Support Selection Beyond Smooth DAG Exactness: Completion Geometry,Score Margins, and Selective Certificates

# Support Selection Beyond Smooth DAG Exactness: Completion Geometry, Score Margins, and Selective Certificates
Source: [https://arxiv.org/html/2608.08103](https://arxiv.org/html/2608.08103)
Rui Wu Zongyuan Chen Hong Xie School of Computer Science and Engineering, University of Science and Technology of China \{wurui22, chenzongyuan\}@mail\.ustc\.edu\.cn hongx87@ustc\.edu\.cn

###### Abstract

Smooth acyclicity constraints answer whether a weighted support is a DAG\. Structure learning asks a different question: which support change should be made? Existing analyses establish degeneracy for particular constraint formulas, but do not isolate what follows from smooth exactness itself\. At a DAG boundary, we show that minimal cycle completions generate a squarefree monomial ideal containing every restricted Taylor jet of an exact representation\. If the smallest completion hasqqedges, the first possible response has orderqqfor a vector residual and2​q2qfor a nonnegative scalar\. Exponentially many constant\-scale cyclic manifolds exhibit the same lack of ranking away from the boundary for NOTEARS and DAGMA\. To connect this representation limit to optimization, we derive the exact selection time for an isolated cycle\. WhenΨ′​\(h\)≍hν\\Psi^\{\\prime\}\(h\)\\asymp h^\{\\nu\}, the feasibility\-only time isT0​\(ε\)=Θ​\(ε−\(2​ν\+1\)\)T\_\{0\}\(\\varepsilon\)=\\Theta\(\\varepsilon^\{\-\(2\\nu\+1\)\}\)\. A score margin changes the leading dynamics at scaleT0−1T\_\{0\}^\{\-1\}forν\>0\\nu\>0\. The endpointν=0\\nu=0has a logarithmic boundary layer: the unperturbed limit requiresγ​T0​log⁡\(1/ε\)→0\\gamma T\_\{0\}\\log\(1/\\varepsilon\)\\to 0\. Controlled experiments check the law, and a separation statistic computed without the generating graph predicts selection time on 320 official NOTEARS/DAGMA trajectories \(Spearman−0\.52\-0\.52and−0\.66\-0\.66, permutationp<10−4p<10^\{\-4\}\)\. The score margin is unknown in finite samples\. A parent\-set confidence family and forced\-opposite queries certify skeleton and unshielded\-collider labels shared by every population optimum of a frozen score\. In a 320\-run audit with four frontends, four graph families, and four SEM regimes, every regret bound covers an independent oracle\-score audit\. None of 3,042 certified skeleton or 2,396 collider labels disagrees with the oracle\-score optimum, although4\.4%4\.4\\%and5\.5%5\.5\\%, respectively, disagree with the generating graph\. The results separate three claims often conflated in continuous DAG learning: DAG feasibility, score\-based support selection, and causal identification\.

## 1Introduction

A weighted matrix represents a directed graph through its zero pattern\. Edge magnitudes determine statistical fit, whereas acyclicity changes only when an entry crosses zero\. Continuous DAG learning must therefore make a discrete support decision while optimizing over a Euclidean space\. A typical estimator combines

min𝐖⁡ℒ​\(𝐖;X\)\+λ​ℛ​\(𝐖\)\+Ψ​\(h​\(𝐖\)\),\\min\_\{\\mathbf\{W\}\}\\;\\mathcal\{L\}\(\\mathbf\{W\};X\)\+\\lambda\\,\\mathcal\{R\}\(\\mathbf\{W\}\)\+\\Psi\(h\(\\mathbf\{W\}\)\),\(1\)whereℒ\\mathcal\{L\}measures fit,ℛ\\mathcal\{R\}promotes sparsity, andhhenforces acyclicity\. A score may rank competing graphs, and a nonsmooth update or threshold may create zeros\. Exactness ofhh, however, only identifies the acyclic supports\. We ask what support\-ranking information, if any, follows from exactness itself\.

NOTEARS supplies the smooth equality

hexp​\(𝐖\)=tr⁡\(exp⁡\(𝐖∘𝐖\)\)−d=0,h\_\{\\exp\}\(\\mathbf\{W\}\)=\\operatorname\{tr\}\\\!\\left\(\\exp\(\\mathbf\{W\}\\circ\\mathbf\{W\}\)\\right\)\-d=0,\(2\)where𝐖∈ℝd×d\\mathbf\{W\}\\in\\mathbb\{R\}^\{d\\times d\}is a weighted adjacency matrix\(Zheng et al\.,[2018](https://arxiv.org/html/2608.08103#bib.bib30)\)\. Polynomial, log\-determinant, and spectral alternatives change the geometry of the optimization problem\(Yu et al\.,[2019](https://arxiv.org/html/2608.08103#bib.bib27); Bello et al\.,[2022](https://arxiv.org/html/2608.08103#bib.bib1); Nazaret et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib19); Zhang et al\.,[2025](https://arxiv.org/html/2608.08103#bib.bib29)\)\. Their far\-field landscapes differ substantially, but their exactness statements make the same promise:h​\(𝐖\)=0h\(\\mathbf\{W\}\)=0characterizes DAG support\. That promise does not imply that∇h\\nabla hdistinguishes two deletions that both restore feasibility\. Rescaling a tied or absent local signal cannot supply the missing ranking\.

The obstruction comes from support geometry\. At a DAG boundary, suppose a setFFof absent edges creates a cycle but no proper subset does\. A smooth exact constraint vanishes on every coordinate face obtained by omitting one edge ofFF\. Those faces forbid low\-order Taylor terms according to the number of edges needed to complete a cycle\. Conditioning can change coefficients and far\-field behavior, but not a term excluded by this support geometry\.

The boundary calculation alone does not describe ordinary\-sized iterates\. Relabeling symmetry supplies the missing link: it creates constant\-weight cyclic manifolds on which the feasibility gradient ties competing deletions\. A perturbation of sizeε\\varepsilonbreaks the tie, but the time to reach a fixed edge ratio diverges asε−1\\varepsilon^\{\-1\}for a linear feasibility force andε−3\\varepsilon^\{\-3\}for a cold quadratic penalty\. These are worst\-case conditioning statements, not a claim that observational data are usually symmetric\. They raise the statistical question that closes the paper: when the score supplies the ranking, do finite data resolve it?

The paper develops this argument in three steps\.

- •Earlier degeneracy results are tied to a chosen entrywise power\. We define the cycle\-completion ideal and its initial degreeq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\), which depend on how several absent edges jointly close a cycle\. Every restricted Taylor jet of a smooth exact representation lies in this ideal, forcing response order at leastqqfor vector residuals and2​q2qfor nonnegative scalars\.
- •A local Taylor barrier need not govern finite\-scale optimization\. We close this gap with20\.332​d2^\{0\.332d\}constant\-scale cyclic manifolds in the NOTEARS and DAGMA domains and an exact isolated\-cycle hitting\-time law\. The law identifies the score scale that changes support selection and exposes a logarithmic transition layer atν=0\\nu=0\. Controlled flows and 320 official trajectories test these predictions\.
- •Feasibility does not reveal whether data resolve the resulting score margin\. We construct a parent\-set confidence family whose forced\-opposite queries certify skeleton and collider labels shared by all population optima of a frozen score\. The audit reports both oracle\-score agreement and disagreement with the generating graph, separating statistical selection from causal identification\.

No Fears studies degeneracy for positive Hadamard\-power constraints\(Wei et al\.,[2020](https://arxiv.org/html/2608.08103#bib.bib26)\); the present order is determined instead by the joint completion statisticq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)\. DAGMA improves the far\-field path\(Bello et al\.,[2022](https://arxiv.org/html/2608.08103#bib.bib1)\), while remaining subject to the local representation limit\. Acyclic parameterizations, boundary\-constrained formulations, and discrete graph operations lie outside our assumptions and can break the symmetry\(Yu et al\.,[2021](https://arxiv.org/html/2608.08103#bib.bib28); Massidda et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib16); Gillot & Parviainen,[2022](https://arxiv.org/html/2608.08103#bib.bib10); Rey et al\.,[2026](https://arxiv.org/html/2608.08103#bib.bib22)\)\. The first two results are therefore not global iteration lower bounds for Eq\. \([1](https://arxiv.org/html/2608.08103#S1.E1)\)\. The certificate is likewise conditional on a training\-frozen candidate family and bounded decomposable score; it certifies score selection, not causal identification\. Appendix[C\.5](https://arxiv.org/html/2608.08103#A3.SS5)gives a claim\-by\-claim comparison and the precise boundaries\.

## 2Setup and Scope

Let𝒵=\{𝐖∈ℝd×d:diag⁡\(𝐖\)=0\}\\mathcal\{Z\}=\\\{\\mathbf\{W\}\\in\\mathbb\{R\}^\{d\\times d\}:\\operatorname\{diag\}\(\\mathbf\{W\}\)=0\\\}, and let𝒟⊂𝒵\\mathcal\{D\}\\subset\\mathcal\{Z\}be the matrices whose nonzero support is a DAG\. We work on an open domainΩ⊆𝒵\\Omega\\subseteq\\mathcal\{Z\}containing the candidate subspaces of interest; a proper domain accommodates log\-determinant constraints\.

###### Definition 2\.1\(Exact representations\)\.

A smooth mapH:Ω→ℝrH:\\Omega\\to\\mathbb\{R\}^\{r\}is a*signed/vector exact representation*if, for every𝐖∈Ω\\mathbf\{W\}\\in\\Omega,

H​\(𝐖\)=0⟺𝐖∈𝒟\.H\(\\mathbf\{W\}\)=0\\quad\\Longleftrightarrow\\quad\\mathbf\{W\}\\in\\mathcal\{D\}\.\(3\)A smooth scalarh:Ω→ℝh:\\Omega\\to\\mathbb\{R\}is*nonnegative exact*if, for every𝐖∈Ω\\mathbf\{W\}\\in\\Omega,

h​\(𝐖\)≥0,h​\(𝐖\)=0⟺𝐖∈𝒟\.h\(\\mathbf\{W\}\)\\geq 0,\\qquad h\(\\mathbf\{W\}\)=0\\quad\\Longleftrightarrow\\quad\\mathbf\{W\}\\in\\mathcal\{D\}\.\(4\)

The scalar definition includes EXP, trace polynomials, and DAGMA on its domain; the vector definition also permits signed systems\. Nonsmooth functions, discrete order searches, acyclic parameterizations, and boundary\-constrained domains fall outside these assumptions\.

A pathwise quantity has*exact orderpp*ifR​\(t\)=\|t\|p​v\+o​\(\|t\|p\)R\(t\)=\|t\|^\{p\}v\+o\(\|t\|^\{p\}\)for nonzerovv\. Our barriers lower\-bound the first possible order; exactness need not attain it\. Likelihood choice and sparsity regularization affect which member of a Markov equivalence class a differentiable program selects\(Deng et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib6); Jin et al\.,[2026](https://arxiv.org/html/2608.08103#bib.bib13)\); the next four sections isolate how feasibility enters a local optimizer before Section[7](https://arxiv.org/html/2608.08103#S7)returns to finite\-data score selection\.

## 3Cycle\-Completion Complexity

Fix a DAG𝐖∈𝒟\\mathbf\{W\}\\in\\mathcal\{D\}and a finite setF=\{e1,…,em\}F=\\\{e\_\{1\},\\ldots,e\_\{m\}\\\}of absent off\-diagonal coordinates\. Let𝐄s\\mathbf\{E\}\_\{s\}be a signed coordinate matrix atese\_\{s\}\. ForG∈\{H,h\}G\\in\\\{H,h\\\}define the local restriction

VF=\{x∈ℝm:𝐖\+∑s=1mxs​𝐄s∈Ω\},GF​\(x\)=G​\(𝐖\+∑s=1mxs​𝐄s\),x∈VF\.V\_\{F\}=\\left\\\{x\\in\\mathbb\{R\}^\{m\}:\\mathbf\{W\}\+\\sum\_\{s=1\}^\{m\}x\_\{s\}\\mathbf\{E\}\_\{s\}\\in\\Omega\\right\\\},\\qquad G\_\{F\}\(x\)=G\\\!\\left\(\\mathbf\{W\}\+\\sum\_\{s=1\}^\{m\}x\_\{s\}\\mathbf\{E\}\_\{s\}\\right\),\\quad x\\in V\_\{F\}\.\(5\)BecauseΩ\\Omegais open,VFV\_\{F\}is an open neighborhood of the origin\.

###### Definition 3\.1\(Completion number\)\.

The candidate\-subspace completion number is

q𝐖​\(F\)=min⁡\{\|S\|:S⊆F,supp⁡\(𝐖\)∪S​is cyclic\},q\_\{\\mathbf\{W\}\}\(F\)=\\min\\left\\\{\|S\|:S\\subseteq F,\\ \\operatorname\{supp\}\(\\mathbf\{W\}\)\\cup S\\text\{ is cyclic\}\\right\\\},\(6\)withq𝐖​\(F\)=∞q\_\{\\mathbf\{W\}\}\(F\)=\\inftyif no subset completes a cycle\.

Geometrically, every coordinate face with fewer thanq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)candidates lies in the exact zero set; only afterq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)coordinates are active can the subspace leave the acyclic set\. At𝐖=0\\mathbf\{W\}=0, this is the directed girth of\(V,F\)\(V,F\)\. The definition also covers overlapping cycles and nonempty base DAGs\.

Candidate edges need not approach zero at the same rate\. For positive weightsa=\(a1,…,am\)∈ℝ\>0ma=\(a\_\{1\},\\ldots,a\_\{m\}\)\\in\\mathbb\{R\}\_\{\>0\}^\{m\}, define the weighted analogue

τ𝐖​\(F;a\)=minS⊆F:supp⁡\(𝐖\)∪S​cyclic​∑es∈Sas\.\\tau\_\{\\mathbf\{W\}\}\(F;a\)=\\min\_\{\\begin\{subarray\}\{c\}S\\subseteq F:\\ \\operatorname\{supp\}\(\\mathbf\{W\}\)\\cup S\\text\{ cyclic\}\\end\{subarray\}\}\\sum\_\{e\_\{s\}\\in S\}a\_\{s\}\.\(7\)Alongxs=cs​tasx\_\{s\}=c\_\{s\}t^\{a\_\{s\}\}ast↓0t\\downarrow 0,τ𝐖​\(F;a\)\\tau\_\{\\mathbf\{W\}\}\(F;a\)is the cheapest cycle\-completion exponent\. At the empty graph it is weighted directed girth\.

###### Definition 3\.2\(Cycle\-completion ideal\)\.

The acyclic candidate supports form a restricted directed\-subgraph complex\(Hultman,[2004](https://arxiv.org/html/2608.08103#bib.bib12)\),

Δ𝐖,F=\{S⊆\[m\]:supp⁡\(𝐖\)∪\{es:s∈S\}​is acyclic\}\.\\Delta\_\{\\mathbf\{W\},F\}=\\\{S\\subseteq\[m\]:\\operatorname\{supp\}\(\\mathbf\{W\}\)\\cup\\\{e\_\{s\}:s\\in S\\\}\\text\{ is acyclic\}\\\}\.\(8\)Let𝒞min​\(𝐖,F\)\\mathcal\{C\}\_\{\\rm min\}\(\\mathbf\{W\},F\)collect its inclusion\-minimal nonfaces\. Its Stanley–Reisner ideal is

I𝐖,F=⟨xC:C∈𝒞min\(𝐖,F\)⟩⊂ℝ\[x1,…,xm\],I\_\{\\mathbf\{W\},F\}=\\left\\langle x^\{C\}:C\\in\\mathcal\{C\}\_\{\\rm min\}\(\\mathbf\{W\},F\)\\right\\rangle\\subset\\mathbb\{R\}\[x\_\{1\},\\ldots,x\_\{m\}\],\(9\)wherexC=∏s∈Cxsx^\{C\}=\\prod\_\{s\\in C\}x\_\{s\}\.

Thusq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)andτ𝐖​\(F;a\)\\tau\_\{\\mathbf\{W\}\}\(F;a\)are the least ordinary and weighted degrees inI𝐖,FI\_\{\\mathbf\{W\},F\}\. When this ideal is nonzero, callr𝐖​\(F\)=maxC∈𝒞min​\(𝐖,F\)⁡\|C\|r\_\{\\mathbf\{W\}\}\(F\)=\\max\_\{C\\in\\mathcal\{C\}\_\{\\rm min\}\(\\mathbf\{W\},F\)\}\|C\|its*completion width*\. The two unweighted degrees can differ:q𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)controls the earliest Taylor signal, whiler𝐖​\(F\)r\_\{\\mathbf\{W\}\}\(F\)controls the worst local error bound\.

###### Theorem 3\.3\(Cycle\-completion jet ideal\)\.

LetGFG\_\{F\}be the restriction of either exact representation in Eqs\. \([3](https://arxiv.org/html/2608.08103#S2.E3)\)–\([4](https://arxiv.org/html/2608.08103#S2.E4)\), and supposeGFG\_\{F\}isCkC^\{k\}\. Every component of its order\-kkTaylor polynomial at zero belongs toI𝐖,FI\_\{\\mathbf\{W\},F\}\. Conversely, the vector of minimal generators

M𝐖,F​\(x\)=\(xC\)C∈𝒞min​\(𝐖,F\)M\_\{\\mathbf\{W\},F\}\(x\)=\(x^\{C\}\)\_\{C\\in\\mathcal\{C\}\_\{\\rm min\}\(\\mathbf\{W\},F\)\}\(10\)vanishes exactly on the acyclic candidate supports\.

###### Proof\.

Ifsupp⁡\(α\)∈Δ𝐖,F\\operatorname\{supp\}\(\\alpha\)\\in\\Delta\_\{\\mathbf\{W\},F\}, exactness makesGFG\_\{F\}identically zero near the origin on that coordinate subspace, soDα​GF​\(0\)=0D^\{\\alpha\}G\_\{F\}\(0\)=0\. Every surviving Taylor monomial therefore contains a minimal nonface and belongs toI𝐖,FI\_\{\\mathbf\{W\},F\}\. The generator vector is zero exactly when the active support contains no minimal nonface, which is precisely membership inΔ𝐖,F\\Delta\_\{\\mathbf\{W\},F\}\. ∎

## 4Signed and Vector Representations

The coordinate\-face observation already constrains a vector representation\. No sign or nonnegativity assumption is needed: exactness alone forces the low\-order Taylor coefficients to disappear\.

###### Theorem 4\.1\(Vector cycle\-completion barrier\)\.

Assume Eq\. \([3](https://arxiv.org/html/2608.08103#S2.E3)\), letq=q𝐖​\(F\)<∞q=q\_\{\\mathbf\{W\}\}\(F\)<\\infty, and supposeHHisCq−1C^\{q\-1\}near𝐖\\mathbf\{W\}\. Then

Dα​HF​\(0\)=0for every multi\-index​\|α\|≤q−1\.D^\{\\alpha\}H\_\{F\}\(0\)=0\\qquad\\text\{for every multi\-index \}\|\\alpha\|\\leq q\-1\.\(11\)IfHHisCqC^\{q\}, then for every𝐔\\mathbf\{U\}supported onFF,

‖H​\(𝐖\+t​𝐔\)‖2=O​\(\|t\|q\),‖D​HF​\(t​u\)‖op=O​\(\|t\|q−1\)\.\\left\\lVert H\(\\mathbf\{W\}\+t\\mathbf\{U\}\)\\right\\rVert\_\{2\}=O\(\|t\|^\{q\}\),\\qquad\\left\\lVert DH\_\{F\}\(tu\)\\right\\rVert\_\{\\rm op\}=O\(\|t\|^\{q\-1\}\)\.\(12\)

###### Proof\.

Consider a Taylor coefficient indexed byα\\alphaand letS=supp⁡\(α\)S=\\operatorname\{supp\}\(\\alpha\)\. If\|α\|<q\|\\alpha\|<q, then\|S\|<q\|S\|<qandsupp⁡\(𝐖\)∪S\\operatorname\{supp\}\(\\mathbf\{W\}\)\\cup Sis acyclic\. Exactness makesHFH\_\{F\}identically zero near the origin in the coordinate subspace indexed bySS, so the coefficient must vanish\. Taylor remainder bounds give Eq\. \([12](https://arxiv.org/html/2608.08103#S4.E12)\)\. ∎

The restriction in Eq\. \([12](https://arxiv.org/html/2608.08103#S4.E12)\) matters\. The full Jacobian may contain directions outsideFFthat close an already existing path at lower order; the theorem controls the chosen candidate subspace\.

###### Corollary 4\.2\(Weighted vector barrier\)\.

Leta∈ℕma\\in\\mathbb\{N\}^\{m\},τ=τ𝐖​\(F;a\)<∞\\tau=\\tau\_\{\\mathbf\{W\}\}\(F;a\)<\\infty,cs≠0c\_\{s\}\\neq 0, and𝐖a​\(t\)=𝐖\+∑scs​tas​𝐄s\\mathbf\{W\}\_\{a\}\(t\)=\\mathbf\{W\}\+\\sum\_\{s\}c\_\{s\}t^\{a\_\{s\}\}\\mathbf\{E\}\_\{s\}\. IfHHisCτ−1C^\{\\tau\-1\}, every derivative ofH​\(𝐖a​\(t\)\)H\(\\mathbf\{W\}\_\{a\}\(t\)\)below orderτ\\tauvanishes at zero\. IfHHisCτC^\{\\tau\}, then

‖H​\(𝐖a​\(t\)\)‖2=O​\(\|t\|τ\)\.\\left\\lVert H\(\\mathbf\{W\}\_\{a\}\(t\)\)\\right\\rVert\_\{2\}=O\(\|t\|^\{\\tau\}\)\.\(13\)

The proof lifts edgeese\_\{s\}toasa\_\{s\}auxiliary coordinates whose product activates the edge, then applies the local coordinate\-subspace jet lemma underlying Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1); Appendix[B\.2](https://arxiv.org/html/2608.08103#A2.SS2)gives details\.

The bound is sharp\. To see this directly, let𝒞d\\mathcal\{C\}\_\{d\}be the set of all simple directed cycles and define

Hcyc​\(𝐖\)=\(∏\(i,j\)∈CWi​j\)C∈𝒞d\.H\_\{\\rm cyc\}\(\\mathbf\{W\}\)=\\left\(\\prod\_\{\(i,j\)\\in C\}W\_\{ij\}\\right\)\_\{C\\in\\mathcal\{C\}\_\{d\}\}\.\(14\)This polynomial vector is zero exactly on DAG support\. A minimalqq\-edge completion produces a component of order exactlyqq, and a minimum\-weight completion produces orderτ\\tau\. Its representation size is

\|𝒞d\|=∑k=2d\(dk\)​\(k−1\)\!\.\|\\mathcal\{C\}\_\{d\}\|=\\sum\_\{k=2\}^\{d\}\\binom\{d\}\{k\}\(k\-1\)\!\.\(15\)Equation \([14](https://arxiv.org/html/2608.08103#S4.E14)\) gives an explicit sharp representation, but Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1)does not require a representation to enumerate cycles\.

Theorem[3\.3](https://arxiv.org/html/2608.08103#S3.Thmtheorem3)is the finite\-jet form of the classical Stanley–Reisner correspondence\(Miller & Sturmfels,[2005](https://arxiv.org/html/2608.08103#bib.bib18)\)\. The same geometry also limits succinct vector residuals: uniformqqth\-order sensitivity on an explicit channel family requiresΘ​\(d2\)\\Theta\(d^\{2\}\)outputs for fixedqq\. This dimension statement concerns uniform directional sharpness, not convergence of an arbitrary lower\-dimensional map; Appendix[A](https://arxiv.org/html/2608.08103#A1)states and proves the near\-tight bound\.

## 5Nonnegative Scalar Representations

The scalar case contains one additional restriction\. A vector\-valued leading term may change sign, whereas the leading Taylor polynomial of a nonnegative scalar must itself be nonnegative\. This parity requirement doubles the first possible order\.

###### Lemma 5\.1\(Newton vertices\)\.

Every vertex exponent of the Newton polytope of a globally nonnegative real polynomial is coordinatewise even, and its coefficient is positive\(Reznick,[1978](https://arxiv.org/html/2608.08103#bib.bib23)\)\.

###### Theorem 5\.2\(Nonnegative scalar barrier\)\.

Assume Eq\. \([4](https://arxiv.org/html/2608.08103#S2.E4)\), letq=q𝐖​\(F\)<∞q=q\_\{\\mathbf\{W\}\}\(F\)<\\infty, and supposehhisC2​q−1C^\{2q\-1\}near𝐖\\mathbf\{W\}\. Then

Dα​hF​\(0\)=0for every​\|α\|≤2​q−1\.D^\{\\alpha\}h\_\{F\}\(0\)=0\\qquad\\text\{for every \}\|\\alpha\|\\leq 2q\-1\.\(16\)IfhhisC2​qC^\{2q\}, then

h​\(𝐖\+t​𝐔\)=O​\(\|t\|2​q\),‖D​hF​\(t​u\)‖=O​\(\|t\|2​q−1\)\.h\(\\mathbf\{W\}\+t\\mathbf\{U\}\)=O\(\|t\|^\{2q\}\),\\qquad\\left\\lVert Dh\_\{F\}\(tu\)\\right\\rVert=O\(\|t\|^\{2q\-1\}\)\.\(17\)

###### Proof\.

Suppose the first nonzero homogeneous Taylor termPrP\_\{r\}has degreer<2​qr<2q\. Nonnegativity ofhhmakesPrP\_\{r\}globally nonnegative\. Exactness on acyclic coordinate subspaces implies that every monomial ofPrP\_\{r\}uses at leastqqdistinct candidates\. Choose a vertex exponentα\\alphaof its Newton polytope\. Lemma[5\.1](https://arxiv.org/html/2608.08103#S5.Thmtheorem1)makes every nonzeroαs\\alpha\_\{s\}at least two, sor=\|α\|≥2​\|supp⁡\(α\)\|≥2​qr=\|\\alpha\|\\geq 2\|\\operatorname\{supp\}\(\\alpha\)\|\\geq 2q, a contradiction\. ∎

###### Corollary 5\.3\(Newton\-square geometry\)\.

LetPrP\_\{r\}be the first nonzero homogeneous Taylor form ofhFh\_\{F\}\. Every vertex exponentα\\alphaof its Newton polytope satisfiesxα∈I𝐖,F2x^\{\\alpha\}\\in I\_\{\\mathbf\{W\},F\}^\{2\}\. Hence, for every positive weight vectoraa, the minimumaa\-weight at a Newton vertex is at least2​τ𝐖​\(F;a\)2\\tau\_\{\\mathbf\{W\}\}\(F;a\)\.

###### Proof\.

Theorem[3\.3](https://arxiv.org/html/2608.08103#S3.Thmtheorem3)givesPr∈I𝐖,FP\_\{r\}\\in I\_\{\\mathbf\{W\},F\}\. Lemma[5\.1](https://arxiv.org/html/2608.08103#S5.Thmtheorem1)makesα\\alphaeven, whilesupp⁡\(α/2\)=supp⁡\(α\)\\operatorname\{supp\}\(\\alpha/2\)=\\operatorname\{supp\}\(\\alpha\)is a nonface\. Thusxα/2∈I𝐖,Fx^\{\\alpha/2\}\\in I\_\{\\mathbf\{W\},F\}andxα∈I𝐖,F2x^\{\\alpha\}\\in I\_\{\\mathbf\{W\},F\}^\{2\}\. ∎

Fora∈ℕma\\in\\mathbb\{N\}^\{m\}, the same auxiliary\-coordinate argument gives a weighted2​τ𝐖​\(F;a\)2\\tau\_\{\\mathbf\{W\}\}\(F;a\)barrier with the required regularity made explicit\. IfhhisC2​τ−1C^\{2\\tau\-1\}near𝐖\\mathbf\{W\}, every derivative ofh​\(𝐖a​\(t\)\)h\(\\mathbf\{W\}\_\{a\}\(t\)\)below order2​τ2\\tauvanishes; ifhhisC2​τC^\{2\\tau\}, thenh​\(𝐖a​\(t\)\)=O​\(\|t\|2​τ\)h\(\\mathbf\{W\}\_\{a\}\(t\)\)=O\(\|t\|^\{2\\tau\}\)\. For one minimal completion, a stronger local factorization holds:

hF​\(x\)=\(∏s=1qxs2\)​Ψ​\(x\),Ψ​\(x\)≥0locally,h\_\{F\}\(x\)=\\left\(\\prod\_\{s=1\}^\{q\}x\_\{s\}^\{2\}\\right\)\\Psi\(x\),\\qquad\\Psi\(x\)\\geq 0\\quad\\text\{locally\},\(18\)underC2​qC^\{2q\}regularity\. The positive walk families used by standard continuous methods attain this lower bound\.

###### Corollary 5\.4\(Local order of positive walk constraints\)\.

LetFFbe a minimalqq\-edge completion at a DAGWWand letUUactivate every edge inFFwith a nonzero equal\-scale coefficient\. Then NOTEARS EXP, the usual positive trace polynomial, and DAGMA’s log determinant whileρ​\(W∘W\)<s\\rho\(W\\circ W\)<ssatisfy

h​\(W\+t​U\)=Θ​\(\|t\|2​q\),‖D​hF​\(t​u\)‖2=Θ​\(\|t\|2​q−1\)\.h\(W\+tU\)=\\Theta\(\|t\|^\{2q\}\),\\qquad\\left\\lVert Dh\_\{F\}\(tu\)\\right\\rVert\_\{2\}=\\Theta\(\|t\|^\{2q\-1\}\)\.\(19\)The same conclusion holds for every convergent positive walk sum with a positive coefficient at each simple\-cycle length, and for smooth entrywise squashes with a positive quadratic leading term\.

The proof is Proposition[B\.3](https://arxiv.org/html/2608.08103#A2.Thmtheorem3)in Appendix[B\.5](https://arxiv.org/html/2608.08103#A2.SS5)\. For DAGMA,−log​det\(s​I−A\)\+d​log⁡s=∑k≥1tr⁡\(Ak\)/\(k​sk\)\-\\log\\det\(sI\-A\)\+d\\log s=\\sum\_\{k\\geq 1\}\\operatorname\{tr\}\(A^\{k\}\)/\(ks^\{k\}\)inside its M\-matrix domain\. The log\-determinant can therefore improve coefficients and the landscape away from the boundary without changing the topology\-controlled order at the boundary itself\.

##### Consequences for optimization\.

The corollary does not say that all smooth constraints optimize equally well\. Coefficient size and conditioning still matter, and the log\-determinant path used by DAGMA can improve both\. It says only that these changes do not remove the local support\-boundary order\. In practice, exact support may instead be selected after optimization by thresholding\(Ng et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib20)\), or enforced through absolute\-value reformulations, higher\-order local search, acyclic parameterizations, discrete graph operations, or boundary\-constrained domains\(Wei et al\.,[2020](https://arxiv.org/html/2608.08103#bib.bib26); Shridharan & Iyengar,[2025](https://arxiv.org/html/2608.08103#bib.bib24); Yu et al\.,[2021](https://arxiv.org/html/2608.08103#bib.bib28); Massidda et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib16); Gillot & Parviainen,[2022](https://arxiv.org/html/2608.08103#bib.bib10); Rey et al\.,[2026](https://arxiv.org/html/2608.08103#bib.bib22)\)\. Appendix[C\.5](https://arxiv.org/html/2608.08103#A3.SS5)compares these assumptions, and Appendix[B\.5](https://arxiv.org/html/2608.08103#A2.SS5)gives the scalar proofs\.

