当参数梯度流在表示空间中闭合时

# When Parameter Gradient Flows Close in Representation Space
Source: [https://arxiv.org/html/2610.27474](https://arxiv.org/html/2610.27474)
Shengye TaoAffiliation:Beijing Institute of Mathematical Sciences and Applications \(BIMSA\)Affiliation:Beijing, ChinaEmail:[mailto:](mailto:)[ShengyeTao@bimsa\.cn](mailto:[email protected])

October 20, 2026

###### Abstract

The represented gradient flow of a parameterized model depends on a mobility that can retain hidden parameter information\. We study when one entire represented mobility map works for every autonomous loss from one fixed initialization\. For two\-factor matrix representations wider than both output dimensions, with individually full\-row\-rank factors, we classify all admissible joint ranks: closure occurs precisely at effective scalar\-balanced minimal joint rank or a two\-value signed spectral family at full joint rank\. All intermediate ranks are excluded; positive represented flows belong to known square\-root families\. For general representations with constant ambient representation and gradient\-Lie ranks, ordinary closure is equivalent to a local invariant\-level metric in the analytic case and specified smooth regimes\. An endpoint section forces the initially unrestricted mobility map to admit a locally smooth representative on the represented image\. We also separate testing families: linear losses locally admit closure at every submersion, whereas one fixed translated strongly convex quadratic family fails it on a delayed\-bracket example\. A fixed output readout gives an ordinary squared\-loss witness\. These supporting results do not characterize quadratic\-only closure in general\. We retain the unresolved smooth output\-rank\-two case with Lie rank at least four\. A frozen deterministic experiment checks a separate engineered\-loss construction; it is distinguished from the squared\-loss proof, universal theorems and empirical learning performance\.

## 1Introduction

For an open parameter domainΩ⊂ℝD\\Omega\\subset\\mathbb\{R\}^\{D\}and a smooth representationϕ:Ω→ℝd\\phi:\\Omega\\to\\mathbb\{R\}^\{d\}, parameter gradient flow of a represented lossffgives

θ˙=−D​ϕ​\(θ\)T​grad⁡f⁡\(ϕ⁡\(θ\)\),z˙=−M⁡\(θ\)​grad⁡f⁡\(z\),M=D​ϕ​D​ϕT\.\\dot\{\\theta\}=\-D\\phi\(\\theta\)^\{T\}\\grad f\(\\phi\(\\theta\)\),\\qquad\\dot\{z\}=\-M\(\\theta\)\\grad f\(z\),\\qquad M=D\\phi D\\phi^\{T\}\.\(1\)States with the samezzmay have differentMM\. We ask when the entire mobility, rather than only its action on the gradient of a particular loss, is determined byzzalong all loss trajectories from one fixed initialization\. This is the ordinary intrinsic\-dynamic contract of\[[1](https://arxiv.org/html/2610.27474#bib.bib1)\], distinct from its stronger piecewise\-loss testing convention\.

The two results proved below are an initialization classification for wide two\-factor matrix representations and a converse from ordinary trajectory tests to invariant\-level metric factorization for regular representations in stated analytic and smooth regimes\. All positive matrix mobilities reduce to known scalar\-balanced flows; no new optimizer, optimization rate or task gain is claimed\. The distinction is a necessity theory for eliminating hidden mobility, not a new formula for the already known positive flow\. The loss\-family quantifier is substantive: Proposition[6\.1](https://arxiv.org/html/2610.27474#S6.Thmtheorem1)shows that all linear tests can pass locally at a submersion, while Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3)and Corollary[6\.4](https://arxiv.org/html/2610.27474#S6.Thmtheorem4)give exact nonclosure with familiar output least\-squares losses in a fixed model\. The latter are supporting exclusion controls, not new learning algorithms\. Generic continuous initializations in the nonscalar wide model fail this exact closure, as a direct consequence of the classification\. This does not exclude approximate closure or a simpler description under one particular loss\.

The work uses the conservation\-law and Lie\-distribution framework of\[[2](https://arxiv.org/html/2610.27474#bib.bib2),[1](https://arxiv.org/html/2610.27474#bib.bib1)\]\. In particular the gradient Lie algebra and maximal\-stack factor\-orbit dimensions in the matrix proof are prior ingredients, explicitly attributed below\. The additional inference under study is how single\-initialization, autonomous\-loss tests constrain Gram fibers and a local invariant\-level mobility\. Familiar Frobenius, constant\-rank, inverse\-function, orbit and ODE results are used as standard local theorems, with their hypotheses identified at each application\.

#### Results and reading guide\.

Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)gives an initialization classification, including the distinction between minimal, intermediate, and full joint rank\. Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)supplies the fixed\-start converse in its stated regular regimes\. Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3)separates linear testing from a fixed\-Hessian quadratic family; Corollary[6\.4](https://arxiv.org/html/2610.27474#S6.Thmtheorem4)transfers the witness to squared error after one fixed readout\. The scalar\-output case and positive square\-root flows are known\. Section[7](https://arxiv.org/html/2610.27474#S7)compares these statements with specific predecessor results\. Complete proofs are in the appendices\.

## 2Problem Setting and Definitions

Fixθ0∈Ω\\theta\_\{0\}\\in\\Omega\. A solution on an open set means its forward maximal local interval before exiting that set; no global\-time existence is assumed\. Losses are global𝒞2\\mathcal\{C\}^\{2\}mapsf:ℝd→ℝf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}without a common bound on their derivatives\. Neighborhoods can depend on initialization but not on the tested loss\.

###### Definition 2\.1\(Ordinary entire\-mobility closure\)\.

Property \(O\) holds atθ0\\theta\_\{0\}if there exist an open neighborhoodU∋θ0U\\ni\\theta\_\{0\}and a single mapK:ℝd→ℝd×dK:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}such that every solution of \([1](https://arxiv.org/html/2610.27474#S1.E1)\) fromθ0\\theta\_\{0\}, for every global𝒞2\\mathcal\{C\}^\{2\}loss, satisfiesM⁡\(θ⁡\(t\)\)=K⁡\(ϕ⁡\(θ⁡\(t\)\)\)M\(\\theta\(t\)\)=K\(\\phi\(\\theta\(t\)\)\)on its interval inUU\. The mapKKis independent of the loss and is initially unrestricted: continuity, boundedness and differentiability are not assumptions\.

###### Definition 2\.2\(Local invariant\-level metric\)\.

Property \(I\) holds if, on a parameter neighborhood, there are finitely many common𝒞1\\mathcal\{C\}^\{1\}conservation lawsh=\(h1,…,hk\)h=\(h\_\{1\},\\ldots,h\_\{k\}\)and one globally𝒞1\\mathcal\{C\}^\{1\}mapK¯:ℝd→ℝd×d\\overline\{K\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}such that

M\(θ\)=K¯\(ϕ\(θ\)\)for everyθ∈V∩\{h=h\(θ0\)\}M\(\\theta\)=\\overline\{K\}\(\\phi\(\\theta\)\)\\quad\\hbox\{for every \}\\theta\\in V\\cap\\\{h=h\(\\theta\_\{0\}\)\\\}for some openV∋θ0V\\ni\\theta\_\{0\}\. Common conservation meansD​ϕ​grad⁡hj=0D\\phi\\,\\grad h\_\{j\}=0on the domain ofhh\.

The lawshhare local, as in the conservation\-law convention of\[[1](https://arxiv.org/html/2610.27474#bib.bib1)\]; the represented extensionK¯\\overline\{K\}is global\. Equality is required on the actual local level, not merely on its tangent\. Conservation and substitution immediately give \(I\)⇒\\Rightarrow\(O\)\. The strong intrinsic dynamic property of\[[1](https://arxiv.org/html/2610.27474#bib.bib1), Definition I\.2\]instead tests piecewise𝒞2\\mathcal\{C\}^\{2\}represented lossesff, requiring a continuous trajectory fromθ0\\theta\_\{0\}to follow−∇\(f∘ϕ\)\-\\nabla\(f\\circ\\phi\)whereverffis differentiable along its represented path, with one common neighborhood and one commonKK\. This is a spatial piecewise\-loss convention; arbitrary time\-dependent switching is a separate testing class\. Strong closure implies ordinary closure, so an obstruction to strong closure alone does not prove failure of ordinary closure\. Finite flow concatenations below are auxiliary geometric constructions; each ordinary witness is realized by an actual smooth autonomous loss from the fixed initial state\.

No positive definiteness on all target space is part of these definitions: the mobility may be semidefinite when the representation rank is less thandd\.

LetXi=grad⁡ϕiX\_\{i\}=\\grad\\phi\_\{i\}and letEθE\_\{\\theta\}be the span of the values atθ\\thetaof their generated Lie algebra\. We use the bracket\[X,Y\]=D​Y​X−D​X​Y\[X,Y\]=DY\\,X\-DX\\,Y\. WhendimE=p\\dim E=pis constant on an ambient neighborhood, selecting a nonzero minor of finitely many bracket fields gives a smooth frame forEE\. The frame brackets lie inEE, so Frobenius applies\. It supplies a local product box with common first integralshhand a connected initial plaqueL=\{h=h\(θ0\)\}L=\\\{h=h\(\\theta\_\{0\}\)\\\}of dimensionpp\. Ifp=Dp=D, takehhconstant\. Ifrank⁡D​ϕ=q\\rank D\\phi=qis constant as well, then

rank⁡D⁡\(ϕ\|L\)=q≤p,\\rank D\(\\phi\|\_\{L\}\)=q\\leq p,\(2\)becauseim⁡D​ϕT⊂T​L\\operatorname\{im\}D\\phi^\{T\}\\subset TLandim⁡\(D​ϕ​D​ϕT\)=im⁡D​ϕ\\operatorname\{im\}\(D\\phi D\\phi^\{T\}\)=\\operatorname\{im\}D\\phi\. The analytic versions follow by selecting analytic frames and using analytic Frobenius\. These statements concern one local plaque and require neither globally embedded leaves nor a global quotient\.

###### Lemma 2\.3\(Compatible loss on an embedded represented arc\)\.

Let a smooth parameter curve satisfyθ˙=−D​ϕ​\(θ\)T​g​\(t\)\\dot\{\\theta\}=\-D\\phi\(\\theta\)^\{T\}g\(t\)with smoothgg, and suppose its represented curveγ=ϕ∘θ\\gamma=\\phi\\circ\\thetais a regular embedded compact arc admitting a slightly extended embedded arc\. Then this curve is a trajectory of one global smooth autonomous represented loss from its given initial point\.

Proof\.See Appendix[A\.1](https://arxiv.org/html/2610.27474#A1.SS1)\.

###### Lemma 2\.4\(Clock\-positive endpoint patch\)\.

Suppose a local embedded manifoldL∋θ0L\\ni\\theta\_\{0\}is invariant for all fieldsYg=−D​ϕT​gY\_\{g\}=\-D\\phi^\{T\}g,g∈ℝdg\\in\\mathbb\{R\}^\{d\}, and their Lie values spanTθ​LT\_\{\\theta\}LonLL\. AssumeM⁡\(θ0\)≠0M\(\\theta\_\{0\}\)\\neq 0\. Every sufficiently small ambient neighborhood admits a nonemptyLL\-open set of endpoints fromθ0\\theta\_\{0\}, each realized by one global smooth autonomous represented loss with its whole path in that neighborhood\. Consequently\(O\)impliesM=K∘ϕM=K\\circ\\phion this set\. Constant Lie rank in the ambient parameter space is not required\.

Proof\.See Appendix[A\.2](https://arxiv.org/html/2610.27474#A1.SS2)\.

###### Lemma 2\.5\(Linear losses force local regularity\)\.

Supposeϕ\\phiis smooth with constant ambient rankq\>0q\>0nearθ0\\theta\_\{0\}\. Under\(O\), its common arbitraryKKagrees on the whole local represented image nearz0=ϕ⁡\(θ0\)z\_\{0\}=\\phi\(\\theta\_\{0\}\)with a smooth function, admitting a smooth extensionKlocK\_\{\\rm loc\}to a target neighborhood\. Ifϕ\\phiis analytic the section and local extension can be analytic\. No Lie\-rank assumption is needed\.

Proof\.See Appendix[A\.3](https://arxiv.org/html/2610.27474#A1.SS3)\.

###### Lemma 2\.6\(Local descent and global represented extension\)\.

LetLLbe a local plaque on whichϕ\\phihas constant rankqq\. IfD⁡\(M\|L\)D\(M\|\_\{L\}\)annihilatesker⁡D⁡\(ϕ\|L\)\\ker D\(\\phi\|\_\{L\}\)throughout a neighborhood ofθ0\\theta\_\{0\}inLL, thenM=k∘ϕM=k\\circ\\phion a smaller plaque neighborhood for a smooth local representedkk\. IfLLis the exact local level of common laws, this gives\(I\)after shrinking the parameter box\.

