and throughoutdA​\(x,y\):=∥clrm⁡\(x\)−clrm⁡\(y\)∥2d\_\{A\}\(x,y\):=\\lVert\\clr\_\{m\}\(x\)\-\\clr\_\{m\}\(y\)\\rVert\_\{2\}forx,y∈𝒮mx,y\\in\\mathcal\{S\}^\{m\}, with no dimension\-dependent rescaling\. ForD≥2D\\geq 2andp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}with designated sink coordinate00,

p′:=𝒞⁡\(p1,…,pD−1\),b⁡\(p\):=D−1D​log⁡p0g⁡\(p1,…,pD−1\),p^\{\\prime\}:=\\mathcal\{C\}\(p\_\{1\},\\ldots,p\_\{D\-1\}\),\\qquad b\(p\):=\\sqrt\{\\tfrac\{D\-1\}\{D\}\}\\log\\tfrac\{p\_\{0\}\}\{g\(p\_\{1\},\\ldots,p\_\{D\-1\}\)\},dA⟂​\(p,q\):=dA​\(p′,q′\),CA⟂​\(p\):=∥clrD−1⁡\(p′\)∥2\.d\_\{A\}^\{\\perp\}\(p,q\):=d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\),\\qquad C\_\{A\}^\{\\perp\}\(p\):=\\lVert\\clr\_\{D\-1\}\(p^\{\\prime\}\)\\rVert\_\{2\}\.Perturbation, powering, and the Aitchison center arex⊕y:=𝒞⁡\(\(xi​yi\)i\)x\\oplus y:=\\mathcal\{C\}\(\(x\_\{i\}y\_\{i\}\)\_\{i\}\),α⊙x:=𝒞⁡\(\(xiα\)i\)\\alpha\\odot x:=\\mathcal\{C\}\(\(x\_\{i\}^\{\\alpha\}\)\_\{i\}\), andcen⁡\(x1,…,xn\):=𝒞⁡\(exp⁡\{1n​∑tlog⁡xt\}\)\\cen\(x\_\{1\},\\ldots,x\_\{n\}\):=\\mathcal\{C\}\(\\exp\\\{\\frac\{1\}\{n\}\\sum\_\{t\}\\log x\_\{t\}\\\}\), all componentwise\. Whenx≠emx\\neq e\_\{m\}andy≠emy\\neq e\_\{m\},ρA​\(x,y\):=⟨clrm⁡\(x\),clrm⁡\(y\)⟩/\(∥clrm⁡\(x\)∥2​∥clrm⁡\(y\)∥2\)\\rho\_\{A\}\(x,y\):=\\langle\\clr\_\{m\}\(x\),\\clr\_\{m\}\(y\)\\rangle/\(\\lVert\\clr\_\{m\}\(x\)\\rVert\_\{2\}\\lVert\\clr\_\{m\}\(y\)\\rVert\_\{2\}\), and for nonzero Euclidean vectorscos⁡\(x,y\):=x⊤​y/\(∥x∥2​∥y∥2\)\\cos\(x,y\):=x^\{\\top\}y/\(\\lVert x\\rVert\_\{2\}\\lVert y\\rVert\_\{2\}\)\. Form=1m=1, set𝒮1=\{\(1\)\}\\mathcal\{S\}^\{1\}=\\\{\(1\)\\\}and regard its CLR space and ILR coordinate vector as zero\-dimensional, so its CLR vector, norm, and all distances are zero\.

###### Proof of Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)\.

Form≥1m\\geq 1, let

ℋm:=\{x∈ℝm:𝟏m⊤​x=0\}\.\\mathcal\{H\}\_\{m\}:=\\left\\\{x\\in\\mathbb\{R\}^\{m\}:\\mathbf\{1\}\_\{m\}^\{\\top\}x=0\\right\\\}\.The mapclrm\\clr\_\{m\}takes values inℋm\\mathcal\{H\}\_\{m\}\. Define

ub:=D−1D​\(1,−1D−1,…,−1D−1\)∈ℋDu\_\{b\}:=\\sqrt\{\\frac\{D\-1\}\{D\}\}\\left\(1,\-\\frac\{1\}\{D\-1\},\\ldots,\-\\frac\{1\}\{D\-1\}\\right\)\\in\\mathcal\{H\}\_\{D\}and

W:=\{\(0,z\):z∈ℋD−1\}⊂ℋD,J:ℋD−1⟶W,J​z=\(0,z\)\.W:=\\\{\(0,z\):z\\in\\mathcal\{H\}\_\{D\-1\}\\\}\\subset\\mathcal\{H\}\_\{D\},\\qquad J:\\mathcal\{H\}\_\{D\-1\}\\longrightarrow W,\\qquad Jz=\(0,z\)\.The mapJJis a linear isometry\. Moreover,

∥ub∥22=D−1D​\(1\+D−1\(D−1\)2\)=1,\\lVert u\_\{b\}\\rVert\_\{2\}^\{2\}=\\frac\{D\-1\}\{D\}\\left\(1\+\\frac\{D\-1\}\{\(D\-1\)^\{2\}\}\\right\)=1,and for everyz∈ℋD−1z\\in\\mathcal\{H\}\_\{D\-1\},

⟨ub,Jz⟩=−1D⁡\(D−1\)∑i=1D−1zi=0\.\\langle u\_\{b\},Jz\\rangle=\-\\frac\{1\}\{\\sqrt\{D\(D\-1\)\}\}\\sum\_\{i=1\}^\{D\-1\}z\_\{i\}=0\.SincedimW=D−2\\dim W=D\-2anddimℋD=D−1\\dim\\mathcal\{H\}\_\{D\}=D\-1, it follows that

ℋD=span⁡\{ub\}⊕W,span⁡\{ub\}⟂W\.\\mathcal\{H\}\_\{D\}=\\operatorname\{span\}\\\{u\_\{b\}\\\}\\oplus W,\\qquad\\operatorname\{span\}\\\{u\_\{b\}\\\}\\perp W\.\(9\)
Write uniquely

p=\(s,\(1−s\)​π\),s=p0∈\(0,1\),π=p′∈𝒮D−1\.p=\(s,\(1\-s\)\\pi\),\\qquad s=p\_\{0\}\\in\(0,1\),\\qquad\\pi=p^\{\\prime\}\\in\\mathcal\{S\}^\{D\-1\}\.Because the entries ofubu\_\{b\}sum to zero, the common centering term inclrD⁡\(p\)\\clr\_\{D\}\(p\)vanishes in its inner product withubu\_\{b\}\. Hence

⟨clrD⁡\(p\),ub⟩\\displaystyle\\langle\\clr\_\{D\}\(p\),u\_\{b\}\\rangle=D−1D​\(log⁡p0−1D−1​∑i=1D−1log⁡pi\)\\displaystyle=\\sqrt\{\\frac\{D\-1\}\{D\}\}\\left\(\\log p\_\{0\}\-\\frac\{1\}\{D\-1\}\\sum\_\{i=1\}^\{D\-1\}\\log p\_\{i\}\\right\)=D−1D​log⁡p0g⁡\(p1,…,pD−1\)=b⁡\(p\)\.\\displaystyle=\\sqrt\{\\frac\{D\-1\}\{D\}\}\\log\\frac\{p\_\{0\}\}\{g\(p\_\{1\},\\ldots,p\_\{D\-1\}\)\}=b\(p\)\.\(10\)
LetPWP\_\{W\}denote Euclidean orthogonal projection ontoWW\. Forx∈ℋDx\\in\\mathcal\{H\}\_\{D\},

\(PW​x\)0=0,\(PW​x\)i=xi−1D−1​∑j=1D−1xj,1≤i≤D−1\.\(P\_\{W\}x\)\_\{0\}=0,\\qquad\(P\_\{W\}x\)\_\{i\}=x\_\{i\}\-\\frac\{1\}\{D\-1\}\\sum\_\{j=1\}^\{D\-1\}x\_\{j\},\\quad 1\\leq i\\leq D\-1\.Applying this tox=clrD⁡\(p\)x=\\clr\_\{D\}\(p\)gives, fori≥1i\\geq 1,

\(PW​clrD⁡\(p\)\)i\\displaystyle\(P\_\{W\}\\clr\_\{D\}\(p\)\)\_\{i\}=log⁡pi−1D−1​∑j=1D−1log⁡pj\\displaystyle=\\log p\_\{i\}\-\\frac\{1\}\{D\-1\}\\sum\_\{j=1\}^\{D\-1\}\\log p\_\{j\}=log⁡πi−1D−1​∑j=1D−1log⁡πj=\(clrD−1⁡\(π\)\)i\.\\displaystyle=\\log\\pi\_\{i\}\-\\frac\{1\}\{D\-1\}\\sum\_\{j=1\}^\{D\-1\}\\log\\pi\_\{j\}=\(\\clr\_\{D\-1\}\(\\pi\)\)\_\{i\}\.Therefore

PW​clrD⁡\(p\)=J​clrD−1⁡\(p′\)\.P\_\{W\}\\clr\_\{D\}\(p\)=J\\clr\_\{D\-1\}\(p^\{\\prime\}\)\.\(11\)Equations \([9](https://arxiv.org/html/2608.14712#A3.E9)\)–\([11](https://arxiv.org/html/2608.14712#A3.E11)\) yield the exact orthogonal identity

clrD⁡\(p\)=b⁡\(p\)​ub\+J​clrD−1⁡\(p′\)\.\\boxed\{\\;\\clr\_\{D\}\(p\)=b\(p\)u\_\{b\}\+J\\clr\_\{D\-1\}\(p^\{\\prime\}\)\.\\;\}\(12\)
Apply \([12](https://arxiv.org/html/2608.14712#A3.E12)\) toppandqq, subtract, and use the orthogonality in \([9](https://arxiv.org/html/2608.14712#A3.E9)\) and the isometry ofJJ\. Then

dA​\(p,q\)2\\displaystyle d\_\{A\}\(p,q\)^\{2\}=∥clrD⁡\(p\)−clrD⁡\(q\)∥22\\displaystyle=\\lVert\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)\\rVert\_\{2\}^\{2\}=\(b⁡\(p\)−b⁡\(q\)\)2\+∥clrD−1⁡\(p′\)−clrD−1⁡\(q′\)∥22\\displaystyle=\\bigl\(b\(p\)\-b\(q\)\\bigr\)^\{2\}\+\\lVert\\clr\_\{D\-1\}\(p^\{\\prime\}\)\-\\clr\_\{D\-1\}\(q^\{\\prime\}\)\\rVert\_\{2\}^\{2\}=\(b⁡\(p\)−b⁡\(q\)\)2\+dA⟂​\(p,q\)2,\\displaystyle=\\bigl\(b\(p\)\-b\(q\)\\bigr\)^\{2\}\+d\_\{A\}^\{\\perp\}\(p,q\)^\{2\},which proves the Pythagorean identity\.

To verify the ILR assertion, letw2,…,wD−1w\_\{2\},\\ldots,w\_\{D\-1\}be any orthonormal basis ofWW; the list is empty whenD=2D=2\. Then\(ub,w2,…,wD−1\)\(u\_\{b\},w\_\{2\},\\ldots,w\_\{D\-1\}\)is an orthonormal contrast basis ofℋD\\mathcal\{H\}\_\{D\}and hence defines an ILR map\. Its first coordinate isb⁡\(p\)b\(p\)by \([10](https://arxiv.org/html/2608.14712#A3.E10)\)\. The vectorsJ−1​w2,…,J−1​wD−1J^\{\-1\}w\_\{2\},\\ldots,J^\{\-1\}w\_\{D\-1\}form an orthonormal basis ofℋD−1\\mathcal\{H\}\_\{D\-1\}, and \([11](https://arxiv.org/html/2608.14712#A3.E11)\) gives

⟨clrD⁡\(p\),wj⟩=⟨clrD−1⁡\(p′\),J−1​wj⟩,2≤j≤D−1\.\\langle\\clr\_\{D\}\(p\),w\_\{j\}\\rangle=\\langle\\clr\_\{D\-1\}\(p^\{\\prime\}\),J^\{\-1\}w\_\{j\}\\rangle,\\qquad 2\\leq j\\leq D\-1\.Thus the remaining coordinates are precisely an ILR coordinate system forp′p^\{\\prime\}\. In particular,

dA⟂\(p,q\)=∥PW\(clrD\(p\)−clrD\(q\)\)∥2,CA⟂\(p\)=∥PWclrD\(p\)∥2\.d\_\{A\}^\{\\perp\}\(p,q\)=\\bigl\\lVert P\_\{W\}\(\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)\)\\bigr\\rVert\_\{2\},\\qquad C\_\{A\}^\{\\perp\}\(p\)=\\lVert P\_\{W\}\\clr\_\{D\}\(p\)\\rVert\_\{2\}\.The first expression depends only onp′p^\{\\prime\}andq′q^\{\\prime\}, and the second only onp′p^\{\\prime\}; neither depends on a sink mass\. Equation \([11](https://arxiv.org/html/2608.14712#A3.E11)\) also makes the projection claim precise: it is a statement in CLR space underJJ, not a Euclidean projection of probability vectors\. ∎

#### Exact evaluation of the Table[1](https://arxiv.org/html/2608.14712#S4.T1)witness\.

For

p=\(910,225,150\),q=\(710,625,350\),r=\(910,150,225\),p=\\left\(\\frac\{9\}\{10\},\\frac\{2\}\{25\},\\frac\{1\}\{50\}\\right\),\\qquad q=\\left\(\\frac\{7\}\{10\},\\frac\{6\}\{25\},\\frac\{3\}\{50\}\\right\),\\qquad r=\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{50\},\\frac\{2\}\{25\}\\right\),interpreting the decimal entries in Table[1](https://arxiv.org/html/2608.14712#S4.T1)as these exact rational numbers, one has

p′=q′=\(45,15\),r′=\(15,45\),p^\{\\prime\}=q^\{\\prime\}=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\),\\qquad r^\{\\prime\}=\\left\(\\frac\{1\}\{5\},\\frac\{4\}\{5\}\\right\),and

b⁡\(p\)=b⁡\(r\)\\displaystyle b\(p\)=b\(r\)=23​log⁡452≈2\.542174604636,\\displaystyle=\\sqrt\{\\frac\{2\}\{3\}\}\\log\\frac\{45\}\{2\}\\approx 2\.542174604636,b⁡\(q\)\\displaystyle b\(q\)=23​log⁡356≈1\.439964055745\.\\displaystyle=\\sqrt\{\\frac\{2\}\{3\}\}\\log\\frac\{35\}\{6\}\\approx 1\.439964055745\.Consequently, the pure\-sink and pure\-content identities are

dA⟂​\(p,q\)\\displaystyle d\_\{A\}^\{\\perp\}\(p,q\)=0,\\displaystyle=0,dA​\(p,q\)2\\displaystyle d\_\{A\}\(p,q\)^\{2\}=23​log2​277≈1\.214868094088,\\displaystyle=\\frac\{2\}\{3\}\\log^\{2\}\\frac\{27\}\{7\}\\approx 1\.214868094088,b⁡\(p\)−b⁡\(r\)\\displaystyle b\(p\)\-b\(r\)=0,\\displaystyle=0,dA​\(p,r\)2\\displaystyle d\_\{A\}\(p,r\)^\{2\}=dA​\(p′,r′\)2=2​log2​4=8​log2​2≈3\.843624111346,\\displaystyle=d\_\{A\}\(p^\{\\prime\},r^\{\\prime\}\)^\{2\}=2\\log^\{2\}4=8\\log^\{2\}2\\approx 3\.843624111346,with

dA​\(p,r\)=2​log⁡4≈1\.960516286937\.d\_\{A\}\(p,r\)=\\sqrt\{2\}\\log 4\\approx 1\.960516286937\.Thus rounded decimal displays must use≈\\approx; the exact equalities are the logarithmic identities above\.

### Full statement of Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2)

LetD≥2D\\geq 2, letℓ,ℓ′∈ℝD\\ell,\\ell^\{\\prime\}\\in\\mathbb\{R\}^\{D\}be finite logit rows, and putp=𝒞⁡\(eℓ\)p=\\mathcal\{C\}\(e^\{\\ell\}\),q=𝒞⁡\(eℓ′\)q=\\mathcal\{C\}\(e^\{\\ell^\{\\prime\}\}\), where exponentials, products, and powers are componentwise\.

1. \(i\)*Shared finite logit bias equals perturbation\.*For every finiteu∈ℝDu\\in\\mathbb\{R\}^\{D\}, withcu=𝒞⁡\(eu\)c\_\{u\}=\\mathcal\{C\}\(e^\{u\}\), 𝒞⁡\(eℓ\+u\)=p⊕cu,𝒞⁡\(eℓ′\+u\)=q⊕cu,\\mathcal\{C\}\(e^\{\\ell\+u\}\)=p\\oplus c\_\{u\},\\qquad\\mathcal\{C\}\(e^\{\\ell^\{\\prime\}\+u\}\)=q\\oplus c\_\{u\},anddA​\(p⊕cu,q⊕cu\)=dA​\(p,q\)d\_\{A\}\(p\\oplus c\_\{u\},q\\oplus c\_\{u\}\)=d\_\{A\}\(p,q\)\. For a fixed designated sink,dA⟂d\_\{A\}^\{\\perp\}has the same shared\-perturbation invariance\. IfD≥3D\\geq 3,CA⟂C\_\{A\}^\{\\perp\}is not perturbation\-invariant in general; whenD=2D=2, it is identically zero\. Euclidean distance, Jensen–Shannon divergence, and cosine similarity likewise fail to be invariant under every shared perturbation\. The directional similarityρA\\rho\_\{A\}is also not perturbation\-invariant in general, even when it is defined both before and after perturbation\.
2. \(ii\)*Temperature equals powering\.*For every finite temperatureτ\>0\\tau\>0,𝒞⁡\(eℓ/τ\)=1τ⊙p\\mathcal\{C\}\(e^\{\\ell/\\tau\}\)=\\frac\{1\}\{\\tau\}\\odot pand𝒞⁡\(eℓ′/τ\)=1τ⊙q\\mathcal\{C\}\(e^\{\\ell^\{\\prime\}/\\tau\}\)=\\frac\{1\}\{\\tau\}\\odot q, and hence dA​\(1τ⊙p,1τ⊙q\)=1τ​dA​\(p,q\),dA⟂​\(1τ⊙p,1τ⊙q\)=1τ​dA⟂​\(p,q\),CA⟂​\(1τ⊙p\)=1τ​CA⟂​\(p\)\.d\_\{A\}\\left\(\\tfrac\{1\}\{\\tau\}\\odot p,\\tfrac\{1\}\{\\tau\}\\odot q\\right\)=\\tfrac\{1\}\{\\tau\}d\_\{A\}\(p,q\),\\qquad d\_\{A\}^\{\\perp\}\\left\(\\tfrac\{1\}\{\\tau\}\\odot p,\\tfrac\{1\}\{\\tau\}\\odot q\\right\)=\\tfrac\{1\}\{\\tau\}d\_\{A\}^\{\\perp\}\(p,q\),\\qquad C\_\{A\}^\{\\perp\}\\left\(\\tfrac\{1\}\{\\tau\}\\odot p\\right\)=\\tfrac\{1\}\{\\tau\}C\_\{A\}^\{\\perp\}\(p\)\.Thus a single common temperature applied to every member of a family preserves all weak and strict rankings among the resulting distances or concentration values; these are equivariances, not value invariances\. For a nonempty finite collectionP=\(p1,…,pn\)P=\(p\_\{1\},\\ldots,p\_\{n\}\), defineVarA⁡\(P\):=1n​∑tdA​\(pt,cen⁡\(P\)\)2\\operatorname\{Var\}\_\{A\}\(P\):=\\frac\{1\}\{n\}\\sum\_\{t\}d\_\{A\}\(p\_\{t\},\\cen\(P\)\)^\{2\}\. Then VarA⁡\(1τ⊙p1,…,1τ⊙pn\)=1τ2​VarA⁡\(P\),\\operatorname\{Var\}\_\{A\}\\left\(\\tfrac\{1\}\{\\tau\}\\odot p\_\{1\},\\ldots,\\tfrac\{1\}\{\\tau\}\\odot p\_\{n\}\\right\)=\\tfrac\{1\}\{\\tau^\{2\}\}\\operatorname\{Var\}\_\{A\}\(P\),where the center on the left is recomputed after powering\. Wheneverp≠eDp\\neq e\_\{D\}andq≠eDq\\neq e\_\{D\}, so thatρA​\(p,q\)\\rho\_\{A\}\(p,q\)is defined, separate positive temperatures cancel: for everyτh,τh′\>0\\tau\_\{h\},\\tau\_\{h^\{\\prime\}\}\>0, ρA​\(1τh⊙p,1τh′⊙q\)=ρA​\(p,q\)\.\\rho\_\{A\}\\left\(\\tfrac\{1\}\{\\tau\_\{h\}\}\\odot p,\\tfrac\{1\}\{\\tau\_\{h^\{\\prime\}\}\}\\odot q\\right\)=\\rho\_\{A\}\(p,q\)\.More formally, forh∈\{1,2\}h\\in\\\{1,2\\\}letnh≥1n\_\{h\}\\geq 1andph​t∈𝒮Dp\_\{ht\}\\in\\mathcal\{S\}^\{D\}, putPh:=\(ph​1,…,ph​nh\)P\_\{h\}:=\(p\_\{h1\},\\ldots,p\_\{hn\_\{h\}\}\)andsh:=cen⁡\(Ph\)≠eDs\_\{h\}:=\\cen\(P\_\{h\}\)\\neq e\_\{D\}, and give every row in headhhthe same exponentαh\>0\\alpha\_\{h\}\>0\. Ifsh\(αh\):=cen⁡\(αh⊙ph​1,…,αh⊙ph​nh\)s\_\{h\}^\{\(\\alpha\_\{h\}\)\}:=\\cen\(\\alpha\_\{h\}\\odot p\_\{h1\},\\ldots,\\alpha\_\{h\}\\odot p\_\{hn\_\{h\}\}\), then sh\(αh\)=αh⊙sh,ρA​\(s1\(α1\),s2\(α2\)\)=ρA​\(s1,s2\)\.s\_\{h\}^\{\(\\alpha\_\{h\}\)\}=\\alpha\_\{h\}\\odot s\_\{h\},\\qquad\\rho\_\{A\}\\bigl\(s\_\{1\}^\{\(\\alpha\_\{1\}\)\},s\_\{2\}^\{\(\\alpha\_\{2\}\)\}\\bigr\)=\\rho\_\{A\}\(s\_\{1\},s\_\{2\}\)\.Thus one positive temperature per head cancels inρA\\rho\_\{A\}when the rows have a common support and the two resulting signatures are nonuniform\. This conclusion can fail for row\-dependent temperatures\. Under unequal head temperatures or row\-dependent temperatures, the separate\-scaling cancellation asserted here is specific toρA\\rho\_\{A\}\. No general invariance or ranking\-preservation claim is made fordAd\_\{A\},dA⟂d\_\{A\}^\{\\perp\},CA⟂C\_\{A\}^\{\\perp\}, or Aitchison variance\.
3. \(iii\)*Sink coherence\.*For a fixed designated sink, writep=\(s,\(1−s\)​π\)p=\(s,\(1\-s\)\\pi\)andq=\(t,\(1−t\)​η\)q=\(t,\(1\-t\)\\eta\)withπ,η∈𝒮D−1\\pi,\\eta\\in\\mathcal\{S\}^\{D\-1\}\. ThendA⟂​\(p,q\)=dA​\(π,η\)d\_\{A\}^\{\\perp\}\(p,q\)=d\_\{A\}\(\\pi,\\eta\)andCA⟂​\(p\)=∥clrD−1⁡\(π\)∥2C\_\{A\}^\{\\perp\}\(p\)=\\lVert\\clr\_\{D\-1\}\(\\pi\)\\rVert\_\{2\}\. These values are independent ofssandttand are identical whether computed in the balance\-orthogonal coordinates of the full compositions or after dropping the sink and re\-closing\. FulldAd\_\{A\}is not itself sink\-invariant; Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)gives its exact sink–content decomposition\.
4. \(iv\)*Subcomposition and masking\.*LetS⊆\{0,…,D−1\}S\\subseteq\\\{0,\\ldots,D\-1\\\}with\|S\|≥2\|S\|\\geq 2, and definep\(S\)=𝒞⁡\(\(pi\)i∈S\)p^\{\(S\)\}=\\mathcal\{C\}\(\(p\_\{i\}\)\_\{i\\in S\}\)andq\(S\)=𝒞⁡\(\(qi\)i∈S\)q^\{\(S\)\}=\\mathcal\{C\}\(\(q\_\{i\}\)\_\{i\\in S\}\)\. After canonically embedding the\|S\|\|S\|\-part CLR space intoℋD\\mathcal\{H\}\_\{D\}, subcomposition is an orthogonal projection and dA​\(p\(S\),q\(S\)\)≤dA​\(p,q\)\.d\_\{A\}\\bigl\(p^\{\(S\)\},q^\{\(S\)\}\\bigr\)\\leq d\_\{A\}\(p,q\)\.Coordinate deletion is literal in anSS\-adapted ILR basis; in an arbitrary fixed ILR basis the same map is generally a non\-diagonal orthogonal projection\. Outside the standing finite\-logit hypothesis, letp~\\widetilde\{p\}andq~\\widetilde\{q\}be hard\-masked probability rows in the closed simplex with positive supportsTp~T\_\{\\widetilde\{p\}\}andTq~T\_\{\\widetilde\{q\}\}\. IfS=Tp~∩Tq~S=T\_\{\\widetilde\{p\}\}\\cap T\_\{\\widetilde\{q\}\}has at least two parts,dA​\(p~\(S\),q~\(S\)\)d\_\{A\}\(\\widetilde\{p\}^\{\(S\)\},\\widetilde\{q\}^\{\(S\)\}\)\(of the re\-closed restrictions\) is a well\-defined within\-common\-support comparison\. If the two supports differ, this comparison discards the support mismatch\. Since structural\-zero rows do not lie in the open simplex, their full Aitchison distance is undefined, and no contraction inequality relative to that nonexistent full distance is asserted\. If positive pre\-mask rows in𝒮D\\mathcal\{S\}^\{D\}are available and the same mask retainsSS, the preceding contraction inequality does apply to those pre\-mask rows and theirSS\-subcompositions\.
5. \(v\)*Dominance\.*Part \(iv\) shows thatdAd\_\{A\}is subcompositionally dominant\. For everyD≥3D\\geq 3, Euclidean distance, Jensen–Shannon divergenceJS\\mathrm\{JS\}\(not its square root\), and cosine dissimilarity1−cos1\-\\cosare not subcompositionally dominant\. ForD=2D=2, there is no nontrivial proper subcomposition containing at least two parts\.