## 6From Constraint Symmetry to the Score–Topology Crossover

The far\-field construction uses a small ordinary\-coordinate gadget\. For eachi∈\[m\]i\\in\[m\], introduce nodesai,bia\_\{i\},b\_\{i\}, a candidate edgeai→bia\_\{i\}\\to b\_\{i\}, and fixed cross\-edgesbi→ajb\_\{i\}\\to a\_\{j\}for everyi≠ji\\neq j\. The fixed graph is acyclic, while any two active candidates form a directed four\-cycle\. We takettdisjoint copies\. Appendix[A](https://arxiv.org/html/2608.08103#A1)shows that the same reduction has a broader topological consequence: local candidate sections can realize any finite homotopy type and illustrates the construction\. That result is not needed for the selection theorem below\.

Callhh*square\-symmetric*if

h​\(𝐖\)=Φ​\(𝐖∘𝐖\),h​\(P​𝐖​P⊤\)=h​\(𝐖\)h\(\\mathbf\{W\}\)=\\Phi\(\\mathbf\{W\}\\circ\\mathbf\{W\}\),\\qquad h\(P\\mathbf\{W\}P^\{\\top\}\)=h\(\\mathbf\{W\}\)\(20\)for every permutation matrixPP\. NOTEARS, positive trace polynomials, and DAGMA are square\-symmetric on their domains\.

###### Theorem 6\.1\(Far\-field support\-selection blindness\)\.

For every integerm≥2m\\geq 2and integert≥1t\\geq 1, there is a base DAG ond=2​m​td=2mtnodes withm​tmtindependent candidate entries and

\(m2\)t\\binom\{m\}\{2\}^\{t\}\(21\)disjoint positive cyclic support manifolds\. Leth:Ω→ℝh:\\Omega\\to\\mathbb\{R\}be aC1C^\{1\}square\-symmetric exact scalar, and suppose thatΩ\\Omegacontains a neighborhood of these manifolds\. On each manifold,

∇xbh​\(𝐖0\+x\)∈span⁡\{xb\},b=1,…,t,\\nabla\_\{x\_\{b\}\}h\(\\mathbf\{W\}\_\{0\}\+x\)\\in\\operatorname\{span\}\\\{x\_\{b\}\\\},\\qquad b=1,\\ldots,t,\(22\)so the constraint gives zero first\-order signal in every within\-block edge\-redistribution direction\.

If∇h\\nabla his locally Lipschitz, every scalarized flowx˙=−ω​\(x\)​∇h​\(𝐖0\+x\)\\dot\{x\}=\-\\omega\(x\)\\nabla h\(\\mathbf\{W\}\_\{0\}\+x\)with locally Lipschitz scalarω\\omega, when initialized on one of these manifolds, preserves it on the solution’s maximal interval of existence\. Positive active coefficients cannot reach zero at a finite time in that interval, so the support remains cyclic whenever the solution exists\. Withm=5m=5there are10t=20\.332​…​d10^\{t\}=2^\{0\.332\\ldots d\}such manifolds, and every active coefficient may be fixed at1/21/2\. The resulting manifolds lie in the domains of NOTEARS and DAGMA withs=1s=1\.

An automorphism ties the two active derivatives, while evenness zeros every inactive derivative\. The flow claim then follows from uniqueness and Gronwall’s inequality\.

###### Proposition 6\.2\(Full\-cycle scalarization and score perturbation\)\.

LetW​\(z\)W\(z\)be supported on an isolated directedLL\-cycle with positive edge weightsz=\(z1,…,zL\)z=\(z\_\{1\},\\ldots,z\_\{L\}\)\. Suppose

h​\(W​\(z\)\)=ϕ​\(p\),p=∏i=1Lzi2,ϕ​\(0\)=0,ϕ′​\(0\)\>0,h\(W\(z\)\)=\\phi\(p\),\\qquad p=\\prod\_\{i=1\}^\{L\}z\_\{i\}^\{2\},\\qquad\\phi\(0\)=0,\\quad\\phi^\{\\prime\}\(0\)\>0,whereϕ\\phiisC1C^\{1\}andϕ′\>0\\phi^\{\\prime\}\>0on the relevant range\. LetΨ\\PsibeC1C^\{1\}, withΨ′\>0\\Psi^\{\\prime\}\>0on that range andΨ′​\(s\)=c​sν​\(1\+o​\(1\)\)\\Psi^\{\\prime\}\(s\)=cs^\{\\nu\}\(1\+o\(1\)\)ass↓0s\\downarrow 0, wherec\>0c\>0andν≥0\\nu\\geq 0\. Underz˙=−∇z\(Ψ∘h\)\\dot\{z\}=\-\\nabla\_\{z\}\(\\Psi\\circ h\), initialize two tracked edges as\(x0,y0\)=\(a\+ε,a−ε\)\(x\_\{0\},y\_\{0\}\)=\(a\+\\varepsilon,a\-\\varepsilon\)and initialize the remaining edges atbj\>ab\_\{j\}\>a\. Put

C=∏j=1L−2\(bj2−a2\),Iν​\(r\)=∫r2/\(1−r2\)∞\[u​\(u\+1\)\]−\(ν\+1\)​𝑑u\.C=\\prod\_\{j=1\}^\{L\-2\}\(b\_\{j\}^\{2\}\-a^\{2\}\),\\qquad I\_\{\\nu\}\(r\)=\\int\_\{r^\{2\}/\(1\-r^\{2\}\)\}^\{\\infty\}\[u\(u\+1\)\]^\{\-\(\\nu\+1\)\}\\,du\.For fixed0<r<10<r<1, letTr​\(ε\)T\_\{r\}\(\\varepsilon\)be the first timey/x=ry/x=r\. Then every differencezi2−zj2z\_\{i\}^\{2\}\-z\_\{j\}^\{2\}is conserved and

Tr​\(ε\)∼Iν​\(r\)4​c​\[ϕ′​\(0\)​C\]ν\+1​\(4​a​ε\)2​ν\+1\.T\_\{r\}\(\\varepsilon\)\\sim\\frac\{I\_\{\\nu\}\(r\)\}\{4c\[\\phi^\{\\prime\}\(0\)C\]^\{\\nu\+1\}\(4a\\varepsilon\)^\{2\\nu\+1\}\}\.\(23\)For a separate flow with a smooth scoreSS, allow arbitrary positive tracked initial values withy0/x0\>ry\_\{0\}/x\_\{0\}\>r, and define the total logarithmic deletion margin

msc​\(z\)=∂y\[S\+Ψ​\(h\)\]y−∂x\[S\+Ψ​\(h\)\]x\.m\_\{\\rm sc\}\(z\)=\\frac\{\\partial\_\{y\}\[S\+\\Psi\(h\)\]\}\{y\}\-\\frac\{\\partial\_\{x\}\[S\+\\Psi\(h\)\]\}\{x\}\.Ifmsc​\(z\)≥γflow\>0m\_\{\\rm sc\}\(z\)\\geq\\gamma\_\{\\rm flow\}\>0untily/x=ry/x=r, then

Tr​\(ε\)≤1γflow​log⁡y0r​x0,T\_\{r\}\(\\varepsilon\)\\leq\\frac\{1\}\{\\gamma\_\{\\rm flow\}\}\\log\\frac\{y\_\{0\}\}\{rx\_\{0\}\},\(24\)so a fixed score margin removes the divergence and can overturn anO​\(ε\)O\(\\varepsilon\)reversed initial ranking\. More precisely, add the smooth edge scoreSγ​\(z\)=γ​y2/2S\_\{\\gamma\}\(z\)=\\gamma y^\{2\}/2, letTr​\(ε,γ\)T\_\{r\}\(\\varepsilon,\\gamma\)be the new hitting time, and writeT0​\(ε\)=Tr​\(ε,0\)T\_\{0\}\(\\varepsilon\)=T\_\{r\}\(\\varepsilon,0\)\. For any positive scheduleγ=γ​\(ε\)\\gamma=\\gamma\(\\varepsilon\),

γ​T0​\(ε\)→0,ν\>0,γ​T0​\(ε\)​log⁡\(1/ε\)→0,ν=0\}⟹Tr​\(ε,γ\)T0​\(ε\)→1,\\left\.\\begin\{array\}\[\]\{ll\}\\gamma T\_\{0\}\(\\varepsilon\)\\to 0,&\\nu\>0,\\\\ \\gamma T\_\{0\}\(\\varepsilon\)\\log\(1/\\varepsilon\)\\to 0,&\\nu=0\\end\{array\}\\right\\\}\\quad\\Longrightarrow\\quad\\frac\{T\_\{r\}\(\\varepsilon,\\gamma\)\}\{T\_\{0\}\(\\varepsilon\)\}\\to 1,\(25\)whereas, for everyν≥0\\nu\\geq 0,

γ​T0​\(ε\)→∞⟹Tr​\(ε,γ\)T0​\(ε\)→0\.\\gamma T\_\{0\}\(\\varepsilon\)\\to\\infty\\quad\\Longrightarrow\\quad\\frac\{T\_\{r\}\(\\varepsilon,\\gamma\)\}\{T\_\{0\}\(\\varepsilon\)\}\\to 0\.\(26\)Thus, whenν\>0\\nu\>0, the two regimes meet at the power scale

γc​\(ε\)=T0​\(ε\)−1=Θ​\(ε2​ν\+1\)\.\\gamma\_\{c\}\(\\varepsilon\)=T\_\{0\}\(\\varepsilon\)^\{\-1\}=\\Theta\(\\varepsilon^\{2\\nu\+1\}\)\.\(27\)Atν=0\\nu=0, the scale\[T0​\(ε\)​log⁡\(1/ε\)\]−1=Θ​\(ε/log⁡\(1/ε\)\)\[T\_\{0\}\(\\varepsilon\)\\log\(1/\\varepsilon\)\]^\{\-1\}=\\Theta\(\\varepsilon/\\log\(1/\\varepsilon\)\)is sufficient for asymptotically negligible score perturbations, whileγ​T0→∞\\gamma T\_\{0\}\\to\\inftyis sufficient for score domination\. In general a logarithmic transition layer lies between these statements\.

The endpoint logarithm is structural rather than an artifact of the proof\. For the critical limiting system, after a constant time rescaling,

Q′=−4​Q​V,V′=−4​Q​V−2​η​V,Q^\{\\prime\}=\-4QV,\\qquad V^\{\\prime\}=\-4QV\-2\\eta V,the quantityV−Q−\(η/2\)​log⁡QV\-Q\-\(\\eta/2\)\\log Qis conserved\. There are schedules withη→0\\eta\\to 0butη​log⁡\(1/Δ\)→∞\\eta\\log\(1/\\Delta\)\\to\\infty, whereΔ=x02−y02\\Delta=x\_\{0\}^\{2\}\-y\_\{0\}^\{2\}, for which the normalized hitting time tends to zero\. Thus the implicationγ​T0→0⇒Tr/T0→1\\gamma T\_\{0\}\\to 0\\Rightarrow T\_\{r\}/T\_\{0\}\\to 1is false atν=0\\nu=0without additional entrance control\.

The pathwise quantityγflow\\gamma\_\{\\rm flow\}and the completion\-exchange marginκ𝒞\\kappa\_\{\\mathcal\{C\}\}introduced below measure different objects\. The former lower\-bounds a gradient ratio along one trajectory; the latter is a discrete score gap between repairs\. Neither bounds the other without additional assumptions on the score and path\.

The conserved squared differences echo balancing invariants for homogeneous models\(Du et al\.,[2018](https://arxiv.org/html/2608.08103#bib.bib8)\)\. Here the product is an exact DAG\-cycle restriction; the new consequence is the scalarization\-dependent hitting\-time law and its score\-margin counterpart, rather than the invariant by itself\. For anLL\-cycle,ϕEXP​\(p\)=L​∑ℓ≥1pℓ/\(ℓ​L\)\!\\phi\_\{\\rm EXP\}\(p\)=L\\sum\_\{\\ell\\geq 1\}p^\{\\ell\}/\(\\ell L\)\!, soϕEXP′​\(0\)=1/\(L−1\)\!\\phi\_\{\\rm EXP\}^\{\\prime\}\(0\)=1/\(L\-1\)\!, whereasϕDAGMA​\(p\)=−log⁡\(1−p\)\\phi\_\{\\rm DAGMA\}\(p\)=\-\\log\(1\-p\)andϕDAGMA′​\(0\)=1\\phi\_\{\\rm DAGMA\}^\{\\prime\}\(0\)=1\. Thus the proposition applies exactly to both constraints\. Linear feasibility and seeded ALM haveν=0\\nu=0, while a cold quadratic penalty hasν=1\\nu=1, yielding the feasibility\-only exponents1,3,11,3,1\. The logarithmic endpoint qualification applies to the first and third scalarizations\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x1.png)Figure 1:Finite\-range score perturbation experiment\. After normalizing the margin by the feasibility\-only timeT0T\_\{0\}, 480 controlled NOTEARS/DAGMA flows separate aroundγ​T0=1\\gamma T\_\{0\}=1on the plotted grid \(left\)\. The fitted power scale isT0−1≍ε2​ν\+1T\_\{0\}^\{\-1\}\\asymp\\varepsilon^\{2\\nu\+1\}\(right\)\. This is the asymptotic crossover scale forν\>0\\nu\>0; at the linear endpointν=0\\nu=0, the theorem contains the additional logarithmic entrance factor\.##### Numerical checks\.

The frozen grid supports each link in the argument\. Over 72 controlled cycle settings, the largest exponent error in Eq\. \([23](https://arxiv.org/html/2608.08103#S6.E23)\) is 0\.044, and direct matrix values and gradients agree with the closed form to1\.1×10−151\.1\\times 10^\{\-15\}\. The 480\-run phase experiment in Figure[1](https://arxiv.org/html/2608.08103#S6.F1)givesTγ/T0∈\[0\.960,0\.978\]T\_\{\\gamma\}/T\_\{0\}\\in\[0\.960,0\.978\]atγ​T0=0\.01\\gamma T\_\{0\}=0\.01,\[0\.236,0\.365\]\[0\.236,0\.365\]at11, and\[0\.0069,0\.0102\]\[0\.0069,0\.0102\]at100100; over this finite grid, the largest fitted power\-exponent error is 0\.034\. These ratios check the normalization and the finite\-range phase change; forν=0\\nu=0, they are not evidence for the invalid fixed\-γ​T0\\gamma T\_\{0\}endpoint limit\. A population linear\-Gaussian audit selects the false back edge first in all 216 dense flows, while 91\.25% of 720 finite\-sample runs retain the required score margin\. On 320 disjoint official NOTEARS/DAGMA trajectories, first\-stage separation correlates with selection time at−0\.521/−0\.664\-0\.521/\-0\.664\(Spearman,p<10−4p<10^\{\-4\}\)\. Appendix[C\.3](https://arxiv.org/html/2608.08103#A3.SS3)gives the full trajectories and Appendix[E\.9](https://arxiv.org/html/2608.08103#A5.SS9)gives held\-out errors and boundary cases\.

## 7Interaction\-Aware Score Certificates

Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)makes support selection depend on a score margin that is unknown in finite samples\. Certifying that margin with edge\-additive deletion costs would be convenient, but would discard interactions among parents of the same child\. We retain those interactions by freezing candidate parent sets𝒫j\\mathcal\{P\}\_\{j\}, refitting each set on training data, and evaluating bounded lossesℓj,S\\ell\_\{j,S\}on an independent holdout\. LetG0∈𝒢​\(𝒫\)G\_\{0\}\\in\\mathcal\{G\}\(\\mathcal\{P\}\)be a reference DAG from any continuous, constraint\-based, or score\-based procedure; the audit uses the holdout\-score minimizer\. For a candidate DAGGG, let

Q​\(G\)=∑j\{Rj,paG⁡\(j\)\+aj,paG⁡\(j\)\}\.Q\(G\)=\\sum\_\{j\}\\\{R\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\}\+a\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\}\\\}\.Simultaneous paired intervals give, for every local alternativeSS, a lower boundgj,Sg\_\{j,S\}on its population score difference from the parent setS0​j=paG0⁡\(j\)S\_\{0j\}=\\operatorname\{pa\}\_\{G\_\{0\}\}\(j\)\. Put

Γ​\(G\)\\displaystyle\\Gamma\(G\)=∑jgj,paG⁡\(j\),\\displaystyle=\\sum\_\{j\}g\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\},L\\displaystyle L=minG⁡Γ​\(G\),\\displaystyle=\\min\_\{G\}\\Gamma\(G\),L−\\displaystyle L\_\{\-\}=minG≠G0⁡Γ​\(G\),\\displaystyle=\\min\_\{G\\neq G\_\{0\}\}\\Gamma\(G\),\(28\)𝒜α​\(G0\)\\displaystyle\\mathcal\{A\}\_\{\\alpha\}\(G\_\{0\}\)=\{G∈𝒢​\(𝒫\):Γ​\(G\)≤0\}\.\\displaystyle=\\\{G\\in\\mathcal\{G\}\(\\mathcal\{P\}\):\\Gamma\(G\)\\leq 0\\\}\.\(29\)For a binary graph featureφ\\varphi, define

Lφopp=minG:φ​\(G\)≠φ​\(G0\)⁡Γ​\(G\)\.L\_\{\\varphi\}^\{\\rm opp\}=\\min\_\{G:\\varphi\(G\)\\neq\\varphi\(G\_\{0\}\)\}\\Gamma\(G\)\.\(30\)
###### Theorem 7\.1\(Interaction\-aware selective feature certificate\)\.

Condition on the training sample\. Suppose the holdout observations are i\.i\.d\. and the paired intervals used to constructgj,Sg\_\{j,S\}cover all local score differences simultaneously with probability at least1−α1\-\\alpha\. The referenceG0G\_\{0\}may depend on this holdout\. Then every population minimizer ofQQbelongs to𝒜α​\(G0\)\\mathcal\{A\}\_\{\\alpha\}\(G\_\{0\}\), and

Q​\(G0\)−minG∈𝒢​\(𝒫\)⁡Q​\(G\)≤−L\.Q\(G\_\{0\}\)\-\\min\_\{G\\in\\mathcal\{G\}\(\\mathcal\{P\}\)\}Q\(G\)\\leq\-L\.\(31\)IfL−\>0L\_\{\-\}\>0,G0G\_\{0\}is the unique population minimizer\. IfLφopp\>0L\_\{\\varphi\}^\{\\rm opp\}\>0, every population minimizer shares the labelφ​\(G0\)\\varphi\(G\_\{0\}\)\. The feature statement applies to skeleton adjacencies, directed edges, and unshielded colliders\.

The parent\-set score retains collider interactions\. Linear rows in the parent\-set integer program force skeleton and unshielded\-collider labels, so one opposite\-label solve certifies each feature\. These features characterize Markov equivalence\. Exact search is combinatorial; for edge\-modular scores, repairs reduce to transversals and a cycle\-cover LP gives a polynomial\-time conservative certificate\. A bounded\-parent audit runs the exact core and predeclared feature queries throughd=200d=200, without claiming worst\-case tractability\. Appendices[F](https://arxiv.org/html/2608.08103#A6)and[G](https://arxiv.org/html/2608.08103#A7)give the modular specialization and main proof\.

Table 1:Unified 320\-run pipeline with a shared top\-2​d2dscreen\. “Oracle parent” uses 20,000 independent observations; labels are certified feature percentages\.Every regret bound covers the independent oracle\-score audit across the 320 frontend–dataset pairs\. None of 3,042 certified skeleton or 2,396 collider labels disagrees with its oracle\-score optimum\. Parent\-set SHD is 14\.63, compared with 14\.56 for the 20,000\-sample oracle parent\-set solution, and has correlation−0\.886\-0\.886with screen recall\. Finite holdout variance therefore does not explain the weaker DAGMA/GOLEM parent\-set results; the candidate family and score target do\. Increasing the frozen screen from top\-2​d2dto top\-6​d6draises family recall only from66\.4%66\.4\\%to68\.5%68\.5\\%and worsens oracle\-parent SHD from 14\.56 to 15\.67\. Modular, parent\-set, and reported frontend SHDs average 12\.91, 14\.63, and 16\.87\. Against the generating graph,4\.4%4\.4\\%of certified skeleton labels and5\.5%5\.5\\%of collider labels disagree\. The certificate quantifies uncertainty for its score target; it does not establish causal identification\. Appendix[G](https://arxiv.org/html/2608.08103#A7)gives stratified results and runtimes\.

## 8Implications

The analysis assigns separate roles to the terms in Eq\. \([1](https://arxiv.org/html/2608.08103#S1.E1)\)\. A smooth constraint can shape the far\-field landscape and restrict the search to DAGs, but its exactness does not rank support changes\. When the ranking comes from data, the relevant quantity is a score margin together with its uncertainty, not acyclicity residual alone\.

This distinction leads to a concrete reporting rule\. Freeze a candidate parent\-set family, evaluate its score on independent data, report the skeleton and collider labels shared by all interval\-compatible DAGs, and leave the remaining labels unresolved\. A modular cycle\-cover relaxation exchanges parent interactions for scalability\. The reference graph may come from NOTEARS, PC, GES, or another screened search; continuous optimization is one way to reduce the candidate family, not a source of statistical certification\.

The guarantee remains conditional on the chosen screen and score\. It cannot recover omitted edges, prove observational identifiability, or remove the combinatorial cost of exact parent\-set search\. These limits are part of the main conclusion: DAG feasibility, score\-based support selection, and causal identification require different arguments\.

### AI use statement

In this work, generative AI tools were used to assist with drafting and editing portions of the manuscript for clarity, grammar, and readability\. They were not used to formulate the scientific claims, develop theoretical results or proofs, design the methodology or experiments, or interpret the experimental results\. All AI\-assisted text was reviewed, verified, and revised by the authors\. We take full responsibility for the final content of this work\.

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##### Appendix guide\.

The supplement is organized around the three claims in the main text\. Sections[A](https://arxiv.org/html/2608.08103#A1)–[B](https://arxiv.org/html/2608.08103#A2)develop completion geometry and its proofs\. The symmetry, crossover, and numerical audits follow, after which Sections[F](https://arxiv.org/html/2608.08103#A6)and[G](https://arxiv.org/html/2608.08103#A7)give the two score certificates and their frozen protocols\.

## Appendix AAdditional Geometry and Consequences

The main text uses only the completion order and the far\-field symmetry construction\. Two additional consequences describe the output dimension needed for uniform sharpness and the topology of local candidate sections\.

### A\.1Uniform sharpness and residual dimension

WriteJq​\(U\)=Dq​H​\(0\)​\[U,…,U\]/q\!J\_\{q\}\(U\)=D^\{q\}H\(0\)\[U,\\ldots,U\]/q\!for the degree\-qqTaylor jet\. CallHH*directionwiseqq\-sharp*on a familyU​\(z\)U\(z\)ifJq​\(U​\(z\)\)≠0J\_\{q\}\(U\(z\)\)\\neq 0for everyz≠0z\\neq 0in that family\. Its*qq\-sharp residual complexity*sq​\(U\)s\_\{q\}\(U\)is the minimum output dimension of aCqC^\{q\}mapH:𝒵→ℝrH:\\mathcal\{Z\}\\to\\mathbb\{R\}^\{r\}that is globally exact and has this property\.

###### Theorem A\.1\(Near\-tight sensitivity–succinctness law\)\.

For every3≤q≤d3\\leq q\\leq d, there is an explicit channel familyUqU\_\{q\}with

Mq​\(d\)=⌊\(d−q\+2\)24⌋M\_\{q\}\(d\)=\\left\\lfloor\\frac\{\(d\-q\+2\)^\{2\}\}\{4\}\\right\\rfloor\(32\)independentqq\-cycle channels satisfying

Mq​\(d\)≤sq​\(Uq\)≤Mq​\(d\)\+1\.M\_\{q\}\(d\)\\leq s\_\{q\}\(U\_\{q\}\)\\leq M\_\{q\}\(d\)\+1\.\(33\)Forq=2q=2, there is a family with one channel per unordered node pair and

\(d2\)≤s2​\(U2\)≤\(d2\)\+1\.\\binom\{d\}\{2\}\\leq s\_\{2\}\(U\_\{2\}\)\\leq\\binom\{d\}\{2\}\+1\.\(34\)Consequently,sq​\(Uq\)=Θ​\(d2\)s\_\{q\}\(U\_\{q\}\)=\\Theta\(d^\{2\}\)for every fixedqq\. Below either lower bound, some cyclic mixture has zero degree\-qqjet; the upper bound is globally exact and directionwise sharp\.

The lower bound is a rank argument on a layered funnel\. The upper bound collects the selected cycle products and appends one globally exact scalar\. Section[B\.4](https://arxiv.org/html/2608.08103#A2.SS4)gives the construction and proof\. The theorem requires sharpness in every channel mixture; it is not a convergence lower bound for arbitrary lower\-dimensional residuals\.

### A\.2Coordinate topology

![Refer to caption](https://arxiv.org/html/2608.08103v1/x2.png)Figure 2:Ordinary\-coordinate construction\. AK5K\_\{5\}gadget realizes all pairwise cyclic candidate supports \(left\), producing exponentially many cyclic angular critical points under disjoint union \(right\)\.The positive normalized candidate section is

𝒜𝐖,F\+=\{x≥0:‖x‖2=1,𝐖\+∑e∈Fxe​𝐄e∈𝒟\}≅\|Δ𝐖,F\|\.\\mathcal\{A\}^\{\+\}\_\{\\mathbf\{W\},F\}=\\left\\\{x\\geq 0:\\\|x\\\|\_\{2\}=1,\\ \\mathbf\{W\}\+\\sum\_\{e\\in F\}x\_\{e\}\\mathbf\{E\}\_\{e\}\\in\\mathcal\{D\}\\right\\\}\\cong\|\\Delta\_\{\\mathbf\{W\},F\}\|\.\(35\)
###### Theorem A\.2\(Coordinate DAG\-germ universality\)\.

For every finite simplicial complexKK, there are a base DAG𝐖0\\mathbf\{W\}\_\{0\}and independently parameterized absent adjacency entriesFFsuch that

Δ𝐖0,F≅sd⁡K\.\\Delta\_\{\\mathbf\{W\}\_\{0\},F\}\\cong\\operatorname\{sd\}K\.\(36\)Consequently𝒜𝐖0,F\+\\mathcal\{A\}^\{\+\}\_\{\\mathbf\{W\}\_\{0\},F\}is homeomorphic to\|K\|\|K\|\.

The construction represents the vertices of an independence complex by candidate adjacency entries\. This differs from the full acyclic\-subgraph complex, whose homotopy type is restricted\(Hultman,[2004](https://arxiv.org/html/2608.08103#bib.bib12)\): conditioning on a base DAG and then selecting candidate coordinates changes the local topology\.

###### Corollary A\.3\(Exponential local homology\)\.

There is a family withd=10​td=10tgraph nodes and5​t5tindependent candidates such that

Δ𝐖t,Ft≃⋁4tSt−1,∑jβ~j​\(Δ𝐖t,Ft\)=4t=2d/5\.\\Delta\_\{\\mathbf\{W\}\_\{t\},F\_\{t\}\}\\simeq\\bigvee\_\{4^\{t\}\}S^\{t\-1\},\\qquad\\sum\_\{j\}\\widetilde\{\\beta\}\_\{j\}\(\\Delta\_\{\\mathbf\{W\}\_\{t\},F\_\{t\}\}\)=4^\{t\}=2^\{d/5\}\.\(37\)

###### Corollary A\.4\(Exponential angular criticality\)\.

On oneKmK\_\{m\}gadget, the positive and signed candidate unit spheres contain at least2m−m−12^\{m\}\-m\-1and3m−2​m−13^\{m\}\-2m\-1distinct cyclic critical points, respectively\. All corresponding DAGMA points lie in itss=1s=1M\-matrix domain\.

The proof uses the fact that every finite complex becomes an independence complex after barycentric subdivision\(Ehrenborg & Hetyei,[2006](https://arxiv.org/html/2608.08103#bib.bib9)\); the DAG contribution is an ordinary\-adjacency realization of that complex\.

### A\.3Metric consequences

Derivative order also controls how reliably a residual measures distance to the DAG set\. Write

𝒟F=\{x∈VF:𝐖\+∑sxs​𝐄s∈𝒟\}\.\\mathcal\{D\}\_\{F\}=\\left\\\{x\\in V\_\{F\}:\\mathbf\{W\}\+\\sum\_\{s\}x\_\{s\}\\mathbf\{E\}\_\{s\}\\in\\mathcal\{D\}\\right\\\}\.\(38\)A local Hölder error bound with exponentθ\>0\\theta\>0is an inequality

dist⁡\(x,𝒟F\)≤κ​‖RF​\(x\)‖θ\\operatorname\{dist\}\(x,\\mathcal\{D\}\_\{F\}\)\\leq\\kappa\\left\\lVert R\_\{F\}\(x\)\\right\\rVert^\{\\theta\}\(39\)for allxxnear zero and someκ\>0\\kappa\>0; this is also called Hölder metric subregularity\(Li & Mordukhovich,[2012](https://arxiv.org/html/2608.08103#bib.bib15)\)\.

###### Theorem A\.5\(Sharp topological error\-bound exponents\)\.

Supposeq𝐖​\(F\)<∞q\_\{\\mathbf\{W\}\}\(F\)<\\inftyand letr=r𝐖​\(F\)r=r\_\{\\mathbf\{W\}\}\(F\)\. For anyCrC^\{r\}signed/vector exact residualHFH\_\{F\}, every exponent in Eq\. \([39](https://arxiv.org/html/2608.08103#A1.E39)\) satisfiesθ≤1/r\\theta\\leq 1/r\. For anyC2​rC^\{2r\}nonnegative scalar exact residualhFh\_\{F\}, every exponent satisfiesθ≤1/\(2​r\)\\theta\\leq 1/\(2r\)\. Both bounds are attained: the simple\-cycle product vector has optimal exponent1/r1/r, while the positive walk constraints in Corollary[5\.4](https://arxiv.org/html/2608.08103#S5.Thmtheorem4)have optimal exponent1/\(2​r\)1/\(2r\)\. Consequently, squaring these sharp residuals gives optimal exponents1/\(2​r\)1/\(2r\)and1/\(4​r\)1/\(4r\)\.