Proof\.See Appendix[A\.4](https://arxiv.org/html/2610.27474#A1.SS4)\.

## 3Complete classification for wide two\-factor representations

We identify matrix spaces with Euclidean spaces using the Frobenius inner product\. Put

N=n\+m,J=diag⁡\(In,−Im\),F=\[UV\],C=F​F⊤=\[AZZ⊤B\],S=F⊤​J​F\.N=n\+m,\\qquad J=\\diag\(I\_\{n\},\-I\_\{m\}\),\\qquad F=\\begin\{bmatrix\}U\\\\ V\\end\{bmatrix\},\\qquad C=FF^\{\\top\}=\\begin\{bmatrix\}A&Z\\\\ Z^\{\\top\}&B\\end\{bmatrix\},\\qquad S=F^\{\\top\}JF\.HereU∈ℝn×rU\\in\\mathbb\{R\}^\{n\\times r\},V∈ℝm×rV\\in\\mathbb\{R\}^\{m\\times r\}, andϕ⁡\(F\)=Z=U​V⊤\\phi\(F\)=Z=UV^\{\\top\}\. Direct differentiation gives

D​ϕ​\(F\)​\[δ​U,δ​V\]=δ​U​V⊤\+U​δ​V⊤,D​ϕ​\(F\)⊤​G=\[G​VG⊤​U\]\.D\\phi\(F\)\[\\delta U,\\delta V\]=\\delta UV^\{\\top\}\+U\\delta V^\{\\top\},\\qquad D\\phi\(F\)^\{\\top\}G=\\begin\{bmatrix\}GV\\\\ G^\{\\top\}U\\end\{bmatrix\}\.Consequently the full represented mobility is the linear operator

MF​\[G\]=A​G\+G​B,A=U​U⊤,B=V​V⊤\.M\_\{F\}\[G\]=AG\+GB,\\qquad A=UU^\{\\top\},\\quad B=VV^\{\\top\}\.\(3\)ForG∈ℝn×mG\\in\\mathbb\{R\}^\{n\\times m\}, write

HG=\[0GG⊤0\],XG​\(F\)=−HG​F\.H\_\{G\}=\\begin\{bmatrix\}0&G\\\\ G^\{\\top\}&0\\end\{bmatrix\},\\qquad X\_\{G\}\(F\)=\-H\_\{G\}F\.These are the parameter gradient fields of the linear represented losses\. An arbitrary represented lossf∈C2​\(ℝn×m,ℝ\)f\\in C^\{2\}\(\\mathbb\{R\}^\{n\\times m\},\\mathbb\{R\}\)gives

F˙=−Hgrad⁡f⁡\(Z\)​F\.\\dot\{F\}=\-H\_\{\\grad f\(Z\)\}F\.\(4\)
For clarity, property\(O\)atF0F\_\{0\}means that one mapK:ℝn×m→ℝ\(n​m\)×\(n​m\)K:\\mathbb\{R\}^\{n\\times m\}\\to\\mathbb\{R\}^\{\(nm\)\\times\(nm\)\}, independent offf, equals the entireMFM\_\{F\}along every such autonomous flow fromF0F\_\{0\}, until first exit from one common parameter neighborhood\. No regularity ofKKis assumed\. Property\(I\)means that a finite family of common localC1C^\{1\}conservation lawshh, together with a globallyC1C^\{1\}represented mapK¯\\overline\{K\}, givesMF=K¯​\(ϕ​\(F\)\)M\_\{F\}=\\overline\{K\}\(\\phi\(F\)\)on the exact local levelh⁡\(F\)=h⁡\(F0\)h\(F\)=h\(F\_\{0\}\)\. The conservation laws need only be defined on a parameter neighborhood\. Conservation and the chain rule show that\(I\)implies\(O\)\.

The gradient Lie algebra and the factor\-orbit dimensions used below are prior ingredients of the conservation\-law framework of Marcotte, Gribonval, and Peyré\[[2](https://arxiv.org/html/2610.27474#bib.bib2)\], Propositions H\.3 and H\.5\. We include their elementary linear\-algebra derivations to make the present argument complete\. Likewise the positive scalar\-balanced square\-root dynamics are known\[[1](https://arxiv.org/html/2610.27474#bib.bib1),[3](https://arxiv.org/html/2610.27474#bib.bib3)\]; the classification below concerns their initialization conditions\.

###### Theorem 3\.1\(Wide initialization classification\)\.

Assumen,m≥1n,m\\geq 1,r\>max⁡\(n,m\)r\>\\max\(n,m\), andrank⁡U0=n\\rank U\_\{0\}=n,rank⁡V0=m\\rank V\_\{0\}=m\. SetL0=max⁡\(n,m\)L\_\{0\}=\\max\(n,m\),s=rank⁡F0s=\\rank F\_\{0\}, andC0=F0​F0⊤C\_\{0\}=F\_\{0\}F\_\{0\}^\{\\top\}\. Properties\(O\)and\(I\), with the latter interpreted in the original width\-rrparameter space, are equivalent to the union of the following disjoint cases:

s=L0,C0​J​C0=λ​C0for some​λ∈ℝ;\\displaystyle s=L\_\{0\},\\qquad C\_\{0\}JC\_\{0\}=\\lambda C\_\{0\}\\quad\\hbox\{for some \}\\lambda\\in\\mathbb\{R\};\(5\)s=N,spec\(JC0\)=\{awith multiplicityn,−bwith multiplicitym\},a,b\>0\.\\displaystyle s=N,\\qquad\\operatorname\{spec\}\(JC\_\{0\}\)=\\\{a\\text\{ with multiplicity \}n,\\,\-b\\text\{ with multiplicity \}m\\\},\\quad a,b\>0\.\(6\)In \([5](https://arxiv.org/html/2610.27474#S3.E5)\),n\>mn\>mforcesλ\>0\\lambda\>0,m\>nm\>nforcesλ<0\\lambda<0, andn=mn=mpermits anyλ\\lambda\. There are no positive cases withL0<s<NL\_\{0\}<s<N\. The mobility in either positive case is

Kσ​\(Z\)​\[G\]=σ2​In\+Z​Z⊤​G\+G​σ2​Im\+Z⊤​Z,K\_\{\\sigma\}\(Z\)\[G\]=\\sqrt\{\\sigma^\{2\}I\_\{n\}\+ZZ^\{\\top\}\}\\,G\+G\\sqrt\{\\sigma^\{2\}I\_\{m\}\+Z^\{\\top\}Z\},\(7\)whereσ=\|λ\|/2\\sigma=\|\\lambda\|/2in \([5](https://arxiv.org/html/2610.27474#S3.E5)\) andσ=\(a\+b\)/2\\sigma=\(a\+b\)/2in \([6](https://arxiv.org/html/2610.27474#S3.E6)\)\. Forσ=0\\sigma=0, the formula is used on an invertible represented neighborhood and extended smoothly by a cutoff\.

Figure 1:Initialization classification in Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)\. HereL0=max⁡\(n,m\)L\_\{0\}=\\max\(n,m\),N=n\+mN=n\+m,s=rank⁡F0s=\\rank F\_\{0\},C0=F0​F0⊤C\_\{0\}=F\_\{0\}F\_\{0\}^\{\\top\}, andJ=diag⁡\(In,−Im\)J=\\diag\(I\_\{n\},\-I\_\{m\}\)\. Only rankss≤rs\\leq rare attainable\. The assumptionsr\>L0r\>L\_\{0\},rank⁡U0=n\\rank U\_\{0\}=n, andrank⁡V0=m\\rank V\_\{0\}=mapply to every row\. At minimal joint rank,n\>mn\>mforcesλ\>0\\lambda\>0andm\>nm\>nforcesλ<0\\lambda<0; whenn=mn=m, either sign or zero is allowed\. This is a theorem summary, not a numerical experiment\.### 3\.1Factor orbits and the ordinary\-closure obstruction

###### Lemma 3\.2\(Boost algebra and factor\-orbit dimension\)\.

The fieldsXGX\_\{G\}generate the left action of

𝔬⁡\(n,m\)=\{T∈ℝN×N:T⊤​J\+J​T=0\}\.\\mathfrak\{o\}\(n,m\)=\\\{T\\in\\mathbb\{R\}^\{N\\times N\}:T^\{\\top\}J\+JT=0\\\}\.At a factor matrix of rankss, its local orbit has dimension

dF=\(N2\)−\(N−s2\)=s⁡\(2​N−s−1\)2\.d\_\{F\}=\\binom\{N\}\{2\}\-\\binom\{N\-s\}\{2\}=\\frac\{s\(2N\-s\-1\)\}\{2\}\.\(8\)This formula does not requireJJrestricted toim⁡F\\operatorname\{im\}Fto be nondegenerate\. Every commonC1C^\{1\}conservation law is constant on the connected local orbit\.

Proof\.See Appendix[B\.1](https://arxiv.org/html/2610.27474#A2.SS1)\.

###### Lemma 3\.3\(A necessary Gram\-orbit dimension bound\)\.

IfF0≠0F\_\{0\}\\neq 0satisfies\(O\), then the congruence orbit ofC0=F0​F0⊤C\_\{0\}=F\_\{0\}F\_\{0\}^\{\\top\}under theJJ\-orthogonal group has dimensiondC≤n​md\_\{C\}\\leq nm\. The same bound follows from\(I\), for arbitrary finite families of commonC1C^\{1\}conservation laws\.

Proof\.See Appendix[B\.2](https://arxiv.org/html/2610.27474#A2.SS2)\.

### 3\.2The full\-column and degenerate\-inertia calculations

###### Lemma 3\.4\(Full\-column Gram rank\)\.

SupposeF∈ℝN×sF\\in\\mathbb\{R\}^\{N\\times s\}has full column rank, and letc1,…,cℓc\_\{1\},\\ldots,c\_\{\\ell\}be the multiplicities of all distinct eigenvalues of the symmetricS=F⊤​J​FS=F^\{\\top\}JF, including zero when present\. Then

dC=s⁡\(2​N−s−1\)2−∑j=1ℓ\(cj2\)\.d\_\{C\}=\\frac\{s\(2N\-s\-1\)\}\{2\}\-\\sum\_\{j=1\}^\{\\ell\}\\binom\{c\_\{j\}\}\{2\}\.\(9\)

Proof\.See Appendix[B\.3](https://arxiv.org/html/2610.27474#A2.SS3)\.

###### Lemma 3\.5\(Strict excess at intermediate width\)\.

Formax⁡\(n,m\)<s<N\\max\(n,m\)<s<Nand full\-columnF∈ℝN×sF\\in\\mathbb\{R\}^\{N\\times s\}, one hasdC\>n​md\_\{C\}\>nm, whether or notSSis singular\.

Proof\.See Appendix[B\.4](https://arxiv.org/html/2610.27474#A2.SS4)\.

###### Lemma 3\.6\(The minimal full\-column width\)\.

Supposes=L0=max⁡\(n,m\)s=L\_\{0\}=\\max\(n,m\),FFis full column rank, and both blocks have their full row ranks\. ThendC≤n​md\_\{C\}\\leq nmif and only ifS=λ​IsS=\\lambda I\_\{s\}\. This is equivalent toC​J​C=λ​CCJC=\\lambda C, with the sign conditions in Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)\.

Proof\.See Appendix[B\.5](https://arxiv.org/html/2610.27474#A2.SS5)\.

### 3\.3Full joint rank and the signed spectral family

###### Lemma 3\.7\(Full\-row Gram orbit and its positive family\)\.

SupposeF∈ℝN×rF\\in\\mathbb\{R\}^\{N\\times r\}has full row rank, soC=F​F⊤\>0C=FF^\{\\top\}\>0\. Letcic\_\{i\}anddjd\_\{j\}be the multiplicities of the distinct positive and negative eigenvalue values, respectively, ofJ​CJC\. Then

dC−n​m=n2−∑ici2\+m2−∑jdj22\.d\_\{C\}\-nm=\\frac\{n^\{2\}\-\\sum\_\{i\}c\_\{i\}^\{2\}\+m^\{2\}\-\\sum\_\{j\}d\_\{j\}^\{2\}\}\{2\}\.\(10\)ThusdC≤n​md\_\{C\}\\leq nmprecisely whenJ​CJChas one positive valueaaand one negative value−b\-b, with multiplicitiesn,mn,m\. At such an initialization the original ambient canonical levelF⊤​J​F=F0⊤​J​F0F^\{\\top\}JF=F\_\{0\}^\{\\top\}JF\_\{0\}has mobility \([7](https://arxiv.org/html/2610.27474#S3.E7)\), withσ=\(a\+b\)/2\>0\\sigma=\(a\+b\)/2\>0\.

Proof\.See Appendix[B\.6](https://arxiv.org/html/2610.27474#A2.SS6)\.