###### Proof of Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2)\.

For \(i\), closure is unchanged by multiplication by a common positive scalar\. Therefore

𝒞⁡\(eℓ\+u\)\\displaystyle\\mathcal\{C\}\(e^\{\\ell\+u\}\)=𝒞⁡\(\(eℓi​eui\)i=0D−1\)\\displaystyle=\\mathcal\{C\}\\bigl\(\(e^\{\\ell\_\{i\}\}e^\{u\_\{i\}\}\)\_\{i=0\}^\{D\-1\}\\bigr\)=𝒞⁡\(\(pi​\(cu\)i\)i=0D−1\)=p⊕cu,\\displaystyle=\\mathcal\{C\}\\bigl\(\(p\_\{i\}\(c\_\{u\}\)\_\{i\}\)\_\{i=0\}^\{D\-1\}\\bigr\)=p\\oplus c\_\{u\},and similarly forqq\. Sinceclr⁡\(a⊕b\)=clr⁡\(a\)\+clr⁡\(b\)\\clr\(a\\oplus b\)=\\clr\(a\)\+\\clr\(b\), we obtain

dA​\(p⊕cu,q⊕cu\)\\displaystyle d\_\{A\}\(p\\oplus c\_\{u\},q\\oplus c\_\{u\}\)=∥clr\(p\)\+clr\(cu\)−clr\(q\)−clr\(cu\)∥2\\displaystyle=\\bigl\\lVert\\clr\(p\)\+\\clr\(c\_\{u\}\)\-\\clr\(q\)\-\\clr\(c\_\{u\}\)\\bigr\\rVert\_\{2\}=dA​\(p,q\)\.\\displaystyle=d\_\{A\}\(p,q\)\.
For the content distance, letcu′c\_\{u\}^\{\\prime\}be the sink\-dropped and re\-closed composition\. Directly from the definition of perturbation,

\(p⊕cu\)′=p′⊕cu′,\(q⊕cu\)′=q′⊕cu′\.\(p\\oplus c\_\{u\}\)^\{\\prime\}=p^\{\\prime\}\\oplus c\_\{u\}^\{\\prime\},\\qquad\(q\\oplus c\_\{u\}\)^\{\\prime\}=q^\{\\prime\}\\oplus c\_\{u\}^\{\\prime\}\.Applying the same CLR perturbation identity in dimensionD−1D\-1\(including the stated zero\-dimensional convention whenD=2D=2\) givesdA⟂​\(p⊕cu,q⊕cu\)=dA⟂​\(p,q\)d\_\{A\}^\{\\perp\}\(p\\oplus c\_\{u\},q\\oplus c\_\{u\}\)=d\_\{A\}^\{\\perp\}\(p,q\)\. In contrast,

CA⟂\(p⊕cu\)=∥clrD−1\(p′\)\+clrD−1\(cu′\)∥2\.C\_\{A\}^\{\\perp\}\(p\\oplus c\_\{u\}\)=\\bigl\\lVert\\clr\_\{D\-1\}\(p^\{\\prime\}\)\+\\clr\_\{D\-1\}\(c\_\{u\}^\{\\prime\}\)\\bigr\\rVert\_\{2\}\.ForD≥3D\\geq 3, this equalsCA⟂​\(p\)C\_\{A\}^\{\\perp\}\(p\)for everyppif and only ifclrD−1⁡\(cu′\)=0\\clr\_\{D\-1\}\(c\_\{u\}^\{\\prime\}\)=0, equivalently, the content coordinates ofuuare all equal\. Thus the analogous invariance under arbitrary shared perturbations is false forCA⟂C\_\{A\}^\{\\perp\}\. Indeed, letp=eDp=e\_\{D\}andc=𝒞⁡\(1,2,1,…,1\)c=\\mathcal\{C\}\(1,2,1,\\ldots,1\), where coordinate00is the sink and the factor22is on a content coordinate\. Then

CA⟂​\(p\)=0,\(p⊕c\)′=𝒞⁡\(2,1,…,1\),CA⟂​\(p⊕c\)\>0\.C\_\{A\}^\{\\perp\}\(p\)=0,\\qquad\(p\\oplus c\)^\{\\prime\}=\\mathcal\{C\}\(2,1,\\ldots,1\),\\qquad C\_\{A\}^\{\\perp\}\(p\\oplus c\)\>0\.AtD=3D=3, the last value is exactly\(log⁡2\)/2\(\\log 2\)/\\sqrt\{2\}\. WhenD=2D=2, the content simplex has one part andCA⟂≡0C\_\{A\}^\{\\perp\}\\equiv 0\.

To prove failure of the corresponding classical invariances, fix distinctp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}, choose a coordinatejj, letut=t​𝐞ju\_\{t\}=t\\mathbf\{e\}\_\{j\}, and putct=𝒞⁡\(eut\)c\_\{t\}=\\mathcal\{C\}\(e^\{u\_\{t\}\}\)\. Since all entries ofppandqqare positive,

ct⊕p⟶𝐞j,ct⊕q⟶𝐞j\(t→∞\)\.c\_\{t\}\\oplus p\\longrightarrow\\mathbf\{e\}\_\{j\},\\qquad c\_\{t\}\\oplus q\\longrightarrow\\mathbf\{e\}\_\{j\}\\qquad\(t\\to\\infty\)\.Using the standard continuous extension ofJS\\mathrm\{JS\}to the closed simplex \(with0​log⁡0=00\\log 0=0\), and continuity of Euclidean distance and cosine similarity at probability vectors, gives

∥ct⊕p−ct⊕q∥2\\displaystyle\\lVert c\_\{t\}\\oplus p\-c\_\{t\}\\oplus q\\rVert\_\{2\}⟶0,\\displaystyle\\longrightarrow 0,JS⁡\(ct⊕p,ct⊕q\)\\displaystyle\\mathrm\{JS\}\(c\_\{t\}\\oplus p,c\_\{t\}\\oplus q\)⟶0,\\displaystyle\\longrightarrow 0,cos⁡\(ct⊕p,ct⊕q\)\\displaystyle\\cos\(c\_\{t\}\\oplus p,c\_\{t\}\\oplus q\)⟶1\.\\displaystyle\\longrightarrow 1\.Forp≠qp\\neq q, the original Euclidean distance andJS\\mathrm\{JS\}divergence are strictly positive, whereas the original cosine similarity is strictly less than one\. Consequently none is invariant under every shared perturbation\.

The same conclusion holds forρA\\rho\_\{A\}\. Fix0≠v∈ℋD0\\neq v\\in\\mathcal\{H\}\_\{D\}and set

p=𝒞⁡\(ev\),q=𝒞⁡\(e−v\),c=𝒞⁡\(e2​v\)\.p=\\mathcal\{C\}\(e^\{v\}\),\\qquad q=\\mathcal\{C\}\(e^\{\-v\}\),\\qquad c=\\mathcal\{C\}\(e^\{2v\}\)\.Thenclr⁡\(p\)=v\\clr\(p\)=v,clr⁡\(q\)=−v\\clr\(q\)=\-v, andclr⁡\(c\)=2​v\\clr\(c\)=2v, so all relevant directional similarities are defined and

ρA​\(p,q\)=−1,ρA​\(p⊕c,q⊕c\)=ρA​\(𝒞⁡\(e3​v\),𝒞⁡\(ev\)\)=1\.\\rho\_\{A\}\(p,q\)=\-1,\\qquad\\rho\_\{A\}\(p\\oplus c,q\\oplus c\)=\\rho\_\{A\}\\bigl\(\\mathcal\{C\}\(e^\{3v\}\),\\mathcal\{C\}\(e^\{v\}\)\\bigr\)=1\.
For \(ii\), setα=1/τ\>0\\alpha=1/\\tau\>0\. Then𝒞⁡\(eℓ/τ\)=𝒞⁡\(\(eℓi\)α\)=α⊙p\\mathcal\{C\}\(e^\{\\ell/\\tau\}\)=\\mathcal\{C\}\(\(e^\{\\ell\_\{i\}\}\)^\{\\alpha\}\)=\\alpha\\odot p\. The elementary identity

clr⁡\(α⊙p\)=α​clr⁡\(p\)\\clr\(\\alpha\\odot p\)=\\alpha\\clr\(p\)\(13\)impliesdA​\(α⊙p,α⊙q\)=α​dA​\(p,q\)d\_\{A\}\(\\alpha\\odot p,\\alpha\\odot q\)=\\alpha d\_\{A\}\(p,q\)\. Moreover, dropping and re\-closing commutes with powering:\(α⊙p\)′=α⊙p′\(\\alpha\\odot p\)^\{\\prime\}=\\alpha\\odot p^\{\\prime\}\. Applying \([13](https://arxiv.org/html/2608.14712#A3.E13)\) in the\(D−1\)\(D\-1\)\-part content simplex proves the displayed scaling identities fordA⟂d\_\{A\}^\{\\perp\}andCA⟂C\_\{A\}^\{\\perp\}\.

For the variance assertion, putxt=clr⁡\(pt\)x\_\{t\}=\\clr\(p\_\{t\}\)andx¯=n−1​∑t=1nxt\\bar\{x\}=n^\{\-1\}\\sum\_\{t=1\}^\{n\}x\_\{t\}\. From the definition of the Aitchison center,clr⁡\(cen⁡\(P\)\)=x¯\\clr\(\\cen\(P\)\)=\\bar\{x\}\. After common powering byα\\alpha, the CLR vectors and their recomputed center areα​xt\\alpha x\_\{t\}andα​x¯\\alpha\\bar\{x\}\. Hence

1n​∑t=1n∥α​xt−α​x¯∥22=α2​1n​∑t=1n∥xt−x¯∥22,\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\lVert\\alpha x\_\{t\}\-\\alpha\\bar\{x\}\\rVert\_\{2\}^\{2\}=\\alpha^\{2\}\\frac\{1\}\{n\}\\sum\_\{t=1\}^\{n\}\\lVert x\_\{t\}\-\\bar\{x\}\\rVert\_\{2\}^\{2\},which proves the factor1/τ21/\\tau^\{2\}\.

Ifp,q≠eDp,q\\neq e\_\{D\}andα,β\>0\\alpha,\\beta\>0, then

⟨α​clr⁡\(p\),β​clr⁡\(q\)⟩∥α​clr⁡\(p\)∥2​∥β​clr⁡\(q\)∥2=ρA​\(p,q\),\\frac\{\\langle\\alpha\\clr\(p\),\\beta\\clr\(q\)\\rangle\}\{\\lVert\\alpha\\clr\(p\)\\rVert\_\{2\}\\lVert\\beta\\clr\(q\)\\rVert\_\{2\}\}=\\rho\_\{A\}\(p,q\),which proves invariance ofρA\\rho\_\{A\}to separate positive temperatures\.

For the aggregation qualification, letsh=cen⁡\(ph​1,…,ph​nh\)s\_\{h\}=\\cen\(p\_\{h1\},\\ldots,p\_\{hn\_\{h\}\}\)and letαh=1/τh\>0\\alpha\_\{h\}=1/\\tau\_\{h\}\>0be common to all rows in headhh\. Applying \([13](https://arxiv.org/html/2608.14712#A3.E13)\) and the center identity above gives

cen⁡\(αh⊙ph​1,…,αh⊙ph​nh\)=αh⊙sh\.\\cen\(\\alpha\_\{h\}\\odot p\_\{h1\},\\ldots,\\alpha\_\{h\}\\odot p\_\{hn\_\{h\}\}\)=\\alpha\_\{h\}\\odot s\_\{h\}\.Thus separate positive head temperatures cancel inρA\\rho\_\{A\}whenever the resulting signatures are nonuniform\. With row\-dependent coefficientsαh​t\\alpha\_\{ht\}, the new center has CLR vectornh−1​∑tαh​t​clr⁡\(ph​t\)n\_\{h\}^\{\-1\}\\sum\_\{t\}\\alpha\_\{ht\}\\clr\(p\_\{ht\}\), which need not be a scalar multiple ofclr⁡\(sh\)\\clr\(s\_\{h\}\)\. An explicit failure is available in everyD≥2D\\geq 2\. Fix0≠v∈ℋD0\\neq v\\in\\mathcal\{H\}\_\{D\}\. Let the first head contain

p11=𝒞⁡\(e2​v\),p12=𝒞⁡\(e−v\),p\_\{11\}=\\mathcal\{C\}\(e^\{2v\}\),\\qquad p\_\{12\}=\\mathcal\{C\}\(e^\{\-v\}\),and let the second head consist ofp21=𝒞⁡\(ev\)p\_\{21\}=\\mathcal\{C\}\(e^\{v\}\)\. Before row\-dependent powering, their signatures satisfy

clr⁡\(s1\)=12​v,clr⁡\(s2\)=v,ρA​\(s1,s2\)=1\.\\clr\(s\_\{1\}\)=\\frac\{1\}\{2\}v,\\qquad\\clr\(s\_\{2\}\)=v,\\qquad\\rho\_\{A\}\(s\_\{1\},s\_\{2\}\)=1\.Now use exponentsα11=1/4\\alpha\_\{11\}=1/4andα12=1\\alpha\_\{12\}=1in the first head, leaving the second head unchanged\. The new first\-head signatures~1\\widetilde\{s\}\_\{1\}has

clr⁡\(s~1\)=12​\(14​\(2​v\)−v\)=−14​v,ρA​\(s~1,s2\)=−1\.\\clr\(\\widetilde\{s\}\_\{1\}\)=\\frac\{1\}\{2\}\\left\(\\frac\{1\}\{4\}\(2v\)\-v\\right\)=\-\\frac\{1\}\{4\}v,\\qquad\\rho\_\{A\}\(\\widetilde\{s\}\_\{1\},s\_\{2\}\)=\-1\.Hence row\-dependent positive temperatures can changeρA\\rho\_\{A\}\.

For \(iii\), closure of the content block givesp′=πp^\{\\prime\}=\\piandq′=ηq^\{\\prime\}=\\eta\. The two displayed identities follow immediately from the definitionsdA⟂​\(p,q\):=dA​\(p′,q′\)d\_\{A\}^\{\\perp\}\(p,q\):=d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\)andCA⟂​\(p\):=∥clrD−1⁡\(p′\)∥2C\_\{A\}^\{\\perp\}\(p\):=\\lVert\\clr\_\{D\-1\}\(p^\{\\prime\}\)\\rVert\_\{2\}\. Their equality with the full balance\-orthogonal calculation is \([11](https://arxiv.org/html/2608.14712#A3.E11)\)\. The final qualification follows from the Pythagorean identity: the balance term generally changes when sink masses change\.

For \(iv\), define

WS:=\{w∈ℝD:wi=0fori∉S,∑i∈Swi=0\}⊂ℋD,W\_\{S\}:=\\left\\\{w\\in\\mathbb\{R\}^\{D\}:w\_\{i\}=0\\text\{ for \}i\\notin S,\\ \\sum\_\{i\\in S\}w\_\{i\}=0\\right\\\}\\subset\\mathcal\{H\}\_\{D\},and letIS:ℋ\|S\|→WSI\_\{S\}:\\mathcal\{H\}\_\{\|S\|\}\\to W\_\{S\}be zero\-extension, using the inherited coordinate order onSS\. This is a linear isometry\. IfPWSP\_\{W\_\{S\}\}is orthogonal projection ontoWSW\_\{S\}, then for everyx∈ℋDx\\in\\mathcal\{H\}\_\{D\},

\(PWS​x\)i=\{xi−\|S\|−1​∑j∈Sxj,i∈S,0,i∉S\.\(P\_\{W\_\{S\}\}x\)\_\{i\}=\\begin\{cases\}x\_\{i\}\-\|S\|^\{\-1\}\\sum\_\{j\\in S\}x\_\{j\},&i\\in S,\\\\ 0,&i\\notin S\.\\end\{cases\}Applying this tox=clrD⁡\(p\)x=\\clr\_\{D\}\(p\)shows that

IS​clr\|S\|⁡\(p\(S\)\)=PWS​clrD⁡\(p\),IS​clr\|S\|⁡\(q\(S\)\)=PWS​clrD⁡\(q\)\.I\_\{S\}\\clr\_\{\|S\|\}\(p^\{\(S\)\}\)=P\_\{W\_\{S\}\}\\clr\_\{D\}\(p\),\\qquad I\_\{S\}\\clr\_\{\|S\|\}\(q^\{\(S\)\}\)=P\_\{W\_\{S\}\}\\clr\_\{D\}\(q\)\.\(14\)Subtracting the identities in \([14](https://arxiv.org/html/2608.14712#A3.E14)\), using the isometry ofISI\_\{S\}, and using contractivity of an orthogonal projection gives

dA​\(p\(S\),q\(S\)\)\\displaystyle d\_\{A\}\(p^\{\(S\)\},q^\{\(S\)\}\)=∥PWS\(clrD\(p\)−clrD\(q\)\)∥2\\displaystyle=\\bigl\\lVert P\_\{W\_\{S\}\}\(\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)\)\\bigr\\rVert\_\{2\}≤∥clrD⁡\(p\)−clrD⁡\(q\)∥2=dA​\(p,q\)\.\\displaystyle\\leq\\lVert\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)\\rVert\_\{2\}=d\_\{A\}\(p,q\)\.This proves both the projection identity and dominance\. The masking qualifications in the statement follow because CLR coordinates exist only for strictly positive parts\.

For \(v\), first considerD=3D=3and use the Table[1](https://arxiv.org/html/2608.14712#S4.T1)compositions

p=\(910,225,150\),r=\(910,150,225\),S=\{1,2\}\.p=\\left\(\\frac\{9\}\{10\},\\frac\{2\}\{25\},\\frac\{1\}\{50\}\\right\),\\qquad r=\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{50\},\\frac\{2\}\{25\}\\right\),\\qquad S=\\\{1,2\\\}\.Then

p\(S\)=\(45,15\),r\(S\)=\(15,45\)\.p^\{\(S\)\}=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\),\\qquad r^\{\(S\)\}=\\left\(\\frac\{1\}\{5\},\\frac\{4\}\{5\}\\right\)\.For Euclidean distance,

∥p−r∥2=3​250<3​25=∥p\(S\)−r\(S\)∥2\.\\lVert p\-r\\rVert\_\{2\}=\\frac\{3\\sqrt\{2\}\}\{50\}<\\frac\{3\\sqrt\{2\}\}\{5\}=\\lVert p^\{\(S\)\}\-r^\{\(S\)\}\\rVert\_\{2\}\.For cosine dissimilarity, direct calculation gives

∥p∥22=∥r∥22=20422500,p⊤​r=20332500,\\lVert p\\rVert\_\{2\}^\{2\}=\\lVert r\\rVert\_\{2\}^\{2\}=\\frac\{2042\}\{2500\},\\qquad p^\{\\top\}r=\\frac\{2033\}\{2500\},and

∥p\(S\)∥22=∥r\(S\)∥22=1725,\(p\(S\)\)⊤​r\(S\)=825\.\\lVert p^\{\(S\)\}\\rVert\_\{2\}^\{2\}=\\lVert r^\{\(S\)\}\\rVert\_\{2\}^\{2\}=\\frac\{17\}\{25\},\\qquad\(p^\{\(S\)\}\)^\{\\top\}r^\{\(S\)\}=\\frac\{8\}\{25\}\.Consequently,

1−cos⁡\(p,r\)=92042<917=1−cos⁡\(p\(S\),r\(S\)\)\.1\-\\cos\(p,r\)=\\frac\{9\}\{2042\}<\\frac\{9\}\{17\}=1\-\\cos\(p^\{\(S\)\},r^\{\(S\)\}\)\.
Fora,ba,bin a common simplex, let

JS⁡\(a,b\):=12​KL​\(a∥a\+b2\)\+12​KL​\(b∥a\+b2\)\.\\mathrm\{JS\}\(a,b\):=\\frac\{1\}\{2\}\\mathrm\{KL\}\\left\(a\\middle\\\|\\frac\{a\+b\}\{2\}\\right\)\+\\frac\{1\}\{2\}\\mathrm\{KL\}\\left\(b\\middle\\\|\\frac\{a\+b\}\{2\}\\right\)\.Put

π=\(45,15\),π~=\(15,45\),mc=\(12,12\),\\pi=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\),\\qquad\\widetilde\{\\pi\}=\\left\(\\frac\{1\}\{5\},\\frac\{4\}\{5\}\\right\),\\qquad m\_\{c\}=\\left\(\\frac\{1\}\{2\},\\frac\{1\}\{2\}\\right\),and let

Jπ:=45​log⁡85\+15​log⁡25\>0\.J\_\{\\pi\}:=\\frac\{4\}\{5\}\\log\\frac\{8\}\{5\}\+\\frac\{1\}\{5\}\\log\\frac\{2\}\{5\}\>0\.By symmetry,