###### Proof sketch\.

A largest minimal completion gives necessity alongt​𝟏Ct\\mathbf\{1\}\_\{C\}\. Conversely, zeroing one smallest coordinate from each minimal completion gives a hitting set anddist⁡\(x,𝒟F\)≤\|𝒞min\|​‖M𝐖,F​\(x\)‖21/r\\operatorname\{dist\}\(x,\\mathcal\{D\}\_\{F\}\)\\leq\\sqrt\{\|\\mathcal\{C\}\_\{\\rm min\}\|\}\\left\\lVert M\_\{\\mathbf\{W\},F\}\(x\)\\right\\rVert\_\{2\}^\{1/r\}\. Cycle products dominateM𝐖,FM\_\{\\mathbf\{W\},F\}and positive walk scalars dominate its squared norm; see Proposition[B\.4](https://arxiv.org/html/2608.08103#A2.Thmtheorem4)\. ∎

Two further invariants describe the size of neighborhoods rather than their worst direction\. Let

b𝐖​\(F\)\\displaystyle b\_\{\\mathbf\{W\}\}\(F\)=m−maxS∈Δ𝐖,F⁡\|S\|,\\displaystyle=m\-\\max\_\{S\\in\\Delta\_\{\\mathbf\{W\},F\}\}\|S\|,λ𝐖​\(F\)\\displaystyle\\lambda\_\{\\mathbf\{W\}\}\(F\)=minz≥0∑s∈Czs≥1,C∈𝒞min​∑s=1mzs\.\\displaystyle=\\min\_\{\\begin\{subarray\}\{c\}z\\geq 0\\\\ \\sum\_\{s\\in C\}z\_\{s\}\\geq 1,\\ C\\in\\mathcal\{C\}\_\{\\rm min\}\\end\{subarray\}\}\\sum\_\{s=1\}^\{m\}z\_\{s\}\.\(40\)Thusbbis the minimum integral completion cover \(the height ofI𝐖,FI\_\{\\mathbf\{W\},F\}\), whileλ\\lambdais its fractional relaxation\.

###### Theorem A\.6\(Completion\-volume geometry\)\.

For a sufficiently small boxBρ=\[−ρ,ρ\]mB\_\{\\rho\}=\[\-\\rho,\\rho\]^\{m\}and0≤ε≤ρ0\\leq\\varepsilon\\leq\\rho, the normalizedℓ∞\\ell\_\{\\infty\}tube volume is exactly

∑S∈Δ𝐖,F\(1−ε/ρ\)\|S\|​\(ε/ρ\)m−\|S\|;\\sum\_\{S\\in\\Delta\_\{\\mathbf\{W\},F\}\}\(1\-\\varepsilon/\\rho\)^\{\|S\|\}\(\\varepsilon/\\rho\)^\{m\-\|S\|\};\(41\)its small\-ε\\varepsilonexponent isbb\. If𝒱R​\(η\)=vol⁡\{x∈Bρ:‖R​\(x\)‖≤η\}\\mathcal\{V\}\_\{R\}\(\\eta\)=\\operatorname\{vol\}\\\{x\\in B\_\{\\rho\}:\\\|R\(x\)\\\|\\leq\\eta\\\}, then

limη↓0log⁡𝒱Hcyc​\(η\)log⁡η=λ,limη↓0log⁡𝒱hf​\(η\)log⁡η=λ2\\lim\_\{\\eta\\downarrow 0\}\\frac\{\\log\\mathcal\{V\}\_\{H\_\{\\rm cyc\}\}\(\\eta\)\}\{\\log\\eta\}=\\lambda,\\qquad\\lim\_\{\\eta\\downarrow 0\}\\frac\{\\log\\mathcal\{V\}\_\{h\_\{f\}\}\(\\eta\)\}\{\\log\\eta\}=\\frac\{\\lambda\}\{2\}\(42\)for the simple\-cycle vector and every locally convergent positive\-walk scalarhfh\_\{f\}\. Moreover,b/r≤λ≤bb/r\\leq\\lambda\\leq b\.

The logarithmic substitution\|xs\|=e−us\|x\_\{s\}\|=e^\{\-u\_\{s\}\}turns a generator sublevel set into the fractional\-cover inequalities∑s∈Cus≥log⁡\(1/η\)\\sum\_\{s\\in C\}u\_\{s\}\\geq\\log\(1/\\eta\)\. LP primal and dual solutions give matching exponential bounds; Section[B\.8](https://arxiv.org/html/2608.08103#A2.SS8)also proves the required residual comparisons\. The formula forλ\\lambdais the classical monomial\-ideal threshold\(Howald,[2001](https://arxiv.org/html/2608.08103#bib.bib11)\); its completion\-graph interpretation and separation fromq,r,bq,r,bare specific to this setting\.

Thus a smooth nonnegative exact scalar is never Lipschitz metrically subregular at such a support boundary, and a vector residual is not Lipschitz metrically subregular whenr≥2r\\geq 2\. It also makes finite smooth penalties non\-exact against first\-order score descent \(Corollary[C\.6](https://arxiv.org/html/2608.08103#A3.Thmtheorem6)\)\.

Smooth scalarization adds a second loss of sensitivity\. On paths where a vector residual isqq\-sharp and a nonnegative scalar residual has exact order2​q2q, the common squared\-penalty constructions obey

H⏟q⟶12​‖H‖2⏟2​qandh⏟2​q⟶12​h2⏟4​q\.\\underbrace\{H\}\_\{q\}\\quad\\longrightarrow\\quad\\underbrace\{\\tfrac\{1\}\{2\}\\left\\lVert H\\right\\rVert^\{2\}\}\_\{2q\}\\quad\\text\{and\}\\quad\\underbrace\{h\}\_\{2q\}\\quad\\longrightarrow\\quad\\underbrace\{\\tfrac\{1\}\{2\}h^\{2\}\}\_\{4q\}\.\(43\)The subscripts are exact orders under sharpness and lower bounds otherwise\. Increasing an augmented\-Lagrangian penalty changes coefficients, not these missing derivatives; a residual\-sized multiplier also leaves a cold start at the squared order\. Section[C\.7](https://arxiv.org/html/2608.08103#A3.SS7)gives the full sensitivity–smoothness frontier, ALM statements, and finite\-precision calculations\.

### A\.4Statistical consequences

The derivative orders in the main paper also determine the null scale of a residual evaluated at a noisy estimator\. For fixeddd, letFFcontain all coordinates absent from a fixed DAGWW, and suppose

n​\(W^n−W\)⇒Z\\sqrt\{n\}\(\\widehat\{W\}\_\{n\}\-W\)\\ \\Rightarrow\\ Z\(44\)for a nondegenerate Gaussian matrixZZ\. This assumption is deliberately modular: it can arise from randomized interventions, temporal information, or another model that identifies the directed parameter\.

###### Theorem A\.7\(Topology\-indexed delta law\)\.

Letq=qW​\(F\)<∞q=q\_\{W\}\(F\)<\\inftyand consider a locally convergent positive\-walk constraint

hf​\(A\)=∑k≥1ck​tr⁡\(\(A∘A\)k\),ck\>0\.h\_\{f\}\(A\)=\\sum\_\{k\\geq 1\}c\_\{k\}\\operatorname\{tr\}\(\(A\\circ A\)^\{k\}\),\\qquad c\_\{k\}\>0\.\(45\)Then

nq​hf​\(W^n\)⇒PW,q​\(ZF\),PW,q​\(z\)=∑C∈𝒞min​\(W,F\)\|C\|=qaC​∏e∈Cze2,n^\{q\}h\_\{f\}\(\\widehat\{W\}\_\{n\}\)\\Rightarrow P\_\{W,q\}\(Z\_\{F\}\),\\qquad P\_\{W,q\}\(z\)=\\sum\_\{\\begin\{subarray\}\{c\}C\\in\\mathcal\{C\}\_\{\\rm min\}\(W,F\)\\\\ \|C\|=q\\end\{subarray\}\}a\_\{C\}\\prod\_\{e\\in C\}z\_\{e\}^\{2\},\(46\)where everyaCa\_\{C\}is positive\. UnderWn=W\+U/nW\_\{n\}=W\+U/\\sqrt\{n\}, the limit isPW,q​\(ZF\+UF\)P\_\{W,q\}\(Z\_\{F\}\+U\_\{F\}\)\.

Theorem[A\.7](https://arxiv.org/html/2608.08103#A1.Thmtheorem7)is a higher\-order delta\-method calculation at a singular restriction, closely related to singular Wald asymptotics and inference for irregular polynomial constraints\(Drton & Xiao,[2016](https://arxiv.org/html/2608.08103#bib.bib7); Sturma et al\.,[2024](https://arxiv.org/html/2608.08103#bib.bib25)\)\. The completion ideal supplies the part that is specific to acyclicity: the degree, the monomials that survive, and their positive coefficients\. Fluctuations of the fixed active edges enter one order later\. Section[B\.9](https://arxiv.org/html/2608.08103#A2.SS9)gives the proof\.

###### Corollary A\.8\(No universal scalar tolerance\)\.

For a rule that accepts whenhf​\(W^n\)≤c​n−ah\_\{f\}\(\\widehat\{W\}\_\{n\}\)\\leq cn^\{\-a\},

Pr⁡\(accept\)⟶\{0,a\>q,Pr⁡\{PW,q​\(ZF\)≤c\},a=q,1,a<q\.\\Pr\(\\mathrm\{accept\}\)\\longrightarrow\\begin\{cases\}0,&a\>q,\\\\ \\Pr\\\{P\_\{W,q\}\(Z\_\{F\}\)\\leq c\\\},&a=q,\\\\ 1,&a<q\.\\end\{cases\}\(47\)Thus one exponentaacannot be nondegenerately calibrated at two DAG strata with different completion numbers\. Moreover, if one completion complex has minimal generators of sizesq<rq<r, the limit in Eq\. \([46](https://arxiv.org/html/2608.08103#A1.E46)\) retains only size\-qqgenerators\. A local shift on a disjoint size\-rrgenerator is invisible to the shortest\-order scalar limit\.

## Appendix BProofs of Completion\-Order Results

All derivatives below are taken in the Euclidean space𝒵=\{W∈ℝd×d:diag⁡\(W\)=0\}\\mathcal\{Z\}=\\\{W\\in\\mathbb\{R\}^\{d\\times d\}:\\operatorname\{diag\}\(W\)=0\\\}\. For a multi\-indexα\\alpha,supp⁡\(α\)=\{s:αs\>0\}\\operatorname\{supp\}\(\\alpha\)=\\\{s:\\alpha\_\{s\}\>0\\\}\.

### B\.1Coordinate\-subspace vanishing

We first record the elementary mechanism behind the vector result\.

###### Lemma B\.1\(Local coordinate\-subspace jet\)\.

LetV⊆ℝmV\\subseteq\\mathbb\{R\}^\{m\}be an open neighborhood of zero and letG:V→ℝrG:V\\to\\mathbb\{R\}^\{r\}beCkC^\{k\}\. Suppose that, for every coordinate setSSwith\|S\|≤k\|S\|\\leq k,GGis zero on a neighborhood of zero inV∩ℝSV\\cap\\mathbb\{R\}^\{S\}\. Then

Dα​G​\(0\)=0for every​\|α\|≤k\.D^\{\\alpha\}G\(0\)=0\\qquad\\text\{for every \}\|\\alpha\|\\leq k\.\(48\)

###### Proof\.

Fixα\\alphawith\|α\|≤k\|\\alpha\|\\leq kand setS=supp⁡\(α\)S=\\operatorname\{supp\}\(\\alpha\)\. Since\|S\|≤\|α\|≤k\|S\|\\leq\|\\alpha\|\\leq k, the local restrictionGSG\_\{S\}obtained by setting all coordinates outsideSSto zero is identically zero near the origin\. The derivativeDα​G​\(0\)D^\{\\alpha\}G\(0\)is the corresponding derivative ofGSG\_\{S\}and is therefore zero, component by component\. ∎

##### Proof of Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1)\.

For everyS⊆FS\\subseteq Fwith\|S\|<qW​\(F\)\|S\|<q\_\{W\}\(F\), the supportsupp⁡\(W\)∪S\\operatorname\{supp\}\(W\)\\cup Sis acyclic\. Exactness therefore makesHFH\_\{F\}zero on a neighborhood of the origin in the coordinate subspace indexed bySS\. Lemma[B\.1](https://arxiv.org/html/2608.08103#A2.Thmtheorem1), withk=q−1k=q\-1, proves Eq\. \([11](https://arxiv.org/html/2608.08103#S4.E11)\)\. IfHHisCqC^\{q\}, the multivariate Taylor formula with integral remainder gives, uniformly foruuin a compact set,

‖HF​\(t​u\)‖2≤C​\|t\|q​‖u‖2q\.\\\|H\_\{F\}\(tu\)\\\|\_\{2\}\\leq C\|t\|^\{q\}\\\|u\\\|\_\{2\}^\{q\}\.\(49\)Apply the same formula toD​HFDH\_\{F\}\. Its derivatives through orderq−2q\-2are derivatives ofHFH\_\{F\}through orderq−1q\-1and vanish at zero, giving

‖D​HF​\(t​u\)‖op≤C′​\|t\|q−1\.\\\|DH\_\{F\}\(tu\)\\\|\_\{\\rm op\}\\leq C^\{\\prime\}\|t\|^\{q\-1\}\.\(50\)This is the Jacobian of the restriction toFF\. A coordinate outsideFFmay close an existing path with fewer edges, so the same order need not hold for the full Jacobian\. ∎

The theorem gives a lower bound on the first possible nonzero order, not an equality\. Smooth nonanalytic exact maps may be flat to all orders, and signed components can cancel on a particular path\.

### B\.2Weighted paths

We prove both weighted statements by a clone\-variable lift, using the local coordinate\-subspace argument directly rather than treating clones as graph edges\. Letas∈ℕa\_\{s\}\\in\\mathbb\{N\}, introduce variableszs,1,…,zs,asz\_\{s,1\},\\ldots,z\_\{s,a\_\{s\}\}, and define

Γ​\(z\)=W\+∑s=1mcs​\(∏j=1aszs,j\)​Es\.\\Gamma\(z\)=W\+\\sum\_\{s=1\}^\{m\}c\_\{s\}\\left\(\\prod\_\{j=1\}^\{a\_\{s\}\}z\_\{s,j\}\\right\)E\_\{s\}\.\(51\)Original edgeese\_\{s\}is active exactly when all of its clones are nonzero\. Ifsupp⁡\(W\)\\operatorname\{supp\}\(W\)together with the active original edges is cyclic, the support ofzzcontains at least

minS:supp⁡\(W\)∪S​cyclic​∑es∈Sas=τW​\(F;a\)\\min\_\{S:\\operatorname\{supp\}\(W\)\\cup S\\text\{ cyclic\}\}\\sum\_\{e\_\{s\}\\in S\}a\_\{s\}=\\tau\_\{W\}\(F;a\)\(52\)clone coordinates\. Consequently, every clone\-coordinate subspace of dimension belowτ\\taumaps to DAG support\. Conversely, activating all clones of a minimizing setSSuses exactlyτ\\tauclone coordinates and maps to cyclic support\. Thusτ\\tauis exactly the coordinate\-subspace completion number of the lifted composite\.

LetVΓ=\{z:Γ​\(z\)∈Ω\}V\_\{\\Gamma\}=\\\{z:\\Gamma\(z\)\\in\\Omega\\\}, an open neighborhood of zero, and setG=H∘ΓG=H\\circ\\Gamma\. IfHHisCτ−1C^\{\\tau\-1\}, then so isGG, andGGis zero near the origin on every clone\-coordinate subspace of dimension belowτ\\tau\. Lemma[B\.1](https://arxiv.org/html/2608.08103#A2.Thmtheorem1)makes all derivatives ofGGbelow orderτ\\tauvanish\. IfHHisCτC^\{\\tau\}, Taylor’s formula also givesG​\(z\)=O​\(‖z‖τ\)G\(z\)=O\(\\\|z\\\|^\{\\tau\}\)\. On the diagonal clone pathzs,j=tz\_\{s,j\}=t, Eq\. \([51](https://arxiv.org/html/2608.08103#A2.E51)\) is preciselyWa​\(t\)W\_\{a\}\(t\), proving Corollary[4\.2](https://arxiv.org/html/2608.08103#S4.Thmtheorem2)\.

For the scalar statement setg=h∘Γg=h\\circ\\Gamma\. Assume first thathhisC2​τ−1C^\{2\\tau\-1\}\. The functionggis locally nonnegative and is zero on every clone\-coordinate subspace of dimension belowτ\\tau\. If its first nonzero homogeneous Taylor term had degreer<2​τr<2\\tau, the proof of Theorem[5\.2](https://arxiv.org/html/2608.08103#S5.Thmtheorem2)would apply verbatim: every monomial would use at leastτ\\tauclones, while a Newton vertex of a nonnegative polynomial has even exponents, forcingr≥2​τr\\geq 2\\tau\. Hence all derivatives below2​τ2\\tauvanish\. UnderC2​τC^\{2\\tau\}regularity, Taylor’s formula yields

h​\(Wa​\(t\)\)=O​\(\|t\|2​τ\)\.h\(W\_\{a\}\(t\)\)=O\(\|t\|^\{2\\tau\}\)\.\(53\)The same one\-sided asymptotic bounds for positive rational exponents follow after multiplying all exponents by a common denominator and reparameterizingt↓0t\\downarrow 0; derivative statements at zero require the resulting path to possess the corresponding regularity\.

### B\.3A sharp vector representation

We verify exactness and count the components of Eq\. \([14](https://arxiv.org/html/2608.08103#S4.E14)\)\.

###### Proposition B\.2\(Simple\-cycle vector\)\.

Hcyc​\(W\)=0H\_\{\\rm cyc\}\(W\)=0if and only ifsupp⁡\(W\)\\operatorname\{supp\}\(W\)is acyclic\. For a minimalqq\-edge completion, its restricted residual and Jacobian orders are exactlyqqandq−1q\-1, respectively\. Along a weighted path the residual order is exactlyτ\\tau, provided the coefficients on a cheapest completion are nonzero\.

###### Proof\.

Every finite directed graph is cyclic if and only if it contains a simple directed cycle\. The component indexed by such a cycle is nonzero exactly when all its edges are present, proving exactness\. A cycle created by a minimal completion must use allqqcompletion edges; otherwise a proper subset would already be cyclic\. Its component is a nonzero fixed\-edge coefficient times the product of thoseqqcandidate coordinates\. This has equal\-scale orderqq, and differentiating in a completion coordinate has orderq−1q\-1\. No component has lower order by Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1)\. The same argument chooses a cycle whose completion cost isτ\\tauon a weighted path\. ∎

There are\(dk\)\\binom\{d\}\{k\}choices of nodes for a length\-kkcycle\. Fixing the smallest selected node as the first node leaves\(k−1\)\!\(k\-1\)\!directed orders, each representing one cyclic\-rotation class\. Summing overk=2,…,dk=2,\\ldots,dproves Eq\. \([15](https://arxiv.org/html/2608.08103#S4.E15)\)\.

### B\.4Uniform sharpness and residual dimension

We prove Theorem[A\.1](https://arxiv.org/html/2608.08103#A1.Thmtheorem1)\. Let

Pq​\(U\)=1q\!​Dq​H​\(0\)​\[U,…,U\]P\_\{q\}\(U\)=\\frac\{1\}\{q\!\}D^\{q\}H\(0\)\[U,\\ldots,U\]\(54\)be the homogeneous degree\-qqTaylor jet ofHHat the empty graph\.

Fix3≤q≤d3\\leq q\\leq dand setn=d−q\+2n=d\-q\+2\. Partitionnnendpoint nodes into disjoint setsS,TS,Tof sizesa=⌊n/2⌋a=\\lfloor n/2\\rfloorandb=⌈n/2⌉b=\\lceil n/2\\rceil\. The remainingq−2q\-2nodes form singleton layersv1,…,vq−2v\_\{1\},\\ldots,v\_\{q\-2\}\. LetU​\(z\)U\(z\)contain all forward funnel edges

s→v1,v1→⋯→vq−2,vq−2→ts\\to v\_\{1\},\\quad v\_\{1\}\\to\\cdots\\to v\_\{q\-2\},\\quad v\_\{q\-2\}\\to t\(55\)with weight one fors∈S,t∈Ts\\in S,t\\in T, and let the reverse edget→st\\to shave weightzs​tz\_\{st\}\. The forward graph is acyclic\. Every nonzero reverse channel completes the simpleqq\-cycles→v1→⋯→vq−2→t→ss\\to v\_\{1\}\\to\\cdots\\to v\_\{q\-2\}\\to t\\to s; hencet​U​\(z\)tU\(z\)is cyclic for everyt≠0t\\neq 0andz≠0z\\neq 0\.

Any cyclic support in this funnel must contain one reverse edge and allq−1q\-1edges of its associated forward path\. A degree\-qqTaylor monomial with any other support is therefore acyclic and its coefficient vanishes by exactness on that coordinate subspace\. Total degreeqqalso forces exponent one on each cycle edge\. It follows that

Pq\(U\(z\)\)=∑s∈S,t∈Tvs​tzs​t=Lz,L:ℝa​b→ℝr\.P\_\{q\}\(U\(z\)\)=\\sum\_\{s\\in S,t\\in T\}v\_\{st\}z\_\{st\}=Lz,\\qquad L:\\mathbb\{R\}^\{ab\}\\to\\mathbb\{R\}^\{r\}\.\(56\)Directionwiseqq\-sharpness requiresL​z≠0Lz\\neq 0for everyz≠0z\\neq 0, soLLis injective and

r≥a​b=⌊\(d−q\+2\)24⌋\.r\\geq ab=\\left\\lfloor\\frac\{\(d\-q\+2\)^\{2\}\}\{4\}\\right\\rfloor\.\(57\)Conversely, ifr<a​br<ab, rank–nullity givesz≠0z\\neq 0withPq​\(U​\(z\)\)=0P\_\{q\}\(U\(z\)\)=0\. Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1)removes every lower jet, soH​\(t​U​\(z\)\)=o​\(\|t\|q\)H\(tU\(z\)\)=o\(\|t\|^\{q\}\)even though exactness requires it to be nonzero for all sufficiently smallt≠0t\\neq 0\.

It remains to prove the upper bound\. LetCq​\(W\)∈ℝa​bC\_\{q\}\(W\)\\in\\mathbb\{R\}^\{ab\}collect the cycle products associated with the selected funnel channels, and define

Hq\+​\(W\)=\(Cq​\(W\),hexp​\(W\)\)∈ℝa​b\+1\.H\_\{q\}^\{\+\}\(W\)=\\left\(C\_\{q\}\(W\),h\_\{\\exp\}\(W\)\\right\)\\in\\mathbb\{R\}^\{ab\+1\}\.\(58\)The EXP component is zero exactly on DAGs, soHq\+H\_\{q\}^\{\+\}is globally exact\. On the funnel family,Cq​\(t​U​\(z\)\)=tq​zC\_\{q\}\(tU\(z\)\)=t^\{q\}zup to fixed nonzero forward\-edge coefficients\. Hence its degree\-qqjet is nonzero for everyz≠0z\\neq 0, provingsq​\(Uq\)≤a​b\+1s\_\{q\}\(U\_\{q\}\)\\leq ab\+1\.

Forq=2q=2, label nodes by a total order\. LetU​\(z\)i​j=1U\(z\)\_\{ij\}=1for everyi<ji<j, and letU​\(z\)j​i=zi​jU\(z\)\_\{ji\}=z\_\{ij\}\. Any two directed edges form a cycle if and only if they are the two opposite orientations of one unordered pair\. Thus exactness eliminates every quadratic Taylor monomial exceptWi​j​Wj​iW\_\{ij\}W\_\{ji\}\. Consequently

P2​\(U​\(z\)\)=∑i<jvi​j​zi​j=L​z\.P\_\{2\}\(U\(z\)\)=\\sum\_\{i<j\}v\_\{ij\}z\_\{ij\}=Lz\.\(59\)Directionwise second\-order sharpness makesL:ℝ\(d2\)→ℝrL:\\mathbb\{R\}^\{\\binom\{d\}\{2\}\}\\to\\mathbb\{R\}^\{r\}injective, provingr≥\(d2\)r\\geq\\binom\{d\}\{2\}\. Conversely, ifr<\(d2\)r<\\binom\{d\}\{2\}, rank–nullity supplies a nonzerozzwithP2​\(U​\(z\)\)=0P\_\{2\}\(U\(z\)\)=0\. The support oft​U​\(z\)tU\(z\)contains a two\-cycle for everyt≠0t\\neq 0, so exactness still requiresH​\(t​U​\(z\)\)≠0H\(tU\(z\)\)\\neq 0; the first two Taylor jets vanish andH​\(t​U​\(z\)\)=o​\(t2\)H\(tU\(z\)\)=o\(t^\{2\}\)\. For the upper bound, collect the\(d2\)\\binom\{d\}\{2\}productsWi​j​Wj​iW\_\{ij\}W\_\{ji\}and appendhexp​\(W\)h\_\{\\exp\}\(W\)as in Eq\. \([58](https://arxiv.org/html/2608.08103#A2.E58)\)\. This map is globally exact and its quadratic jet on the pair\-channel family is the coefficient vectorzz\.

### B\.5Nonnegative scalar representations

We give a proof of the Newton\-polytope fact used in the main text and then prove the scalar barrier in full\.

###### Proof of Lemma[5\.1](https://arxiv.org/html/2608.08103#S5.Thmtheorem1)\.

LetP​\(x\)=∑βcβ​xβP\(x\)=\\sum\_\{\\beta\}c\_\{\\beta\}x^\{\\beta\}be a nonzero globally nonnegative polynomial and letα\\alphabe a vertex of its Newton polytope\. A vectorw∈ℝmw\\in\\mathbb\{R\}^\{m\}uniquely exposesα\\alpha; changing its sign if necessary, take⟨w,α⟩<⟨w,β⟩\\langle w,\\alpha\\rangle<\\langle w,\\beta\\ranglefor every other exponent\. For a sign vectors∈\{−1,1\}ms\\in\\\{\-1,1\\\}^\{m\}, evaluatePPon the monomial curvexi=si​twix\_\{i\}=s\_\{i\}t^\{w\_\{i\}\}\. Although coordinates withwi<0w\_\{i\}<0diverge ast↓0t\\downarrow 0, the polynomial remains nonnegative\. Unique exposure gives

t−⟨w,α⟩​P​\(s1​tw1,…,sm​twm\)⟶cα​sα\.t^\{\-\\langle w,\\alpha\\rangle\}P\(s\_\{1\}t^\{w\_\{1\}\},\\ldots,s\_\{m\}t^\{w\_\{m\}\}\)\\longrightarrow c\_\{\\alpha\}s^\{\\alpha\}\.\(60\)The limit is nonnegative for everyss\. If someαi\\alpha\_\{i\}were odd, flippingsis\_\{i\}would flip the nonzero limit\. Thusα\\alphais coordinatewise even, and takings=𝟏s=\\mathbf\{1\}givescα\>0c\_\{\\alpha\}\>0\. ∎

###### Proof of Theorem[5\.2](https://arxiv.org/html/2608.08103#S5.Thmtheorem2)\.

Suppose, for contradiction, that the Taylor jet ofhFh\_\{F\}has a first nonzero homogeneous termPrP\_\{r\}withr<2​qr<2q\. Local nonnegativity ofhFh\_\{F\}is sufficient to makePrP\_\{r\}globally nonnegative: for fixedxxandt↓0t\\downarrow 0,

t−r​hF​\(t​x\)⟶Pr​\(x\)≥0\.t^\{\-r\}h\_\{F\}\(tx\)\\longrightarrow P\_\{r\}\(x\)\\geq 0\.\(61\)Every coordinate subspace supported on fewer thanqqcandidates is acyclic, so the restriction ofhFh\_\{F\}, and hence ofPrP\_\{r\}, to that subspace is zero\. Therefore every monomial ofPrP\_\{r\}uses at leastqqdistinct coordinates\. Choose any vertex exponentα\\alphaof the Newton polytope ofPrP\_\{r\}\. Lemma[5\.1](https://arxiv.org/html/2608.08103#S5.Thmtheorem1)makes every nonzeroαs\\alpha\_\{s\}at least two\. Hence

r=\|α\|≥2​\|supp⁡\(α\)\|≥2​q,r=\|\\alpha\|\\geq 2\|\\operatorname\{supp\}\(\\alpha\)\|\\geq 2q,\(62\)a contradiction\. Thus all derivatives through order2​q−12q\-1vanish\. UnderC2​qC^\{2q\}regularity, Taylor remainder bounds givehF​\(t​u\)=O​\(\|t\|2​q\)h\_\{F\}\(tu\)=O\(\|t\|^\{2q\}\)\. Applying the same argument toD​hFDh\_\{F\}gives the restricted gradient orderO​\(\|t\|2​q−1\)O\(\|t\|^\{2q\-1\}\)\. ∎

Nonnegativity is essential for the factor of two\. Ford=2d=2, the signed scalarW12​W21W\_\{12\}W\_\{21\}is exact and has order two at the empty graph, whereas every smooth nonnegative scalar exact representation has order at least four there\.

### B\.6Minimal\-completion factorization

LetF=\{e1,…,eq\}F=\\\{e\_\{1\},\\ldots,e\_\{q\}\\\}itself be a minimal completion and writeΦ​\(x\)=h​\(W\+∑sxs​Es\)\\Phi\(x\)=h\(W\+\\sum\_\{s\}x\_\{s\}E\_\{s\}\)on a sufficiently small box contained inVFV\_\{F\}\. Ifxs=0x\_\{s\}=0, only a proper subset ofFFis active and exactness givesΦ​\(x\)=0\\Phi\(x\)=0\. For fixedx−sx\_\{\-s\}, the nonnegative differentiable functionu↦Φ​\(x−s,u\)u\\mapsto\\Phi\(x\_\{\-s\},u\)has a minimum at zero, so

Φ​\(x−s,0\)=0,∂sΦ​\(x−s,0\)=0\.\\Phi\(x\_\{\-s\},0\)=0,\\qquad\\partial\_\{s\}\\Phi\(x\_\{\-s\},0\)=0\.\(63\)
The integral form of the second\-order Hadamard lemma is

f​\(u,z\)=u2​∫01\(1−r\)​∂u​uf​\(r​u,z\)​d​rf\(u,z\)=u^\{2\}\\int\_\{0\}^\{1\}\(1\-r\)\\partial\_\{uu\}f\(ru,z\)\\,dr\(64\)wheneverf​\(0,z\)=∂uf​\(0,z\)=0f\(0,z\)=\\partial\_\{u\}f\(0,z\)=0\. We apply it inductively\. Afterk−1k\-1steps, write

Φ​\(x\)=\(∏s<kxs2\)​Φk−1​\(x\),\\Phi\(x\)=\\left\(\\prod\_\{s<k\}x\_\{s\}^\{2\}\\right\)\\Phi\_\{k\-1\}\(x\),\(65\)whereΦk−1\\Phi\_\{k\-1\}is nonnegative andC2​q−2​\(k−1\)C^\{2q\-2\(k\-1\)\}\. On the dense set where the preceding coordinates are nonzero,Φk−1​\(x\)=0\\Phi\_\{k\-1\}\(x\)=0wheneverxk=0x\_\{k\}=0; continuity extends this identity to the divided hyperplanes\. BecauseΦk−1\\Phi\_\{k\-1\}is nonnegative and differentiable, itsxkx\_\{k\}derivative also vanishes there\. Equation \([64](https://arxiv.org/html/2608.08103#A2.E64)\) extracts another factorxk2x\_\{k\}^\{2\}and leaves a nonnegativeC2​q−2​kC^\{2q\-2k\}quotient\. TheC2​qC^\{2q\}assumption permits allqqsteps and yields

Φ​\(x\)=\(∏s=1qxs2\)​Ψ​\(x\)\\Phi\(x\)=\\left\(\\prod\_\{s=1\}^\{q\}x\_\{s\}^\{2\}\\right\)\\Psi\(x\)\(66\)with continuousΨ\\Psi\. Off the coordinate hyperplanes the squared product is positive and exactness givesΦ\>0\\Phi\>0, soΨ\>0\\Psi\>0there; continuity givesΨ≥0\\Psi\\geq 0everywhere locally\.