### 3\.4Effective hidden width and the original ambient level

###### Lemma 3\.8\(Exact width reduction for fixed\-start trajectories\)\.

Lets=rank⁡F0s=\\rank F\_\{0\}, and choose an orthogonal hidden matrixQQwithF0​Q=\[F~0,0\]F\_\{0\}Q=\[\\widetilde\{F\}\_\{0\},0\], whereF~0\\widetilde\{F\}\_\{0\}hasssindependent columns\. For every representedC2C^\{2\}loss, the original flow fromF0F\_\{0\}and the width\-ssflow fromF~0\\widetilde\{F\}\_\{0\}have identicalZ,A,B,MZ,A,B,Mon corresponding existence intervals\. Property\(O\)holds at one of these initializations if and only if it holds at the other\. If both original blocks have full row rank, so do the reduced blocks\.

Proof\.See Appendix[B\.7](https://arxiv.org/html/2610.27474#A2.SS7)\.

###### Lemma 3\.9\(Minimal\-rank sufficiency in the original width\)\.

Under the assumptions of Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1), supposes=L0s=L\_\{0\}andC0​J​C0=λ​C0C\_\{0\}JC\_\{0\}=\\lambda C\_\{0\}\. Then\(I\)holds in the original width\-rrspace, using the canonical common invarianth⁡\(F\)=F⊤​J​Fh\(F\)=F^\{\\top\}JF\.

Proof\.See Appendix[B\.8](https://arxiv.org/html/2610.27474#A2.SS8)\.

Proof of Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)\.See Appendix[B\.9](https://arxiv.org/html/2610.27474#A2.SS9)\.

### 3\.5The matching known positive dynamics

###### Proposition 3\.10\(Exact narrow scalar\-balanced realization\)\.

Forσ\>0\\sigma\>0and everyZ∈ℝn×mZ\\in\\mathbb\{R\}^\{n\\times m\}, the operator \([7](https://arxiv.org/html/2610.27474#S3.E7)\) is realized by widthmin⁡\(n,m\)\\min\(n,m\)factors with scalar hidden imbalance\. This includes rank\-deficientZZ\. Forσ=0\\sigma=0and square invertibleZZ, a balanced realization exists locally as well\. Thus the positive cases in Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)do not provide new represented continuous\-time trajectories\.

Proof\.See Appendix[B\.10](https://arxiv.org/html/2610.27474#A2.SS10)\.

###### Corollary 3\.11\(Generic continuously sampled wide factors\)\.

Letr\>max⁡\(n,m\)r\>\\max\(n,m\)\. For any jointly absolutely continuous initialization law on the full factor space, properties\(O\)and\(I\)have probability zero if\(n,m\)≠\(1,1\)\(n,m\)\\neq\(1,1\)\. For the scalar pair they have probability one\. No independence of the factor entries is required\.

Proof\.See Appendix[B\.11](https://arxiv.org/html/2610.27474#A2.SS11)\.

## 4From initialized trajectory tests to a local metric

We use the ordinary closure and local metric properties in Definitions[2\.1](https://arxiv.org/html/2610.27474#S2.Thmtheorem1)and[2\.2](https://arxiv.org/html/2610.27474#S2.Thmtheorem2)\. Throughout this section the parameter metric is the original Euclidean metric\. Write

M=D​ϕ​D​ϕ𝖳,ℋ=image⁡D​ϕ𝖳,ℰ=Lie⁡\{grad⁡ϕi:1≤i≤d\}\.M=D\\phi D\\phi^\{\\mathsf\{T\}\},\\qquad\\mathcal\{H\}=\\operatorname\{image\}D\\phi^\{\\mathsf\{T\}\},\\qquad\\mathcal\{E\}=\\operatorname\{Lie\}\\\{\\grad\\phi\_\{i\}:1\\leq i\\leq d\\\}\.Here the last expression denotes the pointwise distribution of the generated Lie algebra\. Unless stated otherwise, its rankppand the rankqqofD​ϕD\\phiare constant on an ambient open neighborhood of the initializationθ0\\theta\_\{0\}\. All subsequent shrinkages are inside the common neighborhood in the ordinary closure hypothesis\. Frobenius, the constant\-rank theorem, smooth dependence of ODE solutions, and tubular coordinates are used in their classical local forms\.

###### Lemma 4\.1\(The regular initial plaque\)\.

There is a smooth Frobenius product box with finitely many smooth common conservation lawshh, whose initial level is one connectedpp\-dimensional plaqueLLtangent toℰ\\mathcal\{E\}\. Ifϕ\\phiis real analytic, this box,LL, andhhcan be chosen real analytic\. OnLL,D⁡\(ϕ\|L\)D\(\\phi\|\_\{L\}\)has constant rankqq\.

Proof\.See Appendix[C\.1](https://arxiv.org/html/2610.27474#A3.SS1)\.

###### Theorem 4\.2\(Regular analytic and specified smooth regimes\)\.

For a real\-analytic representation with the two constant ambient ranks, ordinary closure is equivalent to a local invariant\-level metric\. The same equivalence holds for a smooth representation whenq≥3q\\geq 3, whenq≤1q\\leq 1, or whenq=2q=2andp≤3p\\leq 3\. The conservation laws and the global represented extension in the conclusion can be smooth\.

### 4\.1The analytic case and the exact descent

Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)in the analytic case\.See Appendix[C\.2](https://arxiv.org/html/2610.27474#A3.SS2)\.

### 4\.2Smooth horizontal paths in represented rank at least three

###### Lemma 4\.3\(An elementary embedded\-arc approximation\)\.

Letq≥3q\\geq 3and letw:\[0,T\]→ℝqw:\[0,T\]\\to\\mathbb\{R\}^\{q\},T\>0T\>0, be smooth, with compact image in an open coordinate set\. There are vectorsaj→0a\_\{j\}\\to 0for whichwj​\(t\)=w⁡\(t\)\+t​ajw\_\{j\}\(t\)=w\(t\)\+ta\_\{j\}are regular embedded smooth arcs, remain in the same coordinate set, havewj​\(0\)=w​\(0\)w\_\{j\}\(0\)=w\(0\), and converge towwinC1C^\{1\}\. Their other endpoints need only converge\.

Proof\.See Appendix[C\.3](https://arxiv.org/html/2610.27474#A3.SS3)\.

###### Lemma 4\.4\(Closure on a smooth horizontal path\)\.

Assume ordinary closure, the constant representation rank, andq≥3q\\geq 3\. On a sufficiently small parameter neighborhood, every smooth path fromθ0\\theta\_\{0\}tangent toℋ\\mathcal\{H\}satisfiesM=Kloc∘ϕM=K\_\{\\mathrm\{loc\}\}\\circ\\phi, whereKlocK\_\{\\mathrm\{loc\}\}is the forced smooth representative of Lemma[2\.5](https://arxiv.org/html/2610.27474#S2.Thmtheorem5)\.

Proof\.See Appendix[C\.4](https://arxiv.org/html/2610.27474#A3.SS4)\.

###### Lemma 4\.5\(Local finite\-word accessibility\)\.

On the regular plaqueLL, every point of a sufficiently small neighborhood ofθ0\\theta\_\{0\}is the endpoint of a finite horizontal flow word lying in a prescribed slightly larger plaque neighborhood\. Such words admit smooth horizontal time reparametrizations with the same endpoints\.

Proof\.See Appendix[C\.5](https://arxiv.org/html/2610.27474#A3.SS5)\.

Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq≥3q\\geq 3and the elementary low\-rank cases\.See Appendix[C\.6](https://arxiv.org/html/2610.27474#A3.SS6)\.

### 4\.3The rank\-two, three\-dimensional\-leaf case

###### Lemma 4\.6\(A finite horizontal curvature jet\)\.

Supposeq=2,p=3q=2,p=3\. OnLL, choose coordinates\(u,v,η\)\(u,v,\\eta\)with\(u,v\)=π\(u,v\)=\\pirepresented coordinates andθ0=\(0,0,0\)\\theta\_\{0\}=\(0,0,0\)\. Write its horizontal lifts as

H1=∂u\+A∂η,H2=∂v\+B∂η,\[H1,H2\]=c∂η\.H\_\{1\}=\\partial\_\{u\}\+A\\partial\_\{\\eta\},\\qquad H\_\{2\}=\\partial\_\{v\}\+B\\partial\_\{\\eta\},\\qquad\[H\_\{1\},H\_\{2\}\]=c\\partial\_\{\\eta\}\.Some finite ordered horizontal derivative ofccis nonzero atθ0\\theta\_\{0\}\. Ifmmis the least such order, the leading word tensor is symmetric and the homogeneous polynomialPm​\(e\)=Hem​c​\(θ0\)P\_\{m\}\(e\)=H\_\{e\}^\{m\}c\(\\theta\_\{0\}\),He=e1​H1\+e2​H2H\_\{e\}=e\_\{1\}H\_\{1\}\+e\_\{2\}H\_\{2\}, is not identically zero\. Form=0m=0this meansP0=c⁡\(θ0\)≠0P\_\{0\}=c\(\\theta\_\{0\}\)\\neq 0\.

Proof\.See Appendix[C\.7](https://arxiv.org/html/2610.27474#A3.SS7)\.

###### Lemma 4\.7\(Four rectangle families and a common fiber interval\)\.

Under the preceding hypotheses there are two independent unit base directions and their negatives, and one fixed smallε\>0\\varepsilon\>0, such that the signed rectangle loops in any of these four directions have horizontal endpoints covering one common open interval aboutθ0\\theta\_\{0\}in the initial one\-dimensional fiber\.

Proof\.See Appendix[C\.8](https://arxiv.org/html/2610.27474#A3.SS8)\.

For the next lemma, let𝒟=\{±e\(1\),±e\(2\)\}\\mathcal\{D\}=\\\{\\pm e^\{\(1\)\},\\pm e^\{\(2\)\}\\\}andε\>0\\varepsilon\>0be the directions and fixed length supplied by Lemma[4\.7](https://arxiv.org/html/2610.27474#S4.Thmtheorem7)\. Fore∈𝒟e\\in\\mathcal\{D\}, lete⟂e^\{\\perp\}be its counterclockwise rotation throughπ/2\\pi/2\. The rectangle of signed widthδ\\deltais the loop

0⟶ε​e⟶ε​e\+δ​e⟂⟶δ​e⟂⟶0,0\\longrightarrow\\varepsilon e\\longrightarrow\\varepsilon e\+\\delta e^\{\\perp\}\\longrightarrow\\delta e^\{\\perp\}\\longrightarrow 0,withδ\\deltasufficiently small\.

###### Lemma 4\.8\(The required planar approximation\)\.

For any sufficiently small nonzero base pointbb, one may selecte∈𝒟e\\in\\mathcal\{D\}withe⋅b<0e\\cdot b<0\. For fixed sufficiently small nonzeroδ\\delta, the rectangle in Lemma[4\.7](https://arxiv.org/html/2610.27474#S4.Thmtheorem7), followed by the straight tail from zero tobb, is a limit of regular embedded smooth base arcs with initial point exactly zero and terminal pointbb\. Their velocities converge inL1L^\{1\}and their total variations remain bounded\. Forb=0b=0their terminal points may instead converge to zero\. These assertions suffice to pass the corresponding horizontal lift endpoints through the forced smoothKlocK\_\{\\mathrm\{loc\}\}\.

Proof\.See Appendix[C\.9](https://arxiv.org/html/2610.27474#A3.SS9)\.

Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq=2,p=3q=2,p=3\.See Appendix[C\.10](https://arxiv.org/html/2610.27474#A3.SS10)\.

### 4\.4All\-order and Euclidean necessary conditions

###### Proposition 4\.9\(All\-order flatness at the initial plaque\)\.

Under smooth constant ambient ranks and ordinary closure, takeKloc=0K\_\{\\mathrm\{loc\}\}=0whenq=0q=0and otherwise use the smooth representative from Lemma[2\.5](https://arxiv.org/html/2610.27474#S2.Thmtheorem5)\. SetR=M−Kloc∘ϕR=M\-K\_\{\\mathrm\{loc\}\}\\circ\\phi\. Every finite plaque\-coordinate derivative ofR\|LR\|\_\{L\}atθ0\\theta\_\{0\}vanishes\.

Proof\.See Appendix[C\.11](https://arxiv.org/html/2610.27474#A3.SS11)\.

###### Proposition 4\.10\(A Euclidean output obstruction\)\.

Supposeq=2q=2\. Letu=a⋅ϕ\+c0u=a\\cdot\\phi\+c\_\{0\}andv=b⋅ϕv=b\\cdot\\phibe constant output\-linear combinations with independent gradientsX=grad⁡uX=\\grad u,Y=grad⁡vY=\\grad von the chosen neighborhood\. AssumeD2​u​Y∈ℋD^\{2\}u\\,Y\\in\\mathcal\{H\}there\. Ordinary closure holds if and only ifp=2p=2\. In particular this applies to an affineuu, or to an independent isotropic quadraticu=α​‖θ‖2/2\+a0⋅θ\+c0u=\\alpha\\\|\\theta\\\|^\{2\}/2\+a\_\{0\}\\cdot\\theta\+c\_\{0\}that is such an output combination\.