KL\(π∥mc\)=KL\(π~∥mc\)=Jπ,JS\(π,π~\)=Jπ\.\\mathrm\{KL\}\(\\pi\\\|m\_\{c\}\)=\\mathrm\{KL\}\(\\widetilde\{\\pi\}\\\|m\_\{c\}\)=J\_\{\\pi\},\\qquad\\mathrm\{JS\}\(\\pi,\\widetilde\{\\pi\}\)=J\_\{\\pi\}\.Here

p=\(910,110​π\),r=\(910,110​π~\),p=\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{10\}\\pi\\right\),\\qquad r=\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{10\}\\widetilde\{\\pi\}\\right\),and their midpoint is\(9/10,\(1/10\)​mc\)\(9/10,\(1/10\)m\_\{c\}\)\. Therefore

KL\(p∥\(p\+r\)/2\)=KL\(r∥\(p\+r\)/2\)=110Jπ,\\mathrm\{KL\}\\bigl\(p\\\|\(p\+r\)/2\\bigr\)=\\mathrm\{KL\}\\bigl\(r\\\|\(p\+r\)/2\\bigr\)=\\frac\{1\}\{10\}J\_\{\\pi\},and hence

JS⁡\(p,r\)=110​Jπ<Jπ=JS⁡\(p\(S\),r\(S\)\)\.\\mathrm\{JS\}\(p,r\)=\\frac\{1\}\{10\}J\_\{\\pi\}<J\_\{\\pi\}=\\mathrm\{JS\}\(p^\{\(S\)\},r^\{\(S\)\}\)\.
It remains to extend the counterexample to everyD\>3D\>3\. Putk=D−3k=D\-3, chooseε∈\(0,1/k\)\\varepsilon\\in\(0,1/k\), seta=1−k​ε∈\(0,1\)a=1\-k\\varepsilon\\in\(0,1\), and appendkkequal parts:

p¯=\(a​p0,a​p1,a​p2,ε,…,ε⏟k\),r¯=\(a​r0,a​r1,a​r2,ε,…,ε⏟k\)\.\\bar\{p\}=\(ap\_\{0\},ap\_\{1\},ap\_\{2\},\\underbrace\{\\varepsilon,\\ldots,\\varepsilon\}\_\{k\}\),\\qquad\\bar\{r\}=\(ar\_\{0\},ar\_\{1\},ar\_\{2\},\\underbrace\{\\varepsilon,\\ldots,\\varepsilon\}\_\{k\}\)\.These are strictly positiveDD\-part compositions, and their subcompositions onS=\{1,2\}S=\\\{1,2\\\}remainp\(S\)p^\{\(S\)\}andr\(S\)r^\{\(S\)\}\. The appended parts agree, so

∥p¯−r¯∥2=a​∥p−r∥2<∥p\(S\)−r\(S\)∥2\\lVert\\bar\{p\}\-\\bar\{r\}\\rVert\_\{2\}=a\\lVert p\-r\\rVert\_\{2\}<\\lVert p^\{\(S\)\}\-r^\{\(S\)\}\\rVert\_\{2\}and, because the common appended parts contribute zero to both KL terms while the factoraacancels inside each logarithm,

KL⁡\(p¯∥p¯\+r¯2\)\\displaystyle\\mathrm\{KL\}\\left\(\\bar\{p\}\\middle\\\|\\frac\{\\bar\{p\}\+\\bar\{r\}\}\{2\}\\right\)=a​KL​\(p∥p\+r2\),\\displaystyle=a\\mathrm\{KL\}\\left\(p\\middle\\\|\\frac\{p\+r\}\{2\}\\right\),KL⁡\(r¯∥p¯\+r¯2\)\\displaystyle\\mathrm\{KL\}\\left\(\\bar\{r\}\\middle\\\|\\frac\{\\bar\{p\}\+\\bar\{r\}\}\{2\}\\right\)=a​KL​\(r∥p\+r2\)\.\\displaystyle=a\\mathrm\{KL\}\\left\(r\\middle\\\|\\frac\{p\+r\}\{2\}\\right\)\.Consequently,

JS⁡\(p¯,r¯\)=a​JS​\(p,r\)<JS⁡\(p\(S\),r\(S\)\)\.\\mathrm\{JS\}\(\\bar\{p\},\\bar\{r\}\)=a\\mathrm\{JS\}\(p,r\)<\\mathrm\{JS\}\(p^\{\(S\)\},r^\{\(S\)\}\)\.Furthermore,

1−cos⁡\(p¯,r¯\)\\displaystyle 1\-\\cos\(\\bar\{p\},\\bar\{r\}\)=a2​\(∥p∥22−p⊤​r\)a2​∥p∥22\+k​ε2\\displaystyle=\\frac\{a^\{2\}\(\\lVert p\\rVert\_\{2\}^\{2\}\-p^\{\\top\}r\)\}\{a^\{2\}\\lVert p\\rVert\_\{2\}^\{2\}\+k\\varepsilon^\{2\}\}<∥p∥22−p⊤​r∥p∥22=92042<917=1−cos⁡\(p\(S\),r\(S\)\)\.\\displaystyle<\\frac\{\\lVert p\\rVert\_\{2\}^\{2\}\-p^\{\\top\}r\}\{\\lVert p\\rVert\_\{2\}^\{2\}\}=\\frac\{9\}\{2042\}<\\frac\{9\}\{17\}=1\-\\cos\(p^\{\(S\)\},r^\{\(S\)\}\)\.The first inequality is strict becausek​ε2\>0k\\varepsilon^\{2\}\>0\. Thus, for everyD≥3D\\geq 3, each classical dissimilarity can increase strictly after taking a subcomposition, so none is subcompositionally dominant\. ∎

### Full statement and proof of Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)

Here𝒮m\\mathcal\{S\}^\{m\}is the openmm\-part simplex and𝔖m\\mathfrak\{S\}\_\{m\}the symmetric group\. Under the operations above, the mapTm:=clrm:\(𝒮m,⊕,⊙\)→\(ℋm,\+,⋅\)T\_\{m\}:=\\clr\_\{m\}:\(\\mathcal\{S\}^\{m\},\\oplus,\\odot\)\\to\(\\mathcal\{H\}\_\{m\},\+,\\cdot\)is a linear isomorphism with inverseTm−1​\(x\)=𝒞⁡\(exp⁡x\)T\_\{m\}^\{\-1\}\(x\)=\\mathcal\{C\}\(\\exp x\); forσ∈𝔖m\\sigma\\in\\mathfrak\{S\}\_\{m\}we use the action\(σ​p\)i=pσ−1​\(i\)\(\\sigma p\)\_\{i\}=p\_\{\\sigma^\{\-1\}\(i\)\}\. The full statement of Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)is: ifδD:𝒮D×𝒮D→\[0,∞\)\\delta\_\{D\}:\\mathcal\{S\}^\{D\}\\times\\mathcal\{S\}^\{D\}\\to\[0,\\infty\)satisfies \(A1\)δD\\delta\_\{D\}is a metric; \(A2\)δD​\(c⊕p,c⊕q\)=δD​\(p,q\)\\delta\_\{D\}\(c\\oplus p,c\\oplus q\)=\\delta\_\{D\}\(p,q\)for allcc; \(A3\)δD​\(α⊙p,α⊙q\)=\|α\|​δD​\(p,q\)\\delta\_\{D\}\(\\alpha\\odot p,\\alpha\\odot q\)=\|\\alpha\|\\delta\_\{D\}\(p,q\)for allα∈ℝ\\alpha\\in\\mathbb\{R\}; \(A4\) withN⁡\(v\):=δD​\(v,eD\)N\(v\):=\\delta\_\{D\}\(v,e\_\{D\}\),N​\(v⊕w\)2\+N​\(v⊖w\)2=2​N​\(v\)2\+2​N​\(w\)2N\(v\\oplus w\)^\{2\}\+N\(v\\ominus w\)^\{2\}=2N\(v\)^\{2\}\+2N\(w\)^\{2\}for allv,wv,w; and \(A5\)δD​\(σ​p,σ​q\)=δD​\(p,q\)\\delta\_\{D\}\(\\sigma p,\\sigma q\)=\\delta\_\{D\}\(p,q\)for everyσ∈𝔖D\\sigma\\in\\mathfrak\{S\}\_\{D\}; then there is a unique constantκD\>0\\kappa\_\{D\}\>0withδD=κD​dA\\delta\_\{D\}=\\kappa\_\{D\}d\_\{A\}on𝒮D\\mathcal\{S\}^\{D\}, and conversely everyκD​dA\\kappa\_\{D\}d\_\{A\}satisfies \(A1\)–\(A5\)\.

###### Proof of Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)\.

We give the complete argument\.

*Reduction to a norm\.*BecauseT=clrDT=\\clr\_\{D\}is bijective, the identities

clrD⁡\(p⊕q\)=clrD⁡\(p\)\+clrD⁡\(q\),clrD⁡\(α⊙p\)=α​clrD⁡\(p\)\\clr\_\{D\}\(p\\oplus q\)=\\clr\_\{D\}\(p\)\+\\clr\_\{D\}\(q\),\\qquad\\clr\_\{D\}\(\\alpha\\odot p\)=\\alpha\\clr\_\{D\}\(p\)transport the usual vector\-space laws onℋD\\mathcal\{H\}\_\{D\}to\(𝒮D,⊕,⊙\)\(\\mathcal\{S\}^\{D\},\\oplus,\\odot\), whose zero iseDe\_\{D\}\. Applying \(A2\) withc=⊖qc=\\ominus qgives

δD​\(p,q\)=δD​\(p⊖q,eD\)=N⁡\(p⊖q\)\.\\delta\_\{D\}\(p,q\)=\\delta\_\{D\}\(p\\ominus q,e\_\{D\}\)=N\(p\\ominus q\)\.\(15\)By \(A1\),N⁡\(v\)≥0N\(v\)\\geq 0andN⁡\(v\)=0N\(v\)=0if and only ifv=eDv=e\_\{D\}\. Sinceα⊙eD=eD\\alpha\\odot e\_\{D\}=e\_\{D\}, \(A3\) yieldsN⁡\(α⊙v\)=\|α\|​N​\(v\)N\(\\alpha\\odot v\)=\|\\alpha\|N\(v\)\. Finally, the triangle inequality in \(A1\) and perturbation invariance give

N⁡\(v⊕w\)\\displaystyle N\(v\\oplus w\)=δD​\(v⊕w,eD\)\\displaystyle=\\delta\_\{D\}\(v\\oplus w,e\_\{D\}\)≤δD​\(v⊕w,w\)\+δD​\(w,eD\)\\displaystyle\\leq\\delta\_\{D\}\(v\\oplus w,w\)\+\\delta\_\{D\}\(w,e\_\{D\}\)=δD​\(v,eD\)\+δD​\(w,eD\)=N⁡\(v\)\+N⁡\(w\),\\displaystyle=\\delta\_\{D\}\(v,e\_\{D\}\)\+\\delta\_\{D\}\(w,e\_\{D\}\)=N\(v\)\+N\(w\),where the third line follows by perturbing the first distance by⊖w\\ominus w\. HenceNNis a norm on the Aitchison vector space\.

*The parallelogram identity\.*Assumption \(A4\) is exactly the parallelogram identity forNN\. The real Jordan–von Neumann theorem therefore implies that

Bδ​\(v,w\):=14​\{N​\(v⊕w\)2−N​\(v⊖w\)2\}B\_\{\\delta\}\(v,w\):=\\frac\{1\}\{4\}\\left\\\{N\(v\\oplus w\)^\{2\}\-N\(v\\ominus w\)^\{2\}\\right\\\}\(16\)is an inner product and thatN​\(v\)2=Bδ​\(v,v\)N\(v\)^\{2\}=B\_\{\\delta\}\(v,v\)\.

*Transport to CLR coordinates\.*LetT:=clrDT:=\\clr\_\{D\}\. TransportingBδB\_\{\\delta\}through the linear isomorphismT:𝒮D→ℋDT:\\mathcal\{S\}^\{D\}\\to\\mathcal\{H\}\_\{D\}gives an inner product onℋD\\mathcal\{H\}\_\{D\}\. BecauseℋD\\mathcal\{H\}\_\{D\}is finite\-dimensional, there is a unique Euclidean\-self\-adjoint positive\-definite operatorM:ℋD→ℋDM:\\mathcal\{H\}\_\{D\}\\to\\mathcal\{H\}\_\{D\}such that

Bδ​\(v,w\)=⟨T⁡\(v\),M​T​\(w\)⟩2\(v,w∈𝒮D\)\.B\_\{\\delta\}\(v,w\)=\\left\\langle T\(v\),MT\(w\)\\right\\rangle\_\{2\}\\qquad\(v,w\\in\\mathcal\{S\}^\{D\}\)\.\(17\)
*Permutation invariance forces isotropy\.*LetPσP\_\{\\sigma\}denote the permutation matrix corresponding toσ∈𝔖D\\sigma\\in\\mathfrak\{S\}\_\{D\}, and putQσ:=Pσ\|ℋDQ\_\{\\sigma\}:=P\_\{\\sigma\}\|\_\{\\mathcal\{H\}\_\{D\}\}\. The spaceℋD\\mathcal\{H\}\_\{D\}is invariant underPσP\_\{\\sigma\}, andQσQ\_\{\\sigma\}is orthogonal\. Coordinate permutations are Aitchison\-linear: they commute with⊕\\oplus,⊖\\ominus, and⊙\\odot, andT⁡\(σ​v\)=Qσ​T​\(v\)T\(\\sigma v\)=Q\_\{\\sigma\}T\(v\)\. Sinceσ​eD=eD\\sigma e\_\{D\}=e\_\{D\}, assumption \(A5\) givesN⁡\(σ​v\)=N⁡\(v\)N\(\\sigma v\)=N\(v\)\. Polarization in \([16](https://arxiv.org/html/2608.14712#A3.E16)\) consequently givesBδ​\(σ​v,σ​w\)=Bδ​\(v,w\)B\_\{\\delta\}\(\\sigma v,\\sigma w\)=B\_\{\\delta\}\(v,w\), and hence

Qσ∗​M​Qσ=M\.Q\_\{\\sigma\}^\{\*\}MQ\_\{\\sigma\}=M\.AsQσQ\_\{\\sigma\}is orthogonal,MMcommutes with everyQσQ\_\{\\sigma\}\.

For completeness, the coordinate\-permutation representation onℋD\\mathcal\{H\}\_\{D\}is irreducible overℝ\\mathbb\{R\}\. Indeed, letUUbe a nonzero linear subspace ofℋD\\mathcal\{H\}\_\{D\}that is invariant under every coordinate permutation, and choose0≠x∈U0\\neq x\\in U\. Not all coordinates ofxxcan be equal, because𝟏D⊤​x=0\\mathbf\{1\}\_\{D\}^\{\\top\}x=0; hencexi≠xjx\_\{i\}\\neq x\_\{j\}for somei≠ji\\neq j\. If\(i​j\)\(ij\)is the corresponding transposition and𝐞i\\mathbf\{e\}\_\{i\}denotes theiith standard basis vector ofℝD\\mathbb\{R\}^\{D\}, then

x−P\(i​j\)​x=\(xi−xj\)​\(𝐞i−𝐞j\)∈U\.x\-P\_\{\(ij\)\}x=\(x\_\{i\}\-x\_\{j\}\)\(\\mathbf\{e\}\_\{i\}\-\\mathbf\{e\}\_\{j\}\)\\in U\.Thus𝐞i−𝐞j∈U\\mathbf\{e\}\_\{i\}\-\\mathbf\{e\}\_\{j\}\\in U\. Permuting coordinates shows that𝐞k−𝐞ℓ∈U\\mathbf\{e\}\_\{k\}\-\\mathbf\{e\}\_\{\\ell\}\\in Ufor everyk≠ℓk\\neq\\ell, and these vectors spanℋD\\mathcal\{H\}\_\{D\}\. ThereforeU=ℋDU=\\mathcal\{H\}\_\{D\}\.

By the finite\-dimensional spectral theorem,MMhas a real eigenvalueλ\\lambdaand a nonzero eigenspaceEλE\_\{\\lambda\}\. SinceMMcommutes with allQσQ\_\{\\sigma\}, the spaceEλE\_\{\\lambda\}is invariant under every coordinate permutation\. Irreducibility forcesEλ=ℋDE\_\{\\lambda\}=\\mathcal\{H\}\_\{D\}, so

M=λ​IℋD\.M=\\lambda I\_\{\\mathcal\{H\}\_\{D\}\}\.\(18\)Positive definiteness givesλ\>0\\lambda\>0\.

*Conclusion and converse\.*Equations \([15](https://arxiv.org/html/2608.14712#A3.E15)\), \([17](https://arxiv.org/html/2608.14712#A3.E17)\), and \([18](https://arxiv.org/html/2608.14712#A3.E18)\) yield

δD​\(p,q\)2\\displaystyle\\delta\_\{D\}\(p,q\)^\{2\}=N​\(p⊖q\)2\\displaystyle=N\(p\\ominus q\)^\{2\}=λ​‖T⁡\(p⊖q\)‖22\\displaystyle=\\lambda\\left\\\|T\(p\\ominus q\)\\right\\\|\_\{2\}^\{2\}=λ​‖T⁡\(p\)−T⁡\(q\)‖22=λ​dA​\(p,q\)2\.\\displaystyle=\\lambda\\left\\\|T\(p\)\-T\(q\)\\right\\\|\_\{2\}^\{2\}=\\lambda d\_\{A\}\(p,q\)^\{2\}\.ThusδD=κD​dA\\delta\_\{D\}=\\kappa\_\{D\}d\_\{A\}withκD:=λ\>0\\kappa\_\{D\}:=\\sqrt\{\\lambda\}\>0\. SinceD≥2D\\geq 2,dAd\_\{A\}is nonzero for some pair, soκD\\kappa\_\{D\}is unique\.

Conversely, fixκD\>0\\kappa\_\{D\}\>0and defineδD​\(p,q\):=κD​‖T⁡\(p\)−T⁡\(q\)‖2\\delta\_\{D\}\(p,q\):=\\kappa\_\{D\}\\\|T\(p\)\-T\(q\)\\\|\_\{2\}\. SinceTTis injective and Euclidean distance is a metric, \(A1\) holds\. The CLR identities give

δD​\(c⊕p,c⊕q\)\\displaystyle\\delta\_\{D\}\(c\\oplus p,c\\oplus q\)=κD​‖T⁡\(c\)\+T⁡\(p\)−T⁡\(c\)−T⁡\(q\)‖2=δD​\(p,q\),\\displaystyle=\\kappa\_\{D\}\\\|T\(c\)\+T\(p\)\-T\(c\)\-T\(q\)\\\|\_\{2\}=\\delta\_\{D\}\(p,q\),δD​\(α⊙p,α⊙q\)\\displaystyle\\delta\_\{D\}\(\\alpha\\odot p,\\alpha\\odot q\)=κD​‖α⁡\(T⁡\(p\)−T⁡\(q\)\)‖2=\|α\|​δD​\(p,q\),\\displaystyle=\\kappa\_\{D\}\\\|\\alpha\(T\(p\)\-T\(q\)\)\\\|\_\{2\}=\|\\alpha\|\\,\\delta\_\{D\}\(p,q\),which prove \(A2\) and \(A3\)\. Moreover,N⁡\(v\)=κD​‖T⁡\(v\)‖2N\(v\)=\\kappa\_\{D\}\\\|T\(v\)\\\|\_\{2\}, so the Euclidean parallelogram identity gives

N​\(v⊕w\)2\+N​\(v⊖w\)2\\displaystyle N\(v\\oplus w\)^\{2\}\+N\(v\\ominus w\)^\{2\}=κD2​\(‖T⁡\(v\)\+T⁡\(w\)‖22\+‖T⁡\(v\)−T⁡\(w\)‖22\)\\displaystyle=\\kappa\_\{D\}^\{2\}\\bigl\(\\\|T\(v\)\+T\(w\)\\\|\_\{2\}^\{2\}\+\\\|T\(v\)\-T\(w\)\\\|\_\{2\}^\{2\}\\bigr\)=2​N​\(v\)2\+2​N​\(w\)2,\\displaystyle=2N\(v\)^\{2\}\+2N\(w\)^\{2\},which is \(A4\)\. Finally, because every permutation matrix is orthogonal,

δD​\(σ​p,σ​q\)=κD​‖Qσ​\(T⁡\(p\)−T⁡\(q\)\)‖2=δD​\(p,q\),\\delta\_\{D\}\(\\sigma p,\\sigma q\)=\\kappa\_\{D\}\\\|Q\_\{\\sigma\}\(T\(p\)\-T\(q\)\)\\\|\_\{2\}=\\delta\_\{D\}\(p,q\),which is \(A5\)\. ∎

###### Corollary 2\(Consequences under a dimension\-compatible calibration\)\.

AssumeD≥3D\\geq 3and letδD\\delta\_\{D\}satisfy the hypotheses of Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)\. WriteδD=κD​dA\\delta\_\{D\}=\\kappa\_\{D\}d\_\{A\}and, for each2≤m<D2\\leq m<D, define the compatible common\-calibration distanceδm\[κD\]​\(r,s\):=κD​dA​\(r,s\)\\delta\_\{m\}^\{\[\\kappa\_\{D\}\]\}\(r,s\):=\\kappa\_\{D\}d\_\{A\}\(r,s\)forr,s∈𝒮mr,s\\in\\mathcal\{S\}^\{m\}\(withdAd\_\{A\}themm\-part CLR distance\)\. Also putδD\[κD\]:=δD\\delta\_\{D\}^\{\[\\kappa\_\{D\}\]\}:=\\delta\_\{D\}\. Then:

1. \(i\)*Scale invariance\.*Forx,y∈\(0,∞\)Dx,y\\in\(0,\\infty\)^\{D\}, defineδ¯D​\(x,y\):=δD​\(𝒞⁡\(x\),𝒞⁡\(y\)\)\\bar\{\\delta\}\_\{D\}\(x,y\):=\\delta\_\{D\}\(\\mathcal\{C\}\(x\),\\mathcal\{C\}\(y\)\)\. For alla,b\>0a,b\>0,δ¯D​\(a​x,b​y\)=δ¯D​\(x,y\)\\bar\{\\delta\}\_\{D\}\(ax,by\)=\\bar\{\\delta\}\_\{D\}\(x,y\)\. Thusδ¯D\\bar\{\\delta\}\_\{D\}is a scale\-invariant pseudometric on the positive cone; it is not a metric there, since proportional vectors have distance zero\.
2. \(ii\)*Subcompositional dominance\.*IfS⊆\{1,…,D\}S\\subseteq\\\{1,\\ldots,D\\\}hasm:=\|S\|≥2m:=\|S\|\\geq 2, letr\(S\):=𝒞⁡\(\(ri\)i∈S\)r^\{\(S\)\}:=\\mathcal\{C\}\(\(r\_\{i\}\)\_\{i\\in S\}\)forr∈𝒮Dr\\in\\mathcal\{S\}^\{D\}, with the coordinates in their inherited order\. Then δm\[κD\]​\(p\(S\),q\(S\)\)≤δD​\(p,q\)\.\\delta\_\{m\}^\{\[\\kappa\_\{D\}\]\}\(p^\{\(S\)\},q^\{\(S\)\}\)\\leq\\delta\_\{D\}\(p,q\)\.
3. \(iii\)*Exact sink decomposition\.*Relabel theDDcoordinates by0,1,…,D−10,1,\\ldots,D\-1, with00the sink, and for anyr∈𝒮Dr\\in\\mathcal\{S\}^\{D\}putr′:=𝒞⁡\(\(,,,,,\)\)r^\{\\prime\}:=\\mathcal\{C\}\(\(r\_\{1\},\\ldots,r\_\{D\-1\}\)\)\. Definebδ​\(r\):=κD​b​\(r\)b\_\{\\delta\}\(r\):=\\kappa\_\{D\}\\,b\(r\)\(withbbthe sink balance\) and the content pseudometricδD⟂​\(p,q\):=δD−1\[κD\]​\(p′,q′\)=κD​dA​\(p′,q′\)\\delta\_\{D\}^\{\\perp\}\(p,q\):=\\delta\_\{D\-1\}^\{\[\\kappa\_\{D\}\]\}\(p^\{\\prime\},q^\{\\prime\}\)=\\kappa\_\{D\}d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\)\. Then δD​\(p,q\)2=δD⟂​\(p,q\)2\+\(bδ​\(p\)−bδ​\(q\)\)2\.\\delta\_\{D\}\(p,q\)^\{2\}=\\delta\_\{D\}^\{\\perp\}\(p,q\)^\{2\}\+\\bigl\(b\_\{\\delta\}\(p\)\-b\_\{\\delta\}\(q\)\\bigr\)^\{2\}\.\(19\)More precisely, withWc:=\{x∈ℋD:x0=0\}W\_\{c\}:=\\\{x\\in\\mathcal\{H\}\_\{D\}:x\_\{0\}=0\\\}andPWcP\_\{W\_\{c\}\}Euclidean orthogonal projection ontoWcW\_\{c\}, δD⟂​\(p,q\)=κD​‖PWc​\(clrD⁡\(p\)−clrD⁡\(q\)\)‖2=κD​dA​\(p′,q′\)\.\\delta\_\{D\}^\{\\perp\}\(p,q\)=\\kappa\_\{D\}\\left\\\|P\_\{W\_\{c\}\}\\bigl\(\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)\\bigr\)\\right\\\|\_\{2\}=\\kappa\_\{D\}d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\)\.Thus the orthogonal projection recenters the kept CLR coordinates and is not mere deletion of the sink coordinate\. Moreover,δD⟂\\delta\_\{D\}^\{\\perp\}and the content log\-ratio dispersion𝒦δ​\(r\):=δD−1\[κD\]​\(r′,eD−1\)\\mathcal\{K\}\_\{\\delta\}\(r\):=\\delta\_\{D\-1\}^\{\[\\kappa\_\{D\}\]\}\(r^\{\\prime\},e\_\{D\-1\}\)depend only on content log\-ratios and not on the sink mass\. The full distanceδD\\delta\_\{D\}need not be sink\-invariant, because it also contains the balance term in \([19](https://arxiv.org/html/2608.14712#A3.E19)\)\.
4. \(iv\)*Shared coordinate\-bias invariance and temperature equivariance\.*LetsmD⁡\(ℓ\):=𝒞⁡\(exp⁡ℓ\)\\operatorname\{sm\}\_\{D\}\(\\ell\):=\\mathcal\{C\}\(\\exp\\ell\)\. For allℓ,ℓ′,u∈ℝD\\ell,\\ell^\{\\prime\},u\\in\\mathbb\{R\}^\{D\}andτ\>0\\tau\>0, δD​\(smD⁡\(ℓ\+u\),smD⁡\(ℓ′\+u\)\)\\displaystyle\\delta\_\{D\}\\bigl\(\\operatorname\{sm\}\_\{D\}\(\\ell\+u\),\\operatorname\{sm\}\_\{D\}\(\\ell^\{\\prime\}\+u\)\\bigr\)=δD​\(smD⁡\(ℓ\),smD⁡\(ℓ′\)\),\\displaystyle=\\delta\_\{D\}\\bigl\(\\operatorname\{sm\}\_\{D\}\(\\ell\),\\operatorname\{sm\}\_\{D\}\(\\ell^\{\\prime\}\)\\bigr\),δD​\(smD⁡\(ℓ/τ\),smD⁡\(ℓ′/τ\)\)\\displaystyle\\delta\_\{D\}\\bigl\(\\operatorname\{sm\}\_\{D\}\(\\ell/\\tau\),\\operatorname\{sm\}\_\{D\}\(\\ell^\{\\prime\}/\\tau\)\\bigr\)=τ−1​δD​\(smD⁡\(ℓ\),smD⁡\(ℓ′\)\)\.\\displaystyle=\\tau^\{\-1\}\\delta\_\{D\}\\bigl\(\\operatorname\{sm\}\_\{D\}\(\\ell\),\\operatorname\{sm\}\_\{D\}\(\\ell^\{\\prime\}\)\\bigr\)\.Consequently, one common temperature applied to every row preserves all pairwise\-distance rankings and ties\. Forn≥1n\\geq 1and rowsp\(1\),…,p\(n\)∈𝒮Dp^\{\(1\)\},\\ldots,p^\{\(n\)\}\\in\\mathcal\{S\}^\{D\}, withcen\\centhe Aitchison center and Vδ​\(p\(1\),…,p\(n\)\):=1n​∑j=1nδD​\(p\(j\),cen⁡\(p\(1\),…,p\(n\)\)\)2,V\_\{\\delta\}\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\):=\\frac\{1\}\{n\}\\sum\_\{j=1\}^\{n\}\\delta\_\{D\}\\\!\\left\(p^\{\(j\)\},\\cen\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\)\\right\)^\{2\},one has, writingcu:=smD⁡\(u\)c\_\{u\}:=\\operatorname\{sm\}\_\{D\}\(u\), Vδ​\(cu⊕p\(1\),…,cu⊕p\(n\)\)=Vδ​\(p\(1\),…,p\(n\)\),Vδ​\(1τ⊙p\(1\),…,1τ⊙p\(n\)\)=τ−2​Vδ​\(p\(1\),…,p\(n\)\)\.V\_\{\\delta\}\(c\_\{u\}\\oplus p^\{\(1\)\},\\ldots,c\_\{u\}\\oplus p^\{\(n\)\}\)=V\_\{\\delta\}\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\),\\qquad V\_\{\\delta\}\\\!\\left\(\\tfrac\{1\}\{\\tau\}\\odot p^\{\(1\)\},\\ldots,\\tfrac\{1\}\{\\tau\}\\odot p^\{\(n\)\}\\right\)=\\tau^\{\-2\}V\_\{\\delta\}\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\)\.Finally, putNδ​\(r\):=δD​\(r,eD\)N\_\{\\delta\}\(r\):=\\delta\_\{D\}\(r,e\_\{D\}\)and, forv,w∈𝒮Dv,w\\in\\mathcal\{S\}^\{D\}, define Bδ\(v,w\):=14\{Nδ\(v⊕w\)2−Nδ\(v⊖w\)2\},ρδ\(v,w\):=Bδ​\(v,w\)Nδ​\(v\)​Nδ​\(w\)\(v,w≠eD\)\.B\_\{\\delta\}\(v,w\):=\\frac\{1\}\{4\}\\left\\\{N\_\{\\delta\}\(v\\oplus w\)^\{2\}\-N\_\{\\delta\}\(v\\ominus w\)^\{2\}\\right\\\},\\qquad\\rho\_\{\\delta\}\(v,w\):=\\frac\{B\_\{\\delta\}\(v,w\)\}\{N\_\{\\delta\}\(v\)N\_\{\\delta\}\(w\)\}\\quad\(v,w\\neq e\_\{D\}\)\.Then ρδ​\(v,w\)=⟨clrD⁡\(v\),clrD⁡\(w\)⟩2‖clrD⁡\(v\)‖2​‖clrD⁡\(w\)‖2,\\rho\_\{\\delta\}\(v,w\)=\\frac\{\\langle\\clr\_\{D\}\(v\),\\clr\_\{D\}\(w\)\\rangle\_\{2\}\}\{\\\|\\clr\_\{D\}\(v\)\\\|\_\{2\}\\\|\\clr\_\{D\}\(w\)\\\|\_\{2\}\},and, for allα,β\>0\\alpha,\\beta\>0,ρδ​\(α⊙v,β⊙w\)=ρδ​\(v,w\)\\rho\_\{\\delta\}\(\\alpha\\odot v,\\beta\\odot w\)=\\rho\_\{\\delta\}\(v,w\): the cosine is invariant under separate positive powerings, equivalently arbitrary separate temperaturesτv,τw\>0\\tau\_\{v\},\\tau\_\{w\}\>0\. A shared nonconstant coordinate\-bias vector generally does not preserve the cosine\.

###### Proof\.

Part \(i\) follows from𝒞⁡\(a​x\)=𝒞⁡\(x\)\\mathcal\{C\}\(ax\)=\\mathcal\{C\}\(x\)and𝒞⁡\(b​y\)=𝒞⁡\(y\)\\mathcal\{C\}\(by\)=\\mathcal\{C\}\(y\)\.

For part \(ii\), put

ai:=log⁡piqi,a¯D:=1D​∑i=1Dai,a¯S:=1m​∑i∈Sai,a\_\{i\}:=\\log\\frac\{p\_\{i\}\}\{q\_\{i\}\},\\qquad\\bar\{a\}\_\{D\}:=\\frac\{1\}\{D\}\\sum\_\{i=1\}^\{D\}a\_\{i\},\\qquad\\bar\{a\}\_\{S\}:=\\frac\{1\}\{m\}\\sum\_\{i\\in S\}a\_\{i\},so thatz:=clrD⁡\(p\)−clrD⁡\(q\)z:=\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)satisfieszi=ai−a¯Dz\_\{i\}=a\_\{i\}\-\\bar\{a\}\_\{D\}\. Also defineWS:=\{x∈ℋD:xi=0​for​i∉S\}W\_\{S\}:=\\\{x\\in\\mathcal\{H\}\_\{D\}:x\_\{i\}=0\\text\{ for \}i\\notin S\\\}\. The orthogonal projection ofzzontoWSW\_\{S\}is

\(PWS​z\)i=\{ai−a¯S,i∈S,0,i∉S\.\(P\_\{W\_\{S\}\}z\)\_\{i\}=\\begin\{cases\}a\_\{i\}\-\\bar\{a\}\_\{S\},&i\\in S,\\\\\[4\.0pt\] 0,&i\\notin S\.\\end\{cases\}Indeed, the displayed vector belongs toWSW\_\{S\}, and, for everyy∈WSy\\in W\_\{S\},

⟨z−PWS​z,y⟩2=\(a¯S−a¯D\)​∑i∈Syi=0\.\\left\\langle z\-P\_\{W\_\{S\}\}z,y\\right\\rangle\_\{2\}=\(\\bar\{a\}\_\{S\}\-\\bar\{a\}\_\{D\}\)\\sum\_\{i\\in S\}y\_\{i\}=0\.Its restriction to the coordinates inSSis preciselyclrm⁡\(p\(S\)\)−clrm⁡\(q\(S\)\)\\clr\_\{m\}\(p^\{\(S\)\}\)\-\\clr\_\{m\}\(q^\{\(S\)\}\), and it vanishes onScS^\{c\}\. Therefore

dA​\(p\(S\),q\(S\)\)=‖PWS​z‖2≤‖z‖2=dA​\(p,q\)\.d\_\{A\}\(p^\{\(S\)\},q^\{\(S\)\}\)=\\\|P\_\{W\_\{S\}\}z\\\|\_\{2\}\\leq\\\|z\\\|\_\{2\}=d\_\{A\}\(p,q\)\.Multiplication by the common positive constantκD\\kappa\_\{D\}proves part \(ii\)\.

For part \(iii\), after the stated relabeling define

ub:=D−1D​\(1,−1D−1,…,−1D−1\),Wc:=\{x∈ℋD:x0=0\}\.u\_\{b\}:=\\sqrt\{\\frac\{D\-1\}\{D\}\}\\left\(1,\-\\frac\{1\}\{D\-1\},\\ldots,\-\\frac\{1\}\{D\-1\}\\right\),\\qquad W\_\{c\}:=\\\{x\\in\\mathcal\{H\}\_\{D\}:x\_\{0\}=0\\\}\.Then‖ub‖2=1\\\|u\_\{b\}\\\|\_\{2\}=1,ub⟂Wcu\_\{b\}\\perp W\_\{c\}, andℋD=span⁡\{ub\}⊕⟂Wc\\mathcal\{H\}\_\{D\}=\\operatorname\{span\}\\\{u\_\{b\}\\\}\\mathbin\{\\overset\{\\perp\}\{\\oplus\}\}W\_\{c\}\. Direct calculation gives⟨clrD⁡\(r\),ub⟩2=b⁡\(r\)\\left\\langle\\clr\_\{D\}\(r\),u\_\{b\}\\right\\rangle\_\{2\}=b\(r\)and

PWc​clrD⁡\(r\)=\(0,clrD−1⁡\(r′\)1,…,clrD−1⁡\(r′\)D−1\)\.P\_\{W\_\{c\}\}\\clr\_\{D\}\(r\)=\\bigl\(0,\\clr\_\{D\-1\}\(r^\{\\prime\}\)\_\{1\},\\ldots,\\clr\_\{D\-1\}\(r^\{\\prime\}\)\_\{D\-1\}\\bigr\)\.Applying Pythagoras toclrD⁡\(p\)−clrD⁡\(q\)\\clr\_\{D\}\(p\)\-\\clr\_\{D\}\(q\)therefore yields

dA​\(p,q\)2=dA​\(p′,q′\)2\+\(b⁡\(p\)−b⁡\(q\)\)2\.d\_\{A\}\(p,q\)^\{2\}=d\_\{A\}\(p^\{\\prime\},q^\{\\prime\}\)^\{2\}\+\\bigl\(b\(p\)\-b\(q\)\\bigr\)^\{2\}\.Multiplying byκD2\\kappa\_\{D\}^\{2\}and using the definitions ofδD⟂\\delta\_\{D\}^\{\\perp\}andbδb\_\{\\delta\}proves \([19](https://arxiv.org/html/2608.14712#A3.E19)\)\. The displayed formula forPWc​clrD​\(r\)P\_\{W\_\{c\}\}\\clr\_\{D\}\(r\)proves the remaining sink claims\. In particular, the zero\-padded vector on the right is the orthogonal projection; simple deletion of the zeroth CLR coordinate would not give this vector without the displayed recentering\.

For part \(iv\), coordinatewise calculation gives

smD⁡\(ℓ\+u\)=smD⁡\(u\)⊕smD⁡\(ℓ\),smD⁡\(ℓ/τ\)=1τ⊙smD⁡\(ℓ\)\.\\operatorname\{sm\}\_\{D\}\(\\ell\+u\)=\\operatorname\{sm\}\_\{D\}\(u\)\\oplus\\operatorname\{sm\}\_\{D\}\(\\ell\),\\qquad\\operatorname\{sm\}\_\{D\}\(\\ell/\\tau\)=\\frac\{1\}\{\\tau\}\\odot\\operatorname\{sm\}\_\{D\}\(\\ell\)\.The two distance identities follow from \(A2\) and \(A3\)\. Moreover,

clrD⁡\(cen⁡\(p\(1\),…,p\(n\)\)\)=1n​∑j=1nclrD⁡\(p\(j\)\)\.\\clr\_\{D\}\\\!\\left\(\\cen\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\)\\right\)=\\frac\{1\}\{n\}\\sum\_\{j=1\}^\{n\}\\clr\_\{D\}\(p^\{\(j\)\}\)\.Consequently, for everyc∈𝒮Dc\\in\\mathcal\{S\}^\{D\}andα∈ℝ\\alpha\\in\\mathbb\{R\},

cen⁡\(c⊕p\(1\),…,c⊕p\(n\)\)=c⊕cen⁡\(p\(1\),…,p\(n\)\),\\displaystyle\\cen\(c\\oplus p^\{\(1\)\},\\ldots,c\\oplus p^\{\(n\)\}\)=c\\oplus\\cen\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\),cen⁡\(α⊙p\(1\),…,α⊙p\(n\)\)=α⊙cen⁡\(p\(1\),…,p\(n\)\)\.\\displaystyle\\cen\(\\alpha\\odot p^\{\(1\)\},\\ldots,\\alpha\\odot p^\{\(n\)\}\)=\\alpha\\odot\\cen\(p^\{\(1\)\},\\ldots,p^\{\(n\)\}\)\.Applying the two distance identities term by term proves the formulas forVδV\_\{\\delta\}\.

Finally, Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)and \([18](https://arxiv.org/html/2608.14712#A3.E18)\) give

Bδ​\(v,w\)=κD2​⟨clrD⁡\(v\),clrD⁡\(w\)⟩2\.B\_\{\\delta\}\(v,w\)=\\kappa\_\{D\}^\{2\}\\left\\langle\\clr\_\{D\}\(v\),\\clr\_\{D\}\(w\)\\right\\rangle\_\{2\}\.Forα,β\>0\\alpha,\\beta\>0, bilinearity and absolute homogeneity give

Bδ​\(α⊙v,β⊙w\)=α​β​Bδ​\(v,w\),Nδ​\(α⊙v\)=α​Nδ​\(v\)\.B\_\{\\delta\}\(\\alpha\\odot v,\\beta\\odot w\)=\\alpha\\beta B\_\{\\delta\}\(v,w\),\\qquad N\_\{\\delta\}\(\\alpha\\odot v\)=\\alpha N\_\{\\delta\}\(v\)\.The factorκD2\\kappa\_\{D\}^\{2\}, and then the positive powering factors, cancel upon normalizing, proving the cosine claims\. To see that every nonconstant shared coordinate bias can change a cosine, fixc≠eDc\\neq e\_\{D\}and puty:=clrD⁡\(c\)≠0y:=\\clr\_\{D\}\(c\)\\neq 0\. Surjectivity of the CLR map providesv,w∈𝒮Dv,w\\in\\mathcal\{S\}^\{D\}with CLR vectorsy/2y/2and−y/2\-y/2, respectively\. Thenρδ​\(v,w\)=−1\\rho\_\{\\delta\}\(v,w\)=\-1, whereasρδ​\(c⊕v,c⊕w\)=1\\rho\_\{\\delta\}\(c\\oplus v,c\\oplus w\)=1\. ∎

## Appendix DProofs for Section[4\.3](https://arxiv.org/html/2608.14712#S4.SS3)

Throughout this section, coordinate00is the designated sink,x′x^\{\\prime\}denotes the sink\-dropped, re\-closed content composition ofx∈𝒮Dx\\in\\mathcal\{S\}^\{D\}\(D≥3D\\geq 3\),H\(x\):=−∑jxjlogxjH\(x\):=\-\\sum\_\{j\}x\_\{j\}\\log x\_\{j\}with0​log⁡0:=00\\log 0:=0, anddA⟂​\(x,y\):=dA​\(x′,y′\)d\_\{A\}^\{\\perp\}\(x,y\):=d\_\{A\}\(x^\{\\prime\},y^\{\\prime\}\),CA⟂​\(x\):=∥clrD−1⁡\(x′\)∥2C\_\{A\}^\{\\perp\}\(x\):=\\lVert\\clr\_\{D\-1\}\(x^\{\\prime\}\)\\rVert\_\{2\}with no dimension\-dependent rescaling\.

### Full statement of Theorem[3](https://arxiv.org/html/2608.14712#Thmtheorem3)

Fix an integerD≥3D\\geq 3ands0∈\(12,1\)s\_\{0\}\\in\(\\tfrac\{1\}\{2\},1\), and designate coordinate00as the sink coordinate\. \(i\) For nonzerox,y∈ℝDx,y\\in\\mathbb\{R\}^\{D\}, writec⁡\(x,y\):=⟨x,y⟩/\(∥x∥2​∥y∥2\)c\(x,y\):=\\langle x,y\\rangle/\(\\lVert x\\rVert\_\{2\}\\lVert y\\rVert\_\{2\}\), and letJS\\mathrm\{JS\}denote the equal\-weight Jensen–Shannon divergence formed from Kullback–Leibler divergence with natural logarithms\. Ifp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}satisfyp0,q0≥s0p\_\{0\},q\_\{0\}\\geq s\_\{0\}, then

1−c⁡\(p,q\)\\displaystyle 1\-c\(p,q\)<\(1−s0\)2s02\+\(1−s0\)2≤\(1−s0s0\)2,\\displaystyle<\\frac\{\(1\-s\_\{0\}\)^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\}\\leq\\left\(\\frac\{1\-s\_\{0\}\}\{s\_\{0\}\}\\right\)^\{2\},∥p−q∥2\\displaystyle\\lVert p\-q\\rVert\_\{2\}<2​\(1−s0\)≤5​\(1−s0\),\\displaystyle<\\sqrt\{2\}\\,\(1\-s\_\{0\}\)\\leq\\sqrt\{5\}\\,\(1\-s\_\{0\}\),JS⁡\(p,q\)\\displaystyle\\mathrm\{JS\}\(p,q\)<\(1−s0\)​log⁡2≤32​\(1−s0\)\.\\displaystyle<\(1\-s\_\{0\}\)\\log 2\\leq\\frac\{3\}\{2\}\(1\-s\_\{0\}\)\.All three sharp endpoints are nevertheless approached from within the open simplex\. More precisely,

supp,q∈𝒮Dp0,q0≥s0\{1−c⁡\(p,q\)\}\\displaystyle\\sup\_\{\\begin\{subarray\}\{c\}p,q\\in\\mathcal\{S\}^\{D\}\\\\ p\_\{0\},q\_\{0\}\\geq s\_\{0\}\\end\{subarray\}\}\\\{1\-c\(p,q\)\\\}=\(1−s0\)2s02\+\(1−s0\)2,\\displaystyle=\\frac\{\(1\-s\_\{0\}\)^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\},supp,q∈𝒮Dp0,q0≥s0∥p−q∥2\\displaystyle\\sup\_\{\\begin\{subarray\}\{c\}p,q\\in\\mathcal\{S\}^\{D\}\\\\ p\_\{0\},q\_\{0\}\\geq s\_\{0\}\\end\{subarray\}\}\\lVert p\-q\\rVert\_\{2\}=2​\(1−s0\),\\displaystyle=\\sqrt\{2\}\\,\(1\-s\_\{0\}\),supp,q∈𝒮Dp0,q0≥s0JS⁡\(p,q\)\\displaystyle\\sup\_\{\\begin\{subarray\}\{c\}p,q\\in\\mathcal\{S\}^\{D\}\\\\ p\_\{0\},q\_\{0\}\\geq s\_\{0\}\\end\{subarray\}\}\\mathrm\{JS\}\(p,q\)=\(1−s0\)​log⁡2\.\\displaystyle=\(1\-s\_\{0\}\)\\log 2\.Thus none of the three suprema is attained\. In fact, their respective ranges are exactly

\[0,\(1−s0\)2s02\+\(1−s0\)2\),\[0,2​\(1−s0\)\),\[0,\(1−s0\)​log⁡2\),\\left\[0,\\frac\{\(1\-s\_\{0\}\)^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\}\\right\),\\qquad\\left\[0,\\sqrt\{2\}\(1\-s\_\{0\}\)\\right\),\\qquad\\left\[0,\(1\-s\_\{0\}\)\\log 2\\right\),and their upper endpoints are sharp\. In particular, all three range widths tend to zero ass0↑1s\_\{0\}\\uparrow 1, while the cosine range has the quadratic asymptotic behavior\(1−s0\)2s02\+\(1−s0\)2∼\(1−s0\)2\\frac\{\(1\-s\_\{0\}\)^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\}\\sim\(1\-s\_\{0\}\)^\{2\}ass0↑1s\_\{0\}\\uparrow 1\. \(ii\) Forr∈𝒮Dr\\in\\mathcal\{S\}^\{D\}, define the conditional content composition and the content\-preserving sink\-mass transformation by

r′:=\(r1,…,rD−1\)1−r0,Tt​r:=\(t,\(1−t\)​r′\),0<t<1\.r^\{\\prime\}:=\\frac\{\(r\_\{1\},\\ldots,r\_\{D\-1\}\)\}\{1\-r\_\{0\}\},\\qquad T\_\{t\}r:=\\bigl\(t,\(1\-t\)r^\{\\prime\}\\bigr\),\\qquad 0<t<1\.Then, for arbitraryr,z∈𝒮Dr,z\\in\\mathcal\{S\}^\{D\}andt,τ∈\(0,1\)t,\\tau\\in\(0,1\),

dA⟂​\(Tt​r,Tτ​z\)=dA⟂​\(r,z\),CA⟂​\(Tt​r\)=CA⟂​\(r\),CA⟂​\(Tτ​z\)=CA⟂​\(z\)\.d\_\{A\}^\{\\perp\}\(T\_\{t\}r,T\_\{\\tau\}z\)=d\_\{A\}^\{\\perp\}\(r,z\),\\qquad C\_\{A\}^\{\\perp\}\(T\_\{t\}r\)=C\_\{A\}^\{\\perp\}\(r\),\\qquad C\_\{A\}^\{\\perp\}\(T\_\{\\tau\}z\)=C\_\{A\}^\{\\perp\}\(z\)\.Consequently, for every integern≥1n\\geq 1, everyr\(1\),…,r\(n\)∈𝒮Dr^\{\(1\)\},\\ldots,r^\{\(n\)\}\\in\\mathcal\{S\}^\{D\}, and everyt1,…,tn∈\(0,1\)t\_\{1\},\\ldots,t\_\{n\}\\in\(0,1\),

dA⟂​\(Tti​r\(i\),Ttj​r\(j\)\)=dA⟂​\(r\(i\),r\(j\)\)\(1≤i,j≤n\),CA⟂​\(Tti​r\(i\)\)=CA⟂​\(r\(i\)\)\(1≤i≤n\)\.d\_\{A\}^\{\\perp\}\\bigl\(T\_\{t\_\{i\}\}r^\{\(i\)\},T\_\{t\_\{j\}\}r^\{\(j\)\}\\bigr\)=d\_\{A\}^\{\\perp\}\\bigl\(r^\{\(i\)\},r^\{\(j\)\}\\bigr\)\\quad\(1\\leq i,j\\leq n\),\\qquad C\_\{A\}^\{\\perp\}\\bigl\(T\_\{t\_\{i\}\}r^\{\(i\)\}\\bigr\)=C\_\{A\}^\{\\perp\}\\bigl\(r^\{\(i\)\}\\bigr\)\\quad\(1\\leq i\\leq n\)\.Hence composition\-specific changes of sink mass that preserve each conditional content composition leave the entire content\-distance matrix and all content\-concentration values unchanged\. In particular, choosingr0\(i\)<ti<1r^\{\(i\)\}\_\{0\}<t\_\{i\}<1increases every designated sink mass while leaving the conditional\-content geometry exactly unchanged\.