###### Proposition B\.3\(Sharpness for nonnegative walk sums\)\.

LetA=W∘WA=W\\circ Wandhf​\(W\)=∑k≥1ck​tr⁡\(Ak\)h\_\{f\}\(W\)=\\sum\_\{k\\geq 1\}c\_\{k\}\\operatorname\{tr\}\(A^\{k\}\), whereck≥0c\_\{k\}\\geq 0and the coefficient for every simple\-cycle length is positive\. Suppose the series converges in a neighborhood of a minimalqq\-edge completion to aC2​qC^\{2q\}function\. Then

hf​\(W\+t​U\)=C​\|t\|2​q\+o​\(\|t\|2​q\),‖D​\(hf\)F​\(t​u\)‖2=Θ​\(\|t\|2​q−1\)h\_\{f\}\(W\+tU\)=C\|t\|^\{2q\}\+o\(\|t\|^\{2q\}\),\\qquad\\left\\lVert D\(h\_\{f\}\)\_\{F\}\(tu\)\\right\\rVert\_\{2\}=\\Theta\(\|t\|^\{2q\-1\}\)\(67\)for someC\>0C\>0\.

###### Proof\.

Choose a simple cycle in the completed support\. Minimality forces it to use allqqcandidate edges\. Its cyclic rotations contribute a positive constant times the product of theqqsquared candidate coordinates to the appropriate trace power\. All other walk\-sum terms are nonnegative, and the same expansion is zero exactly on DAG support\. Thushf​\(W\+t​U\)≥c​\|t\|2​qh\_\{f\}\(W\+tU\)\\geq c\|t\|^\{2q\}for somec\>0c\>0and all sufficiently small nonzerott\. Theorem[5\.2](https://arxiv.org/html/2608.08103#S5.Thmtheorem2)andC2​qC^\{2q\}regularity givehf​\(W\+t​U\)=C​\|t\|2​q\+o​\(\|t\|2​q\)h\_\{f\}\(W\+tU\)=C\|t\|^\{2q\}\+o\(\|t\|^\{2q\}\); the lower bound forcesC\>0C\>0\. LetP2​qP\_\{2q\}be the first homogeneous Taylor term, soP2​q​\(u\)=C\>0P\_\{2q\}\(u\)=C\>0\. The restricted gradient has leading termt2​q−1​∇P2​q​\(u\)t^\{2q\-1\}\\nabla P\_\{2q\}\(u\)\. Euler’s identity gives⟨∇P2​q​\(u\),u⟩=2​q​P2​q​\(u\)=2​q​C≠0\\langle\\nabla P\_\{2q\}\(u\),u\\rangle=2qP\_\{2q\}\(u\)=2qC\\neq 0, so the gradient norm is bounded below by a positive multiple of\|t\|2​q−1\|t\|^\{2q\-1\}\. Its Taylor expansion gives the matching upper bound\. ∎

This proposition covers EXP and the usual positive trace polynomial\. The same proof covers an entrywise squash satisfyings​\(0\)=0s\(0\)=0,s​\(x\)\>0s\(x\)\>0forx≠0x\\neq 0, ands​\(x\)=γ​x2\+o​\(x2\)s\(x\)=\\gamma x^\{2\}\+o\(x^\{2\}\)withγ\>0\\gamma\>0\. The log\-determinant series has the same positive walk expansion inside its convergence domain\.

### B\.7General completion complexes and exact error bounds

###### Proposition B\.4\(Generator domination and optimal exponents\)\.

Let𝒞min=𝒞min​\(W,F\)\\mathcal\{C\}\_\{\\rm min\}=\\mathcal\{C\}\_\{\\rm min\}\(W,F\)be nonempty,N=\|𝒞min\|N=\|\\mathcal\{C\}\_\{\\rm min\}\|, andr=maxC∈𝒞min⁡\|C\|r=\\max\_\{C\\in\\mathcal\{C\}\_\{\\rm min\}\}\|C\|\. On a sufficiently small candidate box,

dist⁡\(x,𝒟F\)≤N​‖MW,F​\(x\)‖21/r\.\\operatorname\{dist\}\(x,\\mathcal\{D\}\_\{F\}\)\\leq\\sqrt\{N\}\\,\\\|M\_\{W,F\}\(x\)\\\|\_\{2\}^\{1/r\}\.\(68\)Moreover, there are constantsc1,c2\>0c\_\{1\},c\_\{2\}\>0such that

‖\(Hcyc\)F​\(x\)‖2\\displaystyle\\\|\(H\_\{\\rm cyc\}\)\_\{F\}\(x\)\\\|\_\{2\}≥c1​‖MW,F​\(x\)‖2,\\displaystyle\\geq c\_\{1\}\\\|M\_\{W,F\}\(x\)\\\|\_\{2\},\(69\)hf​\(W\+∑sxs​Es\)\\displaystyle h\_\{f\}\\left\(W\+\\sum\_\{s\}x\_\{s\}E\_\{s\}\\right\)≥c2​‖MW,F​\(x\)‖22\\displaystyle\\geq c\_\{2\}\\\|M\_\{W,F\}\(x\)\\\|\_\{2\}^\{2\}\(70\)for every positive walk scalar in Proposition[B\.3](https://arxiv.org/html/2608.08103#A2.Thmtheorem3)\. Consequently, the two families have optimal error\-bound exponents1/r1/rand1/\(2​r\)1/\(2r\), respectively\.

###### Proof\.

For eachC∈𝒞minC\\in\\mathcal\{C\}\_\{\\rm min\}, choosei​\(C\)∈arg⁡mins∈C⁡\|xs\|i\(C\)\\in\\arg\\min\_\{s\\in C\}\|x\_\{s\}\|and set every selected coordinate to zero\. This deletion set meets every minimal nonface, so the remaining candidate support lies inΔW,F\\Delta\_\{W,F\}\. If‖x‖∞≤1\\\|x\\\|\_\{\\infty\}\\leq 1, the distance to this particular acyclic point satisfies

dist\(x,𝒟F\)2\\displaystyle\\operatorname\{dist\}\(x,\\mathcal\{D\}\_\{F\}\)^\{2\}≤∑C∈𝒞minmins∈C⁡\|xs\|2\\displaystyle\\leq\\sum\_\{C\\in\\mathcal\{C\}\_\{\\rm min\}\}\\min\_\{s\\in C\}\|x\_\{s\}\|^\{2\}\(71\)≤∑C∈𝒞min\|\(xC\)\|2/\|C\|≤N​‖MW,F​\(x\)‖22/r\.\\displaystyle\\leq\\sum\_\{C\\in\\mathcal\{C\}\_\{\\rm min\}\}\|\(x^\{C\}\)\|^\{2/\|C\|\}\\leq N\\\|M\_\{W,F\}\(x\)\\\|\_\{2\}^\{2/r\}\.\(72\)The middle inequality is the minimum–geometric\-mean inequality\. The last uses\|C\|≤r\|C\|\\leq rand\|xC\|≤1\|x^\{C\}\|\\leq 1, proving Eq\. \([68](https://arxiv.org/html/2608.08103#A2.E68)\)\.

For every minimal completionCC, choose a simple directed cycle insupp⁡\(W\)∪C\\operatorname\{supp\}\(W\)\\cup C\. Minimality forces this cycle to use every candidate inCC; its other edges have fixed nonzero weights inWW\. The corresponding component ofHcycH\_\{\\rm cyc\}is thereforebC​xCb\_\{C\}x^\{C\}withbC≠0b\_\{C\}\\neq 0\. Taking the minimum of finitely many\|bC\|\|b\_\{C\}\|proves Eq\. \([69](https://arxiv.org/html/2608.08103#A2.E69)\)\. The same selected cycles occur with positive coefficients and squared edge weights inhfh\_\{f\}, and all other walk terms are nonnegative\. Their finite minimum gives Eq\. \([70](https://arxiv.org/html/2608.08103#A2.E70)\)\.

Finally chooseC⋆∈𝒞minC\_\{\\star\}\\in\\mathcal\{C\}\_\{\\rm min\}with\|C⋆\|=r\|C\_\{\\star\}\|=rand setx=t​𝟏C⋆x=t\\mathbf\{1\}\_\{C\_\{\\star\}\}\. Every other minimal generator contains a coordinate outsideC⋆C\_\{\\star\}, while acyclicity requires at least one coordinate ofC⋆C\_\{\\star\}to be zero\. Hence

dist⁡\(x,𝒟F\)=\|t\|,‖MW,F​\(x\)‖2=\|t\|r\.\\operatorname\{dist\}\(x,\\mathcal\{D\}\_\{F\}\)=\|t\|,\\qquad\\\|M\_\{W,F\}\(x\)\\\|\_\{2\}=\|t\|^\{r\}\.\(73\)Proposition[B\.3](https://arxiv.org/html/2608.08103#A2.Thmtheorem3), restricted toC⋆C\_\{\\star\}, gives scalar order2​r2r\. Thus neither exponent can be increased\. ∎

### B\.8Coordinate tubes and residual sublevel volume

The completion ideal also controls two volume notions that are not determined byqqorrr\. Letm=\|F\|m=\|F\|and define

bW​\(F\)\\displaystyle b\_\{W\}\(F\)=m−maxS∈ΔW,F⁡\|S\|\\displaystyle=m\-\\max\_\{S\\in\\Delta\_\{W,F\}\}\|S\|=min⁡\{\|T\|:T∩C≠∅​for every​C∈𝒞min\},\\displaystyle=\\min\\\{\|T\|:T\\cap C\\neq\\varnothing\\text\{ for every \}C\\in\\mathcal\{C\}\_\{\\rm min\}\\\},\(74\)λW​\(F\)\\displaystyle\\lambda\_\{W\}\(F\)=minz∈ℝ\+m∑s∈Czs≥1,C∈𝒞min​∑s=1mzs\.\\displaystyle=\\min\_\{\\begin\{subarray\}\{c\}z\\in\\mathbb\{R\}\_\{\+\}^\{m\}\\\\ \\sum\_\{s\\in C\}z\_\{s\}\\geq 1,\\ C\\in\\mathcal\{C\}\_\{\\rm min\}\\end\{subarray\}\}\\sum\_\{s=1\}^\{m\}z\_\{s\}\.\(75\)The first quantity is the height ofIW,FI\_\{W,F\}and the minimum number of candidate edges whose removal from the full candidate graph makes it acyclic\. The second is the fractional cover number of the minimal\-completion hypergraph\. In the standard algebraic terminology, Eq\. \([75](https://arxiv.org/html/2608.08103#A2.E75)\) is the Newton\-polyhedron formula for the log\-canonical threshold of this monomial ideal\(Howald,[2001](https://arxiv.org/html/2608.08103#bib.bib11)\)\.

###### Proposition B\.5\(Exact coordinate\-tube law\)\.

Chooseρ\>0\\rho\>0such thatBρ=\[−ρ,ρ\]m⊂VFB\_\{\\rho\}=\[\-\\rho,\\rho\]^\{m\}\\subset V\_\{F\}\. For0≤ε≤ρ0\\leq\\varepsilon\\leq\\rho,

vol⁡\{x∈Bρ:dist∞⁡\(x,𝒟F\)≤ε\}\(2​ρ\)m\\displaystyle\\frac\{\\operatorname\{vol\}\\\{x\\in B\_\{\\rho\}:\\operatorname\{dist\}\_\{\\infty\}\(x,\\mathcal\{D\}\_\{F\}\)\\leq\\varepsilon\\\}\}\{\(2\\rho\)^\{m\}\}=∑S∈ΔW,F\(1−ερ\)\|S\|​\(ερ\)m−\|S\|\.\\displaystyle\\qquad=\\sum\_\{S\\in\\Delta\_\{W,F\}\}\\left\(1\-\\frac\{\\varepsilon\}\{\\rho\}\\right\)^\{\|S\|\}\\left\(\\frac\{\\varepsilon\}\{\\rho\}\\right\)^\{m\-\|S\|\}\.\(76\)In particular, its small\-ε\\varepsilonexponent isbW​\(F\)b\_\{W\}\(F\), and the leading coefficient after normalization by\(ε/ρ\)bW​\(F\)\(\\varepsilon/\\rho\)^\{b\_\{W\}\(F\)\}is the number of maximum\-cardinality faces ofΔW,F\\Delta\_\{W,F\}\.

###### Proof\.

Forx∈Bρx\\in B\_\{\\rho\}, setSε​\(x\)=\{s:\|xs\|\>ε\}S\_\{\\varepsilon\}\(x\)=\\\{s:\|x\_\{s\}\|\>\\varepsilon\\\}\. If somey∈𝒟Fy\\in\\mathcal\{D\}\_\{F\}satisfies‖x−y‖∞≤ε\\\|x\-y\\\|\_\{\\infty\}\\leq\\varepsilon, then every coordinate inSε​\(x\)S\_\{\\varepsilon\}\(x\)is nonzero inyy\. HenceSε​\(x\)⊆supp⁡\(y\)∈ΔW,FS\_\{\\varepsilon\}\(x\)\\subseteq\\operatorname\{supp\}\(y\)\\in\\Delta\_\{W,F\}and, by downward closure,Sε​\(x\)∈ΔW,FS\_\{\\varepsilon\}\(x\)\\in\\Delta\_\{W,F\}\. Conversely, ifSε​\(x\)S\_\{\\varepsilon\}\(x\)is a face, retaining those coordinates and setting all others to zero produces an element of𝒟F\\mathcal\{D\}\_\{F\}within distanceε\\varepsilon\.

For a uniform point inBρB\_\{\\rho\}, the probability thatSε​\(x\)=SS\_\{\\varepsilon\}\(x\)=Sis\(1−ε/ρ\)\|S\|​\(ε/ρ\)m−\|S\|\(1\-\\varepsilon/\\rho\)^\{\|S\|\}\(\\varepsilon/\\rho\)^\{m\-\|S\|\}\. The events are disjoint, so summing over the faces proves Eq\. \([76](https://arxiv.org/html/2608.08103#A2.E76)\)\. Its smallest power ofε\\varepsilonism−maxS∈ΔW,F⁡\|S\|=bW​\(F\)m\-\\max\_\{S\\in\\Delta\_\{W,F\}\}\|S\|=b\_\{W\}\(F\); all coefficients at that power are positive\. ∎

###### Theorem B\.6\(Residual tolerance\-volume exponent\)\.

LetM=MW,FM=M\_\{W,F\}be the minimal\-generator vector and

VM​\(η\)=vol⁡\{x∈Bρ:‖M​\(x\)‖∞≤η\}\.V\_\{M\}\(\\eta\)=\\operatorname\{vol\}\\\{x\\in B\_\{\\rho\}:\\\|M\(x\)\\\|\_\{\\infty\}\\leq\\eta\\\}\.Then

limη↓0log⁡VM​\(η\)log⁡η=λW​\(F\)\.\\lim\_\{\\eta\\downarrow 0\}\\frac\{\\log V\_\{M\}\(\\eta\)\}\{\\log\\eta\}=\\lambda\_\{W\}\(F\)\.\(77\)Moreover, on a sufficiently small candidate box there are positive constantsci,Cic\_\{i\},C\_\{i\}such that

c1​‖M​\(x\)‖∞\\displaystyle c\_\{1\}\\\|M\(x\)\\\|\_\{\\infty\}≤‖\(Hcyc\)F​\(x\)‖2≤C1​‖M​\(x\)‖∞,\\displaystyle\\leq\\\|\(H\_\{\\rm cyc\}\)\_\{F\}\(x\)\\\|\_\{2\}\\leq C\_\{1\}\\\|M\(x\)\\\|\_\{\\infty\},\(78\)c2​‖M​\(x\)‖∞2\\displaystyle c\_\{2\}\\\|M\(x\)\\\|\_\{\\infty\}^\{2\}≤hf​\(W\+∑sxs​Es\)≤C2​‖M​\(x\)‖∞2\\displaystyle\\leq h\_\{f\}\\left\(W\+\\sum\_\{s\}x\_\{s\}E\_\{s\}\\right\)\\leq C\_\{2\}\\\|M\(x\)\\\|\_\{\\infty\}^\{2\}\(79\)for every locally uniformly convergent positive\-walk scalar in Proposition[B\.3](https://arxiv.org/html/2608.08103#A2.Thmtheorem3)\. Consequently, the sublevel\-volume exponents of the cycle vector and positive\-walk scalar are respectivelyλW​\(F\)\\lambda\_\{W\}\(F\)andλW​\(F\)/2\\lambda\_\{W\}\(F\)/2\.

###### Proof\.

Fixed coordinate rescaling changes only constants, so first takeρ=1\\rho=1\. By symmetry it suffices to work on\[0,1\]m\[0,1\]^\{m\}\. Putxs=e−usx\_\{s\}=e^\{\-u\_\{s\}\}andL=log⁡\(1/η\)L=\\log\(1/\\eta\)\. The coordinatesusu\_\{s\}have independent unit\-exponential density, and‖M​\(x\)‖∞≤η\\\|M\(x\)\\\|\_\{\\infty\}\\leq\\etais equivalent to

u≥0,∑s∈Cus≥Lfor every​C∈𝒞min\.u\\geq 0,\\qquad\\sum\_\{s\\in C\}u\_\{s\}\\geq L\\quad\\text\{for every \}C\\in\\mathcal\{C\}\_\{\\rm min\}\.\(80\)Letz⋆z^\{\\star\}solve Eq\. \([75](https://arxiv.org/html/2608.08103#A2.E75)\)\. The orthant eventus≥L​zs⋆u\_\{s\}\\geq Lz^\{\\star\}\_\{s\}for allssis contained in Eq\. \([80](https://arxiv.org/html/2608.08103#A2.E80)\) and has probabilitye−L​∑szs⋆=e−λ​Le^\{\-L\\sum\_\{s\}z^\{\\star\}\_\{s\}\}=e^\{\-\\lambda L\}\.

For the reverse bound, linear\-programming duality givesyC≥0y\_\{C\}\\geq 0satisfying

∑CyC=λ,∑C∋syC≤1for every​s\.\\sum\_\{C\}y\_\{C\}=\\lambda,\\qquad\\sum\_\{C\\ni s\}y\_\{C\}\\leq 1\\quad\\text\{for every \}s\.\(81\)On the event in Eq\. \([80](https://arxiv.org/html/2608.08103#A2.E80)\),

∑sus≥∑sus​∑C∋syC=∑CyC​∑s∈Cus≥λ​L\.\\sum\_\{s\}u\_\{s\}\\geq\\sum\_\{s\}u\_\{s\}\\sum\_\{C\\ni s\}y\_\{C\}=\\sum\_\{C\}y\_\{C\}\\sum\_\{s\\in C\}u\_\{s\}\\geq\\lambda L\.\(The first inequality usesus≥0u\_\{s\}\\geq 0\.\) Since∑sus\\sum\_\{s\}u\_\{s\}has a Gamma\(m,1\)\(m,1\)distribution,

e−λ​L≤Pr⁡\{‖M​\(x\)‖∞≤e−L\}≤e−λ​L​∑j=0m−1\(λ​L\)jj\!\.e^\{\-\\lambda L\}\\leq\\Pr\\\{\\\|M\(x\)\\\|\_\{\\infty\}\\leq e^\{\-L\}\\\}\\leq e^\{\-\\lambda L\}\\sum\_\{j=0\}^\{m\-1\}\\frac\{\(\\lambda L\)^\{j\}\}\{j\!\}\.\(82\)Orthant factors and the fixed radiusρ\\rhodo not affect the logarithmic limit\. Dividing logarithms bylog⁡η=−L\\log\\eta=\-Lproves Eq\. \([77](https://arxiv.org/html/2608.08103#A2.E77)\)\.

The lower inequalities in Eqs\. \([78](https://arxiv.org/html/2608.08103#A2.E78)\) and \([79](https://arxiv.org/html/2608.08103#A2.E79)\) were proved in Proposition[B\.4](https://arxiv.org/html/2608.08103#A2.Thmtheorem4)\. For the cycle\-vector upper bound, every nonzero restricted simple\-cycle product contains a minimal completionCCamong its candidate edges\. On a box with‖x‖∞≤1\\\|x\\\|\_\{\\infty\}\\leq 1, its absolute value is at most a fixed base\-edge coefficient times\|xC\|\|x^\{C\}\|\. There are finitely many simple cycles, which givesC1C\_\{1\}\.

For the scalar upper bound, every nonzero closed\-walk monomial likewise contains a minimal completion and is bounded by a fixed coefficient times\|xC\|2\|x^\{C\}\|^\{2\}\. More explicitly, choose a slightly larger candidate polydisc whose closure remains in the convergence domain and assign each walk to one minimal completion dividing its monomial\. Absolute convergence on the larger polydisc implies absolute convergence of the finitely many termwise\-divided series on the smaller box\. Their suprema sum to a finiteC2C\_\{2\}\. The two\-sided comparisons sandwich each residual sublevel set between generator sublevel sets with constant\-rescaled tolerances\. Equation \([77](https://arxiv.org/html/2608.08103#A2.E77)\) then yieldsλ\\lambdafor the vector andλ/2\\lambda/2for the scalar\. ∎

###### Corollary B\.8\(Four scales of the completion ideal\)\.

The invariants satisfy

bW​\(F\)rW​\(F\)≤λW​\(F\)≤bW​\(F\)\.\\frac\{b\_\{W\}\(F\)\}\{r\_\{W\}\(F\)\}\\leq\\lambda\_\{W\}\(F\)\\leq b\_\{W\}\(F\)\.\(83\)Hereqqis the first possible jet degree,rrcontrols the worst Hölder error\-bound exponent,bbis the coordinate\-tube codimension, andλ\\lambdais the residual tolerance\-volume exponent\.

###### Proof\.

The indicator of an integral completion cover is feasible in Eq\. \([75](https://arxiv.org/html/2608.08103#A2.E75)\), soλ≤b\\lambda\\leq b\. Conversely, given any feasiblezz, the setT=\{s:zs≥1/r\}T=\\\{s:z\_\{s\}\\geq 1/r\\\}meets every minimal completion: otherwise someCCof size at mostrrwould satisfy∑s∈Czs<1\\sum\_\{s\\in C\}z\_\{s\}<1\. Thusb≤\|T\|≤r​∑szsb\\leq\|T\|\\leq r\\sum\_\{s\}z\_\{s\}\. Minimize overzz\. ∎

### B\.9Statistical desingularization

##### Proof of Theorem[A\.7](https://arxiv.org/html/2608.08103#A1.Thmtheorem7)\.

WriteA=W∘WA=W\\circ W\. Every term oftr⁡\(Ak\)\\operatorname\{tr\}\(A^\{k\}\)is a positive closed\-walk monomial\. After restricting to the absent coordinatesFF, its candidate degree is twice the number of candidate edges traversed with multiplicity\. Theorem[3\.3](https://arxiv.org/html/2608.08103#S3.Thmtheorem3)rules out candidate degree below2​q2q\. At degree2​q2q, the candidate support of a closed walk contains a cycle completion of size at leastqq, so it has exactlyqqdistinct candidate edges and traverses each once\. Removing any one of them breaks all cycles in that support; otherwise a smaller completion would occur\. Its candidate factor is therefore∏e∈Cxe2\\prod\_\{e\\in C\}x\_\{e\}^\{2\}for a size\-qqminimal completionCC\.

Conversely, every minimal completionCCcontains a simple directed cycle insupp⁡\(W\)∪C\\operatorname\{supp\}\(W\)\\cup C\. The cycle uses every edge ofCC, since otherwise a proper subset would complete a cycle\. Its fixed\-edge product is nonzero, and the coefficientckc\_\{k\}at its length is positive\. Summing all degree\-2​q2qclosed walks therefore gives

hf,F​\(x\)=∑C∈𝒞min​\(W,F\)\|C\|=qaC​∏e∈Cxe2\+o​\(‖x‖2​q\),aC\>0\.h\_\{f,F\}\(x\)=\\sum\_\{\\begin\{subarray\}\{c\}C\\in\\mathcal\{C\}\_\{\\rm min\}\(W,F\)\\\\ \|C\|=q\\end\{subarray\}\}a\_\{C\}\\prod\_\{e\\in C\}x\_\{e\}^\{2\}\+o\(\\\|x\\\|^\{2q\}\),\\qquad a\_\{C\}\>0\.\(84\)LetZn=n​\(W^n−W\)Z\_\{n\}=\\sqrt\{n\}\(\\widehat\{W\}\_\{n\}\-W\)\. Substitution into Eq\. \([84](https://arxiv.org/html/2608.08103#A2.E84)\) yields

nq​hf​\(W^n\)=PW,q​\(\(Zn\)F\)\+op​\(1\)\.n^\{q\}h\_\{f\}\(\\widehat\{W\}\_\{n\}\)=P\_\{W,q\}\(\(Z\_\{n\}\)\_\{F\}\)\+o\_\{p\}\(1\)\.\(85\)The continuous mapping theorem gives Eq\. \([46](https://arxiv.org/html/2608.08103#A1.E46)\)\. The fluctuations on fixed active edges perturb the coefficientsaCa\_\{C\}byop​\(1\)o\_\{p\}\(1\)and hence disappear by Slutsky’s theorem\. Under the local sequence,n​\(W^n−W\)=Zn\+U\+op​\(1\)\\sqrt\{n\}\(\\widehat\{W\}\_\{n\}\-W\)=Z\_\{n\}\+U\+o\_\{p\}\(1\), proving the shifted claim\.

##### Proof of Corollary[A\.8](https://arxiv.org/html/2608.08103#A1.Thmtheorem8)\.

The acceptance event is

nq​hf​\(W^n\)≤c​nq−a\.n^\{q\}h\_\{f\}\(\\widehat\{W\}\_\{n\}\)\\leq cn^\{q\-a\}\.\(86\)The right side tends to zero,cc, or infinity according asa\>qa\>q,a=qa=q, ora<qa<q\. The limiting polynomial is positive almost surely because its coefficients are positive and a nondegenerate Gaussian has no zero coordinate\. This proves Eq\. \([47](https://arxiv.org/html/2608.08103#A1.E47)\)\. The degree\-2​q2qinitial form contains only size\-qqgenerators\. A local shift supported on a disjoint size\-rrgenerator withr\>qr\>qleaves this initial form unchanged, which proves the final statement\. Disjointness is sufficient, not necessary; without it, a shifted coordinate may also belong to a shortest generator\.

## Appendix CSymmetry and the Score–Topology Crossover

### C\.1Coordinate realization and homology

##### Proof of Theorem[A\.2](https://arxiv.org/html/2608.08103#A1.Thmtheorem2)\.

LetL=sd⁡KL=\\operatorname\{sd\}K\. Its vertices are the nonempty faces ofKK, and its simplices are chains under inclusion\. In particular,LLis flag: a set of vertices is a face exactly when every pair is comparable\. LetHHbe the graph whose vertices are those ofLL, with an edge between two incomparable faces\. Then

Ind⁡\(H\)=L\.\\operatorname\{Ind\}\(H\)=L\.\(87\)
For eachi∈V​\(H\)i\\in V\(H\), create graph nodesai,bia\_\{i\},b\_\{i\}and candidate edgeei:ai→bie\_\{i\}:a\_\{i\}\\to b\_\{i\}\. For every\{i,j\}∈E​\(H\)\\\{i,j\\\}\\in E\(H\), add fixed edgesbi→ajb\_\{i\}\\to a\_\{j\}andbj→aib\_\{j\}\\to a\_\{i\}\. Every fixed edge goes from thebb\-layer to theaa\-layer, so the fixed graphW0W\_\{0\}is acyclic\.

Ifei,eje\_\{i\},e\_\{j\}are active for\{i,j\}∈E​\(H\)\\\{i,j\\\}\\in E\(H\), then

ai→bi→aj→bj→aia\_\{i\}\\to b\_\{i\}\\to a\_\{j\}\\to b\_\{j\}\\to a\_\{i\}\(88\)is a directed cycle\. Conversely, every directed cycle must alternate between a candidate edgeai→bia\_\{i\}\\to b\_\{i\}and a fixed edgebi→ajb\_\{i\}\\to a\_\{j\}\. Any such fixed edge certifies\{i,j\}∈E​\(H\)\\\{i,j\\\}\\in E\(H\), while the cycle certifies that both candidates are active\. Thus an active candidate set is acyclic if and only if it is independent inHH\. Equation \([87](https://arxiv.org/html/2608.08103#A3.E87)\) provesΔW0,F≅sd⁡K\\Delta\_\{W\_\{0\},F\}\\cong\\operatorname\{sd\}K\. Radial projection identifies the positive normalized coordinate faces with the usual geometric realization, and barycentric subdivision preserves that realization\.

##### Proof of Corollary[A\.3](https://arxiv.org/html/2608.08103#A1.Thmtheorem3)\.

LetHt=⨆b=1tK5H\_\{t\}=\\bigsqcup\_\{b=1\}^\{t\}K\_\{5\}and apply the preceding construction\. There are5​t5tcandidates and10​t10tgraph nodes\. Independence complexes turn disjoint graph unions into simplicial joins:

Ind\(Ht\)=Ind\(K5\)∗t\.\\operatorname\{Ind\}\(H\_\{t\}\)=\\operatorname\{Ind\}\(K\_\{5\}\)^\{\*t\}\.\(89\)The complexInd⁡\(K5\)\\operatorname\{Ind\}\(K\_\{5\}\)is a set of five points, so its only nonzero reduced homology has rank four in degree zero\. The reduced join formula

H~r​\(A∗B\)≅⨁i\+j=r−1H~i​\(A\)⊗H~j​\(B\)\\widetilde\{H\}\_\{r\}\(A\*B\)\\cong\\bigoplus\_\{i\+j=r\-1\}\\widetilde\{H\}\_\{i\}\(A\)\\otimes\\widetilde\{H\}\_\{j\}\(B\)\(90\)shows inductively that thett\-fold join has reduced homology of rank4t4^\{t\}in degreet−1t\-1and zero in all other degrees\. Equivalently it is homotopy equivalent to a wedge of4t4^\{t\}copies ofSt−1S^\{t\-1\}\.

### C\.2Far\-field symmetry and invariant flows

##### Proof of Theorem[6\.1](https://arxiv.org/html/2608.08103#S6.Thmtheorem1)\.

Usettdisjoint coordinate gadgets forKmK\_\{m\}\. Write the candidate restriction as

g​\(x\)=h​\(W0\+∑b=1t∑i=1mxb​i​Eb​i\)\.g\(x\)=h\\\!\\left\(W\_\{0\}\+\\sum\_\{b=1\}^\{t\}\\sum\_\{i=1\}^\{m\}x\_\{bi\}E\_\{bi\}\\right\)\.\(91\)For each block choose a pairSb=\{ib,jb\}S\_\{b\}=\\\{i\_\{b\},j\_\{b\}\\\}and define

ℳS=\{x:xb​ib=xb​jb=ab\>0,xb​ℓ=0​for​ℓ∉Sb,b∈\[t\]\}\.\\mathcal\{M\}\_\{S\}=\\left\\\{x:\\ x\_\{bi\_\{b\}\}=x\_\{bj\_\{b\}\}=a\_\{b\}\>0,\\ x\_\{b\\ell\}=0\\ \\text\{for \}\\ell\\notin S\_\{b\},\\ b\\in\[t\]\\right\\\}\.\(92\)Each block contains the four\-cycle in Eq\. \([88](https://arxiv.org/html/2608.08103#A3.E88)\), hence every point ofℳS\\mathcal\{M\}\_\{S\}is cyclic\. The theorem’s domain assumption ensures that the restriction and its derivatives are defined on a neighborhood of every such manifold\.