Proof\.See Appendix[C\.12](https://arxiv.org/html/2610.27474#A3.SS12)\.

Figure 2:An algebraic illustration forϕ2​\(x,y,z\)=\(x,y\+x2​z\)\\phi\_\{2\}\(x,y,z\)=\(x,y\+x^\{2\}z\)\. Two states have the same representation and different mobility\. These are exact rational states on one fiber; this panel does not assert that they are endpoints of two losses from the same initialization\. The actual fixed\-start witnesses are proved in Proposition[5\.1](https://arxiv.org/html/2610.27474#S5.Thmtheorem1)and Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3)\. No integration or learning benchmark is shown\.

## 5Exact autonomous\-loss exclusion controls

The following constructions illustrate the logical strength of the ordinary premise\. They are algebraic witnesses, not experiments or separate novelty claims\. They do not refute the bridge theorem, because they fail its ordinary closure assumption\.

###### Proposition 5\.1\(Arbitrarily delayed brackets and a two\-loss witness\)\.

For each integerk≥2k\\geq 2, letϕk​\(x,y,z\)=\(x,y\+xk​z\)\\phi\_\{k\}\(x,y,z\)=\(x,y\+x^\{k\}z\)\. Its representation rank is two and its gradient Lie rank is three everywhere\. At the origin all iterated brackets of lengths two throughkkvanish, while a bracket of lengthk\+1k\+1supplies the third direction\. The first differential mobility test passes at the origin, yet ordinary closure fails there, as witnessed by two global smooth autonomous losses\.

Proof\.See Appendix[D\.1](https://arxiv.org/html/2610.27474#A4.SS1)\.

###### Proposition 5\.2\(A rank\-four affine\-output exclusion\)\.

Forϕ⁡\(x,y,z,w\)=\(x,y\+x​z\+z​w\)\\phi\(x,y,z,w\)=\(x,y\+xz\+zw\),q=2q=2andp=4p=4everywhere, and ordinary closure fails at every initialization\.

Proof\.See Appendix[D\.2](https://arxiv.org/html/2610.27474#A4.SS2)\.

## 6What restricted loss families test

The arbitrary\-loss quantifier in property \(O\) is substantive\. For a familyℱ\\mathcal\{F\}of global output losses, write\(O\)ℱ\\textup\{\(O\)\}\_\{\\mathcal\{F\}\}for the same property with onlyf∈ℱf\\in\\mathcal\{F\}tested: one parameter neighborhoodUUand one entire matrix mapK:ℝd→ℝd×dK:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}, independent offf, must satisfy

M⁡\(θf​\(t\)\)=K⁡\(ϕ⁡\(θf​\(t\)\)\)M\(\\theta\_\{f\}\(t\)\)=K\(\\phi\(\\theta\_\{f\}\(t\)\)\)on every forward local solution interval before exit fromUU, from the same fixed initialization\. The matrixKKis initially arbitrary\. In particular the equality concerns the entire mobility, rather than only its product with the tested loss gradient\. Property \(O\) restricts to every smaller family, but its necessity conclusions cannot be transferred in the opposite direction without proof\. The following supporting results make this distinction explicit\.

###### Proposition 6\.1\(Linear tests at a submersion\)\.

Supposeϕ\\phiis smooth andD​ϕ​\(θ0\)D\\phi\(\\theta\_\{0\}\)has full row rankdd\. Then\(O\)ℱlin\\textup\{\(O\)\}\_\{\\mathcal\{F\}\_\{\\rm lin\}\}holds forℱlin=\{z↦gT​z:g∈ℝd\}\\mathcal\{F\}\_\{\\rm lin\}=\\\{z\\mapsto g^\{T\}z:g\\in\\mathbb\{R\}^\{d\}\\\}\. A commonKKcan be chosen globally smooth after shrinking the parameter neighborhood\. This holds for allgg, without a common bound on its norm\.

Proof\.See Appendix[E\.1](https://arxiv.org/html/2610.27474#A5.SS1)\.

###### Theorem 6\.3\(One fixed strongly convex quadratic family\)\.

Fix an integerk≥2k\\geq 2, a number0<γ<10<\\gamma<1, and

H=\(1γγ1\),fb​\(w\)=12​wT​H​w−bT​w,b∈ℝ2\.H=\\begin\{pmatrix\}1&\\gamma\\\\ \\gamma&1\\end\{pmatrix\},\\qquad f\_\{b\}\(w\)=\\tfrac\{1\}\{2\}w^\{T\}Hw\-b^\{T\}w,\\quad b\\in\\mathbb\{R\}^\{2\}\.Forϕk​\(x,y,z\)=\(x,y\+xk​z\)\\phi\_\{k\}\(x,y,z\)=\(x,y\+x^\{k\}z\)initialized at the origin,\(O\)\{fb:b∈ℝ2\}\\textup\{\(O\)\}\_\{\\\{f\_\{b\}:b\\in\\mathbb\{R\}^\{2\}\\\}\}fails\. More precisely, for any fixed0<c1<c20<c\_\{1\}<c\_\{2\}there area∗\>0a\_\{\*\}\>0and analytic parameter functionsbi​\(a\)b\_\{i\}\(a\)neara=0a=0, withbi​\(0\)=\(1/ci,0\)b\_\{i\}\(0\)=\(1/c\_\{i\},0\), such that the actual flows satisfy, for every0<a<a∗0<a<a\_\{\*\},

ϕk​\(θi​\(ci​a\)\)=\(a,0\),\\phi\_\{k\}\(\\theta\_\{i\}\(c\_\{i\}a\)\)=\(a,0\),and

\[M⁡\(θ2​\(c2​a\)\)−M⁡\(θ1​\(c1​a\)\)\]12=−γ​k2​\(c2−c1\)2​\(k\+1\)​\(k\+2\)​a2​k\+1\+O⁡\(a2​k\+2\)≠0\.\\bigl\[M\(\\theta\_\{2\}\(c\_\{2\}a\)\)\-M\(\\theta\_\{1\}\(c\_\{1\}a\)\)\\bigr\]\_\{12\}=\-\\frac\{\\gamma k^\{2\}\(c\_\{2\}\-c\_\{1\}\)\}\{2\(k\+1\)\(k\+2\)\}a^\{2k\+1\}\+O\(a^\{2k\+2\}\)\\neq 0\.\(11\)Both complete parameter paths converge uniformly to the origin asa↓0a\\downarrow 0\. The Hessian is fixed for both losses, with eigenvalues1−γ1\-\\gammaand1\+γ1\+\\gamma; the translations and endpoint times differ\.

Proof\.See Appendix[E\.2](https://arxiv.org/html/2610.27474#A5.SS2)\.

There is a second check on the coefficient in \([11](https://arxiv.org/html/2610.27474#S6.E11)\)\. For the shooting trajectories constructed in Appendix[E\.2](https://arxiv.org/html/2610.27474#A5.SS2), fixc∈\{c1,c2\}c\\in\\\{c\_\{1\},c\_\{2\}\\\}and writes=t/\(c​a\)s=t/\(ca\)andx=a​X​\(s\)x=aX\(s\)\. WriteF⁡\(x\)=xkF\(x\)=x^\{k\}, setv=y\+F⁡\(x\)​zv=y\+F\(x\)z, and letr1,r2r\_\{1\},r\_\{2\}be the negative output\-gradient components along any horizontal trajectory\. Then

x˙=r1\+F′​z​r2,z˙=F​r2,v˙=F′​z​x˙\+\(1\+F2\)​r2\.\\dot\{x\}=r\_\{1\}\+F^\{\\prime\}zr\_\{2\},\\quad\\dot\{z\}=Fr\_\{2\},\\quad\\dot\{v\}=F^\{\\prime\}z\\dot\{x\}\+\(1\+F^\{2\}\)r\_\{2\}\.For a path with increasingxx, elimination of time gives

dd​x​\{z​1\+F2\}=F1\+F2​vx\.\\frac\{d\}\{dx\}\\\{z\\sqrt\{1\+F^\{2\}\}\\\}=\\frac\{F\}\{\\sqrt\{1\+F^\{2\}\}\}v\_\{x\}\.Since the shooting endpoints havev⁡\(0\)=v⁡\(a\)=0v\(0\)=v\(a\)=0, integration by parts gives the exact hidden\-coordinate identity

z\(a\)=−11\+F​\(a\)2∫0av⁡\(x\)​F′​\(x\)\(1\+F​\(x\)2\)3/2dx\.z\(a\)=\-\\frac\{1\}\{\\sqrt\{1\+F\(a\)^\{2\}\}\}\\int\_\{0\}^\{a\}\\frac\{v\(x\)F^\{\\prime\}\(x\)\}\{\(1\+F\(x\)^\{2\}\)^\{3/2\}\}\\,dx\.The shooting solution hasx˙=1/c\+O⁡\(a\)\>0\\dot\{x\}=1/c\+O\(a\)\>0uniformly\. From the limiting equations and smooth inversion ofs↦X⁡\(s\)s\\mapsto X\(s\),

v⁡\(x\)=c​γ2​x​\(a−x\)\+O⁡\(a3\),0≤x≤a\.v\(x\)=\\frac\{c\\gamma\}\{2\}x\(a\-x\)\+O\(a^\{3\}\),\\qquad 0\\leq x\\leq a\.The uniform remainder contributesO⁡\(ak\+3\)O\(a^\{k\+3\}\)toz⁡\(a\)z\(a\); denominator corrections contributeO⁡\(a3​k\+2\)O\(a^\{3k\+2\}\), which is absorbed fork≥2k\\geq 2\. The leading integral again gives−cγkak\+2/\[2\(k\+1\)\(k\+2\)\]\-c\\gamma ka^\{k\+2\}/\[2\(k\+1\)\(k\+2\)\]\.

###### Corollary 6\.4\(Ordinary squared error with one fixed readout\)\.

LetW=H1/2W=H^\{1/2\}be the symmetric positive\-definite square root of the fixed Hessian in Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3), and fix the representationψk=W​ϕk\\psi\_\{k\}=W\\phi\_\{k\}\. Its ordinary entire\-mobility closure fails even when the tested losses are only

ℓτ​\(w\)=12​‖w−τ‖22,τ∈ℝ2\.\\ell\_\{\\tau\}\(w\)=\\tfrac\{1\}\{2\}\\\|w\-\\tau\\\|\_\{2\}^\{2\},\\qquad\\tau\\in\\mathbb\{R\}^\{2\}\.It has representation rank two and gradient Lie rank three everywhere, andD​Mψk​\(0\)=0DM\_\{\\psi\_\{k\}\}\(0\)=0\. The compared targets and endpoint times differ; the representation is fixed\.

Proof\.See Appendix[E\.3](https://arxiv.org/html/2610.27474#A5.SS3)\.

###### Proposition 6\.6\(Quadratic tests force section regularity\)\.

Supposeϕ\\phiis smooth and has constant ambient rankq\>0q\>0on a neighborhood ofθ0\\theta\_\{0\}\. Setz0=ϕ⁡\(θ0\)z\_\{0\}=\\phi\(\\theta\_\{0\}\)and test only

fb​\(z\)=12​‖z−z0‖22\+bT​\(z−z0\),b∈ℝd\.f\_\{b\}\(z\)=\\tfrac\{1\}\{2\}\\\|z\-z\_\{0\}\\\|\_\{2\}^\{2\}\+b^\{T\}\(z\-z\_\{0\}\),\\qquad b\\in\\mathbb\{R\}^\{d\}\.If\(O\)\{fb:b∈ℝd\}\\textup\{\(O\)\}\_\{\\\{f\_\{b\}:b\\in\\mathbb\{R\}^\{d\}\\\}\}holds, then its arbitrary commonKKagrees on a local represented image with a smooth representative\. That representative has a local ambient extension, and after shrinking the image it has a global smooth extension\. Ifϕ\\phiis analytic, the representative on the image and its local extension can be analytic\. This is an image\-section conclusion, not an invariant\-level factorization\.

Proof\.See Appendix[E\.4](https://arxiv.org/html/2610.27474#A5.SS4)\.

Proposition[6\.6](https://arxiv.org/html/2610.27474#S6.Thmtheorem6)identifiesKKon an actual endpoint section\. It says nothing by itself about other parameter points of the same represented fiber\. Proposition[6\.1](https://arxiv.org/html/2610.27474#S6.Thmtheorem1)and Remark[6\.2](https://arxiv.org/html/2610.27474#S6.Thmtheorem2)explain why section regularity alone cannot substitute for the full arbitrary\-loss necessity arguments\. The supporting results here remove a loss\-family limitation of the explicit negative control while retaining that distinction\. They imply no numerical value ofa∗a\_\{\*\}, empirical convergence order, universal quadratic\-family converse, or learning\-performance claim\.