###### Proof of Theorem[3](https://arxiv.org/html/2608.14712#Thmtheorem3)\.

Fixp,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}withp0,q0≥s0p\_\{0\},q\_\{0\}\\geq s\_\{0\}\. Putε:=1−s0\\varepsilon:=1\-s\_\{0\}, and set

sp:=p0,sq:=q0,πp:=\(p1,…,pD−1\)1−sp,πq:=\(q1,…,qD−1\)1−sq\.s\_\{p\}:=p\_\{0\},\\qquad s\_\{q\}:=q\_\{0\},\\qquad\\pi\_\{p\}:=\\frac\{\(p\_\{1\},\\ldots,p\_\{D\-1\}\)\}\{1\-s\_\{p\}\},\\qquad\\pi\_\{q\}:=\\frac\{\(q\_\{1\},\\ldots,q\_\{D\-1\}\)\}\{1\-s\_\{q\}\}\.Becausep,q∈𝒮Dp,q\\in\\mathcal\{S\}^\{D\}, the vectorsπp,πq\\pi\_\{p\},\\pi\_\{q\}are strictly positive probability vectors withD−1≥2D\-1\\geq 2coordinates\. Hence

p=\(sp,\(1−sp\)​πp\),q=\(sq,\(1−sq\)​πq\),p=\\bigl\(s\_\{p\},\(1\-s\_\{p\}\)\\pi\_\{p\}\\bigr\),\\qquad q=\\bigl\(s\_\{q\},\(1\-s\_\{q\}\)\\pi\_\{q\}\\bigr\),and

∥πp∥2<1,∥πq∥2<1,⟨πp,πq⟩\>0\.\\lVert\\pi\_\{p\}\\rVert\_\{2\}<1,\\qquad\\lVert\\pi\_\{q\}\\rVert\_\{2\}<1,\\qquad\\langle\\pi\_\{p\},\\pi\_\{q\}\\rangle\>0\.
*Cosine dissimilarity\.*We have

⟨p,q⟩\\displaystyle\\langle p,q\\rangle=sp​sq\+\(1−sp\)​\(1−sq\)​⟨πp,πq⟩,\\displaystyle=s\_\{p\}s\_\{q\}\+\(1\-s\_\{p\}\)\(1\-s\_\{q\}\)\\langle\\pi\_\{p\},\\pi\_\{q\}\\rangle,∥p∥22\\displaystyle\\lVert p\\rVert\_\{2\}^\{2\}=sp2\+\(1−sp\)2​∥πp∥22≤sp2\+\(1−sp\)2,\\displaystyle=s\_\{p\}^\{2\}\+\(1\-s\_\{p\}\)^\{2\}\\lVert\\pi\_\{p\}\\rVert\_\{2\}^\{2\}\\leq s\_\{p\}^\{2\}\+\(1\-s\_\{p\}\)^\{2\},and the analogous norm inequality holds forqq\. Define

h⁡\(s\):=ss2\+\(1−s\)2,0<s<1\.h\(s\):=\\frac\{s\}\{\\sqrt\{s^\{2\}\+\(1\-s\)^\{2\}\}\},\\qquad 0<s<1\.Sincesp,sq<1s\_\{p\},s\_\{q\}<1and⟨πp,πq⟩\>0\\langle\\pi\_\{p\},\\pi\_\{q\}\\rangle\>0,

c⁡\(p,q\)\\displaystyle c\(p,q\)\>sp​sq∥p∥2​∥q∥2\\displaystyle\>\\frac\{s\_\{p\}s\_\{q\}\}\{\\lVert p\\rVert\_\{2\}\\lVert q\\rVert\_\{2\}\}≥sp​sqsp2\+\(1−sp\)2​sq2\+\(1−sq\)2=h⁡\(sp\)​h​\(sq\)\.\\displaystyle\\geq\\frac\{s\_\{p\}s\_\{q\}\}\{\\sqrt\{s\_\{p\}^\{2\}\+\(1\-s\_\{p\}\)^\{2\}\}\\sqrt\{s\_\{q\}^\{2\}\+\(1\-s\_\{q\}\)^\{2\}\}\}=h\(s\_\{p\}\)h\(s\_\{q\}\)\.Moreover,

h′​\(s\)=1−s\{s2\+\(1−s\)2\}3/2\>0\.h^\{\\prime\}\(s\)=\\frac\{1\-s\}\{\\\{s^\{2\}\+\(1\-s\)^\{2\}\\\}^\{3/2\}\}\>0\.Sincesp,sq≥s0s\_\{p\},s\_\{q\}\\geq s\_\{0\}, it follows that

c⁡\(p,q\)\>h​\(s0\)2=s02s02\+\(1−s0\)2,c\(p,q\)\>h\(s\_\{0\}\)^\{2\}=\\frac\{s\_\{0\}^\{2\}\}\{s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\},which proves the first strict bound\. The coarser bound follows becauses02\+\(1−s0\)2≥s02s\_\{0\}^\{2\}\+\(1\-s\_\{0\}\)^\{2\}\\geq s\_\{0\}^\{2\}\.

*Euclidean distance\.*Leta:=1−spa:=1\-s\_\{p\}andb:=1−sqb:=1\-s\_\{q\}\. Then0<a,b≤ε0<a,b\\leq\\varepsilon, and

∥p−q∥22\\displaystyle\\lVert p\-q\\rVert\_\{2\}^\{2\}=\(a−b\)2\+∥a​πp−b​πq∥22\\displaystyle=\(a\-b\)^\{2\}\+\\lVert a\\pi\_\{p\}\-b\\pi\_\{q\}\\rVert\_\{2\}^\{2\}=\(a−b\)2\+a2​∥πp∥22\+b2​∥πq∥22−2​a​b​⟨πp,πq⟩\\displaystyle=\(a\-b\)^\{2\}\+a^\{2\}\\lVert\\pi\_\{p\}\\rVert\_\{2\}^\{2\}\+b^\{2\}\\lVert\\pi\_\{q\}\\rVert\_\{2\}^\{2\}\-2ab\\langle\\pi\_\{p\},\\pi\_\{q\}\\rangle<\(a−b\)2\+a2\+b2\\displaystyle<\(a\-b\)^\{2\}\+a^\{2\}\+b^\{2\}=2​\(a2\+b2−a​b\)\.\\displaystyle=2\(a^\{2\}\+b^\{2\}\-ab\)\.Ifa≥ba\\geq b, then

a2\+b2−a​b=a2−b⁡\(a−b\)≤a2;a^\{2\}\+b^\{2\}\-ab=a^\{2\}\-b\(a\-b\)\\leq a^\{2\};ifb≥ab\\geq a, the symmetric argument givesa2\+b2−a​b≤b2a^\{2\}\+b^\{2\}\-ab\\leq b^\{2\}\. Therefore

∥p−q∥22<2​max⁡\{a,b\}2≤2​ε2,\\lVert p\-q\\rVert\_\{2\}^\{2\}<2\\max\\\{a,b\\\}^\{2\}\\leq 2\\varepsilon^\{2\},which proves the second strict bound; the displayed5\\sqrt\{5\}\-bound is the immediate numerical relaxation2≤5\\sqrt\{2\}\\leq\\sqrt\{5\}\.

*Jensen–Shannon divergence\.*Lete0,…,eD−1e\_\{0\},\\ldots,e\_\{D\-1\}be the standard coordinate vectors inℝD\\mathbb\{R\}^\{D\}, and define

u:=p−s0​e0ε,v:=q−s0​e0ε\.u:=\\frac\{p\-s\_\{0\}e\_\{0\}\}\{\\varepsilon\},\\qquad v:=\\frac\{q\-s\_\{0\}e\_\{0\}\}\{\\varepsilon\}\.The assumptionsp0,q0≥s0p\_\{0\},q\_\{0\}\\geq s\_\{0\}imply thatu,vu,vhave nonnegative coordinates, and

∑i=0D−1ui=∑i=0D−1vi=1−s0ε=1\.\\sum\_\{i=0\}^\{D\-1\}u\_\{i\}=\\sum\_\{i=0\}^\{D\-1\}v\_\{i\}=\\frac\{1\-s\_\{0\}\}\{\\varepsilon\}=1\.Thusu,vu,vare probability vectors in the closed simplex, and

p=s0​e0\+ε​u,q=s0​e0\+ε​v\.p=s\_\{0\}e\_\{0\}\+\\varepsilon u,\\qquad q=s\_\{0\}e\_\{0\}\+\\varepsilon v\.Set

w:=u\+v2,m:=p\+q2=s0​e0\+ε​w\.w:=\\frac\{u\+v\}\{2\},\\qquad m:=\\frac\{p\+q\}\{2\}=s\_\{0\}e\_\{0\}\+\\varepsilon w\.Thensupp⁡\(u\)∪supp⁡\(v\)=supp⁡\(w\)\\operatorname\{supp\}\(u\)\\cup\\operatorname\{supp\}\(v\)=\\operatorname\{supp\}\(w\)\. We use the standard closed\-simplex convention

KL\(x∥y\):=∑i:xi\>0xilogxiyi\\mathrm\{KL\}\(x\\\|y\):=\\sum\_\{i:x\_\{i\}\>0\}x\_\{i\}\\log\\frac\{x\_\{i\}\}\{y\_\{i\}\}whensupp⁡\(x\)⊆supp⁡\(y\)\\operatorname\{supp\}\(x\)\\subseteq\\operatorname\{supp\}\(y\), and set the divergence equal to\+∞\+\\inftyotherwise\. Thus all divergences below are finite\. Joint convexity of Kullback–Leibler divergence gives

KL\(p∥m\)\\displaystyle\\mathrm\{KL\}\(p\\\|m\)≤s0KL\(e0∥e0\)\+εKL\(u∥w\)=εKL\(u∥w\),\\displaystyle\\leq s\_\{0\}\\mathrm\{KL\}\(e\_\{0\}\\\|e\_\{0\}\)\+\\varepsilon\\mathrm\{KL\}\(u\\\|w\)=\\varepsilon\\mathrm\{KL\}\(u\\\|w\),KL\(q∥m\)\\displaystyle\\mathrm\{KL\}\(q\\\|m\)≤s0KL\(e0∥e0\)\+εKL\(v∥w\)=εKL\(v∥w\)\.\\displaystyle\\leq s\_\{0\}\\mathrm\{KL\}\(e\_\{0\}\\\|e\_\{0\}\)\+\\varepsilon\\mathrm\{KL\}\(v\\\|w\)=\\varepsilon\\mathrm\{KL\}\(v\\\|w\)\.Consequently,

JS⁡\(p,q\)≤ε​JS​\(u,v\)\.\\mathrm\{JS\}\(p,q\)\\leq\\varepsilon\\mathrm\{JS\}\(u,v\)\.
For any closed\-simplex probability vectorsx,yx,y, putz=\(x\+y\)/2z=\(x\+y\)/2\. Wheneverxi\>0x\_\{i\}\>0,

log⁡xizi=log⁡2​xixi\+yi≤log⁡2,\\log\\frac\{x\_\{i\}\}\{z\_\{i\}\}=\\log\\frac\{2x\_\{i\}\}\{x\_\{i\}\+y\_\{i\}\}\\leq\\log 2,with equality if and only ifyi=0y\_\{i\}=0\. HenceKL\(x∥z\)≤log2\\mathrm\{KL\}\(x\\\|z\)\\leq\\log 2, and the analogous conclusion holds withx,yx,yinterchanged\. It follows that

JS\(x,y\)≤log2,JS\(x,y\)=log2⟺supp\(x\)∩supp\(y\)=∅\.\\mathrm\{JS\}\(x,y\)\\leq\\log 2,\\qquad\\mathrm\{JS\}\(x,y\)=\\log 2\\quad\\Longleftrightarrow\\quad\\operatorname\{supp\}\(x\)\\cap\\operatorname\{supp\}\(y\)=\\varnothing\.For everyi=1,…,D−1i=1,\\ldots,D\-1, however,

ui=piε\>0,vi=qiε\>0\.u\_\{i\}=\\frac\{p\_\{i\}\}\{\\varepsilon\}\>0,\\qquad v\_\{i\}=\\frac\{q\_\{i\}\}\{\\varepsilon\}\>0\.Thus the supports ofuuandvvoverlap, and therefore

JS⁡\(p,q\)≤ε​JS​\(u,v\)<ε​log⁡2=\(1−s0\)​log⁡2\.\\mathrm\{JS\}\(p,q\)\\leq\\varepsilon\\mathrm\{JS\}\(u,v\)<\\varepsilon\\log 2=\(1\-s\_\{0\}\)\\log 2\.This proves the third strict bound;log⁡2<32\\log 2<\\frac\{3\}\{2\}gives the stated coarser bound\.

*Sharpness\.*The assumptionD≥3D\\geq 3provides two distinct nonsink coordinates\. In the closed simplex, let

p∗:=s0​e0\+ε​e1,q∗:=s0​e0\+ε​e2,m∗:=p∗\+q∗2\.p^\{\*\}:=s\_\{0\}e\_\{0\}\+\\varepsilon e\_\{1\},\\qquad q^\{\*\}:=s\_\{0\}e\_\{0\}\+\\varepsilon e\_\{2\},\\qquad m^\{\*\}:=\\frac\{p^\{\*\}\+q^\{\*\}\}\{2\}\.Direct calculation gives

1−c⁡\(p∗,q∗\)\\displaystyle 1\-c\(p^\{\*\},q^\{\*\}\)=ε2s02\+ε2,\\displaystyle=\\frac\{\\varepsilon^\{2\}\}\{s\_\{0\}^\{2\}\+\\varepsilon^\{2\}\},∥p∗−q∗∥2\\displaystyle\\lVert p^\{\*\}\-q^\{\*\}\\rVert\_\{2\}=2​ε,\\displaystyle=\\sqrt\{2\}\\,\\varepsilon,KL\(p∗∥m∗\)\\displaystyle\\mathrm\{KL\}\(p^\{\*\}\\\|m^\{\*\}\)=KL\(q∗∥m∗\)=εlog2,\\displaystyle=\\mathrm\{KL\}\(q^\{\*\}\\\|m^\{\*\}\)=\\varepsilon\\log 2,and henceJS⁡\(p∗,q∗\)=ε​log⁡2\\mathrm\{JS\}\(p^\{\*\},q^\{\*\}\)=\\varepsilon\\log 2\.

To approximate this pair from within𝒮D\\mathcal\{S\}^\{D\}, fix0<δ<1/\(D−1\)0<\\delta<1/\(D\-1\)and, forj∈\{1,…,D−1\}j\\in\\\{1,\\ldots,D\-1\\\}, define

\(πp,δ\)j:=\{1−\(D−2\)​δ,j=1,δ,j≠1,\(πq,δ\)j:=\{1−\(D−2\)​δ,j=2,δ,j≠2\.\(\\pi\_\{p,\\delta\}\)\_\{j\}:=\\begin\{cases\}1\-\(D\-2\)\\delta,&j=1,\\\\ \\delta,&j\\neq 1,\\end\{cases\}\\qquad\(\\pi\_\{q,\\delta\}\)\_\{j\}:=\\begin\{cases\}1\-\(D\-2\)\\delta,&j=2,\\\\ \\delta,&j\\neq 2\.\\end\{cases\}Set

pδ:=\(s0,ε​πp,δ\),qδ:=\(s0,ε​πq,δ\)\.p\_\{\\delta\}:=\\bigl\(s\_\{0\},\\varepsilon\\pi\_\{p,\\delta\}\\bigr\),\\qquad q\_\{\\delta\}:=\\bigl\(s\_\{0\},\\varepsilon\\pi\_\{q,\\delta\}\\bigr\)\.Eachπp,δ\\pi\_\{p,\\delta\}andπq,δ\\pi\_\{q,\\delta\}is a strictly positive probability vector, sopδ,qδ∈𝒮Dp\_\{\\delta\},q\_\{\\delta\}\\in\\mathcal\{S\}^\{D\}, and

\(pδ,qδ\)⟶\(p∗,q∗\)as​δ↓0\.\(p\_\{\\delta\},q\_\{\\delta\}\)\\longrightarrow\(p^\{\*\},q^\{\*\}\)\\qquad\\text\{as \}\\delta\\downarrow 0\.Cosine dissimilarity and Euclidean distance are continuous at this closed\-simplex limit\. Jensen–Shannon divergence is also continuous there because

JS⁡\(x,y\)=H⁡\(x\+y2\)−H⁡\(x\)\+H⁡\(y\)2,\\mathrm\{JS\}\(x,y\)=H\\\!\\left\(\\frac\{x\+y\}\{2\}\\right\)\-\\frac\{H\(x\)\+H\(y\)\}\{2\},with0​log⁡0:=00\\log 0:=0, andt↦−t​log⁡tt\\mapsto\-t\\log tis continuous on\[0,1\]\[0,1\]under this convention\. The three upper endpoints are therefore the exact suprema\. Their nonattainment follows from the strict bounds already proved\.

Finally, all three dissimilarities are nonnegative: for cosine dissimilarity this follows from Cauchy–Schwarz, for Euclidean distance it is immediate, and for Jensen–Shannon divergence it follows from nonnegativity of Kullback–Leibler divergence\. Each takes the value zero whenp=qp=q\. The feasible set of pairs is convex and hence path connected, and each of the three dissimilarities is continuous on it\. The image of this set under each dissimilarity is therefore an interval\. Because that interval contains zero, has the displayed supremum, and does not contain its supremum, it is exactly the corresponding half\-open interval stated in part \(i\)\. This proves all assertions in part \(i\)\.

For part \(ii\), fixr,z∈𝒮Dr,z\\in\\mathcal\{S\}^\{D\}andt,τ∈\(0,1\)t,\\tau\\in\(0,1\)\. Forj=1,…,D−1j=1,\\ldots,D\-1, one hasrj′\>0r^\{\\prime\}\_\{j\}\>0, and

∑j=1D−1rj′=∑j=1D−1rj1−r0=1\.\\sum\_\{j=1\}^\{D\-1\}r^\{\\prime\}\_\{j\}=\\frac\{\\sum\_\{j=1\}^\{D\-1\}r\_\{j\}\}\{1\-r\_\{0\}\}=1\.Thusr′r^\{\\prime\}is a strictly positive probability vector, and the same holds forz′z^\{\\prime\}\. Moreover, all coordinates ofTt​rT\_\{t\}rare strictly positive and

∑j=0D−1\(Tt​r\)j=t\+\(1−t\)​∑j=1D−1rj′=1\.\\sum\_\{j=0\}^\{D\-1\}\(T\_\{t\}r\)\_\{j\}=t\+\(1\-t\)\\sum\_\{j=1\}^\{D\-1\}r^\{\\prime\}\_\{j\}=1\.HenceTt​r∈𝒮DT\_\{t\}r\\in\\mathcal\{S\}^\{D\}, and similarlyTτ​z∈𝒮DT\_\{\\tau\}z\\in\\mathcal\{S\}^\{D\}\. Directly,

\(Tt​r\)′=\(1−t\)​r′1−t=r′,\(Tτ​z\)′=\(1−τ\)​z′1−τ=z′\.\(T\_\{t\}r\)^\{\\prime\}=\\frac\{\(1\-t\)r^\{\\prime\}\}\{1\-t\}=r^\{\\prime\},\\qquad\(T\_\{\\tau\}z\)^\{\\prime\}=\\frac\{\(1\-\\tau\)z^\{\\prime\}\}\{1\-\\tau\}=z^\{\\prime\}\.Therefore

dA⟂​\(Tt​r,Tτ​z\)\\displaystyle d\_\{A\}^\{\\perp\}\(T\_\{t\}r,T\_\{\\tau\}z\)=dA​\(\(Tt​r\)′,\(Tτ​z\)′\)=dA​\(r′,z′\)=dA⟂​\(r,z\),\\displaystyle=d\_\{A\}\\bigl\(\(T\_\{t\}r\)^\{\\prime\},\(T\_\{\\tau\}z\)^\{\\prime\}\\bigr\)=d\_\{A\}\(r^\{\\prime\},z^\{\\prime\}\)=d\_\{A\}^\{\\perp\}\(r,z\),CA⟂​\(Tt​r\)\\displaystyle C\_\{A\}^\{\\perp\}\(T\_\{t\}r\)=∥clrD−1⁡\(\(Tt​r\)′\)∥2=∥clrD−1⁡\(r′\)∥2=CA⟂​\(r\),\\displaystyle=\\lVert\\clr\_\{D\-1\}\(\(T\_\{t\}r\)^\{\\prime\}\)\\rVert\_\{2\}=\\lVert\\clr\_\{D\-1\}\(r^\{\\prime\}\)\\rVert\_\{2\}=C\_\{A\}^\{\\perp\}\(r\),and the same calculation yieldsCA⟂​\(Tτ​z\)=CA⟂​\(z\)C\_\{A\}^\{\\perp\}\(T\_\{\\tau\}z\)=C\_\{A\}^\{\\perp\}\(z\)\. Applying these identities to every pair\(i,j\)\(i,j\)and every indexiiin a finite family proves the two finite\-family claims\. Finally, ifr0\(i\)<ti<1r^\{\(i\)\}\_\{0\}<t\_\{i\}<1, then\(Tti​r\(i\)\)0=ti\>r0\(i\)\\bigl\(T\_\{t\_\{i\}\}r^\{\(i\)\}\\bigr\)\_\{0\}=t\_\{i\}\>r^\{\(i\)\}\_\{0\}, which proves the last assertion\. The restrictionsti∈\(0,1\)t\_\{i\}\\in\(0,1\)ensure that every transformed vector remains in the open simplex; no endpoint assertion is required\. ∎