Becauseh​\(W\)=Φ​\(W∘W\)h\(W\)=\\Phi\(W\\circ W\),ggis even in every candidate coordinate\. Therefore

∂b​ℓg​\(x\)=0\(ℓ∉Sb,x∈ℳS\)\.\\partial\_\{b\\ell\}g\(x\)=0\\qquad\(\\ell\\notin S\_\{b\},\\ x\\in\\mathcal\{M\}\_\{S\}\)\.\(93\)The node permutation exchanging simultaneouslyaib↔ajba\_\{i\_\{b\}\}\\leftrightarrow a\_\{j\_\{b\}\}andbib↔bjbb\_\{i\_\{b\}\}\\leftrightarrow b\_\{j\_\{b\}\}is an automorphism of theKmK\_\{m\}gadget\. It fixes every point ofℳS\\mathcal\{M\}\_\{S\}\. Relabeling invariance ofhhtherefore gives

∂b​ibg​\(x\)=∂b​jbg​\(x\)\.\\partial\_\{bi\_\{b\}\}g\(x\)=\\partial\_\{bj\_\{b\}\}g\(x\)\.\(94\)Equations \([93](https://arxiv.org/html/2608.08103#A3.E93)\)–\([94](https://arxiv.org/html/2608.08103#A3.E94)\) say exactly that the block gradient is proportional toxbx\_\{b\}\. There are\(m2\)\\binom\{m\}\{2\}choices in each block, and distinct choices have different algebraic supports, proving Eq\. \([21](https://arxiv.org/html/2608.08103#S6.E21)\)\.

Now suppose∇g\\nabla gand the scalar functionω\\omegaare locally Lipschitz\. The vector field−ω​∇g\-\\omega\\nabla gsatisfies the same inactive\-zero and active\-tie identities, hence is tangent toℳS\\mathcal\{M\}\_\{S\}\. On that manifold the common amplitudes solve a locally Lipschitz system

a˙b=Fb​\(a\),Fb​\(0,a−b\)=0\.\\dot\{a\}\_\{b\}=F\_\{b\}\(a\),\\qquad F\_\{b\}\(0,a\_\{\-b\}\)=0\.\(95\)On any compact time interval there isL<∞L<\\inftywith\|Fb​\(a\)\|≤L​\|ab\|\|F\_\{b\}\(a\)\|\\leq L\|a\_\{b\}\|\. Gronwall’s inequality yields

ab​\(t\)≥ab​\(0\)​e−L​t\>0a\_\{b\}\(t\)\\geq a\_\{b\}\(0\)e^\{\-Lt\}\>0\(96\)wheneverab​\(0\)\>0a\_\{b\}\(0\)\>0, on every compact subinterval of the solution’s maximal interval of existence\. No active coefficient reaches zero at a finite time in that interval, and every block retains its four\-cycle while the solution exists\.

Form=5m=5,d=10​td=10tand the number of manifolds is10t10^\{t\}\. Atab=1/2a\_\{b\}=1/2, all active coefficients are fixed away from zero\. ForA=W∘WA=W\\circ W, the restriction ofA2A^\{2\}to the two activeaa\-nodes has eigenvalues±ab2\\pm a\_\{b\}^\{2\}\. Henceρ​\(A\)2=ρ​\(A2\)=ab2\\rho\(A\)^\{2\}=\\rho\(A^\{2\}\)=a\_\{b\}^\{2\}, soρ​\(A\)=ab=1/2<1\\rho\(A\)=a\_\{b\}=1/2<1, strictly inside DAGMA’ss=1s=1domain\.

##### Proof of Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)\.

Putp=∏i=1Lzi2p=\\prod\_\{i=1\}^\{L\}z\_\{i\}^\{2\}\. On the positive orthant,

∂zih=2​p​ϕ′​\(p\)/zi,z˙i=−2​Ψ′​\(ϕ​\(p\)\)​p​ϕ′​\(p\)/zi\.\\partial\_\{z\_\{i\}\}h=2p\\phi^\{\\prime\}\(p\)/z\_\{i\},\\qquad\\dot\{z\}\_\{i\}=\-2\\Psi^\{\\prime\}\(\\phi\(p\)\)p\\phi^\{\\prime\}\(p\)/z\_\{i\}\.\(97\)It follows immediately that

dd​t​zi2=−4​Ψ′​\(ϕ​\(p\)\)​p​ϕ′​\(p\)\\frac\{d\}\{dt\}z\_\{i\}^\{2\}=\-4\\Psi^\{\\prime\}\(\\phi\(p\)\)p\\phi^\{\\prime\}\(p\)\(98\)is independent ofii\. Hence everyzi2−zj2z\_\{i\}^\{2\}\-z\_\{j\}^\{2\}is conserved\. In particular,Δ=x2−y2=4​a​ε\\Delta=x^\{2\}\-y^\{2\}=4a\\varepsilon\. Conservation of squared\-norm differences is a familiar balancing mechanism in homogeneous gradient flows\(Du et al\.,[2018](https://arxiv.org/html/2608.08103#bib.bib8)\); the calculation below uses the exact cycle product to obtain the sharp DAG\-selection time\.

Writeu=y2u=y^\{2\}and

dj,ε=bj2−\(a−ε\)2,pε​\(u\)=u​\(u\+Δ\)​∏j=1L−2\(u\+dj,ε\)\.d\_\{j,\\varepsilon\}=b\_\{j\}^\{2\}\-\(a\-\\varepsilon\)^\{2\},\\qquad p\_\{\\varepsilon\}\(u\)=u\(u\+\\Delta\)\\prod\_\{j=1\}^\{L\-2\}\(u\+d\_\{j,\\varepsilon\}\)\.Equation \([98](https://arxiv.org/html/2608.08103#A3.E98)\) becomes

u˙=−4​Ψ′​\(ϕ​\(pε​\(u\)\)\)​pε​\(u\)​ϕ′​\(pε​\(u\)\)\.\\dot\{u\}=\-4\\Psi^\{\\prime\}\(\\phi\(p\_\{\\varepsilon\}\(u\)\)\)p\_\{\\varepsilon\}\(u\)\\phi^\{\\prime\}\(p\_\{\\varepsilon\}\(u\)\)\.\(99\)The eventy/x=ry/x=roccurs atur=r2​Δ/\(1−r2\)u\_\{r\}=r^\{2\}\\Delta/\(1\-r^\{2\}\)\. Withu=Δ​vu=\\Delta v, separation of variables gives

Tr​\(ε\)=Δ4​∫vr\(a−ε\)2/Δd​vΨ′​\(ϕ​\(pε​\(Δ​v\)\)\)​pε​\(Δ​v\)​ϕ′​\(pε​\(Δ​v\)\),vr=r21−r2\.\\begin\{split\}T\_\{r\}\(\\varepsilon\)=\\frac\{\\Delta\}\{4\}\\int\_\{v\_\{r\}\}^\{\(a\-\\varepsilon\)^\{2\}/\\Delta\}\\frac\{dv\}\{\\Psi^\{\\prime\}\(\\phi\(p\_\{\\varepsilon\}\(\\Delta v\)\)\)p\_\{\\varepsilon\}\(\\Delta v\)\\phi^\{\\prime\}\(p\_\{\\varepsilon\}\(\\Delta v\)\)\},\\\\ v\_\{r\}=\\frac\{r^\{2\}\}\{1\-r^\{2\}\}\.\\end\{split\}\(100\)For fixedvv,

pε​\(Δ​v\)=Δ2​v​\(v\+1\)​∏j=1L−2\(Δ​v\+dj,ε\),dj,ε⟶bj2−a2\.p\_\{\\varepsilon\}\(\\Delta v\)=\\Delta^\{2\}v\(v\+1\)\\prod\_\{j=1\}^\{L\-2\}\(\\Delta v\+d\_\{j,\\varepsilon\}\),\\qquad d\_\{j,\\varepsilon\}\\longrightarrow b\_\{j\}^\{2\}\-a^\{2\}\.Letα=ϕ′​\(0\)\\alpha=\\phi^\{\\prime\}\(0\)andC=∏j\(bj2−a2\)C=\\prod\_\{j\}\(b\_\{j\}^\{2\}\-a^\{2\}\)\. The assumptions onΨ′\\Psi^\{\\prime\}andϕ′\\phi^\{\\prime\}imply

Ψ′​\(ϕ​\(p\)\)​p​ϕ′​\(p\)=c​αν\+1​pν\+1​\(1\+o​\(1\)\)\.\\Psi^\{\\prime\}\(\\phi\(p\)\)p\\phi^\{\\prime\}\(p\)=c\\alpha^\{\\nu\+1\}p^\{\\nu\+1\}\(1\+o\(1\)\)\.After multiplying Eq\. \([100](https://arxiv.org/html/2608.08103#A3.E100)\) byΔ2​ν\+1\\Delta^\{2\\nu\+1\}, its integrand therefore converges pointwise to

14​c​\(α​C\)ν\+1​\[v​\(v\+1\)\]ν\+1\.\\frac\{1\}\{4c\(\\alpha C\)^\{\\nu\+1\}\[v\(v\+1\)\]^\{\\nu\+1\}\}\.
For completeness, this limit does not interchange an uncontrolled moving endpoint\. Choose a small fixedu0\>0u\_\{0\}\>0\. Onu∈\[0,u0\]u\\in\[0,u\_\{0\}\], the asymptotic relations above hold with two\-sided uniform bounds, and the normalized integrand is dominated by a constant multiple of\[v​\(v\+1\)\]−\(ν\+1\)\[v\(v\+1\)\]^\{\-\(\\nu\+1\)\}, which is integrable on\[vr,∞\)\[v\_\{r\},\\infty\)\. Onu∈\[u0,\(a−ε\)2\]u\\in\[u\_\{0\},\(a\-\\varepsilon\)^\{2\}\], all factors in Eq\. \([99](https://arxiv.org/html/2608.08103#A3.E99)\) are bounded away from zero, so that part of the unnormalized hitting time isO​\(1\)O\(1\)and vanishes after multiplication byΔ2​ν\+1\\Delta^\{2\\nu\+1\}\. Dominated convergence now yields

limε↓0Δ2​ν\+1​Tr​\(ε\)=Iν​\(r\)4​c​\[α​C\]ν\+1\.\\lim\_\{\\varepsilon\\downarrow 0\}\\Delta^\{2\\nu\+1\}T\_\{r\}\(\\varepsilon\)=\\frac\{I\_\{\\nu\}\(r\)\}\{4c\[\\alpha C\]^\{\\nu\+1\}\}\.Substitution ofΔ=4​a​ε\\Delta=4a\\varepsilonproves Eq\. \([23](https://arxiv.org/html/2608.08103#S6.E23)\)\.

For the score statement, the full flow satisfies

dd​t​log⁡yx=−\(∂y\[S\+Ψ​\(h\)\]y−∂x\[S\+Ψ​\(h\)\]x\)=−msc​\(z\)\.\\frac\{d\}\{dt\}\\log\\frac\{y\}\{x\}=\-\\left\(\\frac\{\\partial\_\{y\}\[S\+\\Psi\(h\)\]\}\{y\}\-\\frac\{\\partial\_\{x\}\[S\+\\Psi\(h\)\]\}\{x\}\\right\)=\-m\_\{\\rm sc\}\(z\)\.\(101\)Ifmsc​\(z\)≥γflowm\_\{\\rm sc\}\(z\)\\geq\\gamma\_\{\\rm flow\}until the stopping event, integration gives Eq\. \([24](https://arxiv.org/html/2608.08103#S6.E24)\)\. This argument does not assume that the preferred edge is initially smaller\.

It remains to prove the crossover claim forSγ​\(z\)=γ​y2/2S\_\{\\gamma\}\(z\)=\\gamma y^\{2\}/2\. Put

A​\(p\)=Ψ′​\(ϕ​\(p\)\)​p​ϕ′​\(p\)\.A\(p\)=\\Psi^\{\\prime\}\(\\phi\(p\)\)p\\phi^\{\\prime\}\(p\)\.The squared\-weight equations become

dd​t​x2=−4​A​\(p\),dd​t​y2=−4​A​\(p\)−2​γ​y2,\\frac\{d\}\{dt\}x^\{2\}=\-4A\(p\),\\qquad\\frac\{d\}\{dt\}y^\{2\}=\-4A\(p\)\-2\\gamma y^\{2\},\(102\)while every unpenalized cycle edge has the same squared speed asxx\. Consequently,

dd​t​log⁡yx=−γ−2​A​\(p\)​\(1y2−1x2\)≤−γ\.\\frac\{d\}\{dt\}\\log\\frac\{y\}\{x\}=\-\\gamma\-2A\(p\)\\left\(\\frac\{1\}\{y^\{2\}\}\-\\frac\{1\}\{x^\{2\}\}\\right\)\\leq\-\\gamma\.\(103\)This proves

Tr​\(ε,γ\)≤γ−1​log⁡y0r​x0,T\_\{r\}\(\\varepsilon,\\gamma\)\\leq\\gamma^\{\-1\}\\log\\frac\{y\_\{0\}\}\{rx\_\{0\}\},and therefore Eq\. \([26](https://arxiv.org/html/2608.08103#S6.E26)\) wheneverγ​T0→∞\\gamma T\_\{0\}\\to\\infty\.

For the other regime, letΔ=x02−y02=4​a​ε\\Delta=x\_\{0\}^\{2\}\-y\_\{0\}^\{2\}=4a\\varepsilon, writeq=x2q=x^\{2\},v=y2v=y^\{2\}, and note that every other squared cycle weight equalsq\+dj,εq\+d\_\{j,\\varepsilon\}for a constantdj,ε→bj2−a2\>0d\_\{j,\\varepsilon\}\\to b\_\{j\}^\{2\}\-a^\{2\}\>0\. On the bottleneck scale

q=Δ​Q,v=Δ​V,q​q​u​a​d​τ=Δ2​ν\+1​t,q=\\Delta Q,\\qquad v=\\Delta V,qquad\\tau=\\Delta^\{2\\nu\+1\}t,Eq\. \([102](https://arxiv.org/html/2608.08103#A3.E102)\) has, on compact subsets of the positive\(Q,V\)\(Q,V\)quadrant, the uniform limit

d​Qd​τ\\displaystyle\\frac\{dQ\}\{d\\tau\}=−4​c​\[ϕ′​\(0\)​C\]ν\+1​\(Q​V\)ν\+1,\\displaystyle=\-4c\[\\phi^\{\\prime\}\(0\)C\]^\{\\nu\+1\}\(QV\)^\{\\nu\+1\},\(104\)d​Vd​τ\\displaystyle\\frac\{dV\}\{d\\tau\}=−4​c​\[ϕ′​\(0\)​C\]ν\+1​\(Q​V\)ν\+1−2​η​V,\\displaystyle=\-4c\[\\phi^\{\\prime\}\(0\)C\]^\{\\nu\+1\}\(QV\)^\{\\nu\+1\}\-2\\eta V,\(105\)whereη=γ/Δ2​ν\+1\\eta=\\gamma/\\Delta^\{2\\nu\+1\}\. The pure\-flow calculation above shows thatT0​Δ2​ν\+1T\_\{0\}\\Delta^\{2\\nu\+1\}converges to a finite positive constant\. Thusγ​T0→0\\gamma T\_\{0\}\\to 0is equivalent, up to a convergent positive factor, toη→0\\eta\\to 0\.

The entrance fromQ,V→∞Q,V\\to\\inftyis where the casesν\>0\\nu\>0andν=0\\nu=0differ\. FixMM, and letτM\\tau\_\{M\}be the first scaled time at whichQ=MQ=M\. Whenν\>0\\nu\>0, integration of the exact power\-potential identity bounds the score\-induced change ofQ−VQ\-VbeforeτM\\tau\_\{M\}byO​\(η​M−2​ν\)O\(\\eta M^\{\-2\\nu\}\), uniformly in the initial height\. Before the target ratio is reached, the entrance time is bounded by a constant multiple ofM−\(2​ν\+1\)M^\{\-\(2\\nu\+1\)\}\. Hence the entrance state converges to\(M,M−1\)\(M,M\-1\)for every fixedMM, and the time omitted above that section vanishes uniformly asM→∞M\\to\\infty\.

Atν=0\\nu=0, write the exact scaled equations before the section as

Q′=−4​Kε​Q​V,V′=−4​Kε​Q​V−2​η​V,Q^\{\\prime\}=\-4K\_\{\\varepsilon\}QV,\\qquad V^\{\\prime\}=\-4K\_\{\\varepsilon\}QV\-2\\eta V,where the scalar\-speed assumptions giveKε≥k\>0K\_\{\\varepsilon\}\\geq k\>0for all sufficiently smallε\\varepsilon\. WithD=Q−VD=Q\-V,

D′=2​η​V,−\(log⁡Q\)′=4​Kε​V≥4​k​V\.D^\{\\prime\}=2\\eta V,\\qquad\-\(\\log Q\)^\{\\prime\}=4K\_\{\\varepsilon\}V\\geq 4kV\.SinceD​\(0\)=1D\(0\)=1, integration toτM\\tau\_\{M\}gives the entrance estimate

0≤D​\(τM\)−1≤η2​k​log⁡Q​\(0\)M\.0\\leq D\(\\tau\_\{M\}\)\-1\\leq\\frac\{\\eta\}\{2k\}\\log\\frac\{Q\(0\)\}\{M\}\.\(106\)HereQ​\(0\)=x02/Δ=Θ​\(Δ−1\)Q\(0\)=x\_\{0\}^\{2\}/\\Delta=\\Theta\(\\Delta^\{\-1\}\)\. The conditionγ​T0​log⁡\(1/ε\)→0\\gamma T\_\{0\}\\log\(1/\\varepsilon\)\\to 0is therefore equivalent at this scale toη​log⁡\(1/Δ\)→0\\eta\\log\(1/\\Delta\)\\to 0, and Eq\. \([106](https://arxiv.org/html/2608.08103#A3.E106)\) again sends the entrance state to\(M,M−1\)\(M,M\-1\)\. Moreover, whileV/Q≥r2V/Q\\geq r^\{2\},−Q′≥4​k​r2​Q2\-Q^\{\\prime\}\\geq 4kr^\{2\}Q^\{2\}, soτM≤\(4​k​r2​M\)−1\\tau\_\{M\}\\leq\(4kr^\{2\}M\)^\{\-1\}\. This entrance envelope vanishes asM→∞M\\to\\infty\.

On the compact corridor belowQ=MQ=M, uniform convergence of the vector fields, continuous dependence of solutions, and transversality of the target ratio give convergence to theη=0\\eta=0hitting time\. Taking firstε↓0\\varepsilon\\downarrow 0and thenM→∞M\\to\\inftyproves Eq\. \([25](https://arxiv.org/html/2608.08103#S6.E25)\)\. Forν\>0\\nu\>0, combining this result with Eq\. \([23](https://arxiv.org/html/2608.08103#S6.E23)\) gives the power scale in Eq\. \([27](https://arxiv.org/html/2608.08103#S6.E27)\)\.

The additional logarithm atν=0\\nu=0cannot be removed\. In the constant critical systemKε≡1K\_\{\\varepsilon\}\\equiv 1, direct differentiation gives the first integral

V−Q−η2​log⁡Q=constant\.V\-Q\-\\frac\{\\eta\}\{2\}\\log Q=\\text\{constant\}\.For example, takeΔn=exp⁡\(−\(n\+1\)2\)\\Delta\_\{n\}=\\exp\(\-\(n\+1\)^\{2\}\)andηn=\(n\+1\)−1\\eta\_\{n\}=\(n\+1\)^\{\-1\}\. Thenηn→0\\eta\_\{n\}\\to 0, butηn​log⁡\(1/Δn\)→∞\\eta\_\{n\}\\log\(1/\\Delta\_\{n\}\)\\to\\infty\. The first integral forces the target section to be reached atQhit,n→∞Q\_\{\\rm hit,n\}\\to\\infty; sinceTn≤\(4​r2​Qhit,n\)−1T\_\{n\}\\leq\(4r^\{2\}Q\_\{\\rm hit,n\}\)^\{\-1\}, the normalized hitting time tends to zero rather than to the positive feasibility\-only limit\. This proves the claimed endpoint failure under the weaker conditionγ​T0→0\\gamma T\_\{0\}\\to 0\.

Finally, on an isolatedLL\-cycle, the only closed walks have lengthsℓ​L\\ell L, andtr⁡\(\(W∘W\)ℓ​L\)=L​pℓ\\operatorname\{tr\}\(\(W\\circ W\)^\{\\ell L\}\)=Lp^\{\\ell\}\. Hence

ϕEXP​\(p\)=L​∑ℓ≥1pℓ\(ℓ​L\)\!,ϕEXP′​\(0\)=1\(L−1\)\!\.\\phi\_\{\\rm EXP\}\(p\)=L\\sum\_\{\\ell\\geq 1\}\\frac\{p^\{\\ell\}\}\{\(\\ell L\)\!\},\\qquad\\phi\_\{\\rm EXP\}^\{\\prime\}\(0\)=\\frac\{1\}\{\(L\-1\)\!\}\.The same cycle block satisfiesdet\(I−W∘W\)=1−p\\det\(I\-W\\circ W\)=1\-p, soϕDAGMA​\(p\)=−log⁡\(1−p\)\\phi\_\{\\rm DAGMA\}\(p\)=\-\\log\(1\-p\)andϕDAGMA′​\(0\)=1\\phi\_\{\\rm DAGMA\}^\{\\prime\}\(0\)=1\. The linear scalarization hasΨ′​\(s\)=1\\Psi^\{\\prime\}\(s\)=1, the cold quadratic penalty hasΨ′​\(s\)=s\\Psi^\{\\prime\}\(s\)=s, and a seeded ALM termΨ​\(s\)=μ​s\+ρ​s2/2\\Psi\(s\)=\\mu s\+\\rho s^\{2\}/2withμ\>0\\mu\>0hasΨ′​\(s\)∼μ\\Psi^\{\\prime\}\(s\)\\sim\\mu\. ∎

##### Proof of Corollary[A\.4](https://arxiv.org/html/2608.08103#A1.Thmtheorem4)\.

Use oneKmK\_\{m\}gadget\. For everyS⊆\[m\]S\\subseteq\[m\]of sizek≥2k\\geq 2, set

xi=1k​𝟏​\{i∈S\}\.x\_\{i\}=\\frac\{1\}\{\\sqrt\{k\}\}\\mathbf\{1\}\\\{i\\in S\\\}\.\(107\)All active coordinates are related by automorphisms and all inactive derivatives vanish by evenness\. This is the elementary finite\-group case of the symmetric\-criticality principle\(Palais,[1979](https://arxiv.org/html/2608.08103#bib.bib21)\)\. Thus∇g​\(x\)=λ​x\\nabla g\(x\)=\\lambda x, the Lagrange equation on the unit sphere\. Every such support is cyclic, and there are∑k=2m\(mk\)=2m−m−1\\sum\_\{k=2\}^\{m\}\\binom\{m\}\{k\}=2^\{m\}\-m\-1of them\.

Giving each active coordinate an arbitrary sign preserves the Lagrange equation becauseggis coordinatewise even\. This gives

∑k=2m\(mk\)​2k=3m−2​m−1\\sum\_\{k=2\}^\{m\}\\binom\{m\}\{k\}2^\{k\}=3^\{m\}\-2m\-1\(108\)signed critical points\. For DAGMA, withA=W∘WA=W\\circ W, the active block ofA2A^\{2\}isk−1​\(Jk−Ik\)k^\{\-1\}\(J\_\{k\}\-I\_\{k\}\)\. Thusρ​\(A\)=\(k−1\)/k<1\\rho\(A\)=\\sqrt\{\(k\-1\)/k\}<1\.

### C\.3Score–topology flow audit

Figure[3](https://arxiv.org/html/2608.08103#A3.F3)reports the full\-matrix checks that complement the normalized crossover experiment in the main text\. The population score has a positive pathwise margin in every audited setting and selects the false back edge even when that edge starts larger\. Sampling can reverse this margin, which is why the finite\-data experiment does not attain perfect selection\. This distinction motivates the confidence set in Section[7](https://arxiv.org/html/2608.08103#S7): the controlled score isolates the dynamical scale, while the holdout certificate asks whether an empirical score resolves it\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x3.png)Figure 3:Full\-matrix checks of Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)\. Feasibility\-only linear and seeded\-ALM flows scale asε−1\\varepsilon^\{\-1\}, while the cold quadratic scales asε−3\\varepsilon^\{\-3\}\(left\)\. A population linear\-SEM score removes the divergence for reversed initial rankings and dense flows \(center\)\. The same event is not guaranteed from an empirical covariance, although its frequency increases with sample size \(right\)\.
### C\.4Hard\-threshold support stability

###### Proposition C\.1\(Stability of hard\-threshold support\)\.

Forθ\>0\\theta\>0, define

𝒮θ​\(W\)=\{\(i,j\):i≠j,\|Wi​j\|\>θ\},mθ​\(W\)=mini≠j⁡\|\|Wi​j\|−θ\|\.\\mathcal\{S\}\_\{\\theta\}\(W\)=\\\{\(i,j\):i\\neq j,\\ \|W\_\{ij\}\|\>\\theta\\\},\\qquad m\_\{\\theta\}\(W\)=\\min\_\{i\\neq j\}\\bigl\|\|W\_\{ij\}\|\-\\theta\\bigr\|\.\(109\)The support𝒮θ​\(W\)\\mathcal\{S\}\_\{\\theta\}\(W\)is unchanged for every off\-diagonal perturbation‖Δ‖∞<mθ​\(W\)\\\|\\Delta\\\|\_\{\\infty\}<m\_\{\\theta\}\(W\); conversely, for everyη\>0\\eta\>0someΔ\\Deltawith‖Δ‖∞≤mθ​\(W\)\+η\\\|\\Delta\\\|\_\{\\infty\}\\leq m\_\{\\theta\}\(W\)\+\\etachanges it\. Moreover, a positive diagonal rescaling preserves algebraic support but need not preserve𝒮θ​\(W\)\\mathcal\{S\}\_\{\\theta\}\(W\)\.

###### Proposition C\.2\(Scale invariance of profile\-score event paths\)\.

For centered dataXX, define

Δi→j​\(P\)=12​log⁡RSSj​\(P∖\{i\}\)RSSj​\(P\)\.\\Delta\_\{i\\to j\}\(P\)=\\tfrac\{1\}\{2\}\\log\\frac\{\{\\rm RSS\}\_\{j\}\(P\\setminus\\\{i\\\}\)\}\{\{\\rm RSS\}\_\{j\}\(P\)\}\.\(110\)UnderX′=X​DX^\{\\prime\}=XDwith positive diagonalDD, every positive\-RSS deletion score is unchanged\. Conditional on the same starting algebraic support, repeatedly deleting the minimum\-score edge returned by a complete cycle oracle, with fixed tie breaking, produces the same event path in both units\.

We prove Proposition[C\.1](https://arxiv.org/html/2608.08103#A3.Thmtheorem1)\. For every off\-diagonal coordinate, the reverse triangle inequality gives

\|\|Wi​j\+Δi​j\|−\|Wi​j\|\|≤\|Δi​j\|<mθ​\(W\)≤\|\|Wi​j\|−θ\|\.\\bigl\|\|W\_\{ij\}\+\\Delta\_\{ij\}\|\-\|W\_\{ij\}\|\\bigr\|\\leq\|\\Delta\_\{ij\}\|<m\_\{\\theta\}\(W\)\\leq\\bigl\|\|W\_\{ij\}\|\-\\theta\\bigr\|\.\(111\)Thus\|Wi​j\+Δi​j\|−θ\|W\_\{ij\}\+\\Delta\_\{ij\}\|\-\\thetahas the same sign as\|Wi​j\|−θ\|W\_\{ij\}\|\-\\thetafor every coordinate, and the strict threshold support is unchanged\.

Conversely, choose a coordinate attaining the finite minimumm=mθ​\(W\)m=m\_\{\\theta\}\(W\)\. If\|Wi​j\|\>θ\|W\_\{ij\}\|\>\\theta, perturb opposite to its sign bym\+η′m\+\\eta^\{\\prime\}; if\|Wi​j\|≤θ\|W\_\{ij\}\|\\leq\\theta, perturb along its sign \(choosing either sign at zero\) bym\+η′m\+\\eta^\{\\prime\}, where0<η′≤η0<\\eta^\{\\prime\}\\leq\\etais small enough to avoid crossing zero in the first case\. The chosen coordinate crosses the strict threshold and the perturbation norm is at mostm\+ηm\+\\eta\. Hencemmis the supremum of openℓ∞\\ell\_\{\\infty\}balls on which support is constant\. Taking one coefficient equal toθ\\thetashowsinfWmθ​\(W\)=0\\inf\_\{W\}m\_\{\\theta\}\(W\)=0\.

For the scaling claim, letD=diag⁡\(d1,…,dd\)D=\\operatorname\{diag\}\(d\_\{1\},\\ldots,d\_\{d\}\)withdi\>0d\_\{i\}\>0\. Then

\(D−1​W​D\)i​j=djdi​Wi​j\.\(D^\{\-1\}WD\)\_\{ij\}=\\frac\{d\_\{j\}\}\{d\_\{i\}\}W\_\{ij\}\.\(112\)Every multiplier is positive, so exact zero support, directed cycles, and acyclicity are preserved\. For any selected nonzero coefficient, however,dj/did\_\{j\}/d\_\{i\}can be any positive number\. Choosing it above or belowθ/\|Wi​j\|\\theta/\|W\_\{ij\}\|moves that coefficient across the fixed threshold\. This proves non\-invariance\. If standardization is prescribed and repeated after rescaling, this particular source of variation disappears within that fixed pipeline\. The proposition makes the narrower statement that a numerical cutoff depends on the chosen coordinate convention, and that its robustness within any such convention is governed bymθ​\(W\)m\_\{\\theta\}\(W\)\.