## 7Related Work

#### Intrinsic dynamics and initialized tests\.

Marcotte, Peyré, and Gribonval\[[1](https://arxiv.org/html/2610.27474#bib.bib1)\]introduce the ordinary intrinsic\-dynamic and invariant\-metric notions used here, together with a stronger definition for piecewise𝒞2\\mathcal\{C\}^\{2\}autonomous losses\. Their Theorem 3\.8 gives a positive relaxed\-balance branch\. Their Theorem 3\.9 assumes full\-rank factors and hidden width at mostmax⁡\(n,m\)\\max\(n,m\), and excludes their strong intrinsic dynamic property outside relaxed balance\. Proposition 3\.10 already resolvesn=m=1n=m=1at arbitrary width\. Our wide classification addresses ordinary closure with individually full\-row factors; its scalar subcase is inherited\. Their Theorem 2\.14 starts from a differentiable invariant\-level metric and Theorem 2\.17 from recoverability\. The converse here starts from an unrestrictedKKand fixed\-start autonomous tests, including specified cases with Lie rank larger than representation rank\.

The conservation\-law framework of Marcotte, Gribonval, and Peyré\[[2](https://arxiv.org/html/2610.27474#bib.bib2)\], Theorem 3\.3 and Propositions H\.3/H\.5, supplies the Lie distribution, the count of independent common laws, and matrix factor\-orbit ingredients\. Their Proposition 3\.7 concerns recovery from invariants and representation\. The additional necessity step here passes from actual autonomous endpoint patches to mobility constraints on Gram fibers\. A factor\-orbit dimension count alone does not supply that step\.

#### Known positive matrix flows and larger closed states\.

Dominé et al\.\[[3](https://arxiv.org/html/2610.27474#bib.bib3)\], Theorem 5\.2 and Appendix C\.2\.1, recover the factors up to hidden orthogonal transformations, and thereby determine the two mobility Gram blocks, under theirλ\\lambda\-balanced and full\-rank\-product assumptions, giving the square\-root structure\. Bah et al\.\[[4](https://arxiv.org/html/2610.27474#bib.bib4)\], Lemma 2 and Propositions 10–11, give the balanced product flow and its Riemannian interpretation on fixed\-rank matrix manifolds\. These are positive reduction results\. The initialization exclusion and original ambient invariant\-level arguments in Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)answer a different necessity question\. Proposition[3\.10](https://arxiv.org/html/2610.27474#S3.Thmtheorem10)explicitly matches our positive flows to a known narrow family\.

Zhang and Peyré\[[5](https://arxiv.org/html/2610.27474#bib.bib5)\], Proposition 2\.1, close the flow on the full weight covariance, retaining the hidden Gram blocks\. The state here is onlyZ=U​V⊤Z=UV^\{\\top\}\. Importantly, their Proposition E\.2 also gives predictor\-only square\-root closure on the hyperbolically isotropic leafC​J​C=α2​JCJC=\\alpha^\{2\}Jafter the appropriate width normalization; this is thea=ba=bpart of our full\-rank positive branch\. For the unequal pair, the fixed shiftD=C−\(a−b\)​J/2D=C\-\(a\-b\)J/2satisfiesD​J​D=\(\(a\+b\)/2\)2​JDJD=\(\(a\+b\)/2\)^\{2\}Jand has the same covariance flow becauseHG​J\+J​HG=0H\_\{G\}J\+JH\_\{G\}=0\. Thus this positive reduction remains a known\-family consequence\. Their Remark E\.3 shows that conserved spectral values generally do not determine predictor velocity\. That two\-state example does not itself establish ordinary failure from one fixed initialization\. We make no novelty claim for covariance closure or the positive square\-root formula, and do not infer a global priority claim from this comparison\.

#### Geometric criteria and mirror flows\.

Li et al\.\[[6](https://arxiv.org/html/2610.27474#bib.bib6)\], Theorem 4\.9, obtain mirror flow from a regular commuting parametrization under their flow\-domain assumptions\. Theorem 4\.10 gives a necessary Lie\-bracket condition when a mirror representation works for all initializations and time\-dependent losses\. Mirror flow requires an integrable inverse\-Hessian metric, whereas properties \(O\) and \(I\) allow general mobility\. Our initialized autonomous\-loss premise and invariant\-level conclusion therefore have different quantifiers and targets\. Horstmeyer and Atay\[[7](https://arxiv.org/html/2610.27474#bib.bib7)\], Proposition 1 and Theorem 4, characterize exact lumpability by all\-state projectability and invariance of the vertical distribution under Lie differentiation\. Their criterion applies to a fixed vector field on smooth manifolds with the stated submersion/fiber hypotheses\. Applying it backwards here would first require proving nearby\-fiber consistency from initialized tests; this is precisely the extra implication we study\.

Achour, Kohn, and Rauhut\[[8](https://arxiv.org/html/2610.27474#bib.bib8)\], Theorem 4 and Corollary 6 in the April 2026 revision, show that for one\-dimensional linear convolutional networks with the specified strides larger than one, fixed norm\-difference invariants determine the NTK on the smooth neuromanifold, with finite fibers\. Their broader work also treats higher\-dimensional convolutions\. The finite\-fiber recovery geometry differs from the matrix\-factorization fibers andp\>qp\>qregimes considered here\. We do not extend the matrix classification to convolutional architectures\.

#### Optimization and implicit bias\.

Saxe et al\.\[[9](https://arxiv.org/html/2610.27474#bib.bib9)\]provide exact deep\-linear learning solutions for structured settings\. Arora et al\.\[[10](https://arxiv.org/html/2610.27474#bib.bib10)\]interpret depth\-induced preconditioning; Gunasekar et al\.\[[11](https://arxiv.org/html/2610.27474#bib.bib11)\]and Arora et al\.\[[12](https://arxiv.org/html/2610.27474#bib.bib12)\]study implicit regularization in matrix factorization\. These motivate examining represented dynamics, but a convergence or bias result under one loss is not an all\-loss mobility classification\. Min et al\.\[[13](https://arxiv.org/html/2610.27474#bib.bib13)\]bound convergence using imbalance and product singular values\. Nordio\[[14](https://arxiv.org/html/2610.27474#bib.bib14)\]studies training\-set dynamics with fixed readout and quadratic loss, retaining a hierarchy of Gram operators at depth\. Castin et al\.\[[15](https://arxiv.org/html/2610.27474#bib.bib15)\]use balancing to change LoRA conditioning\. Our results concern continuous Euclidean parameter gradient flow and claim neither their discrete\-time guarantees nor an improvement in learning performance\.

## 8Discussion and Limitations

Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)determines when the prediction matrix can serve as a complete mobility state for every autonomous loss from one fixed initialization\. In the nonscalar wide model, its generic nonclosure corollary rules out that exact description under absolutely continuous initialization\. It leaves open approximate closure, loss\-specific reductions, and dynamics on larger states such as a full covariance\.

The factors must be individually full row rank andr\>max⁡\(n,m\)r\>\\max\(n,m\)\. Individually deficient initializations are not classified\. The general bridge requires representation and gradient\-Lie ranks constant on an ambient open neighborhood; singular width reduction does not transfer that hypothesis\. The smooth caseq=2,p≥4q=2,p\\geq 4remains unresolved\. All\-order flatness alone cannot replace equality of a smooth function on a neighborhood\. The fixed\-Hessian quadratic witness is theoretical: no numericala∗a\_\{\*\}, quadratic grid, or empirical convergence order is asserted\. The historical engineered\-loss check in Appendix[G](https://arxiv.org/html/2610.27474#A7)is a different test\. Continuous\-time mobility equality does not imply equality of finite\-step updates\.

#### Use of AI\-assisted tools\.

The research records document AI\-assisted development of mathematical arguments, source comparisons, code, and manuscript text, including automated critiques\. AI tools were also used to prepare this release, check algebra and references, and inspect the build\. These checks do not constitute verification by human experts\. The author is responsible for the mathematical claims, citations, experimental reporting, and the decision to release the work\. No AI system is listed as an author\.

## References

- \[1\]Sibylle Marcotte, Gabriel Peyré, and Rémi Gribonval\.Intrinsic training dynamics of deep neural networks\.arXiv:2508\.07370v2; ICLR 2026, 2026\.[https://arxiv\.org/abs/2508\.07370v2](https://arxiv.org/abs/2508.07370v2)\.
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## Appendix AProofs of the common local constructions

### A\.1Proof of Lemma[2\.3](https://arxiv.org/html/2610.27474#S2.Thmtheorem3)

###### Proof\.

Choose a smooth orthonormal normal frameea​\(t\)e\_\{a\}\(t\)along the extended interval\. In a tubular neighborhood writeΨ⁡\(t,y\)=γ⁡\(t\)\+∑aya​ea​\(t\)\\Psi\(t,y\)=\\gamma\(t\)\+\\sum\_\{a\}y\_\{a\}e\_\{a\}\(t\)and define

A′​\(t\)=⟨g⁡\(t\),γ˙​\(t\)⟩,floc​\(Ψ⁡\(t,y\)\)=A⁡\(t\)\+∑aya​⟨g⁡\(t\),ea​\(t\)⟩\.A^\{\\prime\}\(t\)=\\langle g\(t\),\\dot\{\\gamma\}\(t\)\\rangle,\\qquad f\_\{\\rm loc\}\(\\Psi\(t,y\)\)=A\(t\)\+\\sum\_\{a\}y\_\{a\}\\langle g\(t\),e\_\{a\}\(t\)\\rangle\.Tangential compatibility and the prescribed normal derivatives givegrad⁡floc​\(γ⁡\(t\)\)=g⁡\(t\)\\grad f\_\{\\rm loc\}\(\\gamma\(t\)\)=g\(t\)\. Ford=1d=1there are no normal variables\. Multiply by a smooth cutoff supported inside the extended tube and identically one near the original compact arc, and extend by zero toℝd\\mathbb\{R\}^\{d\}\. This is one global smooth loss\. Uniqueness for its locally Lipschitz parameter ODE identifies the prescribed path with the actual autonomous trajectory\. The slight extension prevents endpoint regularity problems\. The scalar compatibility also givesA′=−gT​M​g=−‖θ˙‖2A^\{\\prime\}=\-g^\{T\}Mg=\-\\\|\\dot\{\\theta\}\\\|^\{2\}along the curve; we have not prescribed inconsistent values around a projected loop\. ∎

### A\.2Proof of Lemma[2\.4](https://arxiv.org/html/2610.27474#S2.Thmtheorem4)

###### Proof\.

Choosegcg\_\{c\}withgcT​M​\(θ0\)​gc\>0g\_\{c\}^\{T\}M\(\\theta\_\{0\}\)g\_\{c\}\>0\. On a smaller state box and a bounded convex open ballBBaboutgcg\_\{c\}, continuity givesgcT​M​\(θ\)​g≥c\>0g\_\{c\}^\{T\}M\(\\theta\)g\\geq c\>0for allg∈Bg\\in B\. Thus along these controls

dd​t​⟨gc,ϕ⁡\(θ⁡\(t\)\)⟩=−gcT​M​\(θ⁡\(t\)\)​g​\(t\)≤−c\.\\frac\{d\}\{dt\}\\langle g\_\{c\},\\phi\(\\theta\(t\)\)\\rangle=\-g\_\{c\}^\{T\}M\(\\theta\(t\)\)g\(t\)\\leq\-c\.Smooth controlled projected paths are regular and embedded\. A compact state box and uniform speed bound give a short horizonε\\varepsilonkeeping every finite word of controls inBBwith total duration less thanε\\varepsiloninside the chosen neighborhood\. The bound depends on total time, not the number of switches\. Differences of members ofBBspan all controls, so their Lie values still spanT​LTL\.

LetQQbe the maximum differential rank of all positive\-duration word endpoint maps fromθ0\\theta\_\{0\}with total time less thanε\\varepsilon\. Integer boundedness ensures a word and a strictly positive duration vector attainQQ\. On a small duration patch its rank is constantlyQQ, giving an embedded local imagePP\. Append a short positive segment of any fixedYgY\_\{g\},g∈Bg\\in B\. The augmented endpoint map has rank at mostQQby maximality\. Its\(Q\+1\)\(Q\+1\)\-minors remain zero as the appended duration decreases to zero, while the originalQQ\-minor persists\. The appended derivative isYgY\_\{g\}, soYgY\_\{g\}is tangent toPPthroughout a local patch\. Brackets of tangent fields are tangent\. The Lie spanning hypothesis forcesQ≥dimLQ\\geq\\dim L, hence equality\. This gives a word endpoint image with nonempty interior inLL, without claiming that it containsθ0\\theta\_\{0\}or arbitrary return fibers\.