### Full statement of Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)

Fix an integerD≥3D\\geq 3, index the sink coordinate by00, and, forx=\(x0,…,xD−1\)∈𝒮Dx=\(x\_\{0\},\\ldots,x\_\{D\-1\}\)\\in\\mathcal\{S\}^\{D\}, writex′:=𝒞⁡\(x1,…,xD−1\)=\(x1,…,xD−1\)/\(1−x0\)∈𝒮D−1x^\{\\prime\}:=\\mathcal\{C\}\(x\_\{1\},\\ldots,x\_\{D\-1\}\)=\(x\_\{1\},\\ldots,x\_\{D\-1\}\)/\(1\-x\_\{0\}\)\\in\\mathcal\{S\}^\{D\-1\}\.*\(i\)*Leth1,…,hn∈𝒮Dh\_\{1\},\\ldots,h\_\{n\}\\in\\mathcal\{S\}^\{D\},n≥2n\\geq 2, be a finite labelled collection of heads\. Define the content\-distance matrix computed without first modifying the stored heads and the Aitchison\-distance matrix computed after explicit sink removal by

Δi​jcontent:=dA⟂​\(hi,hj\),Δi​jexplicit​\-​drop:=dA​\(hi′,hj′\)\.\\Delta^\{\\mathrm\{content\}\}\_\{ij\}:=d\_\{A\}^\{\\perp\}\(h\_\{i\},h\_\{j\}\),\\qquad\\Delta^\{\\mathrm\{explicit\\text\{\-\}drop\}\}\_\{ij\}:=d\_\{A\}\(h\_\{i\}^\{\\prime\},h\_\{j\}^\{\\prime\}\)\.ThenΔcontent=Δexplicit​\-​drop\\Delta^\{\\mathrm\{content\}\}=\\Delta^\{\\mathrm\{explicit\\text\{\-\}drop\}\}entrywise\. Consequently, any deterministic clustering rule of the form𝒜⁡\(Δ,η\)\\mathcal\{A\}\(\\Delta;\\eta\), whose sole data\-dependent input is the labelled dissimilarity matrix and whose ancillary specificationη\\eta, including matrix\-level preprocessing, linkage, hyperparameters, tie\-breaking, and stopping rules, is held fixed, gives identical outputs in the two analyses\. In particular, the rule may not additionally inspect the raw heads, their sink masses, or their ambient dimension\. If the same measurable randomized implementation is written as𝒜⁡\(Δ,U,η\)\\mathcal\{A\}\(\\Delta,U;\\eta\), and in both analysesUUhas conditional lawK⁡\(d​u∣Δ,η\)K\(du\\mid\\Delta,\\eta\), then the outputs have the same distribution: for every measurable output eventBB, both output probabilities equal

∫\{𝒜\(Δ,u;η\)∈B\}K\(du∣Δ,η\)\.\\int\\mathbf\{1\}\\\!\\left\\\{\\mathcal\{A\}\(\\Delta,u;\\eta\)\\in B\\right\\\}K\(du\\mid\\Delta,\\eta\)\.Moreover, there exists a common\-randomness coupling: draw oneU∼K\(⋅∣Δ,η\)U\\sim K\(\\,\\cdot\\mid\\Delta,\\eta\)and use it in both analyses; under this coupling the outputs are equal almost surely\. This identity concernsdA⟂d\_\{A\}^\{\\perp\}; it does not assert that the ordinary full\-composition distancedA​\(hi,hj\)d\_\{A\}\(h\_\{i\},h\_\{j\}\)is unchanged by sink removal\.*\(ii\)*For probability vectorsx,yx,yof the same finite length, possibly with zero coordinates, definec⁡\(x,y\):=⟨x,y⟩/\(‖x‖2​‖y‖2\)c\(x,y\):=\\langle x,y\\rangle/\(\\\|x\\\|\_\{2\}\\\|y\\\|\_\{2\}\),dcos​\(x,y\):=1−c⁡\(x,y\)d\_\{\\cos\}\(x,y\):=1\-c\(x,y\),dE​\(x,y\):=‖x−y‖2d\_\{\\mathrm\{E\}\}\(x,y\):=\\\|x\-y\\\|\_\{2\}, and

JS⁡\(x,y\):=H⁡\(x\+y2\)−12​H​\(x\)−12​H​\(y\)=12​KL​\(x∥x\+y2\)\+12​KL​\(y∥x\+y2\),\\mathrm\{JS\}\(x,y\):=H\\\!\\left\(\\frac\{x\+y\}\{2\}\\right\)\-\\frac\{1\}\{2\}H\(x\)\-\\frac\{1\}\{2\}H\(y\)=\\frac\{1\}\{2\}\\mathrm\{KL\}\\\!\\left\(x\\middle\\\|\\frac\{x\+y\}\{2\}\\right\)\+\\frac\{1\}\{2\}\\mathrm\{KL\}\\\!\\left\(y\\middle\\\|\\frac\{x\+y\}\{2\}\\right\),whereKL\(x∥y\):=∑ℓ:xℓ\>0xℓlog\(xℓ/yℓ\)\\mathrm\{KL\}\(x\\\|y\):=\\sum\_\{\\ell:x\_\{\\ell\}\>0\}x\_\{\\ell\}\\log\(x\_\{\\ell\}/y\_\{\\ell\}\), withKL\(x∥y\)=\+∞\\mathrm\{KL\}\(x\\\|y\)=\+\\inftyifxℓ\>0x\_\{\\ell\}\>0andyℓ=0y\_\{\\ell\}=0for someℓ\\ell; in the displayed Jensen–Shannon formula the midpoint has support containing those of both arguments, so both relative entropies are finite\. There exists a nonempty relatively open set𝒰D⊂\(𝒮D\)3\\mathcal\{U\}\_\{D\}\\subset\(\\mathcal\{S\}^\{D\}\)^\{3\}having positive relative3​\(D−1\)3\(D\-1\)\-dimensional Lebesgue measure in the affine hull, and hence positive3​\(D−1\)3\(D\-1\)\-dimensional Hausdorff measure, such that, for every\(p,q,r\)∈𝒰D\(p,q,r\)\\in\\mathcal\{U\}\_\{D\}, simultaneously forδ∈\{dcos,JS,dE\}\\delta\\in\\\{d\_\{\\cos\},\\mathrm\{JS\},d\_\{\\mathrm\{E\}\}\\\}, the unique closest pair with the sink retained is\(p,r\)\(p,r\), whereas the unique closest pair after sink removal is\(p,q\)\(p,q\)\. Hence, for each of the three dissimilarities, both standard single\-linkage and standard complete\-linkage agglomerative clustering, initialized at singleton clusters and stopped atK=2K=2, return

𝒫keep=\{\{p,r\},\{q\}\},𝒫drop=\{\{p,q\},\{r\}\}\.\\mathcal\{P\}\_\{\\mathrm\{keep\}\}=\\bigl\\\{\\\{p,r\\\},\\\{q\\\}\\bigr\\\},\\qquad\\mathcal\{P\}\_\{\\mathrm\{drop\}\}=\\bigl\\\{\\\{p,q\\\},\\\{r\\\}\\bigr\\\}\.ForD=3D=3, the set𝒰3\\mathcal\{U\}\_\{3\}contains the exact witness

p=150​\(45,4,1\),q=150​\(35,12,3\),r=150​\(45,1,4\)\.p=\\frac\{1\}\{50\}\(45,4,1\),\\qquad q=\\frac\{1\}\{50\}\(35,12,3\),\\qquad r=\\frac\{1\}\{50\}\(45,1,4\)\.
###### Proof of Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)\.

For part*\(i\)*, the standing definition gives, for everyi,ji,j,

Δi​jcontent=dA⟂​\(hi,hj\)=dA​\(hi′,hj′\)=Δi​jexplicit​\-​drop\.\\Delta^\{\\mathrm\{content\}\}\_\{ij\}=d\_\{A\}^\{\\perp\}\(h\_\{i\},h\_\{j\}\)=d\_\{A\}\(h\_\{i\}^\{\\prime\},h\_\{j\}^\{\\prime\}\)=\\Delta^\{\\mathrm\{explicit\\text\{\-\}drop\}\}\_\{ij\}\.Thus both analyses supply exactly the same labelled matrix to the same clustering implementation\. The deterministic outputs are therefore equal\. In the randomized case, the displayed integral in the statement depends only on the common pair\(Δ,η\)\(\\Delta,\\eta\), so the two output laws coincide\. If the same drawUUis used in both analyses, the two arguments of𝒜\\mathcal\{A\}agree pointwise, and hence the coupled outputs are equal almost surely\.

For part*\(ii\)*, first establish nonemptiness in every dimensionD≥3D\\geq 3\. Chooseπ∈𝒮D−1\\pi\\in\\mathcal\{S\}^\{D\-1\}whose first two coordinates are unequal, and letπ~\\widetilde\{\\pi\}be obtained by interchanging those two coordinates\. Fixa∈\(0,1\)a\\in\(0,1\), and, fors∈\(a,1\)s\\in\(a,1\), define

ps=\(s,\(1−s\)​π\),q=\(a,\(1−a\)​π\),rs=\(s,\(1−s\)​π~\)\.p\_\{s\}=\\bigl\(s,\(1\-s\)\\pi\\bigr\),\\qquad q=\\bigl\(a,\(1\-a\)\\pi\\bigr\),\\qquad r\_\{s\}=\\bigl\(s,\(1\-s\)\\widetilde\{\\pi\}\\bigr\)\.Thenps′=q′=πp\_\{s\}^\{\\prime\}=q^\{\\prime\}=\\piandrs′=π~≠πr\_\{s\}^\{\\prime\}=\\widetilde\{\\pi\}\\neq\\pi\.

Each ofdcosd\_\{\\cos\},JS\\mathrm\{JS\}, anddEd\_\{\\mathrm\{E\}\}is nonnegative and vanishes on probability vectors precisely when its two arguments are equal\. This is immediate fordEd\_\{\\mathrm\{E\}\}\. Fordcosd\_\{\\cos\}, it follows from the equality case of Cauchy–Schwarz, because proportional unit\-sum probability vectors must be equal\. ForJS\\mathrm\{JS\}, it follows from the strict concavity ofHH\. Hence, for everyδ∈\{dcos,JS,dE\}\\delta\\in\\\{d\_\{\\cos\},\\mathrm\{JS\},d\_\{\\mathrm\{E\}\}\\\},

δ⁡\(ps′,q′\)=0<δ⁡\(ps′,rs′\)=δ⁡\(q′,rs′\)\.\\delta\(p\_\{s\}^\{\\prime\},q^\{\\prime\}\)=0<\\delta\(p\_\{s\}^\{\\prime\},r\_\{s\}^\{\\prime\}\)=\\delta\(q^\{\\prime\},r\_\{s\}^\{\\prime\}\)\.
Lete0=\(1,0,…,0\)e\_\{0\}=\(1,0,\\ldots,0\)\. Ass↑1s\\uparrow 1, bothpsp\_\{s\}andrsr\_\{s\}converge toe0e\_\{0\}, whereasq≠e0q\\neq e\_\{0\}\. Euclidean and cosine dissimilarities are continuous on the closed simplex; for cosine, the denominator never vanishes there\. The entropy definition ofJS\\mathrm\{JS\}, together with0​log⁡0=00\\log 0=0, shows thatJS\\mathrm\{JS\}is also continuous on the closed simplex\. Therefore, for each suchδ\\delta,

δ⁡\(ps,rs\)⟶0,δ⁡\(ps,q\)⟶δ⁡\(e0,q\)\>0,δ⁡\(q,rs\)⟶δ⁡\(q,e0\)\>0\.\\delta\(p\_\{s\},r\_\{s\}\)\\longrightarrow 0,\\qquad\\delta\(p\_\{s\},q\)\\longrightarrow\\delta\(e\_\{0\},q\)\>0,\\qquad\\delta\(q,r\_\{s\}\)\\longrightarrow\\delta\(q,e\_\{0\}\)\>0\.Because the family of dissimilarities is finite, there exists a commons∗∈\(a,1\)s\_\{\*\}\\in\(a,1\), sufficiently close to11, such that, simultaneously for all three,

δ⁡\(ps∗,rs∗\)<min⁡\{δ⁡\(ps∗,q\),δ⁡\(q,rs∗\)\}\.\\delta\(p\_\{s\_\{\*\}\},r\_\{s\_\{\*\}\}\)<\\min\\bigl\\\{\\delta\(p\_\{s\_\{\*\}\},q\),\\delta\(q,r\_\{s\_\{\*\}\}\)\\bigr\\\}\.
Define𝒰D\\mathcal\{U\}\_\{D\}to be the set of triples\(p,q,r\)∈\(𝒮D\)3\(p,q,r\)\\in\(\\mathcal\{S\}^\{D\}\)^\{3\}satisfying, for everyδ∈\{dcos,JS,dE\}\\delta\\in\\\{d\_\{\\cos\},\\mathrm\{JS\},d\_\{\\mathrm\{E\}\}\\\},

δ⁡\(p,r\)\\displaystyle\\delta\(p,r\)<δ⁡\(p,q\),\\displaystyle<\\delta\(p,q\),δ⁡\(p,r\)\\displaystyle\\delta\(p,r\)<δ⁡\(q,r\),\\displaystyle<\\delta\(q,r\),δ⁡\(p′,q′\)\\displaystyle\\delta\(p^\{\\prime\},q^\{\\prime\}\)<δ⁡\(p′,r′\),\\displaystyle<\\delta\(p^\{\\prime\},r^\{\\prime\}\),δ⁡\(p′,q′\)\\displaystyle\\delta\(p^\{\\prime\},q^\{\\prime\}\)<δ⁡\(q′,r′\)\.\\displaystyle<\\delta\(q^\{\\prime\},r^\{\\prime\}\)\.The preceding construction shows that\(ps∗,q,rs∗\)∈𝒰D\(p\_\{s\_\{\*\}\},q,r\_\{s\_\{\*\}\}\)\\in\\mathcal\{U\}\_\{D\}, so𝒰D\\mathcal\{U\}\_\{D\}is nonempty\. The sink\-removal map is continuous on the open simplex because1−x0\>01\-x\_\{0\}\>0, and all three dissimilarities are continuous there\. Thus𝒰D\\mathcal\{U\}\_\{D\}, being defined by finitely many strict inequalities, is relatively open in\(𝒮D\)3\(\\mathcal\{S\}^\{D\}\)^\{3\}\. Every nonempty relatively open subset of this3​\(D−1\)3\(D\-1\)\-dimensional open set in its affine hull contains a relative open ball\. It therefore has positive relative Lebesgue measure and positive3​\(D−1\)3\(D\-1\)\-dimensional Hausdorff measure\.

It remains to verify the displayedD=3D=3witness exactly\. For Euclidean distance,

‖p−r‖22=182500<1682500=‖p−q‖22<2222500=‖q−r‖22\.\\\|p\-r\\\|\_\{2\}^\{2\}=\\frac\{18\}\{2500\}<\\frac\{168\}\{2500\}=\\\|p\-q\\\|\_\{2\}^\{2\}<\\frac\{222\}\{2500\}=\\\|q\-r\\\|\_\{2\}^\{2\}\.Thus\(p,r\)\(p,r\)is the unique Euclidean closest pair\.

Multiplying all three vectors by5050, which does not change cosine similarity, givesP=\(45,4,1\)P=\(45,4,1\),Q=\(35,12,3\)Q=\(35,12,3\),R=\(45,1,4\)R=\(45,1,4\)\. Consequently,

c⁡\(p,r\)=20332042,c⁡\(p,q\)=16262042⋅1378,c⁡\(q,r\)=15992042⋅1378\.c\(p,r\)=\\frac\{2033\}\{2042\},\\qquad c\(p,q\)=\\frac\{1626\}\{\\sqrt\{2042\\cdot 1378\}\},\\qquad c\(q,r\)=\\frac\{1599\}\{\\sqrt\{2042\\cdot 1378\}\}\.The last two quantities have the same positive denominator\. Moreover, all three displayed cosine similarities are positive, and thereforec⁡\(p,r\)\>c⁡\(p,q\)c\(p,r\)\>c\(p,q\)is equivalent, after squaring, to

20332⋅1378−16262⋅2042=296601850\>0\.2033^\{2\}\\cdot 1378\-1626^\{2\}\\cdot 2042=296601850\>0\.This comparison and1626\>15991626\>1599implyc⁡\(p,r\)\>c⁡\(p,q\)\>c⁡\(q,r\)c\(p,r\)\>c\(p,q\)\>c\(q,r\)\. Thus\(p,r\)\(p,r\)is also the unique closest pair underdcos=1−cd\_\{\\cos\}=1\-c\.

We next record the two Jensen–Shannon mixture identities used below\. Letu,vu,vbe probability vectors of the same size, putw=\(u\+v\)/2w=\(u\+v\)/2, and lets,t∈\(0,1\)s,t\\in\(0,1\)\. Then

KL\(\(s,\(1−s\)u\)∥\(s,\(1−s\)w\)\)=\(1−s\)KL\(u∥w\),\\mathrm\{KL\}\\\!\\left\(\(s,\(1\-s\)u\)\\middle\\\|\(s,\(1\-s\)w\)\\right\)=\(1\-s\)\\mathrm\{KL\}\(u\\\|w\),and the analogous identity holds withvv\. It follows that

JS⁡\(\(s,\(1−s\)​u\),\(s,\(1−s\)​v\)\)=\(1−s\)​JS​\(u,v\)\.\\mathrm\{JS\}\\bigl\(\(s,\(1\-s\)u\),\(s,\(1\-s\)v\)\\bigr\)=\(1\-s\)\\mathrm\{JS\}\(u,v\)\.Next puts¯=\(s\+t\)/2\\bar\{s\}=\(s\+t\)/2\. The midpoint of\(s,\(1−s\)​u\)\(s,\(1\-s\)u\)and\(t,\(1−t\)​u\)\(t,\(1\-t\)u\)is\(s¯,\(1−s¯\)​u\)\(\\bar\{s\},\(1\-\\bar\{s\}\)u\), and direct summation gives

KL\(\(s,\(1−s\)u\)∥\(s¯,\(1−s¯\)u\)\)=KL\(\(s,1−s\)∥\(s¯,1−s¯\)\)\.\\mathrm\{KL\}\\\!\\left\(\(s,\(1\-s\)u\)\\middle\\\|\(\\bar\{s\},\(1\-\\bar\{s\}\)u\)\\right\)=\\mathrm\{KL\}\\bigl\(\(s,1\-s\)\\\|\(\\bar\{s\},1\-\\bar\{s\}\)\\bigr\)\.Applying the analogous identity tottand averaging yields

JS⁡\(\(s,\(1−s\)​u\),\(t,\(1−t\)​u\)\)=JS⁡\(\(s,1−s\),\(t,1−t\)\)\.\\mathrm\{JS\}\\bigl\(\(s,\(1\-s\)u\),\(t,\(1\-t\)u\)\\bigr\)=\\mathrm\{JS\}\\bigl\(\(s,1\-s\),\(t,1\-t\)\\bigr\)\.
For the witness, put

π\+=\(45,15\),π−=\(15,45\)\.\\pi\_\{\+\}=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\),\\qquad\\pi\_\{\-\}=\\left\(\\frac\{1\}\{5\},\\frac\{4\}\{5\}\\right\)\.The first mixture identity and direct calculation give

JS⁡\(p,r\)\\displaystyle\\mathrm\{JS\}\(p,r\)=110​JS​\(π\+,π−\)=110​\(45​log⁡85\+15​log⁡25\)=150​log⁡\(81923125\)<150\.\\displaystyle=\\frac\{1\}\{10\}\\mathrm\{JS\}\(\\pi\_\{\+\},\\pi\_\{\-\}\)=\\frac\{1\}\{10\}\\left\(\\frac\{4\}\{5\}\\log\\frac\{8\}\{5\}\+\\frac\{1\}\{5\}\\log\\frac\{2\}\{5\}\\right\)=\\frac\{1\}\{50\}\\log\\\!\\left\(\\frac\{8192\}\{3125\}\\right\)<\\frac\{1\}\{50\}\.Indeed,

81923125<6524=∑k=041k\!<e\.\\frac\{8192\}\{3125\}<\\frac\{65\}\{24\}=\\sum\_\{k=0\}^\{4\}\\frac\{1\}\{k\!\}<\\mathrm\{e\}\.
Becauseppandqqhave the same conditional content composition, the second mixture identity gives

JS⁡\(p,q\)=JS⁡\(\(910,110\),\(710,310\)\)\.\\mathrm\{JS\}\(p,q\)=\\mathrm\{JS\}\\\!\\left\(\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{10\}\\right\),\\left\(\\frac\{7\}\{10\},\\frac\{3\}\{10\}\\right\)\\right\)\.The midpoint of these two binary probability vectors ism=\(45,15\)m=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\), and each endpoint has total\-variation distance1/101/10frommm, whereTV⁡\(u,v\):=12​‖u−v‖1\\operatorname\{TV\}\(u,v\):=\\frac\{1\}\{2\}\\\|u\-v\\\|\_\{1\}\. Pinsker’s inequality for natural logarithms,KL\(u∥v\)≥2TV\(u,v\)2\\mathrm\{KL\}\(u\\\|v\)\\geq 2\\operatorname\{TV\}\(u,v\)^\{2\}, therefore gives

JS⁡\(p,q\)≥12​\(150\+150\)=150\>JS⁡\(p,r\)\.\\mathrm\{JS\}\(p,q\)\\geq\\frac\{1\}\{2\}\\left\(\\frac\{1\}\{50\}\+\\frac\{1\}\{50\}\\right\)=\\frac\{1\}\{50\}\>\\mathrm\{JS\}\(p,r\)\.
Now define the deterministic coarse\-grainingT⁡\(x0,x1,x2\):=\(x0,x1\+x2\)T\(x\_\{0\},x\_\{1\},x\_\{2\}\):=\(x\_\{0\},x\_\{1\}\+x\_\{2\}\)\. SinceT⁡\(x\+y2\)=T​x\+T​y2T\\\!\\left\(\\frac\{x\+y\}\{2\}\\right\)=\\frac\{Tx\+Ty\}\{2\}, the data\-processing inequality applied to each relative\-entropy term defining Jensen–Shannon divergence gives

JS⁡\(q,r\)\\displaystyle\\mathrm\{JS\}\(q,r\)≥JS⁡\(T​q,T​r\)=JS⁡\(\(710,310\),\(910,110\)\)=JS⁡\(p,q\)\>JS⁡\(p,r\)\.\\displaystyle\\geq\\mathrm\{JS\}\(Tq,Tr\)=\\mathrm\{JS\}\\\!\\left\(\\left\(\\frac\{7\}\{10\},\\frac\{3\}\{10\}\\right\),\\left\(\\frac\{9\}\{10\},\\frac\{1\}\{10\}\\right\)\\right\)=\\mathrm\{JS\}\(p,q\)\>\\mathrm\{JS\}\(p,r\)\.The last equality uses symmetry ofJS\\mathrm\{JS\}and the second mixture identity\. Thus\(p,r\)\(p,r\)is the unique Jensen–Shannon closest pair\.

After sink removal,

p′=q′=\(45,15\),r′=\(15,45\)\.p^\{\\prime\}=q^\{\\prime\}=\\left\(\\frac\{4\}\{5\},\\frac\{1\}\{5\}\\right\),\\qquad r^\{\\prime\}=\\left\(\\frac\{1\}\{5\},\\frac\{4\}\{5\}\\right\)\.The\(p,q\)\(p,q\)dissimilarity is therefore zero\. Sincep′=q′p^\{\\prime\}=q^\{\\prime\}, the common dissimilarity of each of the other two pairs is

dcos​\(p′,r′\)\\displaystyle d\_\{\\cos\}\(p^\{\\prime\},r^\{\\prime\}\)=917\>0,\\displaystyle=\\frac\{9\}\{17\}\>0,JS⁡\(p′,r′\)\\displaystyle\\mathrm\{JS\}\(p^\{\\prime\},r^\{\\prime\}\)=15​log⁡\(81923125\)\>0,\\displaystyle=\\frac\{1\}\{5\}\\log\\\!\\left\(\\frac\{8192\}\{3125\}\\right\)\>0,dE​\(p′,r′\)\\displaystyle d\_\{\\mathrm\{E\}\}\(p^\{\\prime\},r^\{\\prime\}\)=3​25\>0\.\\displaystyle=\\frac\{3\\sqrt\{2\}\}\{5\}\>0\.Hence\(p,q\)\(p,q\)is uniquely closest after sink removal under all three dissimilarities\.