We next prove Proposition[C\.2](https://arxiv.org/html/2608.08103#A3.Thmtheorem2)\. LetPPbe a parent set for nodejj, and letΠP\\Pi\_\{P\}denote orthogonal projection onto the column span ofXPX\_\{P\}\. UnderX′=X​DX^\{\\prime\}=XD, the response becomesXj′=dj​XjX^\{\\prime\}\_\{j\}=d\_\{j\}X\_\{j\}, while the design becomesXP′=XP​DPX^\{\\prime\}\_\{P\}=X\_\{P\}D\_\{P\}\. BecauseDPD\_\{P\}is invertible,col⁡\(XP′\)=col⁡\(XP\)\\operatorname\{col\}\(X^\{\\prime\}\_\{P\}\)=\\operatorname\{col\}\(X\_\{P\}\)and hence the two designs have the same projection operator\. Therefore

RSSj′​\(P\)=‖\(I−ΠP\)​dj​Xj‖22=dj2​RSSj​\(P\)\.\{\\rm RSS\}^\{\\prime\}\_\{j\}\(P\)=\\\|\(I\-\\Pi\_\{P\}\)d\_\{j\}X\_\{j\}\\\|\_\{2\}^\{2\}=d\_\{j\}^\{2\}\{\\rm RSS\}\_\{j\}\(P\)\.\(113\)The same factordj2d\_\{j\}^\{2\}multiplies the numerator and denominator of Eq\. \([110](https://arxiv.org/html/2608.08103#A3.E110)\), so every positive\-RSS deletion score is unchanged\. Conditional on one fixed algebraic support, diagonal rescaling also preserves every directed cycle\. Thus the cycle oracle returns the same candidate edges\. With deterministic tie breaking, equality of all candidate scores implies equality of the first deletion\. Induction gives the same support and candidate set after every later deletion, hence the same event path and final certified DAG\. This proves the proposition\. The conditioning is essential: the claim does not assert that a statistical frontend produces the same support under rescaling\.

### C\.5Relation to prior work and alternative assumptions

Table[2](https://arxiv.org/html/2608.08103#A3.T2)compares the assumptions and conclusions of the closest results\. The algebraic, topological, and inferential tools listed in the middle column are classical\. The right column records what follows after they are tied to cycle\-completion geometry or to the ordinary\-coordinate symmetry construction\.

Table 2:Closest prior results and the boundary of the present claims\.The bounds apply to local Taylor information in an exact representation near a DAG support boundary\. Algorithms that use higher derivatives, change the parameterization, modify support discretely, or stop before exact support fall outside the corresponding assumptions\.

### C\.6Scalarization and augmented Lagrangians

Forp≥1p\\geq 1, definePp​\(W\)=‖H​\(W\)‖2pP\_\{p\}\(W\)=\\left\\lVert H\(W\)\\right\\rVert\_\{2\}^\{p\}\.

###### Proposition C\.3\(Sensitivity–smoothness frontier\)\.

IfH​\(W\+t​U\)=tq​v\+o​\(tq\)H\(W\+tU\)=t^\{q\}v\+o\(t^\{q\}\)withv≠0v\\neq 0, thenPp​\(W\+t​U\)=‖v‖2p​\|t\|p​q\+o​\(\|t\|p​q\)P\_\{p\}\(W\+tU\)=\\left\\lVert v\\right\\rVert\_\{2\}^\{p\}\|t\|^\{pq\}\+o\(\|t\|^\{pq\}\)\. As a function of the residual,z↦‖z‖2pz\\mapsto\\left\\lVert z\\right\\rVert\_\{2\}^\{p\}is nonsmooth at zero forp=1p=1, isC1C^\{1\}but has non\-Lipschitz gradient for1<p<21<p<2, and has locally Lipschitz gradient forp≥2p\\geq 2\.

###### Corollary C\.4\(Augmented\-Lagrangian hierarchy\)\.

For the vector ALMQλ,ρ=λ⊤​H\+ρ​‖H‖2/2Q\_\{\\lambda,\\rho\}=\\lambda^\{\\top\}H\+\\rho\\left\\lVert H\\right\\rVert^\{2\}/2, the zero\-multiplier initialization vanishes to order at least2​q2qalong an equal\-scale completion; on aqq\-sharp path this order is exact, while a generic nonzero multiplier exposes orderqq\. For a nonnegative scalar ALMQμ,ρ=μ​h\+ρ​h2/2Q\_\{\\mu,\\rho\}=\\mu h\+\\rho h^\{2\}/2, the corresponding lower bounds are4​q4qforμ=0\\mu=0and2​q2qforμ≠0\\mu\\neq 0, with equality whenhhhas exact order2​q2q\. In these sharp cases, derivative orders are one lower\.

###### Corollary C\.5\(Order under residual\-scaled multipliers\)\.

LetR​\(t\)=O​\(tp\)R\(t\)=O\(t^\{p\}\)andD​R​\(t\)=O​\(tp−1\)DR\(t\)=O\(t^\{p\-1\}\)\. Ifλ​\(t\)=O​\(tp\)\\lambda\(t\)=O\(t^\{p\}\)andρ=O​\(1\)\\rho=O\(1\), then the primal gradient ofλ⊤​R\+ρ​‖R‖2/2\\lambda^\{\\top\}R\+\\rho\\left\\lVert R\\right\\rVert^\{2\}/2, treating the multiplier as fixed within the primal step, isO​\(t2​p−1\)O\(t^\{2p\-1\}\)\. Any fixed number of updatesλ←λ\+ρ​R\\lambda\\leftarrow\\lambda\+\\rho Rfrom zero, with bounded penalties and residual\-comparable iterates, preserves this lower bound\. Recovering orderp−1p\-1requires a multiplier that does not vanish with the local residual and is not orthogonal to its leading coefficient\.

##### Residual\-power frontier\.

IfH​\(t\)=tq​v\+o​\(tq\)H\(t\)=t^\{q\}v\+o\(t^\{q\}\), continuity of the norm and homogeneity give

‖H​\(t\)‖2p=\|t\|p​q​‖v\+o​\(1\)‖2p=‖v‖2p​\|t\|p​q\+o​\(\|t\|p​q\)\.\\\|H\(t\)\\\|\_\{2\}^\{p\}=\|t\|^\{pq\}\\\|v\+o\(1\)\\\|\_\{2\}^\{p\}=\\\|v\\\|\_\{2\}^\{p\}\|t\|^\{pq\}\+o\(\|t\|^\{pq\}\)\.\(114\)Forz≠0z\\neq 0,∇‖z‖2p=p​‖z‖2p−2​z\\nabla\\\|z\\\|\_\{2\}^\{p\}=p\\\|z\\\|\_\{2\}^\{p\-2\}z\. At zero, the norm is nonsmooth forp=1p=1; for1<p<21<p<2this gradient extends continuously but its derivative scales as‖z‖p−2\\\|z\\\|^\{p\-2\}and is unbounded; forp≥2p\\geq 2it is locally Lipschitz\. These are residual\-space statements\. Composition with a particular high\-orderHHcan add pathwise regularity, but it cannot provide a uniform Lipschitz guarantee over arbitrary residual directions\.

Assume a sharp pathH​\(t\)=tq​v\+o​\(tq\)H\(t\)=t^\{q\}v\+o\(t^\{q\}\)withv≠0v\\neq 0\. Then

12​‖H​\(t\)‖22=12​‖v‖22​t2​q\+o​\(t2​q\)\.\\frac\{1\}\{2\}\\\|H\(t\)\\\|\_\{2\}^\{2\}=\\frac\{1\}\{2\}\\\|v\\\|\_\{2\}^\{2\}t^\{2q\}\+o\(t^\{2q\}\)\.\(115\)More generally, Theorem[4\.1](https://arxiv.org/html/2608.08103#S4.Thmtheorem1)always yields the lower\-order boundO​\(t2​q\)O\(t^\{2q\}\)\. ForQλ,ρ=λ⊤​H\+ρ​‖H‖22/2Q\_\{\\lambda,\\rho\}=\\lambda^\{\\top\}H\+\\rho\\\|H\\\|\_\{2\}^\{2\}/2, zero initializationλ=0\\lambda=0therefore removes the order\-qqterm\. A multiplier withλ⊤​v≠0\\lambda^\{\\top\}v\\neq 0restores it; this is the meaning of “generic” in Corollary[C\.4](https://arxiv.org/html/2608.08103#A3.Thmtheorem4)\.

For a sharp nonnegative scalarh​\(t\)=a​t2​q\+o​\(t2​q\)h\(t\)=at^\{2q\}\+o\(t^\{2q\}\),a\>0a\>0,

Qμ,ρ​\(t\)=μ​a​t2​q\+ρ2​a2​t4​q\+μ​o​\(t2​q\)\+o​\(t4​q\)\.Q\_\{\\mu,\\rho\}\(t\)=\\mu at^\{2q\}\+\\frac\{\\rho\}\{2\}a^\{2\}t^\{4q\}\+\\mu\\,o\(t^\{2q\}\)\+o\(t^\{4q\}\)\.\(116\)Thusμ=0\\mu=0has order4​q4q, while every nonzeroμ\\muhas order2​q2q\. Differentiation lowers each sharp order by one\. If the representation has higher\-than\-minimal order, these statements remain lower bounds on the first possible nonzero term\.

##### Residual\-generated multipliers\.

LetR​\(t\)=O​\(tp\)R\(t\)=O\(t^\{p\}\)andD​R​\(t\)=O​\(tp−1\)DR\(t\)=O\(t^\{p\-1\}\)\. During a primal update the multiplier is fixed, so

∇tQλ,ρ=D​R​\(t\)⊤​λ\+ρ​D​R​\(t\)⊤​R​\(t\)\.\\nabla\_\{t\}Q\_\{\\lambda,\\rho\}=DR\(t\)^\{\\top\}\\lambda\+\\rho DR\(t\)^\{\\top\}R\(t\)\.\(117\)Ifλ=O​\(tp\)\\lambda=O\(t^\{p\}\)andρ=O​\(1\)\\rho=O\(1\), both terms areO​\(t2​p−1\)O\(t^\{2p\-1\}\)\. Starting from zero, a fixed number of updatesλs\+1=λs\+ρs​R​\(ts\)\\lambda\_\{s\+1\}=\\lambda\_\{s\}\+\\rho\_\{s\}R\(t\_\{s\}\)withts=Θ​\(t\)t\_\{s\}=\\Theta\(t\)and boundedρs\\rho\_\{s\}leavesλs=O​\(tp\)\\lambda\_\{s\}=O\(t^\{p\}\)by induction\. An order\-p−1p\-1term can appear only if the multiplier has a nonvanishing leading scale and nonzero inner product with the leading residual coefficient\. Substitutingp=qp=qfor a vector representation andp=2​qp=2qfor a nonnegative scalar proves Corollary[C\.5](https://arxiv.org/html/2608.08103#A3.Thmtheorem5)\. A penalty schedule that diverges ast→0t\\to 0can counteract the coefficient, but then does so by ill\-conditioning, not by restoring a missing Taylor term\.

### C\.7Finite precision and restricted correction time

If\|R​\(t\)\|≤C​\|t\|p\|R\(t\)\|\\leq C\|t\|^\{p\}, then\|R​\(t\)\|<ε\|R\(t\)\|<\\varepsilonwhenever

\|t\|<rε:=\(εC\)1/p\.\|t\|<r\_\{\\varepsilon\}:=\\left\(\\frac\{\\varepsilon\}\{C\}\\right\)^\{1/p\}\.\(118\)This interval follows directly from the stated bound\. Its numerical value depends on the formulation\-specific constantCCand on the comparison rule\. For fixedCCand0<ε<C0<\\varepsilon<C, the radius increases to one asp→∞p\\to\\infty\.

For the gradient\-flow calculation, assume explicitly that

ϕ′​\(t\)=a​p​tp−1\+o​\(tp−1\),a\>0,p≥2,t↓0\.\\phi^\{\\prime\}\(t\)=apt^\{p\-1\}\+o\(t^\{p\-1\}\),\\qquad a\>0,p\\geq 2,t\\downarrow 0\.\(119\)For sufficiently smalltt,ϕ′​\(t\)\\phi^\{\\prime\}\(t\)is bounded above and below by positive constants timestp−1t^\{p\-1\}\. Separating variables int˙=−ϕ′​\(t\)\\dot\{t\}=\-\\phi^\{\\prime\}\(t\)gives

T​\(ε\)≍∫εt0t−\(p−1\)​𝑑t,T\(\\varepsilon\)\\asymp\\int\_\{\\varepsilon\}^\{t\_\{0\}\}t^\{\-\(p\-1\)\}\\,dt,\(120\)and therefore

T​\(ε\)=\{Θ​\(log⁡\(t0/ε\)\),p=2,Θ​\(ε−\(p−2\)\),p\>2\.T\(\\varepsilon\)=\\begin\{cases\}\\Theta\(\\log\(t\_\{0\}/\\varepsilon\)\),&p=2,\\\\ \\Theta\(\\varepsilon^\{\-\(p\-2\)\}\),&p\>2\.\\end\{cases\}\(121\)Thus the sharp vector least\-squares and scalar zero\-multiplier ALM paths have one\-dimensional correction exponents2​q−22q\-2and4​q−24q\-2, respectively\.

###### Corollary C\.6\(Finite smooth penalties are not locally exact\)\.

LetFFbe a minimal cycle completion at a DAGWW, letUUactivate every edge inFF, and letℒ\\mathcal\{L\}beC1C^\{1\}withD​ℒ​\(W\)​\[U\]<0D\\mathcal\{L\}\(W\)\[U\]<0\. For any finiteγ≥0\\gamma\\geq 0,WWis not a local minimizer of

ℒ​\(W~\)\+γ​P​\(W~\)\\mathcal\{L\}\(\\widetilde\{W\}\)\+\\gamma P\(\\widetilde\{W\}\)\(122\)whenP=hP=his a smooth nonnegative exact scalar orP=12​‖H‖22P=\\frac\{1\}\{2\}\\\|H\\\|\_\{2\}^\{2\}is a squared smooth exact vector residual\.

###### Proof\.

AlongW\+t​UW\+tUwitht↓0t\\downarrow 0,

ℒ​\(W\+t​U\)=ℒ​\(W\)\+t​D​ℒ​\(W\)​\[U\]\+o​\(t\)\.\\mathcal\{L\}\(W\+tU\)=\\mathcal\{L\}\(W\)\+tD\\mathcal\{L\}\(W\)\[U\]\+o\(t\)\.\(123\)The scalar and squared\-vector barriers giveP​\(W\+t​U\)=O​\(t2​q\)=o​\(t\)P\(W\+tU\)=O\(t^\{2q\}\)=o\(t\)\. The negative linear score term therefore dominates every finite multiple of the penalty for all sufficiently smallt\>0t\>0\. ∎

### C\.8Universal first\- and second\-order consequences

###### Proposition C\.7\(Gradient and Hessian degeneracy\)\.

Lethhbe differentiable, locally nonnegative, and exact\. At every DAGWW,∇h​\(W\)=0\\nabla h\(W\)=0\. Ifhhis twice differentiable, define

𝒮W=span⁡\{Ei​j:i≠j,j↝̸i​in​supp⁡\(W\)\}\.\\mathcal\{S\}\_\{W\}=\\operatorname\{span\}\\\{E\_\{ij\}:i\\neq j,\\ j\\not\\rightsquigarrow i\\text\{ in \}\\operatorname\{supp\}\(W\)\\\}\.\(124\)Then𝒮W⊆ker​∇2h​\(W\)\\mathcal\{S\}\_\{W\}\\subseteq\\ker\\nabla^\{2\}h\(W\)and

dimker​∇2h​\(W\)≥d​\(d−1\)−r​\(W\)≥d​\(d−1\)2,\\dim\\ker\\nabla^\{2\}h\(W\)\\geq d\(d\-1\)\-r\(W\)\\geq\\frac\{d\(d\-1\)\}\{2\},\(125\)wherer​\(W\)r\(W\)is the number of reachable ordered pairs\. AtW=0W=0the Hessian is zero\.

###### Proof\.

A feasible DAG is a local minimizer ofhh, so Fermat’s condition gives the zero gradient\. Ifj↝̸ij\\not\\rightsquigarrow i, adding only edgei→ji\\to jcannot create a cycle\. Hencet↦h​\(W\+t​Ei​j\)t\\mapsto h\(W\+tE\_\{ij\}\)is identically zero and⟨Ei​j,∇2h​\(W\)​Ei​j⟩=0\\langle E\_\{ij\},\\nabla^\{2\}h\(W\)E\_\{ij\}\\rangle=0\. The Hessian at a local minimum is positive semidefinite; for such an operator, zero quadratic form implies membership in its kernel\. There ared​\(d−1\)−r​\(W\)d\(d\-1\)\-r\(W\)safe coordinates, and a DAG has at most one reachable orientation for each unordered pair, sor​\(W\)≤d​\(d−1\)/2r\(W\)\\leq d\(d\-1\)/2\. ∎

The proposition explains why LICQ and MFCQ fail for a scalar exact equality at every feasible DAG\. It is weaker than the completion theorem whenq\>1q\>1, but makes the topology dependence of the Hessian explicit\.

###### Proof that tolerance is not a support certificate\.

Let a DAGW0W\_\{0\}contain a directed pathj↝ij\\rightsquigarrow iand add the back edgei→ji\\to jwith weighttt\. Everyt≠0t\\neq 0gives cyclic support, whereasWt→W0W\_\{t\}\\to W\_\{0\}\. Exactness givesh​\(Wt\)\>0h\(W\_\{t\}\)\>0and continuity givesh​\(Wt\)→h​\(W0\)=0h\(W\_\{t\}\)\\to h\(W\_\{0\}\)=0\. Therefore every positive tolerance accepts some cyclic support\. Starting from an empty graph and shrinking all edges of a cycle gives the same conclusion without requiring a nonempty base DAG\. ∎

## Appendix DSharpness and Exact Constructions

This construction separates overflow control from low\-order sensitivity\. Lets​\(x\)=x2/\(1\+x2\)s\(x\)=x^\{2\}/\(1\+x^\{2\}\), defineS​\(W\)i​j=s​\(Wi​j\)S\(W\)\_\{ij\}=s\(W\_\{ij\}\)fori≠ji\\neq jandS​\(W\)i​i=0S\(W\)\_\{ii\}=0, and set

C​\(W\)=κd​S​\(W\),hBLD​\(W\)=−\(dκ\)2​log​det\(I−C​\(W\)\),0<κ<1\.C\(W\)=\\frac\{\\kappa\}\{d\}S\(W\),\\qquad h\_\{\\rm BLD\}\(W\)=\-\\left\(\\frac\{d\}\{\\kappa\}\\right\)^\{2\}\\log\\det\(I\-C\(W\)\),\\quad 0<\\kappa<1\.\(126\)
###### Proposition D\.1\(Bounded log\-determinant representation\)\.

For fixedddand0<κ<10<\\kappa<1,hBLDh\_\{\\rm BLD\}isC∞C^\{\\infty\}, nonnegative, exact, and globally bounded\. Its gradient and Hessian are globally bounded, so its gradient is globally Lipschitz\. Nevertheless, it obeys the same2​q2qand2​τ2\\taubarriers as every nonnegative scalar representation\.

###### Proof\.

Every entry ofSSlies in\[0,1\)\[0,1\), so the maximum row and column sums of the nonnegative matrixCCare belowκ\\kappa\. Thusρ​\(C\)<κ<1\\rho\(C\)<\\kappa<1and

−log​det\(I−C\)=∑r≥1tr⁡\(Cr\)r\.\-\\log\\det\(I\-C\)=\\sum\_\{r\\geq 1\}\\frac\{\\operatorname\{tr\}\(C^\{r\}\)\}\{r\}\.\(127\)Every term is nonnegative\. A DAG makesCCnilpotent and every trace zero; a directed cycle contributes a positive product to an appropriate trace\. This proves exactness\. Moreovertr⁡\(Cr\)≤d​κr\\operatorname\{tr\}\(C^\{r\}\)\\leq d\\kappa^\{r\}, so the unscaled series is at most−d​log⁡\(1−κ\)\-d\\log\(1\-\\kappa\)\.

LetR=\(I−C\)−1R=\(I\-C\)^\{\-1\}\. The Neumann series gives‖R‖1,‖R‖∞,‖R‖2≤\(1−κ\)−1\\\|R\\\|\_\{1\},\\\|R\\\|\_\{\\infty\},\\\|R\\\|\_\{2\}\\leq\(1\-\\kappa\)^\{\-1\}\. The first two derivatives ofssare globally bounded\. For perturbationsU,VU,V,

D​h​\[U\]\\displaystyle Dh\[U\]=p​tr⁡\(R​D​C​\[U\]\),\\displaystyle=p\\operatorname\{tr\}\(R\\,DC\[U\]\),\(128\)D2​h​\[U,V\]\\displaystyle D^\{2\}h\[U,V\]=p​tr⁡\(R​D​C​\[V\]​R​D​C​\[U\]\)\+p​tr⁡\(R​D2​C​\[U,V\]\),\\displaystyle=p\\operatorname\{tr\}\(R\\,DC\[V\]R\\,DC\[U\]\)\+p\\operatorname\{tr\}\(R\\,D^\{2\}C\[U,V\]\),\(129\)wherep=\(d/κ\)2p=\(d/\\kappa\)^\{2\}\. Standard Frobenius inequalities therefore give global bounds for the gradient and Hessian\. The final claim follows from Theorem[5\.2](https://arxiv.org/html/2608.08103#S5.Thmtheorem2)and Eq\. \([53](https://arxiv.org/html/2608.08103#A2.E53)\); bounded derivatives change coefficients but not the missing Taylor orders\. ∎

## Appendix ENumerical Verification of Geometry and Dynamics

### E\.1Equal\-scale hierarchy

Table 3:Topology\-dependent order hierarchy\. The last column is the maximum absolute log–log slope error overq=1,…,5q=1,\\ldots,5\.For eachq∈\{1,…,5\}q\\in\\\{1,\\ldots,5\\\}, the diagnostic usesd=max⁡\(3,q\+1\)d=\\max\(3,q\+1\)nodes\. It fixes the firstd−qd\-qedges of the path0→1→⋯→d−10\\to 1\\to\\cdots\\to d\-1at weight0\.70\.7\. The remaining path edges and the closing edged−1→0d\-1\\to 0are theqqcompletion coordinates; the closing edge has negative sign to ensure that signed weights are exercised\. Every proper subset is acyclic\.

The scale grid isgeomspace⁡\(0\.08,0\.30,8\)\\operatorname\{geomspace\}\(0\.08,0\.30,8\)\. We fit the least\-squares slope oflog⁡R​\(t\)\\log R\(t\)againstlog⁡t\\log ton the four smallest positive finite values\. The exact simple\-cycle vector is enumerated ford≤6d\\leq 6\. Its Jacobian is computed by forward automatic differentiation and then restricted to theqqcompletion coordinates before taking the Frobenius norm\. EXP and all gradients use JAX float64 automatic differentiation\. The 320 hierarchy rows contain eight scales for every\(q,method\)\(q,\\text\{method\}\)pair\. A separate 160\-row check applies the same base DAGs to EXP, the trace polynomial, DAGMA log determinant, and BLD ongeomspace⁡\(0\.10,0\.30,8\)\\operatorname\{geomspace\}\(0\.10,0\.30,8\)\.

Table 4:Observed scalar orders forq=1,…,5q=1,\\ldots,5\. Every family is predicted to have order2​q2q\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x4.png)Figure 4:Empirical order hierarchy and its consequences\. Top: residual/penalty and restricted Jacobian/gradient orders\. Bottom left: smallest isolated\-cycle scale visible above10−1210^\{\-12\}in float64\. Bottom right: exact cycle\-vector size and uniform\-sharpness lower bounds\.
### E\.2Conditioned random\-subspace census

We test whether the computed completion number predicts local order away from the controlled families\. For eachq∈\{1,2,3,4\}q\\in\\\{1,2,3,4\\\}and 20 seeds, the generator chooses a random directed cycle ond=8d=8nodes, assigns exactlyqqof its edges to the candidate set, and places the remainder in the base DAG\. It then proposes additional random base edges, retaining one only if the base remains acyclic andq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)is unchanged\. It similarly adds two distractor candidate edges only whenq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)remains unchanged\. Base weights are drawn uniformly in magnitude from\[0\.45,0\.90\]\[0\.45,0\.90\], candidate coefficients from\[0\.70,1\.30\]\[0\.70,1\.30\], and signs are independent\.

As an independent check, the implementation exhaustively enumerates all candidate subsets and recomputesq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)before evaluation\. The resulting instances contain 5–10 base edges, 3–6 candidate edges, and 1–5 active simple cycles\. For eight scales ingeomspace⁡\(0\.006,0\.05,8\)\\operatorname\{geomspace\}\(0\.006,0\.05,8\), we evaluate the norm of all active simple\-cycle products and the positive trace series for EXP\. Slopes use the four smallest scales, exactly as in the controlled diagnostics\. The 1,280 raw rows yield 160 instance/method fits\. The predicted ordersqqand2​q2qhold with maximum absolute error0\.00330\.0033and median error4\.9×10−154\.9\\times 10^\{\-15\}\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x5.png)Figure 5:Conditioned random\-subspace census\. Left: every observed order lies on the prediction from the independently recomputedq𝐖​\(F\)q\_\{\\mathbf\{W\}\}\(F\)\. Right: absolute errors over 20 instances per completion number\.
### E\.3Completion\-volume audit

The scriptpaper\_iclr/explore\_completion\_invariants\.pyperforms a separate finite falsification of Theorem[A\.6](https://arxiv.org/html/2608.08103#A1.Thmtheorem6)\. For every random candidate complex it enumerates all supports, extracts the minimal completions, solves both the integral and fractional cover problems, and evaluates Eq\. \([76](https://arxiv.org/html/2608.08103#A2.E76)\) on 30 values ofε\\varepsilon\. It also normalizes an optimal fractional\-cover vectorz⋆z^\{\\star\}so thatminC​∑s∈Czs⋆=1\\min\_\{C\}\\sum\_\{s\\in C\}z^\{\\star\}\_\{s\}=1and evaluates the logarithmic path\|xs\|=tzs⋆\|x\_\{s\}\|=t^\{z^\{\\star\}\_\{s\}\}\. Along this path, the volume\-density scale ist∑szs⋆=tλt^\{\\sum\_\{s\}z^\{\\star\}\_\{s\}\}=t^\{\\lambda\}, while the generator and squared\-generator scales arettandt2t^\{2\}\.

Table 5:Completion\-volume falsification\. “Ray” lists the fitted vector and scalar exponents\. The tournaments are genuine fractional feedback\-cover gaps, not arbitrary hypergraphs\.For the random suite, the maximum error between the fitted tube slope andbbis1\.7×10−41\.7\\times 10^\{\-4\}; the valuation\-ray errors forλ\\lambdaandλ/2\\lambda/2are below10−810^\{\-8\}\. Thed=7d=7tournament has 21 arcs and 64 simple cycles\. Its minimum feedback\-arc cover has size five, whereas the LP relaxation is4\.54\.5\. Thed=8d=8example has 28 arcs and 288 cycles\. On the same rays, direct positive\-series evaluations give orders\(1\.000000,1\.000001,2\.000001,2\.000003\)\(1\.000000,1\.000001,2\.000001,2\.000003\)for the generator, cycle vector, EXP, and DAGMA in thed=7d=7case; the maximum error is1\.5×10−41\.5\\times 10^\{\-4\}across both tournaments\.

We additionally draw scrambled Sobol points in\[0,1\]m\[0,1\]^\{m\}and directly count generator sublevel events\. At practical thresholds this gives slopes3\.2193\.219and4\.5944\.594for the two tournaments, below the asymptotic values4\.54\.5and6\.6676\.667\. This is expected rather than a contradictory estimate: Eq\. \([82](https://arxiv.org/html/2608.08103#A2.E82)\) permits polynomial factors inlog⁡\(1/η\)\\log\(1/\\eta\), and the high\-dimensional events become too rare before those factors are negligible\. We report this failed finite\-scale extrapolation to distinguish the theorem’s logarithmic limit from a claim about moderate tolerances\. Neither Sobol result is used to select a parameter or establish the theorem\.

### E\.4Weighted overlapping cycles

The weighted graph has candidate cycles

CA\\displaystyle C\_\{A\}:0→1→2→3→0,\\displaystyle:0\\to 1\\to 2\\to 3\\to 0,\(130\)CB\\displaystyle C\_\{B\}:0→1→4→3→0\.\\displaystyle:0\\to 1\\to 4\\to 3\\to 0\.\(131\)They share edges0→10\\to 1and3→03\\to 0\. The four exponent patterns give cycle costs A/B of4/44/4,5/85/8,6/86/8, and8/58/5\. The final pattern switches the cheapest cycle\. Eight scales followgeomspace⁡\(0\.12,0\.30,8\)\\operatorname\{geomspace\}\(0\.12,0\.30,8\)\. A generic weighted shortest\-directed\-cycle routine computesτ\\taufrom the graph rather than reading either displayed cycle cost\. There are 128 raw weighted rows\.

Table 6:Weighted overlapping cycles\. Costs A/B are the two cycle costs andτ\\tauis their minimum\. The final row switches the cheapest cycle\.
### E\.5Subcritical compression diagnostic

Ford=10d=10andq=2,…,6q=2,\\ldots,6, letCq​\(W\)∈ℝMq​\(10\)C\_\{q\}\(W\)\\in\\mathbb\{R\}^\{M\_\{q\}\(10\)\}collect the selected cycle products from Theorem[A\.1](https://arxiv.org/html/2608.08103#A1.Thmtheorem1)\. For a matrixA∈ℝr×Mq​\(10\)A\\in\\mathbb\{R\}^\{r\\times M\_\{q\}\(10\)\}with orthonormal rows, we evaluate the globally exact residual

HA​\(W\)=\(A​Cq​\(W\),hexp​\(W\)\)\.H\_\{A\}\(W\)=\\left\(AC\_\{q\}\(W\),\\ h\_\{\\exp\}\(W\)\\right\)\.\(132\)Because the EXP component is globally exact,HA​\(W\)=0H\_\{A\}\(W\)=0if and only ifWWis a DAG, regardless of the projection\. Along the channel family,Cq​\(t​U​\(z\)\)=tq​zC\_\{q\}\(tU\(z\)\)=t^\{q\}zandhexp​\(t​U​\(z\)\)=Θ​\(t2​q\)h\_\{\\exp\}\(tU\(z\)\)=\\Theta\(t^\{2q\}\)\. A generic unitzztherefore gives‖HA​\(t​U​\(z\)\)‖2=Θ​\(tq\)\\left\\lVert H\_\{A\}\(tU\(z\)\)\\right\\rVert\_\{2\}=\\Theta\(t^\{q\}\), whereas any nonzeroz∈ker⁡Az\\in\\ker AgivesΘ​\(t2​q\)\\Theta\(t^\{2q\}\)\. Rank–nullity supplies such a direction wheneverr<Mq​\(10\)r<M\_\{q\}\(10\)\.

We draw 20 random orthogonal maps at each of ranks⌊Mq/4⌋\\lfloor M\_\{q\}/4\\rfloor,⌊Mq/2⌋\\lfloor M\_\{q\}/2\\rfloor, andMq−2M\_\{q\}\-2\. After adding the fallback, every total residual dimension remains strictly belowMqM\_\{q\}\. An SVD gives the null direction and an independent normalized Gaussian vector gives the generic direction\. Eight local scales followgeomspace⁡\(0\.02,0\.12,8\)\\operatorname\{geomspace\}\(0\.02,0\.12,8\), producing 4,800 raw rows\. EXP is evaluated by its positive trace series rather than subtractingdd, and the validator checks that it is positive on every cyclic mixture\. The diagnostic is a global\-exactness stress test, not a proposal to scale the dense EXP fallback to large graphs\. The largest absolute fitted\-order error is0\.00310\.0031\.