Insert between successive fixed controls smooth flat\-ended convex ramps of one fixed durationδ\\delta, keeping the plateau durations as parameters\. On a compact neighborhood, uniform field and state\-Jacobian bounds imply for each ramp mapTδT\_\{\\delta\}

‖Tδ−I‖≤C​δ,‖D​Tδ−I‖≤eC​δ−1\.\\\|T\_\{\\delta\}\-I\\\|\\leq C\\delta,\\qquad\\\|DT\_\{\\delta\}\-I\\\|\\leq e^\{C\\delta\}\-1\.These follow from the flow integral and variational equations; no bound on the control’s time derivatives is needed\. Finite composition yields convergence of the endpoint maps in state and duration derivatives\. A full\-rank minor therefore persists for one sufficiently small positiveδ\\delta, with the extra times within the strict horizon margin\. The smoothed family itself, not merely its limit, has anLL\-open endpoint image\. The clock inequality still holds and each control is constant near the outer endpoints, so slight extensions remain embedded\. Lemma[2\.3](https://arxiv.org/html/2610.27474#S2.Thmtheorem3)realizes each by one global autonomous smooth loss\. The commonKKin \(O\) then gives the identity on the endpoint patch\. No derivative or limit of an unrestrictedKKhas been used\. ∎

### A\.3Proof of Lemma[2\.5](https://arxiv.org/html/2610.27474#S2.Thmtheorem5)

###### Proof\.

Constant rank gives a single local embedded imageSSof dimensionqq\. Fix a sufficiently shortτ\>0\\tau\>0\. For the global linear lossfg​\(z\)=gT​zf\_\{g\}\(z\)=g^\{T\}z, letθg​\(τ\)\\theta\_\{g\}\(\\tau\)be its actual endpoint fromθ0\\theta\_\{0\}\. Nearg=0g=0all paths remain in the ordinary neighborhood, and smooth dependence gives

Dg​θg​\(τ\)\|g=0=−τ​D​ϕ​\(θ0\)T,Dg​ϕ​\(θg​\(τ\)\)\|g=0=−τ​M​\(θ0\)\.D\_\{g\}\\theta\_\{g\}\(\\tau\)\|\_\{g=0\}=\-\\tau D\\phi\(\\theta\_\{0\}\)^\{T\},\\qquad D\_\{g\}\\phi\(\\theta\_\{g\}\(\\tau\)\)\|\_\{g=0\}=\-\\tau M\(\\theta\_\{0\}\)\.Restricted to theqq\-dimensional spaceim⁡D​ϕ​\(θ0\)\\operatorname\{im\}D\\phi\(\\theta\_\{0\}\), the second derivative is an isomorphism ontoTz0​ST\_\{z\_\{0\}\}S\. The inverse function theorem gives a smooth endpoint sections:S0→Ωs:S\_\{0\}\\to\\Omegawithϕ⁡\(s⁡\(z\)\)=z\\phi\(s\(z\)\)=zon an image neighborhoodS0S\_\{0\}\. Property \(O\) forcesK⁡\(z\)=M⁡\(s⁡\(z\)\)K\(z\)=M\(s\(z\)\)there\. FlattenSSin target coordinates and extend this map constantly in the normal coordinates to obtainKlocK\_\{\\rm loc\}\. Shrink the parameter neighborhood so its whole image lies inS0S\_\{0\}\. Analytic dependence of the linear\-loss flow and the analytic inverse function theorem give the analytic version\. Off\-image values of the originalKKare irrelevant\. ∎

### A\.4Proof of Lemma[2\.6](https://arxiv.org/html/2610.27474#S2.Thmtheorem6)

###### Proof\.

In a constant\-rank product chart onLL, the representation depends only onqqcoordinates and the assumed derivative identity makesMMindependent of the other coordinates\. Restrict to a product with connected fibers to factorMMthrough the local embedded image\. Extend normally in target coordinates to an openW∋z0W\\ni z\_\{0\}, obtainingKlocK\_\{\\rm loc\}\. Choose a smooth cutoffχ\\chicompactly supported inWW, equal to one nearz0z\_\{0\}, and define globally

K¯=χ​Kloc\+\(1−χ\)​Id,\\overline\{K\}=\\chi K\_\{\\rm loc\}\+\(1\-\\chi\)I\_\{d\},with the first term extended by zero off its support\. Shrink the Frobenius product box so its entire initial level lies in the factorization patch and its represented image lies whereχ=1\\chi=1\. This gives the actual local\-level identity and global smooth target extension\. The argument is local and does not identify disconnected global fibers\. The analytic local descent follows in analytic charts; the final cutoff is smooth\. ∎

## Appendix BProofs of the wide matrix classification

### B\.1Proof of Lemma[3\.2](https://arxiv.org/html/2610.27474#S3.Thmtheorem2)

###### Proof\.

The boostsHEi​jH\_\{E\_\{ij\}\}, together with their commutators, span all matrices

T=\[PGG⊤Q\],P⊤=−P,Q⊤=−Q\.T=\\begin\{bmatrix\}P&G\\\\ G^\{\\top\}&Q\\end\{bmatrix\},\\qquad P^\{\\top\}=\-P,\\quad Q^\{\\top\}=\-Q\.Indeed, holding the bottom index fixed in two boosts produces, up to sign, each elementary skew matrix in the top block; holding the top index fixed produces each elementary skew matrix in the bottom block\. A block of dimension one requires no skew generators\.

For the presentJJ\-orthogonal linear action, we use the classical local orbit theorem on a homogeneous orbit restricted to an embedded local orbit chart, with tangent\{T​F:T∈𝔬⁡\(n,m\)\}\\\{TF:T\\in\\mathfrak\{o\}\(n,m\)\\\}, and the short flows of a Lie\-generating family generate its connected local orbit\. Only a local orbit chart is needed, not global closedness of the orbit\. For completeness its tangent rank is computed directly\. The mapT↦J​TT\\mapsto JTidentifies𝔬⁡\(n,m\)\\mathfrak\{o\}\(n,m\)with the skew\-symmetric matrices\. IfW=im⁡FW=\\operatorname\{im\}F, the conditionT​F=0TF=0says that the corresponding alternating form annihilatesWWin either argument\. These are exactly the alternating forms onℝN/W\\mathbb\{R\}^\{N\}/W; their dimension is\(N−s2\)\\binom\{N\-s\}\{2\}\. Subtracting fromdim𝔬⁡\(n,m\)=\(N2\)\\dim\\mathfrak\{o\}\(n,m\)=\\binom\{N\}\{2\}proves \([8](https://arxiv.org/html/2610.27474#S3.E8)\)\. Its rank is constant along the orbit, since left multiplication and conjugation by a group element are invertible\.

A commonC1C^\{1\}conservation law has zero derivative along eachXGX\_\{G\}, so it is constant along every short boost flow, in either time direction\. The local orbit statement therefore makes it constant on the connected local orbit\. Alternatively one may use finite boost products and their commutator limits, passing invariance through continuity\. No second derivative of the conservation law is taken\. ∎

### B\.2Proof of Lemma[3\.3](https://arxiv.org/html/2610.27474#S3.Thmtheorem3)

###### Proof\.

Writeℒ\\mathcal\{L\}for the connected local factor orbit\. It is a locally embedded invariant leaf and the generated Lie values span its tangent space by Lemma[3\.2](https://arxiv.org/html/2610.27474#S3.Thmtheorem2)\. This is a statement on the orbit, even if ambient Lie rank jumps atF0F\_\{0\}\. Moreover

⟨G,MF​\[G\]⟩=‖G⊤​U‖F2\+‖G​V‖F2\.\\langle G,M\_\{F\}\[G\]\\rangle=\\\|G^\{\\top\}U\\\|\_\{\\mathrm\{F\}\}^\{2\}\+\\\|GV\\\|\_\{\\mathrm\{F\}\}^\{2\}\.ForF0≠0F\_\{0\}\\neq 0this positive semidefinite operator is nonzero\. Lemma[2\.4](https://arxiv.org/html/2610.27474#S2.Thmtheorem4), applied on this orbit, supplies a nonemptyℒ\\mathcal\{L\}\-open patch of endpoints of actual single global smooth represented losses, on which

MF=K⁡\(ϕ⁡\(F\)\)\.M\_\{F\}=K\(\\phi\(F\)\)\.Its hypotheses require regularity along this invariant leaf, not constant ambient rank at the singular factor\.

Choose a point in that patch where the rank ofD⁡\(ϕ\|ℒ\)D\(\\phi\|\_\{\\mathcal\{L\}\}\)is maximal within the patch\. A nonzero maximal minor persists, so the rank is constant on a smaller open patch\. The constant\-rank theorem supplies actual local fiber curves there\. Equality through the same, possibly irregularKKmakesMMconstant on these curves\. Consequently every orbit tangent withδ​Z=0\\delta Z=0hasδ​M=0\\delta M=0\. This differentiates the smoothMMalong a fiber, not the irregular functionKK\.

The Gram mapF↦F​F⊤F\\mapsto FF^\{\\top\}intertwines the factor action and the congruence action\. Every Gram\-orbit tangentT​C\+C​T⊤TC\+CT^\{\\top\}lifts to the factor tangentT​FTF\. Its differential rank is constant along the factor orbit by invertibility of both actions; its tangent image is the Gram\-orbit tangent\. Lift any such tangent

δ​C=\[δ​A00δ​B\]\\delta C=\\begin\{bmatrix\}\\delta A&0\\\\ 0&\\delta B\\end\{bmatrix\}to a factor\-orbit tangent\. The preceding fiber statement givesδ​A​G\+G​δ​B=0\\delta AG\+G\\delta B=0for everyGG\. Testing matrix units first eliminates all off\-diagonal entries and then gives

δ​A=t​In,δ​B=−t​Im\\delta A=tI\_\{n\},\\qquad\\delta B=\-tI\_\{m\}for one scalartt\. The congruence orbit preservestr⁡\(J​C\)\\tr\(JC\), becauseQ⊤​J​Q=JQ^\{\\top\}JQ=Jimpliestr⁡\(J​Q​C​Q⊤\)=tr⁡\(J​C\)\\tr\(JQCQ^\{\\top\}\)=\\tr\(JC\)\. Therefore0=tr⁡\(J​δ​C\)=N​t0=\\tr\(J\\delta C\)=Nt, soδ​C=0\\delta C=0\. The Gram tangent projects injectively to then​mnm\-dimensionalZZ\-space at the chosen point\. Its dimension is constant on the orbit, provingdC≤n​md\_\{C\}\\leq nmatC0C\_\{0\}\.

If\(I\)holds, it implies\(O\)as noted above\. One can also work directly on the local orbit, where every chosen common conservation law is constant, and differentiate theC1C^\{1\}represented factorization\. Thus no constant\-rank hypothesis onD​hDhis needed\. ∎

### B\.3Proof of Lemma[3\.4](https://arxiv.org/html/2610.27474#S3.Thmtheorem4)

###### Proof\.

The differential ofF↦SF\\mapsto Sis onto the symmetric matrices\. For any symmetricEE, the choice

δ​F=12​J​F​\(F⊤​F\)−1​E\\delta F=\\tfrac\{1\}\{2\}JF\(F^\{\\top\}F\)^\{\-1\}EgivesF⊤​J​δ​F=E/2F^\{\\top\}J\\delta F=E/2andδ​S=E\\delta S=E\. Thus its kernel has dimensionN​s−s⁡\(s\+1\)/2Ns\-s\(s\+1\)/2, exactly \([8](https://arxiv.org/html/2610.27474#S3.E8)\)\. Every factor\-orbit tangent lies in this kernel, so the two tangent spaces coincide\.

The kernel of the differential ofF↦F​F⊤F\\mapsto FF^\{\\top\}isδ​F=F​Ω\\delta F=F\\OmegawithΩ⊤=−Ω\\Omega^\{\\top\}=\-\\Omega\. To prove the nontrivial direction, projectδ​C=0\\delta C=0ontoW⟂W^\{\\perp\}, whereW=im⁡FW=\\operatorname\{im\}F, and multiply on the right byF​\(F⊤​F\)−1F\(F^\{\\top\}F\)^\{\-1\}\. This givesPW⟂​δ​F=0P\_\{W^\{\\perp\}\}\\delta F=0, soδ​F=F​Ω\\delta F=F\\Omega\. ThenF⁡\(Ω\+Ω⊤\)​F⊤=0F\(\\Omega\+\\Omega^\{\\top\}\)F^\{\\top\}=0and a left inverse ofFFmakesΩ\\Omegaskew\.