Finally, with three singleton clusters, both single linkage and complete linkage assign to a pair of clusters exactly the underlying pairwise dissimilarity\. Their first merge is therefore the unique closest pair\. Stopping immediately after that merge, atK=2K=2, gives the two asserted partitions\. ∎

### Full statement of Corollary[1](https://arxiv.org/html/2608.14712#Thmcorollary1)

Fix an integerD≥3D\\geq 3, let∅≠𝒯\\varnothing\\neq\\mathcal\{T\}be a checkpoint index set, and letpt∈𝒮Dp\_\{t\}\\in\\mathcal\{S\}^\{D\}for everyt∈𝒯t\\in\\mathcal\{T\}\. Write uniquely

pt=\(st,\(1−st\)​πt\),st=\(pt\)0∈\(0,1\),πt=pt′∈𝒮D−1,p\_\{t\}=\\bigl\(s\_\{t\},\(1\-s\_\{t\}\)\\pi\_\{t\}\\bigr\),\\qquad s\_\{t\}=\(p\_\{t\}\)\_\{0\}\\in\(0,1\),\\qquad\\pi\_\{t\}=p\_\{t\}^\{\\prime\}\\in\\mathcal\{S\}^\{D\-1\},and defineHb​\(s\):=−s​log⁡s−\(1−s\)​log⁡\(1−s\)H\_\{b\}\(s\):=\-s\\log s\-\(1\-s\)\\log\(1\-s\)\. Then, for everyt∈𝒯t\\in\\mathcal\{T\},

H⁡\(pt\)=Hb​\(st\)\+\(1−st\)​H​\(πt\)\.H\(p\_\{t\}\)=H\_\{b\}\(s\_\{t\}\)\+\(1\-s\_\{t\}\)H\(\\pi\_\{t\}\)\.\(20\)Suppose that, for someπ∈𝒮D−1\\pi\\in\\mathcal\{S\}^\{D\-1\},πt=π\\pi\_\{t\}=\\pifor everyt∈𝒯t\\in\\mathcal\{T\}\. ThenCA⟂​\(pt\)=‖clrD−1⁡\(π\)‖2C\_\{A\}^\{\\perp\}\(p\_\{t\}\)=\\\|\\clr\_\{D\-1\}\(\\pi\)\\\|\_\{2\}is independent oftt\. If

fπ​\(s\):=H⁡\(s,\(1−s\)​π\),s∈\(0,1\),θπ:=11\+exp⁡\(H⁡\(π\)\),f\_\{\\pi\}\(s\):=H\\bigl\(s,\(1\-s\)\\pi\\bigr\),\\qquad s\\in\(0,1\),\\qquad\\theta\_\{\\pi\}:=\\frac\{1\}\{1\+\\exp\(H\(\\pi\)\)\},thenfπf\_\{\\pi\}is strictly increasing on\(0,θπ\]\(0,\\theta\_\{\\pi\}\]and strictly decreasing on\[θπ,1\)\[\\theta\_\{\\pi\},1\), with unique maximizerθπ\\theta\_\{\\pi\}on\(0,1\)\(0,1\)\. In particular,θπ<1/2\\theta\_\{\\pi\}<1/2, and, for anyt,u∈𝒯t,u\\in\\mathcal\{T\},

12≤st<su<1⟹H⁡\(pu\)<H⁡\(pt\)andCA⟂​\(pu\)=CA⟂​\(pt\)\.\\frac\{1\}\{2\}\\leq s\_\{t\}<s\_\{u\}<1\\quad\\Longrightarrow\\quad H\(p\_\{u\}\)<H\(p\_\{t\}\)\\quad\\text\{and\}\\quad C\_\{A\}^\{\\perp\}\(p\_\{u\}\)=C\_\{A\}^\{\\perp\}\(p\_\{t\}\)\.Moreover, letuD−1:=\(1D−1,…,1D−1\)u\_\{D\-1\}:=\(\\tfrac\{1\}\{D\-1\},\\ldots,\\tfrac\{1\}\{D\-1\}\)andps:=\(s,\(1−s\)​uD−1\)p\_\{s\}:=\\bigl\(s,\(1\-s\)u\_\{D\-1\}\\bigr\)fors∈\(0,1\)s\\in\(0,1\)\. Then

ps′=uD−1,CA⟂​\(ps\)=0,H⁡\(p1/D\)=log⁡D,lims↑1H⁡\(ps\)=0\.p\_\{s\}^\{\\prime\}=u\_\{D\-1\},\\qquad C\_\{A\}^\{\\perp\}\(p\_\{s\}\)=0,\\qquad H\(p\_\{1/D\}\)=\\log D,\\qquad\\lim\_\{s\\uparrow 1\}H\(p\_\{s\}\)=0\.The maps↦H⁡\(ps\)s\\mapsto H\(p\_\{s\}\)is strictly decreasing on\[1/D,1\)\[1/D,1\), and

sups∈\(1/D,1\)\{H⁡\(p1/D\)−H⁡\(ps\)\}=log⁡D\.\\sup\_\{s\\in\(1/D,1\)\}\\left\\\{H\(p\_\{1/D\}\)\-H\(p\_\{s\}\)\\right\\\}=\\log D\.Consequently, for everyε∈\(0,log⁡D\)\\varepsilon\\in\(0,\\log D\), there existssε∈\(1/2,1\)s\_\{\\varepsilon\}\\in\(1/2,1\)such thatH⁡\(p1/D\)−H⁡\(psε\)\>log⁡D−εH\(p\_\{1/D\}\)\-H\(p\_\{s\_\{\\varepsilon\}\}\)\>\\log D\-\\varepsilon, even though the conditional content composition andCA⟂C\_\{A\}^\{\\perp\}are unchanged betweenp1/Dp\_\{1/D\}andpsεp\_\{s\_\{\\varepsilon\}\}\. Thus, variation in sink mass along a path with fixed conditional content composition can produce an apparent entropy collapse relative to the content\-only diagnostic, of size arbitrarily close to the nonattained supremumlog⁡D\\log D\. This is a possibility result; it does not by itself assert that sink mass increases along any empirical training trajectory\.

###### Proof of Corollary[1](https://arxiv.org/html/2608.14712#Thmcorollary1)\.

Sincept∈𝒮Dp\_\{t\}\\in\\mathcal\{S\}^\{D\}, one has0<st<10<s\_\{t\}<1\. Therefore

πt=\(\(pt\)1,…,\(pt\)D−1\)1−st∈𝒮D−1\\pi\_\{t\}=\\frac\{\\bigl\(\(p\_\{t\}\)\_\{1\},\\ldots,\(p\_\{t\}\)\_\{D\-1\}\\bigr\)\}\{1\-s\_\{t\}\}\\in\\mathcal\{S\}^\{D\-1\}is well defined, and the displayed representation ofptp\_\{t\}is unique\.

Using∑i=1D−1πt,i=1\\sum\_\{i=1\}^\{D\-1\}\\pi\_\{t,i\}=1, we obtain

H⁡\(pt\)\\displaystyle H\(p\_\{t\}\)=−st​log⁡st−∑i=1D−1\(1−st\)​πt,i​log⁡\(\(1−st\)​πt,i\)\\displaystyle=\-s\_\{t\}\\log s\_\{t\}\-\\sum\_\{i=1\}^\{D\-1\}\(1\-s\_\{t\}\)\\pi\_\{t,i\}\\log\\bigl\(\(1\-s\_\{t\}\)\\pi\_\{t,i\}\\bigr\)=−st​log⁡st−\(1−st\)​log⁡\(1−st\)−\(1−st\)​∑i=1D−1πt,i​log⁡πt,i\\displaystyle=\-s\_\{t\}\\log s\_\{t\}\-\(1\-s\_\{t\}\)\\log\(1\-s\_\{t\}\)\-\(1\-s\_\{t\}\)\\sum\_\{i=1\}^\{D\-1\}\\pi\_\{t,i\}\\log\\pi\_\{t,i\}=Hb​\(st\)\+\(1−st\)​H​\(πt\),\\displaystyle=H\_\{b\}\(s\_\{t\}\)\+\(1\-s\_\{t\}\)H\(\\pi\_\{t\}\),which proves \([20](https://arxiv.org/html/2608.14712#A4.E20)\)\. This is a pointwise identity, so it requires no topology or ordering on𝒯\\mathcal\{T\}\.

Ifπt=π\\pi\_\{t\}=\\pifor everytt, thenpt′=πp\_\{t\}^\{\\prime\}=\\pi, and the definition of content Aitchison concentration gives

CA⟂​\(pt\)=‖clrD−1⁡\(pt′\)‖2=‖clrD−1⁡\(π\)‖2\.C\_\{A\}^\{\\perp\}\(p\_\{t\}\)=\\\|\\clr\_\{D\-1\}\(p\_\{t\}^\{\\prime\}\)\\\|\_\{2\}=\\\|\\clr\_\{D\-1\}\(\\pi\)\\\|\_\{2\}\.Furthermore,fπ​\(s\)=Hb​\(s\)\+\(1−s\)​H​\(π\)f\_\{\\pi\}\(s\)=H\_\{b\}\(s\)\+\(1\-s\)H\(\\pi\), and hence

fπ′​\(s\)=log⁡1−ss−H⁡\(π\),fπ′′​\(s\)=−1s⁡\(1−s\)<0,0<s<1,f\_\{\\pi\}^\{\\prime\}\(s\)=\\log\\frac\{1\-s\}\{s\}\-H\(\\pi\),\\qquad f\_\{\\pi\}^\{\\prime\\prime\}\(s\)=\-\\frac\{1\}\{s\(1\-s\)\}<0,\\qquad 0<s<1,sofπf\_\{\\pi\}is strictly concave\. Becauses↦log⁡\(\(1−s\)/s\)s\\mapsto\\log\(\(1\-s\)/s\)is strictly decreasing on\(0,1\)\(0,1\),

fπ′​\(s\)=0⇔1−ss=exp⁡\(H⁡\(π\)\)⇔s=11\+exp⁡\(H⁡\(π\)\)=θπ\.f\_\{\\pi\}^\{\\prime\}\(s\)=0\\iff\\frac\{1\-s\}\{s\}=\\exp\(H\(\\pi\)\)\\iff s=\\frac\{1\}\{1\+\\exp\(H\(\\pi\)\)\}=\\theta\_\{\\pi\}\.Consequently,

\{fπ′​\(s\)\>0,0<s<θπ,fπ′​\(s\)=0,s=θπ,fπ′​\(s\)<0,θπ<s<1\.\\begin\{cases\}f\_\{\\pi\}^\{\\prime\}\(s\)\>0,&0<s<\\theta\_\{\\pi\},\\\\ f\_\{\\pi\}^\{\\prime\}\(s\)=0,&s=\\theta\_\{\\pi\},\\\\ f\_\{\\pi\}^\{\\prime\}\(s\)<0,&\\theta\_\{\\pi\}<s<1\.\\end\{cases\}This proves the asserted strict monotonicity and unique maximizer\.

SinceD−1≥2D\-1\\geq 2and every coordinate ofπ\\piis strictly positive,H⁡\(π\)\>0H\(\\pi\)\>0, soθπ<1/2\\theta\_\{\\pi\}<1/2\. Therefore, if12≤st<su<1\\frac\{1\}\{2\}\\leq s\_\{t\}<s\_\{u\}<1, then bothsts\_\{t\}andsus\_\{u\}lie in the strictly decreasing region offπf\_\{\\pi\}, givingH⁡\(pu\)=fπ​\(su\)<fπ​\(st\)=H⁡\(pt\)H\(p\_\{u\}\)=f\_\{\\pi\}\(s\_\{u\}\)<f\_\{\\pi\}\(s\_\{t\}\)=H\(p\_\{t\}\)\. The equality of the two content log\-ratio dispersions was established above\.

Finally,H⁡\(uD−1\)=log⁡\(D−1\)H\(u\_\{D\-1\}\)=\\log\(D\-1\)andclrD−1⁡\(uD−1\)=0\\clr\_\{D\-1\}\(u\_\{D\-1\}\)=0, andps′=uD−1p\_\{s\}^\{\\prime\}=u\_\{D\-1\}\. ThereforeCA⟂​\(ps\)=0C\_\{A\}^\{\\perp\}\(p\_\{s\}\)=0for everys∈\(0,1\)s\\in\(0,1\)\. The corresponding threshold is

θuD−1=11\+exp⁡\(log⁡\(D−1\)\)=1D\.\\theta\_\{u\_\{D\-1\}\}=\\frac\{1\}\{1\+\\exp\(\\log\(D\-1\)\)\}=\\frac\{1\}\{D\}\.The preceding monotonicity result shows thats↦H⁡\(ps\)s\\mapsto H\(p\_\{s\}\)is strictly decreasing on\[1/D,1\)\[1/D,1\)\.

Ats=1/Ds=1/D, every coordinate ofpsp\_\{s\}equals1/D1/D, soH⁡\(p1/D\)=log⁡DH\(p\_\{1/D\}\)=\\log D\. Moreover,

H⁡\(ps\)=Hb​\(s\)\+\(1−s\)​log⁡\(D−1\)⟶0as​s↑1\.H\(p\_\{s\}\)=H\_\{b\}\(s\)\+\(1\-s\)\\log\(D\-1\)\\longrightarrow 0\\qquad\\text\{as \}s\\uparrow 1\.SinceH⁡\(ps\)\>0H\(p\_\{s\}\)\>0for everys∈\(0,1\)s\\in\(0,1\), it follows thatH⁡\(p1/D\)−H⁡\(ps\)<log⁡DH\(p\_\{1/D\}\)\-H\(p\_\{s\}\)<\\log Dfor everys∈\(1/D,1\)s\\in\(1/D,1\)\. On the other hand,H⁡\(p1/D\)−H⁡\(ps\)⟶log⁡DH\(p\_\{1/D\}\)\-H\(p\_\{s\}\)\\longrightarrow\\log Dass↑1s\\uparrow 1\. Therefore

sups∈\(1/D,1\)\{H⁡\(p1/D\)−H⁡\(ps\)\}=log⁡D\.\\sup\_\{s\\in\(1/D,1\)\}\\left\\\{H\(p\_\{1/D\}\)\-H\(p\_\{s\}\)\\right\\\}=\\log D\.The supremum is not attained becauses=1s=1is excluded andH⁡\(ps\)\>0H\(p\_\{s\}\)\>0for every admissibless\.

Givenε∈\(0,log⁡D\)\\varepsilon\\in\(0,\\log D\), choosesε∈\(1/2,1\)s\_\{\\varepsilon\}\\in\(1/2,1\), sufficiently close to11, such thatH⁡\(psε\)<εH\(p\_\{s\_\{\\varepsilon\}\}\)<\\varepsilon\. Then

H⁡\(p1/D\)−H⁡\(psε\)=log⁡D−H⁡\(psε\)\>log⁡D−ε,H\(p\_\{1/D\}\)\-H\(p\_\{s\_\{\\varepsilon\}\}\)=\\log D\-H\(p\_\{s\_\{\\varepsilon\}\}\)\>\\log D\-\\varepsilon,as claimed\. ∎

###### Lemma 2\(Quotient characterization ofdA⟂d\_\{A\}^\{\\perp\}\)\.

Definep∼qp\\sim qiffR0​p=R0​qR\_\{0\}p=R\_\{0\}q\(equal content subcompositions\)\. ThendA⟂d\_\{A\}^\{\\perp\}is the quotient metric induced bydAd\_\{A\}on𝒮/∼\\mathcal\{S\}/\{\\sim\}:dA⟂​\(p,q\)=minp∗∼p,q∗∼q⁡dA​\(p∗,q∗\)d\_\{A\}^\{\\perp\}\(p,q\)=\\min\_\{p^\{\*\}\\sim p,\\;q^\{\*\}\\sim q\}d\_\{A\}\(p^\{\*\},q^\{\*\}\), and Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)applied on the content simplex characterizesdA⟂d\_\{A\}^\{\\perp\}as the unique metric on the quotient satisfying A1–A4 for content transformations\.

###### Proof\.

By Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1),dA​\(p∗,q∗\)2=dA⟂​\(p,q\)2\+\(b⁡\(p∗\)−b⁡\(q∗\)\)2d\_\{A\}\(p^\{\*\},q^\{\*\}\)^\{2\}=d\_\{A\}^\{\\perp\}\(p,q\)^\{2\}\+\\big\(b\(p^\{\*\}\)\-b\(q^\{\*\}\)\\big\)^\{2\}for any representatives, sincedA⟂d\_\{A\}^\{\\perp\}depends only on the content classes\. The balancebbcan be set freely while holding the content fixed \(vary the sink share\), so the minimum over representatives zeroes the second term and equalsdA⟂​\(p,q\)d\_\{A\}^\{\\perp\}\(p,q\); it is attained\. The quotient space with this metric is isometric to the Aitchison geometry of the content simplex, on which Theorem[2](https://arxiv.org/html/2608.14712#Thmtheorem2)applies verbatim\. ∎

## Appendix EExperimental details

Table 5:Measured taxonomy stability across sink conventions\(thedA⟂d\_\{A\}^\{\\perp\}column is omitted: its partitions are identical across conventions by construction, Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)\(i\), with ARI=1=1verified in code\) \(mean ARI overK∈\{4,6,8\}K\\in\\\{4,6,8\\\}, average linkage;reanalyze\_clustering\.py\)\. ThedA⟂d\_\{A\}^\{\\perp\}column is exact by Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)\(i\)\.ViT\-B/16 evaluation uses2,0482\{,\}048COCO val2017 images with a label\-free top\-1 agreement metric against the unpruned model; labeled ImageNet\-val substitutes directly in the released script viaIMAGENET\_DIR\.

### E\.1Are attention rows compositions? Tail mass, rival geometries, and the content channel

A fair objection to the compositional stance is causal: the row’s downstream effect is the value mixture∑iai​vi\\sum\_\{i\}a\_\{i\}v\_\{i\}, so absolute mass matters and a coordinate at10−1710^\{\-17\}moves nothing, while log\-ratios weight it\. Three answers\. First, scope: our claims concern the row as an analysis object, which is what the literature computes cosine, JS, and entropy on; the sink balanceb⁡\(p\)b\(p\)retains the absolute sink share, and functional relevance is tested by the pruning experiments of §[5](https://arxiv.org/html/2608.14712#S5), not assumed\. Second, a direct audit of the estimator\. For each model we decompose the squared content distance between seed\-0 head signatures by coordinate magnitude: coordinate pairs whose smaller entry is below10−610^\{\-6\}carry at most4\.6%4\.6\\%of squareddA⟂d\_\{A\}^\{\\perp\}\(GPT\-2\) and below0\.7%0\.7\\%on every other model, and essentially0%0\\%below10−910^\{\-9\}\(Table[6](https://arxiv.org/html/2608.14712#A5.T6)\)\. No coordinate has median signature value below10−610^\{\-6\}in any model, so no tail\-amalgamation set exists at that threshold and the shares above bound the maximal tail influence; sub\-threshold values are shared across heads, which is why their log\-ratio differences nearly cancel\. Third, rival geometries measured rather than dismissed: Hellinger and Fisher–Rao closer\-head verdicts flip between sink conventions at rates of0\.180\.18to0\.440\.44, the same range as cosine, JS, and Euclidean distance\.

The same signatures address the central methodological question: how much of the observed stability is conditioning out the sink, and how much is Aitchison geometry? Within the content channel, JS on sink\-dropped rows anddA⟂d\_\{A\}^\{\\perp\}agree strongly on coarse ranking \(Spearmanρ≥0\.96\\rho\\geq 0\.96\) but disagree on fine structure: nearest\-neighbor identities differ for1313to27%27\\%of heads, and the induced taxonomies differ with cross\-ARI as low as0\.330\.33on Llama\-3\.2\-1B \(Table[6](https://arxiv.org/html/2608.14712#A5.T6)\)\. Conditioning out the sink is therefore the dominant stabilizer\. The reversal phenomenon is also not an artifact of feeding classical metrics Aitchison\-mean signatures: recomputed on arithmetic\-mean signatures \(their native aggregation, released alongside\), classical reversal rates remain0\.180\.18–0\.450\.45, within0\.150\.15of the Aitchison\-mean rates on every model \(Table[7](https://arxiv.org/html/2608.14712#A5.T7)\)\. The revised protocol ran the functional comparison, and we report it either way it fell: under the global protocol dropped\-row JS matchesdA⟂d\_\{A\}^\{\\perp\}on Llama\-3\.2\-1B and beats it on Llama\-3\.2\-3B; under the same\-layer keep\-one protocoldA⟂d\_\{A\}^\{\\perp\}is best on every strong\-sink model while dropped\-row JS fails catastrophically on Llama\-3\.2\-1B \(Table[3](https://arxiv.org/html/2608.14712#S5.T3)\)\. Two equivalences simplify the rival columns: Hellinger and Fisher–Rao are strictly monotone in the Bhattacharyya coefficient, so all of their ranking verdicts, and hence reversal rates, coincide exactly; and as pruning criteria, Hellinger\-on\-content selects the identical nearest neighbors and prune sets as dropped\-row JS on every model, because for nearby distributions allff\-divergences share the same local quadratic form while the Aitchison metric does not\.

Table 6:Estimator and rival\-geometry audit\(seed\-0 signatures\)\. Hell\./FR==closer\-head reversal rate between sink conventions under Hellinger and Fisher–Rao\. Share==fraction of squareddA⟂d\_\{A\}^\{\\perp\}carried by coordinate pairs with smaller entry below10−610^\{\-6\}\. Last three columns compare dropped\-row JS withdA⟂d\_\{A\}^\{\\perp\}on the content channel: Spearman correlation of pairwise distances, nearest\-neighbor agreement, and mean cross\-ARI of the induced taxonomies overK∈\{4,6,8\}K\\in\\\{4,6,8\\\}\.Table 7:Aggregation robustness: with\-sink versus sink\-dropped reversal rates \(seed 0\) computed on arithmetic\-mean signatures \(the classical metrics’ native aggregation\) and on Aitchison\-mean signatures\.
### E\.2Convention stakes measured: temperature and the sink set

Temperature\.Axiom A3 \(powering\) is the softmax temperature: rescaling the content logits byτ\\taumaps each content rowπ\\pito𝒞⁡\(πτ\)\\mathcal\{C\}\(\\pi^\{\\tau\}\)\. By Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2),dA⟂d\_\{A\}^\{\\perp\}is exactly equivariant \(dA⟂→τ​dA⟂d\_\{A\}^\{\\perp\}\\to\\tau\\,d\_\{A\}^\{\\perp\}\), so every ranking verdict, nearest neighbor, and prune set is invariant; we verify this to the bit on all ten models\. Dropped\-row JS is not: overτ∈\{0\.5,0\.7,1\.5,2\}\\tau\\in\\\{0\.5,0\.7,1\.5,2\\\}its closer\-head verdicts flip by up to6\.6%6\.6\\%and its top\-20%20\\%prune set retains as little as62%62\\%of itsτ=1\\tau=1membership \(Table[8](https://arxiv.org/html/2608.14712#A5.T8)\)\. Effective logit scale is not observable to an analyst comparing models or pipelines, so a criterion whose selections depend on it answers a scale\-relative question;dA⟂d\_\{A\}^\{\\perp\}does not\.