Table 7:Subcritical random compression on thed=10d=10channel families of Theorem[A\.1](https://arxiv.org/html/2608.08103#A1.Thmtheorem1)\. Observed orders are medians over three ranks and 20 seeds\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x6.png)Figure 6:Subcritical compression creates a worst\-case cyclic mixture with doubled residual order \(left\)\. The funnel lower bound is quadratic inddfor each fixedqq\(right\)\.
### E\.6Visibility and cycle\-vector size

For each isolated cycle lengthq=2,…,8q=2,\\ldots,8, the diagnostic evaluates 500 scales from10−410^\{\-4\}to0\.990\.99and records the first residual at least10−810^\{\-8\},10−1210^\{\-12\}, and10−1610^\{\-16\}\. We report the grid\-based value so that EXP cancellation and float64 evaluation are included\. Cycle\-vector dimensions use the closed count in Eq\. \([15](https://arxiv.org/html/2608.08103#S4.E15)\) ford=2,…,12d=2,\\ldots,12\.

Table 8:Smallest isolated\-cycle scale whose residual exceeds10−1210^\{\-12\}in float64\. Lower is better\.
### E\.7Restricted optimization dynamics

The family has one fixed edgec∼Uniform⁡\[0\.55,0\.90\]c\\sim\\operatorname\{Uniform\}\[0\.55,0\.90\]andqqcompletion edges of scalett, givingR​\(t\)=c​tqR\(t\)=ct^\{q\}\. WithL=q\+1L=q\+1andP=c2​t2​qP=c^\{2\}t^\{2q\}, NOTEARS equals

hexp​\(t\)=∑m=1∞L​Pm\(m​L\)\!,tr⁡\(Ak\)=0​unless​L∣k\.h\_\{\\exp\}\(t\)=\\sum\_\{m=1\}^\{\\infty\}\\frac\{LP^\{m\}\}\{\(mL\)\!\},\\qquad\\operatorname\{tr\}\(A^\{k\}\)=0\\ \\text\{unless \}L\\mid k\.\(133\)Twelve terms match the JAX matrix exponential throughq=6q=6within2×10−142\\times 10^\{\-14\}\. Every method uses 1,000 projected\-gradient steps with Armijo parameter10−410^\{\-4\}and maximum step0\.250\.25\. ALM uses 40 updates, 25 primal steps per update,ρ0=1\\rho\_\{0\}=1, and growth1\.21\.2capped at10610^\{6\}; only the seeded variant setsμ0=1\\mu\_\{0\}=1\. Every trajectory starts att=0\.8t=0\.8\. The experiment therefore follows the full restricted correction path into the local regime, but it is not evidence about global parameter learning\.

The coefficient\-controlled protocol rescales each objective to unit initial gradient\. Atq=6q=6, its medians are0\.134/0\.266/0\.357/0\.357/0\.376/0\.3570\.134/0\.266/0\.357/0\.357/0\.376/0\.357forp=1/p=1\.5/p=2p=1/p=1\.5/p=2/EXP/cold/seeded; full records are in the released CSV\.

Table 9:Median completion\-edge scale after 1,000 raw\-objective projected\-gradient steps over 20 seeds\. Lower is better; all runs start at0\.80\.8\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x7.png)Figure 7:Restricted feasibility\-correction trajectories under the fixed protocol\. Lower is better\. The exactp=1p=1residual retains the strongest signal, while smooth scalarization and cold\-start ALM become stiffer as the completion number grows\.
### E\.8Full\-matrix selection and SEM\-score protocol

The experiment supporting Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)uses directed cycles of lengthsL∈\{4,6,8\}L\\in\\\{4,6,8\\\}\. Every cycle edge is optimized\. The two tracked edges start ata±εa\\pm\\varepsilon, the remainingL−2L\-2edges start at0\.80\.8, and we varya∈\{0\.3,0\.5\}a\\in\\\{0\.3,0\.5\\\},r∈\{0\.1,0\.25\}r\\in\\\{0\.1,0\.25\\\}, andε∈\[10−5,10−2\]\\varepsilon\\in\[10^\{\-5\},10^\{\-2\}\]\. We evaluate the exact full\-matrix NOTEARS and DAGMA constraints under

Ψ​\(h\)=h,Ψ​\(h\)=12​h2,Ψ​\(h\)=h\+12​h2\.\\Psi\(h\)=h,\\qquad\\Psi\(h\)=\\frac\{1\}\{2\}h^\{2\},\\qquad\\Psi\(h\)=h\+\\frac\{1\}\{2\}h^\{2\}\.These are the linear feasibility force, a cold quadratic penalty, and an ALM inner objective with unit multiplier and penalty\. Selection times are computed by quadrature of the exact matrix\-flow reduction, not by fitting a surrogate\. We additionally compare the closed values and every cycle\-edge gradient with direct matrix exponential and inverse evaluations at 120 random cycle matrices\.

For the score experiment, the population data distribution is the linear Gaussian SEM whose true graph is the path

0⟶1⟶⋯⟶L−10\\longrightarrow 1\\longrightarrow\\cdots\\longrightarrow L\-1with every coefficient equal to0\.50\.5and unit noise variance\. IfW⋆W\_\{\\star\}is this path matrix, its covariance under the row\-vector convention is

Σ=\(I−W⋆\)−T​\(I−W⋆\)−1\.\\Sigma=\(I\-W\_\{\\star\}\)^\{\-T\}\(I\-W\_\{\\star\}\)^\{\-1\}\.Initialization adds the false back edge\(L−1\)→0\(L\-1\)\\to 0, closing the cycle\. The optimized smooth objective is

FΣ​\(W\)=0\.22​tr⁡\(\(I−W\)T​Σ​\(I−W\)\)\+h​\(W\)\+12​h​\(W\)2\.F\_\{\\Sigma\}\(W\)=\\frac\{0\.2\}\{2\}\\operatorname\{tr\}\\\!\\left\(\(I\-W\)^\{T\}\\Sigma\(I\-W\)\\right\)\+h\(W\)\+\\frac\{1\}\{2\}h\(W\)^\{2\}\.\(134\)We run both a cycle\-support flow and a dense flow in which every off\-diagonal entry may change\. “Aligned” initialization makes the false edge smaller by2​ε2\\varepsilon; the adversarial “reversed” initialization makes it larger\. Correct selection is declared when the false\-to\-true tracked\-edge ratio first reaches1/41/4\. The opposite1/41/4event is recorded as an incorrect selection\. No ground\-truth quantity enters the gradient or stopping rule other than this post\-run correctness label\.

All 216 population flows reach the false\-edge ratio event before its opposite, as do explicit gradient\-descent runs with step0\.010\.01on the predeclared subset ofε\\varepsilonvalues\. The fitted selection\-time slopes range from−0\.0041\-0\.0041to0\.00390\.0039, and the minimum total deletion margin in every population trajectory is positive\. Dense trajectories are genuinely different from the cycle restriction: their off\-cycle Frobenius norm reaches0\.3840\.384\. We enforce the DAGMA domainρ​\(W∘W\)<1\\rho\(W\\circ W\)<1at every gradient evaluation; the largest value over all population and finite\-sample DAGMA trajectories is0\.4530\.453\.

For the finite\-sample check, we replaceΣ\\Sigmaby the empirical covariance fromn∈\{128,512,2048\}n\\in\\\{128,512,2048\\\}observations and use 20 fixed seeds\. All 720 runs use reversed initialization\. Overall correctness is91\.25%91\.25\\%; the per\-setting range is65%65\\%to100%100\\%, increasing toward the population result with sample size\. These failures are retained because Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)is conditional on a deterministic positive margin\. It does not claim that observational samples always supply that margin\.

### E\.9Official optimizer trajectories

The isolated\-cycle experiments verify the asymptotic calculation in Proposition[6\.2](https://arxiv.org/html/2608.08103#S6.Thmtheorem2)\. They do not show, by themselves, that the same geometry is visible along an optimizer trajectory on an ordinary random graph\. We therefore record the unthresholded iterates of the official linear NOTEARS and DAGMA updates\. The update equations, continuation schedules, stopping rules, and regularization parameters are unchanged\. Read\-only callbacks store every fifth L\-BFGS\-B iterate for NOTEARS and every 250th Adam iterate for DAGMA\. On small deterministic tests, the instrumented and uninstrumented implementations agree to relative tolerances10−910^\{\-9\}and10−1210^\{\-12\}, respectively\.

#### E\.9\.1Truth\-free early diagnostics

For an unthresholded matrixWW, define its critical DAG threshold

τ​\(W\)=min⁡\{t≥0:supp⁡\(\|W\|\>t\)​is a DAG\}\.\\tau\(W\)=\\min\\\{t\\geq 0:\\operatorname\{supp\}\(\\lvert W\\rvert\>t\)\\text\{ is a DAG\}\\\}\.\(135\)This quantity is used as a diagnostic, not as the reported graph\. At the first completed continuation stage whose nonzero support is cyclic, letB=supp⁡\(\|W\|\>2​τ\)B=\\operatorname\{supp\}\(\\lvert W\\rvert\>2\\tau\), and treat entries with magnitude in\[τ,2​τ\]\[\\tau,2\\tau\]as completion coordinates\. The baseBBis acyclic\. We compute the smallest numberq2q\_\{2\}of band coordinates needed to complete a cycle and choose a deterministic shortest witnessing cycleCC\. Thusq2q\_\{2\}is a two\-scale operational version of the completion statistic, not a claim that the random trajectory lies in the exact local model of Section[3](https://arxiv.org/html/2608.08103#S3)\.

Three dimensionless quantities describe the early ranking onCC\. Write

ae=\|We\|,be=\|∂eh​\(W\)\|,ce=ℒ​\(W​with​We=0\)−ℒ​\(W\)\.a\_\{e\}=\|W\_\{e\}\|,\\qquad b\_\{e\}=\|\\partial\_\{e\}h\(W\)\|,\\qquad c\_\{e\}=\\mathcal\{L\}\(W\\text\{ with \}W\_\{e\}=0\)\-\\mathcal\{L\}\(W\)\.For the weights and deletion costs, the separation is the relative gap between the smallest and second\-smallest values\. For the constraint force it is the relative gap between the largest and second\-largest values\. A zero gap denotes a first\-order tie\. The outcome is the first later snapshot at which the smallest cycle weight is at most one tenth of the second\-smallest\. Time is divided by the remaining optimizer steps, separately for each method; an unresolved trajectory is right\-censored at one\. No generating edge, coefficient, or SHD enters the anchor, cycle, predictor, or outcome\.

Development seeds 100–104 cover ER2/ER4/SF2/SF4 graphs under standard Gaussian and 40%\-weak\-edge Gaussian SEMs\. They fix one ridge model per optimizer with penalty one\. The baseline useslog⁡h​\(W\)\\log h\(W\),log⁡\(τ​\(W\)/‖W‖max\)\\log\(\\tau\(W\)/\\\|W\\\|\_\{\\max\}\), andlog⁡‖W‖max\\log\\\|W\\\|\_\{\\max\}\. The geometry model addsq2q\_\{2\}and the three separation statistics\. Formal seeds 0–9 use the same two regimes plus exponential and heteroscedastic noise\. There are 160 formal trajectories per optimizer\. Spearman correlations, bootstrap intervals, and permutation tests are computed only after this freeze\.

#### E\.9\.2Formal results

Table 10:Predictive bridge on 320 formal trajectories\. The correlation is between early constraint separation and normalized selection time\. Models are fitted only on disjoint development seeds\.Constraint separation predicts faster selection for both optimizers:ρs=−0\.521\\rho\_\{s\}=\-0\.521for NOTEARS and−0\.664\-0\.664for DAGMA\. The 95% bootstrap intervals exclude zero, and both permutation tests givep<10−4p<10^\{\-4\}\. Ranking observations within each graph–noise stratum gives−0\.564\-0\.564and−0\.649\-0\.649, respectively, so the result is not explained by pooling sparse and dense regimes\. Completion depth has the opposite sign, with correlations0\.4350\.435for NOTEARS and0\.3860\.386for DAGMA\. The development\-frozen geometry model lowers DAGMA’s formal MAE by18\.5%18\.5\\%, from0\.10780\.1078to0\.08790\.0879, and raises its prediction rank correlation from0\.3160\.316to0\.6480\.648\. For NOTEARS the MAE change is negligible \(0\.041120\.04112to0\.041000\.04100\); a local association need not make a richer cross\-regime predictor useful\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x8.png)Figure 8:A truth\-free early diagnostic predicts a later support event\. Left and center: formal trajectories and pooled rank correlations\. Right: MAE of development\-frozen baseline and geometry models\. Prediction improves materially for DAGMA, but not for NOTEARS\.The experiment also locates a limit of the random\-graph diagnostic\. Of the 320 critical cycles, 319 are directed two\-cycles and one is a three\-cycle;q2=1q\_\{2\}=1in 78 runs andq2=2q\_\{2\}=2in 242\. Dense signed frontends therefore expose mostly pairwise orientation conflicts at their first continuation stage\. The higher completion orders in the controlled experiments should not be described as typical of these trajectories\. Score separation is method\-dependent as well: its formal correlation is−0\.304\-0\.304for NOTEARS but0\.1180\.118for DAGMA\. What survives the move from the controlled model is the narrower statement that an early feasibility tie and a larger two\-scale completion depth forecast a slower support decision\. This is evidence that the representation geometry is operationally visible, not a global iteration lower bound\.

## Appendix FModular Certificate: Proof and Scaling Audit

### F\.1The algebraic bridge

###### Lemma F\.1\(Completion\-repair duality\)\.

Let

IE=⟨xC:C∈𝒞min⟩I\_\{E\}=\\langle x^\{C\}:C\\in\\mathcal\{C\}\_\{\\min\}\\ranglebe the squarefree cycle\-completion ideal on an active directed supportEE\. If𝒯min\\mathcal\{T\}\_\{\\min\}is the family of inclusion\-minimal deletion sets that leave a DAG, then

IE∨=⋂C∈𝒞min⟨xe:e∈C⟩=⟨xD:D∈𝒯min⟩\.I\_\{E\}^\{\\vee\}=\\bigcap\_\{C\\in\\mathcal\{C\}\_\{\\min\}\}\\langle x\_\{e\}:e\\in C\\rangle=\\langle x^\{D\}:D\\in\\mathcal\{T\}\_\{\\min\}\\rangle\.\(136\)

###### Proof\.

The Alexander dual of a squarefree monomial ideal is the intersection of the coordinate primes indexed by its minimal generators\(Miller & Sturmfels,[2005](https://arxiv.org/html/2608.08103#bib.bib18)\)\. Hence

IE∨=⋂C∈𝒞min⟨xe:e∈C⟩\.I\_\{E\}^\{\\vee\}=\\bigcap\_\{C\\in\\mathcal\{C\}\_\{\\min\}\}\\langle x\_\{e\}:e\\in C\\rangle\.A squarefree monomialxDx^\{D\}belongs to this intersection exactly whenD∩C≠∅D\\cap C\\neq\\varnothingfor everyC∈𝒞minC\\in\\mathcal\{C\}\_\{\\min\}\. This is the transversal condition, and divisibility\-minimal monomials correspond exactly to inclusion\-minimal transversals\. These transversals are the minimal edge deletions that makeEEacyclic\. ∎

We record the edge\-modular specialization used below\. After a frozen training stage, let each active edge have bounded holdout deletion lossZe∈\[−B,B\]Z\_\{e\}\\in\[\-B,B\]and population meanμe\\mu\_\{e\}\. Writeμ​\(D\)=∑e∈Dμe\\mu\(D\)=\\sum\_\{e\\in D\}\\mu\_\{e\}for a feasible transversalD∈𝒯D\\in\\mathcal\{T\}, and letD∗D^\{\*\}be a minimum\-cost transversal\. When this minimizer is unique, its normalized completion\-exchange margin is

κ𝒞​\(μ\)=minD∈𝒯,D≠D∗⁡μ​\(D\)−μ​\(D∗\)\|D△D∗\|\.\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)=\\min\_\{D\\in\\mathcal\{T\},\\,D\\neq D^\{\*\}\}\\frac\{\\mu\(D\)\-\\mu\(D^\{\*\}\)\}\{\|D\\mathbin\{\\triangle\}D^\{\*\}\|\}\.\(137\)Given simultaneous intervals\[ℓe,ue\]\[\\ell\_\{e\},u\_\{e\}\]and an empirical minimizerD^\\widehat\{D\}, define

Δ¯​\(D;D^\)=∑e∈D∖D^ℓe−∑e∈D^∖Due,𝒜α​\(D^\)=\{D∈𝒯:Δ¯​\(D;D^\)≤0\}\.\\underline\{\\Delta\}\(D;\\widehat\{D\}\)=\\sum\_\{e\\in D\\setminus\\widehat\{D\}\}\\ell\_\{e\}\-\\sum\_\{e\\in\\widehat\{D\}\\setminus D\}u\_\{e\},\\qquad\\mathcal\{A\}\_\{\\alpha\}\(\\widehat\{D\}\)=\\\{D\\in\\mathcal\{T\}:\\underline\{\\Delta\}\(D;\\widehat\{D\}\)\\leq 0\\\}\.\(138\)LetEdel=∩D∈𝒜αDE\_\{\\rm del\}=\\cap\_\{D\\in\\mathcal\{A\}\_\{\\alpha\}\}D,Ekeep=E∖∪D∈𝒜αDE\_\{\\rm keep\}=E\\setminus\\cup\_\{D\\in\\mathcal\{A\}\_\{\\alpha\}\}D,L=minD∈𝒯⁡Δ¯​\(D;D^\)L=\\min\_\{D\\in\\mathcal\{T\}\}\\underline\{\\Delta\}\(D;\\widehat\{D\}\), andL−=minD≠D^⁡Δ¯​\(D;D^\)L\_\{\-\}=\\min\_\{D\\neq\\widehat\{D\}\}\\underline\{\\Delta\}\(D;\\widehat\{D\}\)\.

###### Theorem F\.2\(Selective completion\-repair confidence set\)\.

If all intervals cover simultaneously, every population\-optimal repair belongs to𝒜α​\(D^\)\\mathcal\{A\}\_\{\\alpha\}\(\\widehat\{D\}\)\. Consequently, every population optimum deletesEdelE\_\{\\rm del\}and retainsEkeepE\_\{\\rm keep\}\. On the same event,

μ​\(D^\)−minD∈𝒯⁡μ​\(D\)≤\[−L\]\+\.\\mu\(\\widehat\{D\}\)\-\\min\_\{D\\in\\mathcal\{T\}\}\\mu\(D\)\\leq\[\-L\]\_\{\+\}\.\(139\)Moreover,𝒜α=\{D^\}\\mathcal\{A\}\_\{\\alpha\}=\\\{\\widehat\{D\}\\\}exactly whenL−\>0L\_\{\-\}\>0\. If the intervals have common radiusrrandκ𝒞​\(μ\)\>0\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)\>0, thenr<κ𝒞​\(μ\)r<\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)impliesD^=D∗\\widehat\{D\}=D^\{\*\}, whiler<κ𝒞​\(μ\)/2r<\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)/2makes the confidence set a singleton\.

###### Proposition F\.3\(Conservative cycle\-cover certificate\)\.

For any feasible referenceD0D\_\{0\}, relax repair indicators toP=\{z∈\[0,1\]E:∑e∈Cze≥1​∀C\}P=\\\{z\\in\[0,1\]^\{E\}:\\sum\_\{e\\in C\}z\_\{e\}\\geq 1\\ \\forall C\\\}\. If the robust forced\-opposite LP for edgeeehas lower bound above the robust score ofD0D\_\{0\}, every population optimum has theD0D\_\{0\}label atee\. These labels are a subset of those returned by exact forced\-opposite optimization for the same reference\. Minimum\-weight cycle separation gives a polynomial\-time separation oracle for the LP\.

### F\.2Proof of Theorem[F\.2](https://arxiv.org/html/2608.08103#A6.Thmtheorem2)

A deletion set leaves an acyclic support if and only if it intersects every simple directed cycle\. The simple cycles are precisely the inclusion\-minimal nonfaces of the acyclic edge complex, so the feasible deletion sets are the transversals of𝒞min\\mathcal\{C\}\_\{\\min\}\.

Letm=\|E\|m=\|E\|\. For the common\-radius statement, Hoeffding’s inequality and a union bound give the event

ℰn=\{maxe∈E⁡\|μ^e−μe\|≤rn\},rn=B​2​log⁡\(2​m/α\)/n,Pr⁡\(ℰn\)≥1−α\.\\mathcal\{E\}\_\{n\}=\\left\\\{\\max\_\{e\\in E\}\|\\widehat\{\\mu\}\_\{e\}\-\\mu\_\{e\}\|\\leq r\_\{n\}\\right\\\},\\qquad r\_\{n\}=B\\sqrt\{2\\log\(2m/\\alpha\)/n\},\\qquad\\Pr\(\\mathcal\{E\}\_\{n\}\)\\geq 1\-\\alpha\.\(140\)The first part of the theorem only needs the more general simultaneous eventℰ=\{ℓe≤μe≤ue​∀e\}\\mathcal\{E\}=\\\{\\ell\_\{e\}\\leq\\mu\_\{e\}\\leq u\_\{e\}\\ \\forall e\\\}\. Dependence among the coordinates of one holdout vector is allowed\. Onℰ\\mathcal\{E\}, everyD∈𝒯D\\in\\mathcal\{T\}satisfies

μ​\(D\)−μ​\(D^\)\\displaystyle\\mu\(D\)\-\\mu\(\\widehat\{D\}\)=∑e∈D∖D^μe−∑e∈D^∖Dμe\\displaystyle=\\sum\_\{e\\in D\\setminus\\widehat\{D\}\}\\mu\_\{e\}\-\\sum\_\{e\\in\\widehat\{D\}\\setminus D\}\\mu\_\{e\}\(141\)≥Δ¯​\(D;D^\)\.\\displaystyle\\geq\\underline\{\\Delta\}\(D;\\widehat\{D\}\)\.\(142\)LetD∗D^\{\*\}be any population\-optimal repair\. Sinceμ​\(D∗\)−μ​\(D^\)≤0\\mu\(D^\{\*\}\)\-\\mu\(\\widehat\{D\}\)\\leq 0, Eq\. \([142](https://arxiv.org/html/2608.08103#A6.E142)\) impliesΔ¯​\(D∗;D^\)≤0\\underline\{\\Delta\}\(D^\{\*\};\\widehat\{D\}\)\\leq 0, and henceD∗∈𝒜α​\(D^\)D^\{\*\}\\in\\mathcal\{A\}\_\{\\alpha\}\(\\widehat\{D\}\)\. This holds for every population optimum\. An edge in the intersection of the confidence set is therefore deleted by every optimum; an edge outside its union is retained by every optimum\.

Minimizing Eq\. \([142](https://arxiv.org/html/2608.08103#A6.E142)\) overDDproves Eq\. \([139](https://arxiv.org/html/2608.08103#A6.E139)\)\. By definition,𝒜α=\{D^\}\\mathcal\{A\}\_\{\\alpha\}=\\\{\\widehat\{D\}\\\}if and only ifΔ¯​\(D;D^\)\>0\\underline\{\\Delta\}\(D;\\widehat\{D\}\)\>0for everyD≠D^D\\neq\\widehat\{D\}, which is equivalent toL−\>0L\_\{\-\}\>0\. Equation \([142](https://arxiv.org/html/2608.08103#A6.E142)\) then makesD^\\widehat\{D\}the unique population optimum\.

SupposeD∗D^\{\*\}is unique\. For every other transversal,

μ^​\(D\)−μ^​\(D∗\)≥\(κ𝒞​\(μ\)−rn\)​\|D△D∗\|\.\\widehat\{\\mu\}\(D\)\-\\widehat\{\\mu\}\(D^\{\*\}\)\\geq\\left\(\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)\-r\_\{n\}\\right\)\|D\\mathbin\{\\triangle\}D^\{\*\}\|\.\(143\)Hencern<κ𝒞​\(μ\)r\_\{n\}<\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)makesD∗D^\{\*\}the unique empirical minimizer\. Applying the confidence subtraction once more gives

Δ¯​\(D;D∗\)≥\(κ𝒞​\(μ\)−2​rn\)​\|D△D∗\|,\\underline\{\\Delta\}\(D;D^\{\*\}\)\\geq\\left\(\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)\-2r\_\{n\}\\right\)\|D\\mathbin\{\\triangle\}D^\{\*\}\|,\(144\)which is positive for all competitors whenrn<κ𝒞​\(μ\)/2r\_\{n\}<\\kappa\_\{\\mathcal\{C\}\}\(\\mu\)/2\. This proves the theorem\.

### F\.3Computing the selective labels

The confidence set need not be enumerated\. Define

ce=\{ue,e∈D^,ℓe,e∉D^\.Δ¯​\(D;D^\)=∑e∈Dce−∑e∈D^ue\.c\_\{e\}=\\begin\{cases\}u\_\{e\},&e\\in\\widehat\{D\},\\\\ \\ell\_\{e\},&e\\notin\\widehat\{D\}\.\\end\{cases\}\\qquad\\underline\{\\Delta\}\(D;\\widehat\{D\}\)=\\sum\_\{e\\in D\}c\_\{e\}\-\\sum\_\{e\\in\\widehat\{D\}\}u\_\{e\}\.\(145\)Membership in𝒜α\\mathcal\{A\}\_\{\\alpha\}is therefore a threshold test on a weighted feedback\-edge objective\. For each edge, one solve with its label forced opposite toD^\\widehat\{D\}determines whether any repair in the set can reverse that label\. The whole selective output takes one unconstrained and at most\|E\|\|E\|forced\-opposite solves\. Weighted feedback\-edge optimization is NP\-hard in the worst case, so this identity removes explicit set enumeration but not combinatorial complexity\. The formal main protocol gives each forced query one second; a timeout remains computationally unresolved rather than being treated as evidence\.

### F\.4Proof of Proposition[F\.3](https://arxiv.org/html/2608.08103#A6.Thmtheorem3)

The referenceD0D\_\{0\}need not minimize either the empirical or population score\. Define

𝒜α​\(D0\)=\{D∈𝒯:Δ¯​\(D;D0\)≤0\}\.\\mathcal\{A\}\_\{\\alpha\}\(D\_\{0\}\)=\\\{D\\in\\mathcal\{T\}:\\underline\{\\Delta\}\(D;D\_\{0\}\)\\leq 0\\\}\.Let

P\\displaystyle P=\{z∈\[0,1\]E:∑e∈Cze≥1​for every directed cycle​C\},\\displaystyle=\\left\\\{z\\in\[0,1\]^\{E\}:\\sum\_\{e\\in C\}z\_\{e\}\\geq 1\\text\{ for every directed cycle \}C\\right\\\},ce\\displaystyle c\_\{e\}=\{ue,e∈D0,ℓe,e∉D0,τ0=∑e∈D0ue\.\\displaystyle=\\begin\{cases\}u\_\{e\},&e\\in D\_\{0\},\\\\ \\ell\_\{e\},&e\\notin D\_\{0\},\\end\{cases\}\\qquad\\tau\_\{0\}=\\sum\_\{e\\in D\_\{0\}\}u\_\{e\}\.For an active edgeee, letPeoppP\_\{e\}^\{\\rm opp\}imposeze=0z\_\{e\}=0whene∈D0e\\in D\_\{0\}, andze=1z\_\{e\}=1otherwise\. The relaxation reports theD0D\_\{0\}label precisely when

minz∈Peopp⁡c⊤​z\>τ0\.\\min\_\{z\\in P\_\{e\}^\{\\rm opp\}\}c^\{\\top\}z\>\\tau\_\{0\}\.\(146\)On the simultaneous coverage event, Eq\. \([142](https://arxiv.org/html/2608.08103#A6.E142)\) holds withD0D\_\{0\}in place ofD^\\widehat\{D\}\. IfD∗D^\{\*\}is any population optimum, thenμ​\(D∗\)−μ​\(D0\)≤0\\mu\(D^\{\*\}\)\-\\mu\(D\_\{0\}\)\\leq 0, and henceΔ¯​\(D∗;D0\)≤0\\underline\{\\Delta\}\(D^\{\*\};D\_\{0\}\)\\leq 0\. Thus every population optimum belongs to𝒜α​\(D0\)\\mathcal\{A\}\_\{\\alpha\}\(D\_\{0\}\), even whenD0D\_\{0\}is obtained by a greedy repair\.

For the costs in Proposition[F\.3](https://arxiv.org/html/2608.08103#A6.Thmtheorem3), direct expansion gives

Δ¯​\(D;D0\)=∑e∈Dce−τ0\.\\underline\{\\Delta\}\(D;D\_\{0\}\)=\\sum\_\{e\\in D\}c\_\{e\}\-\\tau\_\{0\}\.\(147\)Every incidence vector𝟏D\\mathbf\{1\}\_\{D\}of a feasible repair intersects every directed cycle, so𝟏D∈P\\mathbf\{1\}\_\{D\}\\in P\. Suppose thatD∈𝒜α​\(D0\)D\\in\\mathcal\{A\}\_\{\\alpha\}\(D\_\{0\}\)gives edgeeethe label opposite toD0D\_\{0\}\. Then𝟏D∈Peopp\\mathbf\{1\}\_\{D\}\\in P\_\{e\}^\{\\rm opp\}, while Eq\. \([147](https://arxiv.org/html/2608.08103#A6.E147)\) givesc⊤​𝟏D≤τ0c^\{\\top\}\\mathbf\{1\}\_\{D\}\\leq\\tau\_\{0\}\. This contradicts Eq\. \([146](https://arxiv.org/html/2608.08103#A6.E146)\)\. The same argument covers an empty forced face by assigning it optimum\+∞\+\\infty\. Therefore every repair in the confidence set, and in particular every population optimum, shares the reported label\.

The exact forced\-opposite problem minimizes the same objective over the integer points ofPeoppP\_\{e\}^\{\\rm opp\}\. Its LP optimum is no larger than its integer optimum\. Any label certified by the relaxation is consequently also certified by an exact forced\-opposite query using the same referenceD0D\_\{0\}; this need not be the confidence set centered at a different empirical minimizer\. The relaxation can lose labels only by abstaining\.

It remains to justify the computational claim\. For a candidate pointz∈\[0,1\]Ez\\in\[0,1\]^\{E\}, a cycle inequality is violated exactly when the directed graph contains a cycle of totalzz\-weight below one\. Since the weights are nonnegative, a minimum\-weight directed cycle can be found by shortest paths, which supplies a polynomial\-time separation oracle\. The equivalence of separation and optimization therefore gives a polynomial\-time algorithm for the rational LP\. Our implementation uses repeated shortest\-cycle separation with a standard LP solver\. This practical cutting\-plane loop is not itself claimed to have a strongly polynomial iteration bound\.

The implementation does not use a primal feasible objective as a lower bound\. For each LP it reconstructs a weak\-duality bound from the row multipliers and box constraints; for each integer query it uses the MILP dual bound\. A floating\-point slack is subtracted before either quantity can certify a label\.

### F\.5Two\-point rate calibration

The following calculation checks the scale of the sufficient condition\. It is not a DAG\-specific lower bound and does not constrain an optimizer with a different output rule\.