For these variations,

δ​S=Ω⊤​S\+S​Ω=S​Ω−Ω​S\.\\delta S=\\Omega^\{\\top\}S\+S\\Omega=S\\Omega\-\\Omega S\.The Gram differential kernel restricted to the orbit is therefore exactly the skew matrices commuting withSS\. An orthogonal spectral decomposition of this real symmetric matrix makes that kernel the direct sum of skew matrices on its eigenspaces, of dimension∑j\(cj2\)\\sum\_\{j\}\\binom\{c\_\{j\}\}\{2\}\. Subtraction proves \([9](https://arxiv.org/html/2610.27474#S3.E9)\), with the null eigenspace included\. ∎

### B\.4Proof of Lemma[3\.5](https://arxiv.org/html/2610.27474#S3.Thmtheorem5)

###### Proof\.

Let\(α,β,κ\)\(\\alpha,\\beta,\\kappa\)be the inertia ofSS\. Thusα\+β\+κ=s\\alpha\+\\beta\+\\kappa=s\. By Sylvester’s law of inertia, remove a nondegenerate subspace of signature\(α,β\)\(\\alpha,\\beta\)fromim⁡F\\operatorname\{im\}F\. ItsJJ\-orthogonal complement has signature\(n−α,m−β\)\(n\-\\alpha,m\-\\beta\)and contains theκ\\kappa\-dimensional totally isotropic radical\. Projection of a totally isotropic subspace into either definite summand is injective\. Henceκ≤min⁡\(n−α,m−β\)\\kappa\\leq\\min\(n\-\\alpha,m\-\\beta\)\. Put

u=n−α−κ≥0,v=m−β−κ≥0\.u=n\-\\alpha\-\\kappa\\geq 0,\\qquad v=m\-\\beta\-\\kappa\\geq 0\.The strict width bounds giveβ\>u\\beta\>u,α\>v\\alpha\>v, andN−s=κ\+u\+v\>0N\-s=\\kappa\+u\+v\>0\. In particularα,β\>0\\alpha,\\beta\>0\. Equal eigenvalues cannot belong to different signs\. Grouping each sign, and the zero cluster, into a single cluster maximizes the possible centralizer dimension:

∑j\(cj2\)≤\(α2\)\+\(β2\)\+\(κ2\)\.\\sum\_\{j\}\\binom\{c\_\{j\}\}\{2\}\\leq\\binom\{\\alpha\}\{2\}\+\\binom\{\\beta\}\{2\}\+\\binom\{\\kappa\}\{2\}\.Substitution into \([9](https://arxiv.org/html/2610.27474#S3.E9)\), withn=α\+κ\+un=\\alpha\+\\kappa\+uandm=β\+κ\+vm=\\beta\+\\kappa\+v, gives

dC−n​m\\displaystyle d\_\{C\}\-nm≥α​u\+β​v\+κ⁡\(α\+β\)−u​v\\displaystyle\\geq\\alpha u\+\\beta v\+\\kappa\(\\alpha\+\\beta\)\-uv=u⁡\(α−v\)\+β​v\+κ⁡\(α\+β\)\>0\.\\displaystyle=u\(\\alpha\-v\)\+\\beta v\+\\kappa\(\\alpha\+\\beta\)\>0\.Ifu\>0u\>0, the first term is positive; ifv\>0v\>0, the second is positive\. If both vanish,κ\>0\\kappa\>0and the last term is positive\. These cover every case, including a nonzero radical\. ∎

### B\.5Proof of Lemma[3\.6](https://arxiv.org/html/2610.27474#S3.Thmtheorem6)

###### Proof\.

Sinces=L0s=L\_\{0\}, direct substitution into \([9](https://arxiv.org/html/2610.27474#S3.E9)\) gives

dC−n​m=\(s2\)−∑j\(cj2\)=∑i<jci​cj\.d\_\{C\}\-nm=\\binom\{s\}\{2\}\-\\sum\_\{j\}\\binom\{c\_\{j\}\}\{2\}=\\sum\_\{i<j\}c\_\{i\}c\_\{j\}\.It is zero precisely whenSShas one eigenvalue value\. AsC​J​C=F​S​F⊤CJC=FSF^\{\\top\}, scalarSSimpliesC​J​C=λ​CCJC=\\lambda C; a left inverse ofFFproves the converse\.

Ifn\>mn\>m, thens=ns=n, the reducedUUis square invertible, and a nonzerox∈ker⁡Vx\\in\\ker Vsatisfiesλ​‖x‖2=‖U​x‖2\>0\\lambda\\\|x\\\|^\{2\}=\\\|Ux\\\|^\{2\}\>0\. The casem\>nm\>nis symmetric and yieldsλ<0\\lambda<0\. Ifn=m=sn=m=s, both blocks are square invertible andZ=U​V⊤Z=UV^\{\\top\}is invertible, without a required sign ofλ\\lambda\. ∎

### B\.6Proof of Lemma[3\.7](https://arxiv.org/html/2610.27474#S3.Thmtheorem7)

###### Proof\.

The symmetric matrixL=C1/2​J​C1/2L=C^\{1/2\}JC^\{1/2\}has inertia\(n,m\)\(n,m\)and is similar toJ​CJC\. Choose an orthogonalOOsuch that

O⊤​L​O=diag⁡\(a1,…,an,−b1,…,−bm\),ai,bj\>0,O^\{\\top\}LO=\\diag\(a\_\{1\},\\ldots,a\_\{n\},\-b\_\{1\},\\ldots,\-b\_\{m\}\),\\qquad a\_\{i\},b\_\{j\}\>0,and putD=diag⁡\(a1,…,an,b1,…,bm\)D=\\diag\(a\_\{1\},\\ldots,a\_\{n\},b\_\{1\},\\ldots,b\_\{m\}\)\. The matrixR=C1/2OD−1/2R=C^\{1/2\}OD^\{\-1/2\}satisfies

R⊤​J​R=J,R​D​R⊤=C\.R^\{\\top\}JR=J,\\qquad RDR^\{\\top\}=C\.We may therefore compute the congruence stabilizer atDD\. ForT=\[PGG⊤Q\]T=\\left\[\\begin\{smallmatrix\}P&G\\\\ G^\{\\top\}&Q\\end\{smallmatrix\}\\right\]in𝔬⁡\(n,m\)\\mathfrak\{o\}\(n,m\), the equationT​D\+D​T⊤=0TD\+DT^\{\\top\}=0requires

\[P,diag⁡\(ai\)\]=0,\[Q,diag⁡\(bj\)\]=0,diag⁡\(ai\)​G\+G​diag⁡\(bj\)=0\.\[P,\\diag\(a\_\{i\}\)\]=0,\\quad\[Q,\\diag\(b\_\{j\}\)\]=0,\\quad\\diag\(a\_\{i\}\)G\+G\\diag\(b\_\{j\}\)=0\.The last equation forcesG=0G=0, entry by entry\. The stabilizer dimension is∑i\(ci2\)\+∑j\(dj2\)\\sum\_\{i\}\\binom\{c\_\{i\}\}\{2\}\+\\sum\_\{j\}\\binom\{d\_\{j\}\}\{2\}\. The classical orbit–stabilizer dimension formula for this smooth finite\-dimensional action now gives

dC=\(N2\)−∑i\(ci2\)−∑j\(dj2\),d\_\{C\}=\\binom\{N\}\{2\}\-\\sum\_\{i\}\\binom\{c\_\{i\}\}\{2\}\-\\sum\_\{j\}\\binom\{d\_\{j\}\}\{2\},which is \([10](https://arxiv.org/html/2610.27474#S3.E10)\)\. Each numeratorn2−∑ici2n^\{2\}\-\\sum\_\{i\}c\_\{i\}^\{2\}orm2−∑jdj2m^\{2\}\-\\sum\_\{j\}d\_\{j\}^\{2\}is nonnegative, and vanishes exactly when that sign has one cluster\.

We next establish sufficiency on the actual ambient invariant level\. The identity

HG​J\+J​HG=0H\_\{G\}J\+JH\_\{G\}=0shows by differentiation of \([4](https://arxiv.org/html/2610.27474#S3.E4)\) thatS=F⊤​J​FS=F^\{\\top\}JFis common to every represented\-loss flow\. For a full\-rowF0F\_\{0\},S0S\_\{0\}has rankNN\. IfF⊤​J​F=S0F^\{\\top\}JF=S\_\{0\}, thenFFalso has full row rank, and

im⁡F⊤=im⁡S0=im⁡F0⊤\.\\operatorname\{im\}F^\{\\top\}=\\operatorname\{im\}S\_\{0\}=\\operatorname\{im\}F\_\{0\}^\{\\top\}\.For example,im⁡\(F⊤​J​F\)=im⁡F⊤\\operatorname\{im\}\(F^\{\\top\}JF\)=\\operatorname\{im\}F^\{\\top\}follows becauseFFis onto andJJis invertible\. Equality of row spaces yieldsF=Q​F0F=QF\_\{0\}for an invertibleQQ; multiplication by a right inverse ofF0F\_\{0\}and its transpose makes the equality of canonical invariants equivalent toQ⊤​J​Q=JQ^\{\\top\}JQ=J\. Conversely every suchQQpreserves that invariant\. Thus the canonical level is precisely theJJ\-orthogonal factor orbit, not only its tangent space\.

SinceJ​C0JC\_\{0\}is diagonalizable with the two stated values, it obeys\(J​C0−a​I\)​\(J​C0\+b​I\)=0\(JC\_\{0\}\-aI\)\(JC\_\{0\}\+bI\)=0\. Multiplication byJJgives

C0​J​C0−\(a−b\)​C0−a​b​J=0\.C\_\{0\}JC\_\{0\}\-\(a\-b\)C\_\{0\}\-abJ=0\.\(12\)This identity is preserved byJJ\-orthogonal congruence, using alsoQ​J​Q⊤=JQJQ^\{\\top\}=J\. Putc=a−bc=a\-bandσ=\(a\+b\)/2\\sigma=\(a\+b\)/2\. On the whole canonical level its diagonal blocks give

A2−c​A=Z​Z⊤\+a​b​In,B2\+c​B=Z⊤​Z\+a​b​Im\.A^\{2\}\-cA=ZZ^\{\\top\}\+abI\_\{n\},\\qquad B^\{2\}\+cB=Z^\{\\top\}Z\+abI\_\{m\}\.The block Grams are positive definite\. Diagonalizing either Gram also diagonalizes the polynomial on the other side\. Each scalar quadratic has exactly one positive root, because the product of its roots is negative\. Sincea​b\+c2/4=σ2ab\+c^\{2\}/4=\\sigma^\{2\}, this yields

A=cIn/2\+σ2​In\+Z​Z⊤,B=−cIm/2\+σ2​Im\+Z⊤​Z\.A=cI\_\{n\}/2\+\\sqrt\{\\sigma^\{2\}I\_\{n\}\+ZZ^\{\\top\}\},\\qquad B=\-cI\_\{m\}/2\+\\sqrt\{\\sigma^\{2\}I\_\{m\}\+Z^\{\\top\}Z\}\.The scalar shifts cancel in \([3](https://arxiv.org/html/2610.27474#S3.E3)\)\. The positive definite square\-root map is smooth, andσ\>0\\sigma\>0makes the displayed represented formula smooth for everyZZ\. The finite entries ofSSand this globally smooth map therefore prove\(I\)\. ∎

### B\.7Proof of Lemma[3\.8](https://arxiv.org/html/2610.27474#S3.Thmtheorem8)

###### Proof\.

The right actionF↦F​QF\\mapsto FQis a Euclidean isometry preservingU​V⊤UV^\{\\top\}andF​F⊤FF^\{\\top\}, hence intertwines the gradient flows\. Along any actual trajectory, \([4](https://arxiv.org/html/2610.27474#S3.E4)\) is a linear left equation with continuous coefficient−Hgrad⁡f⁡\(Z⁡\(t\)\)\-H\_\{\\grad f\(Z\(t\)\)\}\. Its fundamental solutionP⁡\(t\)P\(t\)is invertible at every finite time in the existence interval, soF⁡\(t\)=P⁡\(t\)​F0F\(t\)=P\(t\)F\_\{0\}andker⁡F⁡\(t\)=ker⁡F0\\ker F\(t\)=\\ker F\_\{0\}\. The zero hidden columns remain zero and the other columns satisfy exactly the reduced system\. Their Gram quantities and represented variables coincide\.