The sink set\.Dropping the sink requires deciding which columns are the sink; first token and first four tokens are both used in the literature\. This is a second convention layer: between the two definitions, dropped\-row JS flips11to21%21\\%of verdicts anddA⟂d\_\{A\}^\{\\perp\}flips11to18%18\\%, withdA⟂d\_\{A\}^\{\\perp\}uniformly less sensitive on every model \(Table[8](https://arxiv.org/html/2608.14712#A5.T8)\)\. Both changes are changes of estimand; what distinguishes the Aitchison side is that the definitions are exactly related: the content distances are nested orthogonal projections, and we verify the implied dominance \(dA⟂d\_\{A\}^\{\\perp\}on the smaller content set never exceedsdA⟂d\_\{A\}^\{\\perp\}on the larger\) with no violations across all head pairs of all ten models\. No analogous relation exists for JS\.

Which regime? A dispersion predictor\.Mean sink mass does not predict which pruning regime a model is in \(GPT\-2 and Qwen2\.5 shares=0\.68s=0\.68with opposite verdicts\)\. The cross\-head*dispersion*of the sink share does, for the nine language models: the content\-regime LMs have coefficient of variation at most0\.360\.36\(Llama\-3\.2\-1B0\.110\.11, Llama\-3\.2\-3B0\.080\.08, Qwen2\.50\.360\.36\) and the JS\-regime LMs at least0\.420\.42\(GPT\-20\.420\.42, Pythias0\.430\.43–1\.031\.03\)\. A near\-uniform sink carries no between\-head information, so mixing it into distances only adds noise; a heterogeneous sink marks genuine head classes\. ViT, where the criteria tie within noise, does not fit the LM pattern\. A complementary stability probe: across the three resampling seeds, the top\-20%20\\%prune sets selected bydA⟂d\_\{A\}^\{\\perp\}\-L1 overlap with Jaccard0\.360\.36–0\.770\.77while the JS\-L1 sets are essentially frozen \(Jaccard0\.920\.92–1\.001\.00on LMs\), even though thedA⟂d\_\{A\}^\{\\perp\}\-L1*outcomes*are the stable ones on strong\-sink models\. Each criterion is stable along a different axis:dA⟂d\_\{A\}^\{\\perp\}under the analyst’s conventions \(temperature, sink set\), JS under data resampling; on GPT\-2 thedA⟂d\_\{A\}^\{\\perp\}\-L1 set instability coincides with its one catastrophic cell\. This makes the weighted familydλ2=\(dA⟂\)2\+λ​\(Δ​b\)2d\_\{\\lambda\}^\{2\}=\(d\_\{A\}^\{\\perp\}\)^\{2\}\+\\lambda\\,\(\\Delta b\)^\{2\}natural, withλ\\lambdagrowing with thebb\-channel’s informativeness; fittingλ\\lambdafrom held\-out functional evidence is left to future work, and we note that the with\-sink JS criterion itself lies outside this family\.

Held\-out test, specified before running\.We freeze the rule and the model list here, prior to any evaluation: foropt\-125m,distilgpt2, andSmolLM2\-135Mwe will compute the cross\-head CV of the signature sink share and predict, with the thresholds above, thatCV≤0\.36\\mathrm\{CV\}\\leq 0\.36places the model in the content regime \(dA⟂d\_\{A\}^\{\\perp\}\-L1 best or tied\-best among the six criteria, with at least one classical criterion exceeding four times base perplexity\) andCV≥0\.42\\mathrm\{CV\}\\geq 0\.42in the JS regime \(with\-sink JS\(\-L1\) best\); for0\.36<CV<0\.420\.36<\\mathrm\{CV\}<0\.42the predictor abstains\. The released prediction script logs its output before any pruning evaluation exists, and the outcomes will be reported whichever way they fall\.

Outcomes\.The predictions were logged before any evaluation \(timestamps in the releasedheldout\_predictions\.json\); measured CVs: OPT\-125M0\.280\.28, DistilGPT20\.510\.51, SmolLM2\-135M0\.420\.42\. DistilGPT2 \(JS regime predicted\):*confirmed*\. With\-sink JS wins \(105\.3±0\.8105\.3\{\\pm\}0\.8on base67\.467\.4\) and dropped\-row JS is catastrophic \(1161±241161\{\\pm\}24,17×17\\timesbase\)\. SmolLM2\-135M: the predictor*abstained*at the boundary; the outcome fell on the JS side \(JS\-L1 best at83\.5±0\.283\.5\{\\pm\}0\.2on base24\.824\.8; every content\-conditioned criterion99–54×54\\timesbase\)\. Its sink is layer\-bimodal,0\.010\.01–0\.150\.15in the first twelve layers and0\.680\.68–0\.920\.92thereafter, which is what places its cross\-head CV at the threshold\. OPT\-125M \(content regime predicted\):*the prediction failed*\. Global content criteria do beat with\-sink JS \(59\.259\.2/59\.359\.3versus63\.363\.3on base46\.746\.7\), but JS\-L1 wins outright \(57\.1±0\.557\.1\{\\pm\}0\.5\) whiledA⟂d\_\{A\}^\{\\perp\}\-L1 sits mid\-pack with high seed variance \(68\.5±7\.768\.5\{\\pm\}7\.7\), and no criterion is catastrophic at20%20\\%\(worst2\.1×2\.1\\times; dropped\-JS\-L1 reaches6\.9×6\.9\\timesonly at30%30\\%\)\. The scorecard on non\-abstentions is one confirmation and one failure: the dispersion rule as frozen does not transfer, and OPT exhibits a third behavior the nine\-model map lacks, a uniformly high sink with a benign pruning landscape\. We accordingly present the rule as descriptive of the training fleet, not as a validated decision procedure\. A second\-family checkpoint sweep \(e\.g\. OLMo\) is excluded by compute budget, so the checkpoint analysis of Fig\.[5](https://arxiv.org/html/2608.14712#S5.F5)b currently rests on one model family; we state this as a limitation\.

Exact decomposition of the checkpoint collapse\.WithΔ\\Deltathe change from the first checkpoint to the entropy minimum, the identity gives the signed splitΔ​H=Δ​Hb​\(s\)\+Δ⁡\[\(1−s\)​H​\(π\)\]\\Delta H=\\Delta H\_\{b\}\(s\)\+\\Delta\[\(1\-s\)H\(\\pi\)\]exactly \(residual00to machine precision\)\. Across 70M/160M/410M/1B/1\.4B:ΔH=−0\.78/−1\.55/−2\.11/−1\.58/−2\.01\\Delta H=\-0\.78/\-1\.55/\-2\.11/\-1\.58/\-2\.01,ΔHb=\+0\.18/\+0\.26/\+0\.23/\+0\.29/\+0\.25\\Delta H\_\{b\}=\+0\.18/\+0\.26/\+0\.23/\+0\.29/\+0\.25, andΔ\[\(1−s\)H\(π\)\]=−0\.96/−1\.81/−2\.34/−1\.88/−2\.26\\Delta\[\(1\-s\)H\(\\pi\)\]=\-0\.96/\-1\.81/\-2\.34/\-1\.88/\-2\.26nats\. The binary term rises asssapproaches1/21/2while the weighted content term carries the entire fall, and within that term the conditional entropyH⁡\(π\)H\(\\pi\)itself moves little; the main\-text share1−Δ​H​\(π\)/Δ​H1\-\\Delta H\(\\pi\)/\\Delta Hisolates exactly this, the fraction of the drop not attributable to a drop inH⁡\(π\)H\(\\pi\)\.

Table 8:Convention stakes\(seed\-0 signatures\)\. Temperature: worst\-case dropped\-JS verdict flip rate and top\-20%20\\%prune\-set retention overτ∈\{0\.5,0\.7,1\.5,2\}\\tau\\in\\\{0\.5,0\.7,1\.5,2\\\}\(thedA⟂d\_\{A\}^\{\\perp\}values are exactly00and11on every model, verified to the bit, and are omitted\)\. Sink set: closer\-head verdict flip rate between the first\-token and first\-four\-token sink definitions\.
### E\.3Full\-support Clark reanalysis and window robustness

Clark et al\. on the full support\.We rebuild[4](https://arxiv.org/html/2608.14712#bib.bib10)’s head\-clustering pipeline with no window truncation:512512sequences of exactly128128WikiText tokens \(\[CLS\]\+\+126126wordpieces\+\+\[SEP\]\), so every row shares one support and the sink set is\{\\\{CLS, SEP\}\\\}; per\-head Aitchison\-mean signatures; JS distances and average\-linkage clustering as in the original, plus cosine anddA⟂d\_\{A\}^\{\\perp\}\. Of144144heads,6868place majority mass on SEP and1212on CLS\. Between the keep and drop conventions the JS clusterings agree at chance \(ARI0\.010\.01–0\.050\.05overK∈\{4,5,6,8\}K\\in\\\{4,5,6,8\\\}; cosine0\.000\.00–0\.060\.06\)\. The SEP\-head block is recovered by the with\-sink JS clustering at Jaccard0\.870\.87–0\.920\.92and collapses to0\.460\.46–0\.490\.49after dropping; the content clusterings \(dA⟂d\_\{A\}^\{\\perp\}, convention\-invariant by construction\) recover it only at0\.470\.47–0\.500\.50\. The structure the pipeline is best known for is therefore a property of the sink channel, visible under exactly one convention\.

Table 9:Clark\-style clustering on the full128128\-token support\(BERT\-base; sink set\{\\\{CLS, SEP\}\\\}\)\. Cross==between\-convention ARI; SEP Jac\.==Jaccard of the SEP\-head set with its best\-matching cluster\.Window robustness and mass coverage\.Extracting atW∈\{32,64,128\}W\\in\\\{32,64,128\\\}on GPT\-2, Llama\-3\.2\-1B, and Pythia\-410M: the retained mass of the window before reclosure is0\.490\.49–0\.610\.61\(GPT\-2\),0\.600\.60–0\.710\.71\(Llama\),0\.480\.48–0\.670\.67\(Pythia\-410M\); JS reversal rates move by at most2\.12\.1points across the sweep \(0\.33/0\.34/0\.350\.33/0\.34/0\.35,0\.39/0\.40/0\.400\.39/0\.40/0\.40,0\.40/0\.40/0\.410\.40/0\.40/0\.41respectively\), anddA⟂d\_\{A\}^\{\\perp\}pairwise distances between consecutive windows correlate at Spearmanρ=0\.87\\rho=0\.87–0\.980\.98\. The fixed\-support choice shifts the estimand \(eachWWconditions on a different retained set\) but not the phenomenon\.

### E\.4Synthetic experiments \(Section[5\.1](https://arxiv.org/html/2608.14712#S5.SS1)\)

All synthetic results usenumpywith seed77and run in seconds on CPU \(experiments/run\_synthetic\.py\); the exact witnesses and the machine\-precision checks of Lemma[1](https://arxiv.org/html/2608.14712#Thmlemma1)and Proposition[2](https://arxiv.org/html/2608.14712#Thmproposition2)are inexperiments/exact\_examples\.py\.

Head\-signature sampler\.A row isp=\(s,\(1−s\)​π\)p=\(s,\(1\-s\)\\pi\)with sink masss∼𝒩⁡\(s¯,0\.10\)s\\sim\\mathcal\{N\}\(\\bar\{s\},0\.10\)clipped to\[0\.02,0\.98\]\[0\.02,0\.98\]and contentπ∼Dirichlet⁡\(α​𝟏D−1\)\\pi\\sim\\mathrm\{Dirichlet\}\(\\alpha\\mathbf\{1\}\_\{D\-1\}\),D=64D=64\. The peaked settingα=0\.1\\alpha=0\.1mimics concentrated attention;α=1\.0\\alpha=1\.0\(dashed in Figure[3](https://arxiv.org/html/2608.14712#S5.F3)\) is the diffuse control\.

Ranking\-disagreement Monte Carlo \(Figure[3](https://arxiv.org/html/2608.14712#S5.F3)\)\.For eachs¯\\bar\{s\}on a grid from0\.100\.10to0\.900\.90, sample20,00020\{,\}000independent triples\(p,q,r\)\(p,q,r\), and record whether the verdict of “isqqorrrcloser topp?” differs between the with\-sink and sink\-dropped pipelines, separately for cosine, JS, Euclidean distance, anddAd\_\{A\}\(total\) versusdA⟂d\_\{A\}^\{\\perp\}\. Headline rates ats¯=0\.5/0\.7/0\.9\\bar\{s\}=0\.5/0\.7/0\.9\(peaked content\): cosine36\.4/39\.0/41\.1%36\.4/39\.0/41\.1\\%, JS23\.4/31\.0/39\.7%23\.4/31\.0/39\.7\\%, Euclidean27\.3/36\.8/44\.6%27\.3/36\.8/44\.6\\%;dA⟂d\_\{A\}^\{\\perp\}is0%0\\%by construction, and total\-dAd\_\{A\}\-versus\-dA⟂d\_\{A\}^\{\\perp\}is0\.1%0\.1\\%at every grid point \(see the transparency note in Appendix[A](https://arxiv.org/html/2608.14712#A1)\)\.

Collapse scenarios \(Figure[2](https://arxiv.org/html/2608.14712#S5.F2)\)\.Twelve “layers,”4,0004\{,\}000rows per layer,ss\-noise0\.030\.03\. Scenario \(a\):sℓ=0\.05\+0\.80​ς​\(1\.1​\(ℓ−5\)\)s\_\{\\ell\}=0\.05\+0\.80\\,\\varsigma\\big\(1\.1\(\\ell\-5\)\\big\)withς\\varsigmathe logistic function and content log\-ratio dispersion fixed \(αℓ≡0\.15\\alpha\_\{\\ell\}\\equiv 0\.15\); measuredHHfalls2\.71→0\.822\.71\\to 0\.82nats whileCA⟂C\_\{A\}^\{\\perp\}moves48\.13→47\.9848\.13\\to 47\.98\. Scenario \(b\):sℓs\_\{\\ell\}linear0\.70→0\.050\.70\\to 0\.05whileαℓ\\alpha\_\{\\ell\}decays geometrically1\.0→0\.021\.0\\to 0\.02;HHdrifts1\.73→1\.271\.73\\to 1\.27whileCA⟂C\_\{A\}^\{\\perp\}rises10\.04→70\.4310\.04\\to 70\.43\.

Reversal wedge \(Figure[6](https://arxiv.org/html/2608.14712#A1.F6)\)\.Fix\(p,r\)\(p,r\)of the witness; sample20,00020\{,\}000pointsq∼Dirichlet⁡\(1,1,1\)q\\sim\\mathrm\{Dirichlet\}\(1,1,1\)with all parts\>0\.004\>0\.004; the cosine verdict differs between conventions on77\.9%77\.9\\%of them\.

### E\.5Protocol for pretrained models \(Section[3](https://arxiv.org/html/2608.14712#S5.F3)\)

Implemented inexperiments/run\_real\_models\.py\(PyTorch \+transformers\+datasets\); ten models, three resampling seeds each, on two RTX 4090 GPUs; the measurement passes take minutes per model and the pruning evaluations a few hours in total\. Reported standard deviations are population estimates \(ddof00\) over the three seeds throughout\.

Models and data\.GPT\-2 \(gpt2, 124M, 12 layers×\\times12 heads, fp32\); Llama\-3\.2\-1B \(meta\-llama/Llama\-3\.2\-1B, 16 layers×\\times32 heads, GQA, fp16\); ViT\-B/16 \(google/vit\-base\-patch16\-224, 12 layers×\\times12 heads, fp32\)\. LM data:2,0002\{,\}000sequences of length≥192\\geq 192tokens from the WikiText\-103 validation split; ViT data:2,0482\{,\}048COCO val2017 images \(public, unlabeled; the script accepts any image folder, and the ViT pruning metric is top\-1 agreement with the unpruned model, so labels are not required, labeled ImageNet\-val substitutes directly\)\.

Row extraction on a fixed common support\.Attention probabilities are captured from the softmax output \(viaoutput\_attentions=Trueor forward hooks\)\. For LMs, for every query positiont≥wt\\geq wwe keep the attention over the*firstW=64W=64keys only*and re\-close on that support\. This makes the support identical across all query positions, sequences, and heads, so the main measurements require*no*zero imputation, while retaining the sink column, key00\(the⟨bos⟩\\langle\\mathrm\{bos\}\\rangle/first token; a flag groups the first44keys as a sink block instead, following the StreamingLLM observation, in which case the sink balance is the corresponding SBP balance\)\. For ViT\-B/16 the support is all197197tokens \(no masking\) and the sink group is the CLS column, optionally augmented with the highest\-norm register\-like outlier tokens\[[5](https://arxiv.org/html/2608.14712#bib.bib17)\]\.

Aggregation\.Each head’s signature is the Aitchison mean of its collected rows \(Definition[1](https://arxiv.org/html/2608.14712#Thmdefinition1)\(a\)\); residual numerical zeros \(none on the fixed support, possible under the optional variable\-support ablation\) are multiplicatively replaced atε=10−6\\varepsilon=10^\{\-6\}before taking logs, with the sweepε∈\{10−5,10−6,10−7\}\\varepsilon\\in\\\{10^\{\-5\},10^\{\-6\},10^\{\-7\}\\\}reported\.

Measurements\.\(1\)*Reversal rates:*sample50,00050\{,\}000head triples uniformly \(within and across layers\); report the fraction whose closer\-head verdict flips between the with\-sink and sink\-dropped pipelines, for cosine, JS, Euclidean, anddA⟂d\_\{A\}^\{\\perp\}\(identically00; reported as a check\)\. \(2\)*Collapse curves:*per layer, mean row entropyHH\(with sink\) versus meanCA⟂C\_\{A\}^\{\\perp\}; report the fraction of adjacent\-layer steps on which the two disagree in sign\. \(3\)*Redundancy pruning at matched sparsity:*define a head’s redundancy as its distance to the nearest other head; prune them∈\{10,20,30\}%m\\in\\\{10,20,30\\\}\\%most redundant heads under JS, under the content distancedA⟂d\_\{A\}^\{\\perp\}, and under the fulldAd\_\{A\}including the sink balance \(matched counts\), viahead\_maskwhere supported and output\-zeroing hooks otherwise; evaluate perplexity on100100k held\-out WikiText tokens \(LMs\) and top\-1 agreement with the unpruned model on the2,0482\{,\}048images \(ViT\)\. \(4\)*Sink statistics:*mean sink mass per model and per layer, for context against[3](https://arxiv.org/html/2608.14712#bib.bib20),[10](https://arxiv.org/html/2608.14712#bib.bib19)\.

Runtime\.On two RTX 4090s the full protocol \(all models, all measurements\) completes in roughly1515–2525minutes wall clock with the released parallel launcher; a single comparable GPU takes about one hour end to end\.

### E\.6Protocol for the prospectively specified tests \(Section[5\.3](https://arxiv.org/html/2608.14712#S5.SS3)\)

\(1\) Taxonomies\(reanalyze\_clustering\.py\)\. Objects are per\-head Aitchison\-mean signatures on the fixed6464\-key support, collected exactly as in the main protocol \(decoder models reuse the saved signatures of the §[3](https://arxiv.org/html/2608.14712#S5.F3)runs; for BERT\-base we collect encoder rows for queriest∈\[0,64\)t\\in\[0,64\)renormalized to the first6464keys over1,0001\{,\}000WikiText\-103 sequences of length128128, sink==\[CLS\]\)\. For each dissimilarityδ∈\{1−cos,JS,Euclid\.,dA⟂,dA\}\\delta\\in\\\{1\-\\cos,\\mathrm\{JS\},\\text\{Euclid\.\},d\_\{A\}^\{\\perp\},d\_\{A\}\\\}we form the full pairwise matrix under both conventions and run agglomerative clustering \(average, complete, and single linkage; NumPy implementation released\) cut atK∈\{4,6,8\}K\\in\\\{4,6,8\\\}\. Stability is the Adjusted Rand Index between the two conventions’ partitions; Table[5](https://arxiv.org/html/2608.14712#A5.T5)reports the mean overKKunder average linkage, and the released script prints all linkage×\\timesKKcells\. ThedA⟂d\_\{A\}^\{\\perp\}matrices are asserted equal entrywise at runtime \(Proposition[3](https://arxiv.org/html/2608.14712#Thmproposition3)\(i\)\)\.

\(2\) Scale\(scale\_trend\.pyafterPYTHIA=1 run\_all\.sh\)\. Adds EleutherAI Pythia\{70​M,160​M,410​M,1​B,1\.4​B\}\\\{70\\text\{M\},160\\text\{M\},410\\text\{M\},1\\text\{B\},1\.4\\text\{B\}\\\}under the unchanged main protocol and plots measured sink mass versus parameters and reversal rate versus sink mass over the pre\-computed calibration curves of Figure[3](https://arxiv.org/html/2608.14712#S5.F3)\.

\(3\) Training curves\(checkpoints\_pythia\.py\)\. Pythia\-160M at revisionsstep512,1000,2000,4000,8000,16000,32000,64000,128000,143000;256256held\-out sequences per checkpoint, queriest∈\[64,128\)t\\in\[64,128\)on the fixed6464\-key support; we report mean row entropyHH, mean content log\-ratio dispersionCA⟂C\_\{A\}^\{\\perp\}, and mean sink mass per checkpoint, and shade checkpoint transitions wheresign⁡\(Δ​H\)≠sign⁡\(−Δ​CA⟂\)\\mathrm\{sign\}\(\\Delta H\)\\neq\\mathrm\{sign\}\(\-\\Delta C\_\{A\}^\{\\perp\}\)\(roughly3030–4545minutes on one consumer GPU\)\.

### E\.7Measured results: full pruning sweep andε\\varepsilonsensitivity

Table[10](https://arxiv.org/html/2608.14712#A5.T10)reports the complete pruning sweep behind Table[3](https://arxiv.org/html/2608.14712#S5.T3)\. Two observations beyond the main text: on Llama\-3\.2\-1B, JS\-guided pruning is already catastrophic at10%10\\%sparsity \(15\.3→358\.015\.3\\to 358\.0\), whiledA⟂d\_\{A\}^\{\\perp\}\-guided pruning remains within a factor5\.25\.2of baseline even at30%30\\%\(79\.879\.8\); and the full\-dAd\_\{A\}criterion is the lowest at10%10\\%on Llama \(18\.218\.2\) but degrades sharply at higher sparsity there, consistent with sink\-similar heads being redundant in small numbers but not in bulk\. All reversal rates in Table[3](https://arxiv.org/html/2608.14712#S5.T3)are bit\-identical acrossε∈\{10−5,10−6,10−7\}\\varepsilon\\in\\\{10^\{\-5\},10^\{\-6\},10^\{\-7\}\\\}, as expected: the fixed common support makes multiplicative replacement inactive for these measurements, soε\\varepsilonenters only through numerical clipping\. Figure[4](https://arxiv.org/html/2608.14712#S5.F4)is regenerated from the released per\-model JSONs byexperiments/make\_fig4\_v2\.py\.

Table 10:Full pruning sweepat10/20/30%10/20/30\\%sparsity \(mean±\\pmsd over three seeds\)\. LMs: perplexity \(lower better\); ViT: top\-1 agreement \(higher better; base=1=1by definition\)\. All rates in Table[3](https://arxiv.org/html/2608.14712#S5.T3)are bit\-identical acrossε∈\{10−5,10−6,10−7\}\\varepsilon\\in\\\{10^\{\-5\},10^\{\-6\},10^\{\-7\}\\\}, as the fixed\-support protocol predicts\.Table 11:Additional pruning baselines at20%20\\%sparsity\(mean±\\pmsd over three resampling seeds; two random draws shown individually\)\. TotaldAd\_\{A\}re\-mixes the sink balance; JS′\-L1 is dropped\-row JS under the same\-layer keep\-one protocol; the10/30%10/30\\%values are in the released JSONs\.