Take a directed two\-cycle with edge setE=\{e1,e2\}E=\\\{e\_\{1\},e\_\{2\}\\\}\. Its minimum transversals are\{e1\}\\\{e\_\{1\}\\\}and\{e2\}\\\{e\_\{2\}\\\}\. Fixa\>κa\>\\kappaand consider independent Gaussian score coordinates with mean vectors

μ\+=\(a−κ,a\+κ\),μ−=\(a\+κ,a−κ\)\\mu\_\{\+\}=\(a\-\\kappa,a\+\\kappa\),\\qquad\\mu\_\{\-\}=\(a\+\\kappa,a\-\\kappa\)\(148\)and covarianceσ2​I2\\sigma^\{2\}I\_\{2\}\. The optimal deletion is different under the two models, and Eq\. \([137](https://arxiv.org/html/2608.08103#A6.E137)\) equalsκ\\kappaunder both\. Theirnn\-sample Kullback–Leibler divergence is

KL⁡\(P\+n,P−n\)=4​n​κ2σ2\.\\operatorname\{KL\}\(P\_\{\+\}^\{n\},P\_\{\-\}^\{n\}\)=\\frac\{4n\\kappa^\{2\}\}\{\\sigma^\{2\}\}\.\(149\)Forδ∈\(0,1/4\)\\delta\\in\(0,1/4\), any selector that succeeds under both models induces a test with both error probabilities at mostδ\\delta\. The standard two\-point testing inequality gives2​δ≥12​exp⁡\{−KL⁡\(P\+n,P−n\)\}2\\delta\\geq\\tfrac\{1\}\{2\}\\exp\\\{\-\\operatorname\{KL\}\(P\_\{\+\}^\{n\},P\_\{\-\}^\{n\}\)\\\}, which gives

n≥σ24​κ2​log⁡14​δ\.n\\geq\\frac\{\\sigma^\{2\}\}\{4\\kappa^\{2\}\}\\log\\frac\{1\}\{4\\delta\}\.\(150\)General instance\-dependent best\-set bounds are substantially sharper\(Chen et al\.,[2017](https://arxiv.org/html/2608.08103#bib.bib2)\)\. We use this familiar rate only to calibrate the score margin\.

### F\.6Controlled and random\-graph audits

The two\-cycle calculation has an exact error curve\. UnderP\+P\_\{\+\}, the empirical score difference is Gaussian with mean2​κ2\\kappaand variance2​σ2/n2\\sigma^\{2\}/n, so the wrong\-deletion probability is

Φ​\(−2​n​κ/σ\)\.\\Phi\\\!\\left\(\-\\sqrt\{2n\}\\,\\kappa/\\sigma\\right\)\.\(151\)We use four margins, six values ofn​κ2/σ2n\\kappa^\{2\}/\\sigma^\{2\}, and 200,000 replications per cell\. The simulated curve differs from the formula by at most0\.00160\.0016; the fitted log–log slope of the sample size required for error0\.050\.05is−2\.000\-2\.000\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x9.png)Figure 9:Statistical selection on the directed two\-cycle\. Error curves collapse under the scaled budgetn​κ2/σ2n\\kappa^\{2\}/\\sigma^\{2\}\(left\), and the sample size for fixed error has slope−2\-2inκ\\kappa\(right\)\.The random\-graph audit uses the 160 frozen all\-node Lasso screens from the modular score experiment\. For each screen, an exact weighted feedback\-edge solver finds the best and second\-best deletion transversals under an independent 20,000\-sample oracle proxy\. Their objective gap divided by their symmetric\-difference size estimatesκ𝒞\\kappa\_\{\\mathcal\{C\}\}\. This proxy is used only for the audit, never for selection or certification\.

Table 11:Completion\-margin audit over 160 formal datasets per holdout size\. The margin is measured with an independent oracle proxy\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x10.png)Figure 10:Why unique certificates remain rare\. The median confidence radius is far larger than the oracle\-proxy exchange margin \(left\), even as graph selection and certification improve with holdout size \(right\)\.All 640 regret bounds cover their independent oracle audit\. The median exchange margin is1\.16×10−51\.16\\times 10^\{\-5\}at every nested holdout size\. Atn=20,000n=20\{,\}000, the empirical optimizer selects the oracle deletion set in48\.75%48\.75\\%of runs, but only7\.5%7\.5\\%are uniquely certified\. The median resolution ratio remains95\.995\.9, so the gap between selection and proof is consistent with near\-tied transversals\. Because the oracle margin is itself estimated, the column2​r<κ2r<\\kappais a diagnostic rather than a certificate\.

### F\.7Selective output and score dependence

The selective audit uses the same simultaneous empirical\-Bernstein intervals as the regret experiment\. It records the fraction of active edges with a common label across𝒜α\\mathcal\{A\}\_\{\\alpha\}, along with singleton and oracle\-proxy checks\.

Table 12:Selective repair over 160 formal datasets per holdout size under the one\-second main protocol\. “Statistical” means that an optimally solved forced\-opposite query remains inside the confidence set; “computational” means that the query timed out\. “Proxy retained” checks the independent oracle\-proxy optimum\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x11.png)Figure 11:Selective rather than forced output\. More edges become decidable as the holdout grows \(left\); the independent oracle audit checks set and label coverage \(right\)\.Table 13:Compute\-budget sensitivity atn=20,000n=20\{,\}000over the same 160 datasets\. Solved queries are reused as the cap increases; only previous timeouts are retried\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x12.png)Figure 12:Statistical and computational abstention are reported separately\. A 0\.01\-second cap leaves a visible timeout component; one second eliminates it on the formal suite\.
### F\.8Scaling audit for exact and relaxed queries

Thed=20d=20main certificate suite was chosen to keep the statistical audit from being obscured by a failed combinatorial solve\. We separately test the computational boundary ond∈\{20,40,60,80,100\}d\\in\\\{20,40,60,80,100\\\}\. The frozen grid has ER2 and SF2 graphs, standard and mixed\-strength Gaussian SEMs, three seeds, 500 training observations, and 20,000 independent holdout observations: 60 datasets in total\. A Lasso screen fixes the active directed support\. A deterministic greedy repair supplies the feasibleD0D\_\{0\}, which is permitted by Proposition[F\.3](https://arxiv.org/html/2608.08103#A6.Thmtheorem3); the experiment does not require an exact solve to enter the confidence set\.

The uncapped LP relaxation audits every active edge\. For comparison, each dataset has 20 evenly spaced active\-edge indices fixed before any score is evaluated\. Their forced\-opposite integer queries receive one second each\. Ground truth is not used by either path\.

Table 14:Scaling audit\. Percentages in the upper panel are edge\- or query\-weighted within dimension\. The lower panel separates solved statistical overlap from exact\-solver timeout by candidate\-support size\. LP sec\. is the median time to audit every active edge\.
![Refer to caption](https://arxiv.org/html/2608.08103v1/x13.png)Figure 13:Computational audit\. The relaxed path audits every active edge \(left\)\. Exact\-query outcomes \(right\) show statistical overlap up to 400 candidate edges; the 39 one\-second timeouts all occur in the two supports above 490 edges\.
### F\.9Frontend\-agnostic certificate audit

The certificate is conditional on a fixed candidate support and score; it does not require that the support come from continuous optimization\. We test this point directly using NOTEARS, DAGMA, SDCD, and GOLEM as four interchangeable screens\. Each frontend contributes its top2​d2doff\-diagonal magnitudes with deterministic ties atd=20d=20\. The downstream procedure is otherwise identical: regressors and clipping are frozen onn=500n=500training samples, simultaneous empirical\-Bernstein intervals use an independent holdout of 5,000 samples, and a separate 20,000\-sample draw is consulted only after all labels have been fixed\. The grid contains ER2/ER4/SF2/SF4, Gaussian, non\-Gaussian, mixed\-strength, and heteroscedastic SEMs, and five seeds, for 320 frontend–dataset pairs\.

Table 15:The same selective certificate after four frontend screens\. Coverage refers to simultaneous coverage of all screened deletion costs; errors compare LP\-certified labels with the independent oracle proxy\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x14.png)Figure 14:Frontend\-agnostic audit\. The common certificate resolves different fractions of the four screened supports \(left\)\. Candidate recall and selective resolvability are distinct: a broad screen may retain true edges without creating a large holdout margin between feasible repairs \(right\)\.Regret coverage is100%100\\%for every frontend and no LP\-certified label disagrees with the oracle proxy\. Simultaneous interval coverage is98\.75%98\.75\\%for DAGMA and GOLEM and100%100\\%for NOTEARS and SDCD\. The decided fraction is51\.2%51\.2\\%,46\.1%46\.1\\%,29\.5%29\.5\\%, and13\.3%13\.3\\%for DAGMA, GOLEM, NOTEARS, and SDCD, respectively\. These numbers are not an end\-to\-end causal ranking: the top\-2​d2dscreen itself has different recall\. They isolate the claim needed here—the certificate applies after different frontends and exposes how much of each frontend’s proposed support the frozen score can actually distinguish\.

The relaxed labels are conservative by construction\. On the 1,179 sampled integer queries, they also coincide empirically with all 131 labels certified by completed MILP solves and never certify an edge left unresolved by a completed solve\. Of the remaining queries, 1,009 are statistically unresolved and 39 time out\. The LP path was not assigned a timeout and completed every audit; its median all\-edge runtime ranges from 0\.32 to 0\.74 seconds across the five dimensions, although the two supports above 490 edges take 11\.90 and 20\.35 seconds\. Atd=100d=100, only0\.7%0\.7\\%of active edges are certified\. The relaxation therefore addresses computation, not weak statistical separation\. In this suite, active\-support density is a better warning sign for exact\-query failure than ambient dimension alone; this is an empirical boundary, not an average\-case complexity claim\.

The exchange margin is not a property of the causal graph alone\. It is indexed by the candidate support, deletion score, and fitting protocol\. We therefore repeat the 160\-dataset oracle audit with four dimensionless scores: the frozen clipped score used by the certificate, the same frozen regressors without clipping, oracle\-refitted normalized quadratic loss, and oracle\-refitted Gaussian profile loss\. The oracle\-refitted variants are diagnostics and are not used for selection\.

Table 16:Score sensitivity on the same 160 frozen Lasso supports\. Relativeκ\\kappadivides by the median absolute edge cost\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x15.png)Figure 15:The completion margin is score\-conditional\. Replacing clipping, refitting regressors, or changing the loss moves the margin distribution and can create exact ties\.The one\-second protocol has no query timeouts\. It decides progressively more labels without requiring a unique graph: the mean decided fraction rises from3\.8%3\.8\\%to36\.8%36\.8\\%, while the singleton rate rises from2\.5%2\.5\\%to7\.5%7\.5\\%\. All 640 oracle\-proxy repairs remain in the reported sets, and every reported label agrees with the corresponding proxy\. Atn=20,000n=20\{,\}000, a 0\.01\-second cap classifies34\.6%34\.6\\%of edges, leaves48\.0%48\.0\\%statistically unresolved after completed solves, and times out on17\.3%17\.3\\%\. At 0\.1 seconds, the timeout fraction falls to0\.3%0\.3\\%; at one second it is zero, with36\.8%36\.8\\%decided and63\.2%63\.2\\%statistically unresolved\. The low singleton rate therefore persists after computational abstention is removed\.

The median absolute margin ranges from7\.88×10−87\.88\\times 10^\{\-8\}for refitted profile loss to2\.09×10−52\.09\\times 10^\{\-5\}for the frozen unclipped score\. Exact proxy ties occur in43\.75%43\.75\\%of profile\-score datasets and15\.63%15\.63\\%of refitted\-quadratic datasets, but not for either frozen score\. The associated SHD does not improve monotonically with the margin\. Thus the small1\.16×10−51\.16\\times 10^\{\-5\}value in the primary audit diagnoses one frozen score/frontend pair; it is not evidence for an intrinsic sample complexity of causal discovery\.

## Appendix GInteraction\-Aware Certificate: Proof and Unified Audit

An edge\-modular deletion model does not represent interactions created by refitting several parents jointly\. We therefore treat the entire parent set as the local statistical unit\. This retains collider\-forming interactions and supports selective statements about skeleton and V\-structure features; the resulting exact search is combinatorial\.

For each compatible DAGGG, define

Q​\(G\)=∑j\(Rj,paG⁡\(j\)\+aj,paG⁡\(j\)\),G0∈𝒢​\(𝒫\)\.Q\(G\)=\\sum\_\{j\}\(R\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\}\+a\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\}\),\\qquad G\_\{0\}\\in\\mathcal\{G\}\(\\mathcal\{P\}\)\.\(152\)The referenceG0G\_\{0\}is arbitrary and may be data\-dependent\. The benchmark chooses the empirical holdout minimizer, but the theorem also covers a compatible graph supplied by a continuous, constraint\-based, or score\-based frontend\.

For completeness, we record the data\-dependent quantities suppressed in the main statement\. WriteRj,S=𝔼​ℓj,S​\(X\)R\_\{j,S\}=\\mathbb\{E\}\\ell\_\{j,S\}\(X\),R^j,S=n−1​∑iℓj,S​\(Xi\)\\widehat\{R\}\_\{j,S\}=n^\{\-1\}\\sum\_\{i\}\\ell\_\{j,S\}\(X\_\{i\}\), andM=∑j\|𝒫j\|​\(\|𝒫j\|−1\)M=\\sum\_\{j\}\|\\mathcal\{P\}\_\{j\}\|\(\|\\mathcal\{P\}\_\{j\}\|\-1\)\. For each ordered pairS,T∈𝒫jS,T\\in\\mathcal\{P\}\_\{j\}, letV^j,S,T\\widehat\{V\}\_\{j,S,T\}be the sample variance ofℓj,S​\(Xi\)−ℓj,T​\(Xi\)\\ell\_\{j,S\}\(X\_\{i\}\)\-\\ell\_\{j,T\}\(X\_\{i\}\), and set

rj,S,T=2​V^j,S,T​log⁡\(2​M/α\)n\+28​B​log⁡\(2​M/α\)3​\(n−1\)\.r\_\{j,S,T\}=\\sqrt\{\\frac\{2\\widehat\{V\}\_\{j,S,T\}\\log\(2M/\\alpha\)\}\{n\}\}\+\\frac\{28B\\log\(2M/\\alpha\)\}\{3\(n\-1\)\}\.\(153\)ForS0​j=paG0⁡\(j\)S\_\{0j\}=\\operatorname\{pa\}\_\{G\_\{0\}\}\(j\), define

gj,S0​j=0,gj,S=R^j,S−R^j,S0​j\+aj,S−aj,S0​j−rj,S,S0​j\(S≠S0​j\)\.g\_\{j,S\_\{0j\}\}=0,\\qquad g\_\{j,S\}=\\widehat\{R\}\_\{j,S\}\-\\widehat\{R\}\_\{j,S\_\{0j\}\}\+a\_\{j,S\}\-a\_\{j,S\_\{0j\}\}\-r\_\{j,S,S\_\{0j\}\}\\quad\(S\\neq S\_\{0j\}\)\.\(154\)
The main text definesLL,L−L\_\{\-\},𝒜α​\(G0\)\\mathcal\{A\}\_\{\\alpha\}\(G\_\{0\}\), and the forced\-opposite valueLφoppL\_\{\\varphi\}^\{\\rm opp\}\. We now prove the simultaneous regret, confidence family, and feature statements in Theorem[7\.1](https://arxiv.org/html/2608.08103#S7.Thmtheorem1)\.

### G\.1Proof of Theorem[7\.1](https://arxiv.org/html/2608.08103#S7.Thmtheorem1)

Condition on the training sample\. The parent\-set families, fitted predictors, losses, and penalties are then fixed, while the validation observations remain i\.i\.d\. For an ordered pairS,T∈𝒫jS,T\\in\\mathcal\{P\}\_\{j\}, put

Dij,S,T=ℓj,S​\(Xi\)−ℓj,T​\(Xi\)\.D\_\{i\}^\{j,S,T\}=\\ell\_\{j,S\}\(X\_\{i\}\)\-\\ell\_\{j,T\}\(X\_\{i\}\)\.Because each loss lies in\[−B,B\]\[\-B,B\], the paired difference lies in\[−2​B,2​B\]\[\-2B,2B\], an interval of length4​B4B\. The empirical Bernstein inequality ofMaurer & Pontil \([2009](https://arxiv.org/html/2608.08103#bib.bib17)\), with failure probabilityα/M\\alpha/M, gives

Rj,S−Rj,T≥R^j,S−R^j,T−rj,S,TR\_\{j,S\}\-R\_\{j,T\}\\geq\\widehat\{R\}\_\{j,S\}\-\\widehat\{R\}\_\{j,T\}\-r\_\{j,S,T\}\(155\)with the radius in Eq\. \([153](https://arxiv.org/html/2608.08103#A7.E153)\)\. A union bound over allMMordered local pairs makes Eq\. \([155](https://arxiv.org/html/2608.08103#A7.E155)\) simultaneous with probability at least1−α1\-\\alpha\. Covering the entire ordered\-pair family is essential whenG0G\_\{0\}depends on the same holdout\.

Work on this simultaneous event and setT=S0​jT=S\_\{0j\}\. Adding the deterministic penalty difference to Eq\. \([155](https://arxiv.org/html/2608.08103#A7.E155)\) yields

Rj,S\+aj,S−Rj,S0​j−aj,S0​j≥gj,SR\_\{j,S\}\+a\_\{j,S\}\-R\_\{j,S\_\{0j\}\}\-a\_\{j,S\_\{0j\}\}\\geq g\_\{j,S\}for every local alternative, including equality atS=S0​jS=S\_\{0j\}\. Hence, for every candidate DAGGG,

Q​\(G\)−Q​\(G0\)≥∑jgj,paG⁡\(j\)\.Q\(G\)\-Q\(G\_\{0\}\)\\geq\\sum\_\{j\}g\_\{j,\\operatorname\{pa\}\_\{G\}\(j\)\}\.\(156\)Minimizing both sides overG∈𝒢​\(𝒫\)G\\in\\mathcal\{G\}\(\\mathcal\{P\}\)givesminG⁡Q​\(G\)−Q​\(G0\)≥L\\min\_\{G\}Q\(G\)\-Q\(G\_\{0\}\)\\geq L\. Rearrangement proves Eq\. \([31](https://arxiv.org/html/2608.08103#S7.E31)\)\. Notice thatL≤0L\\leq 0becauseG0G\_\{0\}is feasible and has robust relative cost zero\.

LetG∗G^\{\*\}be any population minimizer\. SinceQ​\(G∗\)−Q​\(G0\)≤0Q\(G^\{\*\}\)\-Q\(G\_\{0\}\)\\leq 0, Eq\. \([156](https://arxiv.org/html/2608.08103#A7.E156)\) gives∑jgj,paG∗⁡\(j\)≤0\\sum\_\{j\}g\_\{j,\\operatorname\{pa\}\_\{G^\{\*\}\}\(j\)\}\\leq 0\. Hence every suchG∗G^\{\*\}lies in Eq\. \([29](https://arxiv.org/html/2608.08103#S7.E29)\)\. IfL−\>0L\_\{\-\}\>0, Eq\. \([156](https://arxiv.org/html/2608.08103#A7.E156)\) is strictly positive for everyG≠G0G\\neq G\_\{0\}; the reference graph is therefore the unique minimizer ofQQ\. Finally, if a population minimizer had the feature label opposite toG0G\_\{0\}, its robust relative cost would be at leastLφopp\>0L\_\{\\varphi\}^\{\\rm opp\}\>0, contradicting its membership in𝒜α​\(G0\)\\mathcal\{A\}\_\{\\alpha\}\(G\_\{0\}\)\. This proves the feature statement\.

### G\.2Optimization and target of the certificate

For each child, the implementation screens at most eight candidate parents on the training sample and fits every subset of those parents\. The validation loss forS∈𝒫jS\\in\\mathcal\{P\}\_\{j\}is

ℓj,S​\(x\)=clip\[−B,B\]⁡\{\(xj−f^j,S​\(x\)\)2−\(xj−f^j,Fj​\(x\)\)22​v^j\},\\ell\_\{j,S\}\(x\)=\\operatorname\{clip\}\_\{\[\-B,B\]\}\\\!\\left\\\{\\frac\{\(x\_\{j\}\-\\widehat\{f\}\_\{j,S\}\(x\)\)^\{2\}\-\(x\_\{j\}\-\\widehat\{f\}\_\{j,F\_\{j\}\}\(x\)\)^\{2\}\}\{2\\widehat\{v\}\_\{j\}\}\\right\\\},\(157\)whereFjF\_\{j\}is the full screened parent set\. We useaj,S=λsp​\|S\|a\_\{j,S\}=\\lambda\_\{\\mathrm\{sp\}\}\|S\|\. The subtraction of the full\-model loss does not affect the graph minimizer but reduces paired variance, and division by the training response variance makes the loss dimensionless\.

Both minimizations in Eq\. \([28](https://arxiv.org/html/2608.08103#S7.E28)\) are standard exact Bayesian\-network structure\-learning problems over a restricted parent\-set family\. We use the parent\-set integer program and add cluster inequalities until the selected support is acyclic\(Cussens,[2011](https://arxiv.org/html/2608.08103#bib.bib4)\)\. A no\-good constraint excludesG0G\_\{0\}when computingL−L\_\{\-\}\. These optimization devices, including parent\-set screening and cluster cuts, are established tools\(Cussens,[2020](https://arxiv.org/html/2608.08103#bib.bib5)\); they are not claimed as contributions\. The new object is the robust objective in Eq\. \([154](https://arxiv.org/html/2608.08103#A7.E154)\) and its conversion of simultaneous local score uncertainty into the global statement of Theorem[7\.1](https://arxiv.org/html/2608.08103#S7.Thmtheorem1)\.

The feature queries use the same parent\-set variables\. Ifxj,Sx\_\{j,S\}selects parent setSSfor childjj, write

au​v=∑S∈𝒫v:u∈Sxv,S,su​v=au​v\+av​u\(u<v\)\.a\_\{uv\}=\\sum\_\{S\\in\\mathcal\{P\}\_\{v\}:u\\in S\}x\_\{v,S\},\\qquad s\_\{uv\}=a\_\{uv\}\+a\_\{vu\}\\quad\(u<v\)\.\(158\)On a DAG,su​vs\_\{uv\}is the skeleton\-adjacency indicator\. Thus an opposite skeleton or directed\-edge label is one linear equality\. An unshielded collideru→v←wu\\to v\\leftarrow wis present exactly whenau​v=aw​v=1a\_\{uv\}=a\_\{wv\}=1andsu​w=0s\_\{uw\}=0\. Its presence is imposed by these three equalities; its absence is the single inequality

au​v\+aw​v−su​w≤1\.a\_\{uv\}\+a\_\{wv\}\-s\_\{uw\}\\leq 1\.\(159\)One forced\-opposite solve therefore computes Eq\. \([30](https://arxiv.org/html/2608.08103#S7.E30)\)\. Skeletons and unshielded colliders characterize a Markov equivalence class, so this projection does not require the data to resolve arbitrary orientations inside the class\(Chickering,[2002](https://arxiv.org/html/2608.08103#bib.bib3)\)\.

The target is deliberately conditional\. Screening can omit a true parent, and even the population minimizer of the clipped regularized score can differ from the causal DAG\. The theorem certifies population regret only within𝒢​\(𝒫\)\\mathcal\{G\}\(\\mathcal\{P\}\)\. It neither assumes nor concludes causal support recovery\.

### G\.3Frozen protocol and results

The formal suite usesd=20d=20and 500 training observations\. A standardized all\-node Lasso frontend uses0\.002​λmax0\.002\\lambda\_\{\\max\}; at most eight incoming parents per node are retained, and all of their subsets are enumerated\. We setB=0\.02B=0\.02andλsp=0\.001\\lambda\_\{\\mathrm\{sp\}\}=0\.001using disjoint development seeds 100–102, then freeze them\. Formal seeds are 0–9\. We test ER2, ER4, SF2, and SF4 graphs under standard Gaussian, exponential\-noise, 40%\-weak\-edge Gaussian, and heteroscedastic Gaussian SEMs\. Independent holdouts contain500,1000,5000500,1000,5000, or20,00020\{,\}000observations; a further 20,000\-sample oracle is used only after selection and certification\. This gives 160 datasets at each holdout size and 640 records in total\.

![Refer to caption](https://arxiv.org/html/2608.08103v1/x16.png)Figure 16:Matched\-budget comparison of the original sequential certificate, a globally optimized modular deletion score, and the parent\-set certificate\. The methods certify different population losses, so regret values should be compared as tightening curves rather than as a common performance metric\. Every reported bound covered its independent oracle audit\.Table 17:Matched validation\-budget audit over 160 formal datasets\. The three rows certify different targets; their regret values are not losses on a common scale\.For the parent\-set target, all 640 bounds cover the independent oracle proxy\. The mean bound decreases from0\.19110\.1911atn=500n=500to0\.006550\.00655atn=20,000n=20\{,\}000, while the mean oracle regret decreases from0\.001910\.00191to0\.000100\.00010\. The selected graph matches the oracle graph in3\.75%3\.75\\%and47\.5%47\.5\\%of runs, respectively, butL−L\_\{\-\}is never positive\. Thus finite samples often select the oracle graph before the simultaneous confidence box is narrow enough to prove uniqueness\.

The parent\-set construction retains deletion interactions and reduces mean SHD from34\.8834\.88for the modular certificate to26\.2226\.22at the largest matched budget\. This remains far behind the strongest frozen frontends in the broad benchmark, so the experiment supports the certificate rather than a recovery claim\. Mean screen recall is0\.6920\.692, and the difficult ER4/SF4 cells account for most remaining errors\. The result also identifies the operative boundary: no validation certificate can recover parents removed by the training screen\.

The modular companion fixes one bounded loss per active\-edge deletion and solves an exact weighted feedback\-edge problem\. Atn=20,000n=20\{,\}000, its mean bound is0\.003720\.00372, its oracle regret is0\.0000690\.000069, and7\.5%7\.5\\%of graphs are uniquely certified\. Its empirical modular objective is lower than that of a same\-cost greedy cycle repair in every formal run\. This verifies the value of global optimization for its stated target, while the parent\-set result shows why an additive single\-edge target is statistically incomplete\.

### G\.4Unified frontend\-to\-feature audit

We next keep the full pipeline fixed within one experiment\. NOTEARS, DAGMA, SDCD, and GOLEM each supply their top\-2​d2dweighted entries\. A deterministic four\-parent\-per\-child cap fixes the parent\-set family before holdout scores are evaluated; every subset is refitted on 500 training observations\. The interaction\-aware score uses a 5,000\-sample holdout, and an independent 20,000\-sample draw is consulted only after the graph and feature labels have been fixed\. The grid contains ER2/ER4/SF2/SF4, Gaussian, non\-Gaussian, mixed\-strength, and heteroscedastic SEMs, and five seeds, for 320 frontend–dataset pairs\. The cap, clipping level, and sparsity penalty are fixed across all cells\.

Table 18:Unified pipeline audit\. “Front\.” is the frontend’s standard reported DAG; “Modular” and “Parent” apply the two repair scores to the same top\-2​d2dscreen\. “Oracle parent” uses 20,000 independent observations; label columns are certified candidate\-feature fractions\.![Refer to caption](https://arxiv.org/html/2608.08103v1/x17.png)Figure 17:One frozen pipeline, reported end to end\. Joint parent\-set refitting does not uniformly improve SHD over modular repair \(left\), but it changes the statistical target and permits selective skeleton and collider labels \(right\)\.All 320 regret bounds cover the independent oracle\-score audit\. Among 3,042 certified skeleton labels and 2,396 certified collider labels, none disagrees with the corresponding oracle\-score optimum\. Decision rates vary with the frontend screen: adjacency labels range from10\.5%10\.5\\%to44\.5%44\.5\\%, and collider labels from4\.4%4\.4\\%to36\.3%36\.3\\%\. The parent\-set score lowers mean SHD relative to the reported frontend for SDCD, but trails modular repair for DAGMA and GOLEM\. Its mean SHD is14\.6314\.63, nearly the14\.5614\.56obtained by replacing the holdout score with a 20,000\-sample oracle score\. Across all 320 cells, parent\-set SHD and screen recall have Pearson correlation−0\.886\-0\.886\. The deterioration is therefore not a finite\-holdout effect that tighter intervals would repair; it reflects the screened family and the score’s population target\. Averaged over frontends, modular and reported frontend SHDs are12\.9112\.91and16\.8716\.87\.

The score certificate is not a causal\-truth certificate\. Final truth\-only evaluation finds disagreement on4\.4%4\.4\\%of certified skeleton labels and5\.5%5\.5\\%of certified collider labels\. Those errors can arise from candidate screen omissions, the finite oracle proxy, or a population score whose minimizer differs from the generating DAG\. Reporting both comparisons keeps score uncertainty separate from causal identification\.

### G\.5Screen\-budget diagnosis

The top\-2​d2dscreen in the primary audit is a protocol choice, not an oracle density estimate\. We rerun all 320 frontend–dataset pairs with predeclared top\-2​d2d, top\-4​d4d, and top\-6​d6dbudgets\. Every frontend fit, data split, four\-parent cap, loss, and penalty is unchanged; no budget is selected using truth\. “Raw recall” is measured before the per\-child cap, whereas “family recall” describes the parent sets actually passed to the certificate\.

Table 19:Frozen screen\-budget audit over 960 cells\. Coverage compares the holdout regret bound with the independent 20,000\-sample score optimum\.The extra candidates raise raw recall by 5\.3 percentage points, but the local cap passes only 2\.1 points to the parent\-set family\. More importantly, the oracle\-score solution does not improve: its mean SHD rises from 14\.56 to 15\.67\. The effect is frontend\-dependent\. At top\-4​d4d, oracle\-parent SHD is 8\.54 for DAGMA and 8\.18 for GOLEM, but 21\.65 for NOTEARS and 23\.28 for SDCD\. Thus the top\-2​d2dresult is not explained by screen omissions alone\. Enlarging the family recovers some omitted edges while admitting alternatives favored by the frozen score but not by the generating DAG\.

### G\.6Bounded\-parent scaling

The unified accuracy audit usesd=20d=20so that statistical and score\-target effects are not mixed with solver failures\. We separately test whether the interaction\-aware implementation itself stops at that dimension\. A truth\-blind absolute\-correlation screen supplies at most two candidate parents per node\. For each instance we solve the global reference certificate and 20 feature queries fixed by index before scores are evaluated: ten skeleton adjacencies and ten collider candidates\. The grid contains ER2 and SF4, standard and mixed\-strength Gaussian SEMs, and five seeds, giving 20 cases at each dimension\.

Table 20:Interaction\-aware computational scaling\. Times are seconds for the global certificate \(core\) and the batch of 20 forced\-opposite queries\.All 80 core solves and all 1,600 feature queries terminate\. Atd=200d=200, the median core time is 1\.80 seconds and the median time for 20 queries is 12\.80 seconds\. This is an empirical bounded\-parent result, not a polynomial\-time guarantee: runtimes grow superlinearly, two\-parent families capture only local pair interactions, and denser candidate families remain subject to the worst\-case complexity of exact Bayesian\-network structure learning\. The edge\-modular LP in Appendix[F](https://arxiv.org/html/2608.08103#A6)is the scalable conservative option when that restriction is unacceptable\.