If\(O\)holds in the original neighborhood, pull that neighborhood back under the continuous embeddingF~↦\[F~,0\]​Q⊤\\widetilde\{F\}\\mapsto\[\\widetilde\{F\},0\]Q^\{\\top\}\. If it holds on a reduced neighborhood, take its inverse image under the projection onto the firstsscolumns ofF​QFQ\. This is an open original neighborhood, and the actual original trajectory always has zero remaining columns\. The sameKKworks in each direction for every loss\. These arguments concern trajectories from the fixed initialization; they make no assertion about arbitrary ambient points on an invariant level\. Finally an orthogonal column action and deletion of identically zero columns preserve each block’s row rank\. ∎

### B\.8Proof of Lemma[3\.9](https://arxiv.org/html/2610.27474#S3.Thmtheorem9)

###### Proof\.

Use the fixed hidden orthogonal coordinates of Lemma[3\.8](https://arxiv.org/html/2610.27474#S3.Thmtheorem8)\. Lemma[3\.6](https://arxiv.org/html/2610.27474#S3.Thmtheorem6)gives

S0=diag⁡\(λ​Is,0\)\.S\_\{0\}=\\diag\(\\lambda I\_\{s\},0\)\.Consider an arbitrary ambientF=\[W,E\]F=\[W,E\]on this exact canonical level\. It satisfies

W⊤​J​W=λ​Is,W⊤​J​E=0,E⊤​J​E=0\.W^\{\\top\}JW=\\lambda I\_\{s\},\\qquad W^\{\\top\}JE=0,\\qquad E^\{\\top\}JE=0\.\(13\)
First supposeλ≠0\\lambda\\neq 0\. The sign restrictions in Lemma[3\.6](https://arxiv.org/html/2610.27474#S3.Thmtheorem6)show thatλ\>0\\lambda\>0entailss=ns=n, andλ<0\\lambda<0entailss=ms=m; this includes the square casen=m=sn=m=s\. Ifλ\>0\\lambda\>0, the subspace spanned byWWis positive definite of dimensionnn, exhausting the positive index ofJJ\. ItsJJ\-orthogonal complement is negative definite\. Each column ofEElies in that complement and has zeroJJ\-square by \([13](https://arxiv.org/html/2610.27474#A2.E13)\), hence is zero\. Forλ<0\\lambda<0the complementary form is positive definite and the same conclusion follows\. Thus every ambient point of this invariant level hasE=0E=0andC​J​C=λ​CCJC=\\lambda C\.

On a small neighborhood ofF0F\_\{0\}, both block Grams remain positive definite, by the assumed separate full row ranks\. The block equations ofC​J​C=λ​CCJC=\\lambda Care

A2−λ​A=Z​Z⊤,B2\+λ​B=Z⊤​Z\.A^\{2\}\-\\lambda A=ZZ^\{\\top\},\\qquad B^\{2\}\+\\lambda B=Z^\{\\top\}Z\.For everyμ≥0\\mu\\geq 0,x2−λ​x−μ=0x^\{2\}\-\\lambda x\-\\mu=0has at most one positive root; any existing positive Gram eigenvalue selects it\. The corresponding statement with−λ\-\\lambdaapplies toBB\. Functional calculus therefore yields

A\\displaystyle A=λ​In/2\+λ2​In/4\+Z​Z⊤,\\displaystyle=\\lambda I\_\{n\}/2\+\\sqrt\{\\lambda^\{2\}I\_\{n\}/4\+ZZ^\{\\top\}\},\(14\)B\\displaystyle B=−λIm/2\+λ2​Im/4\+Z⊤​Z\.\\displaystyle=\-\\lambda I\_\{m\}/2\+\\sqrt\{\\lambda^\{2\}I\_\{m\}/4\+Z^\{\\top\}Z\}\.Their shifts cancel\. Sinceλ≠0\\lambda\\neq 0, the resultingK\|λ\|/2K\_\{\|\\lambda\|/2\}is smooth on the entire represented space\.

Now supposeλ=0\\lambda=0\. The sign restrictions forcen=m=sn=m=s, and the full originalS0S\_\{0\}is zero\. We do not discard or force the extra hidden columns to vanish\. At every point of the actual canonical level,

C​J​C=F⁡\(F⊤​J​F\)​F⊤=0\.CJC=F\(F^\{\\top\}JF\)F^\{\\top\}=0\.ConsequentlyA2=Z​Z⊤A^\{2\}=ZZ^\{\\top\}andB2=Z⊤​ZB^\{2\}=Z^\{\\top\}Z\. Both Grams remain positive definite on a sufficiently small ambient neighborhood, soZZis invertible and their positive roots give \([14](https://arxiv.org/html/2610.27474#A2.E14)\) withλ=0\\lambda=0\. The mapK0K\_\{0\}is smooth onGL⁡\(n,ℝ\)\\operatorname\{GL\}\(n,\\mathbb\{R\}\)\. Choose a smooth cutoffχ\\chicompactly supported in that open set, equal to one nearZ0Z\_\{0\}, and define globally

K¯​\(Z\)=χ⁡\(Z\)​K0​\(Z\)\+\(1−χ⁡\(Z\)\)​Iℝn×n,\\overline\{K\}\(Z\)=\\chi\(Z\)K\_\{0\}\(Z\)\+\(1\-\\chi\(Z\)\)I\_\{\\mathbb\{R\}^\{n\\times n\}\},where the first term is extended by zero off its compact support\. This is globally smooth\. Shrink the ambient parameter neighborhood so its represented image lies in the regionχ=1\\chi=1\. In both cases the finite canonical entries are common smooth conservation laws, and the required identity holds on their actual initial level\. This proves\(I\)in the original space\. ∎

### B\.9Proof of Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)

###### Proof of Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)\.

Separate full row ranks implyL0≤s≤min⁡\(r,N\)L\_\{0\}\\leq s\\leq\\min\(r,N\)\. Apply Lemma[3\.8](https://arxiv.org/html/2610.27474#S3.Thmtheorem8)to property\(O\)\. IfL0<s<NL\_\{0\}<s<N, the reduced factor is full column rank and Lemmas[3\.3](https://arxiv.org/html/2610.27474#S3.Thmtheorem3)and[3\.5](https://arxiv.org/html/2610.27474#S3.Thmtheorem5)contradict each other\. Ifs=L0s=L\_\{0\}, Lemmas[3\.3](https://arxiv.org/html/2610.27474#S3.Thmtheorem3)and[3\.6](https://arxiv.org/html/2610.27474#S3.Thmtheorem6)force \([5](https://arxiv.org/html/2610.27474#S3.E5)\) and its signs\. Ifs=Ns=N, the full\-row calculation \([10](https://arxiv.org/html/2610.27474#S3.E10)\) forces \([6](https://arxiv.org/html/2610.27474#S3.E6)\)\. These exhaust every admissible joint rank\.

Conversely Lemma[3\.9](https://arxiv.org/html/2610.27474#S3.Thmtheorem9)proves\(I\)in \([5](https://arxiv.org/html/2610.27474#S3.E5)\), and Lemma[3\.7](https://arxiv.org/html/2610.27474#S3.Thmtheorem7)proves it in \([6](https://arxiv.org/html/2610.27474#S3.E6)\)\. Both use the actual original canonical level, not a reduced\-space tangent argument\. Conservation gives\(O\)\. The mobility formulas were established in the same lemmas\. ∎

### B\.10Proof of Proposition[3\.10](https://arxiv.org/html/2610.27474#S3.Thmtheorem10)

###### Proof\.

Put

P=σ2​In\+Z​Z⊤,Q=σ2​Im\+Z⊤​Z\.P=\\sqrt\{\\sigma^\{2\}I\_\{n\}\+ZZ^\{\\top\}\},\\qquad Q=\\sqrt\{\\sigma^\{2\}I\_\{m\}\+Z^\{\\top\}Z\}\.Ifn≤mn\\leq m, define

U′=\(P\+σIn\)1/2,V′=Z⊤\(P\+σIn\)−1/2\.U^\{\\prime\}=\(P\+\\sigma I\_\{n\}\)^\{1/2\},\\qquad V^\{\\prime\}=Z^\{\\top\}\(P\+\\sigma I\_\{n\}\)^\{\-1/2\}\.The inverse is lawful sinceP\+σ​In≥2​σ​InP\+\\sigma I\_\{n\}\\geq 2\\sigma I\_\{n\}\. The product isU′​V′⁣⊤=ZU^\{\\prime\}V^\{\\prime\\top\}=Z\. A singular\-value decomposition, including all zero singular values, gives

U′​U′⁣⊤\\displaystyle U^\{\\prime\}U^\{\\prime\\top\}=P\+σ​In,\\displaystyle=P\+\\sigma I\_\{n\},V′​V′⁣⊤\\displaystyle V^\{\\prime\}V^\{\\prime\\top\}=Z⊤​\(P\+σ​In\)−1​Z=Q−σ​Im,\\displaystyle=Z^\{\\top\}\(P\+\\sigma I\_\{n\}\)^\{\-1\}Z=Q\-\\sigma I\_\{m\},U′⁣⊤​U′\\displaystyle U^\{\\prime\\top\}U^\{\\prime\}=P\+σ​In,\\displaystyle=P\+\\sigma I\_\{n\},V′⁣⊤​V′\\displaystyle V^\{\\prime\\top\}V^\{\\prime\}=P−σ​In\.\\displaystyle=P\-\\sigma I\_\{n\}\.For a singular valuedd, the scalar identity used here is

d2σ2\+d2\+σ=σ2\+d2−σ\.\\frac\{d^\{2\}\}\{\\sqrt\{\\sigma^\{2\}\+d^\{2\}\}\+\\sigma\}=\\sqrt\{\\sigma^\{2\}\+d^\{2\}\}\-\\sigma\.It also handles the extra nullspace, where both sides are zero\. The hidden imbalance is2​σ​In2\\sigma I\_\{n\}, and its two scalar Gram shifts cancel in the mobility\.

Ifn\>mn\>m, the transposed construction is

V′=\(Q\+σIm\)1/2,U′=Z\(Q\+σIm\)−1/2\.V^\{\\prime\}=\(Q\+\\sigma I\_\{m\}\)^\{1/2\},\\qquad U^\{\\prime\}=Z\(Q\+\\sigma I\_\{m\}\)^\{\-1/2\}\.It gives

U′​U′⁣⊤=P−σ​In,V′​V′⁣⊤=Q\+σ​Im,U′⁣⊤​U′−V′⁣⊤​V′=−2​σ​Im,U^\{\\prime\}U^\{\\prime\\top\}=P\-\\sigma I\_\{n\},\\quad V^\{\\prime\}V^\{\\prime\\top\}=Q\+\\sigma I\_\{m\},\\quad U^\{\\prime\\top\}U^\{\\prime\}\-V^\{\\prime\\top\}V^\{\\prime\}=\-2\\sigma I\_\{m\},and the same cancellation\.

These scalar hidden invariants remain constant along each narrow factor flow\. In the first constructionU′U^\{\\prime\}stays invertible, sinceU′⁣⊤​U′≥2​σ​InU^\{\\prime\\top\}U^\{\\prime\}\\geq 2\\sigma I\_\{n\}\. Its Gram equation has the unique positive solutionU′​U′⁣⊤=P\+σ​InU^\{\\prime\}U^\{\\prime\\top\}=P\+\\sigma I\_\{n\}; the other block has the unique nonnegative solutionV′​V′⁣⊤=Q−σ​ImV^\{\\prime\}V^\{\\prime\\top\}=Q\-\\sigma I\_\{m\}\. The latter remains unique even at zero singular values, becausex2\+2​σ​x−μ=0x^\{2\}\+2\\sigma x\-\\mu=0,μ≥0\\mu\\geq 0, has a unique nonnegative root\. The transposed construction has the identical argument\. Hence the narrow flow retains exactlyKσK\_\{\\sigma\}\. For square invertibleZZandσ=0\\sigma=0, setU′=P1/2U^\{\\prime\}=P^\{1/2\},V′=Z⊤P−1/2V^\{\\prime\}=Z^\{\\top\}P^\{\-1/2\}; its hidden imbalance is zero and its Grams areP,QP,Q, giving the balanced formula locally\.

For each common representedC2C^\{2\}loss, the wide positive flow and these narrow factors therefore solve the same smooth represented ODE, with the same initial represented matrix\. Local uniqueness identifies their represented trajectories on a common existence interval\. This is a mobility and continuous\-time trajectory identity, not equality of factor states or finite\-step gradient descent\. The scalar\-balanced square\-root dynamics are the known positive family; the construction supplies the exact matching comparison\. ∎

### B\.11Proof of Corollary[3\.11](https://arxiv.org/html/2610.27474#S3.Thmtheorem11)

###### Proof\.

Separate maximal row ranks and maximal joint rank hold almost surely: their failures lie in zero sets of nonzero minor polynomials\. Ifr<Nr<N, the almost\-sure joint rank isrr, strictly betweenL0L\_\{0\}andNN, so Theorem[3\.1](https://arxiv.org/html/2610.27474#S3.Thmtheorem1)excludes closure\. Ifr≥Nr\\geq N, the almost\-sure joint rank isNN\. For\(n,m\)≠\(1,1\)\(n,m\)\\neq\(1,1\), the positive spectral case forces a repeated eigenvalue ofJ​F​F⊤JFF^\{\\top\}\. The discriminant

P⁡\(F\)=disc⁡dett⁡\(t​IN−J​F​F⊤\)P\(F\)=\\operatorname\{disc\}\_\{t\}\\det\(tI\_\{N\}\-JFF^\{\\top\}\)is a polynomial in the factor entries and must vanish there\. It is not identically zero: atF=\[diag⁡\(1,…,N\),0\]F=\[\\diag\(1,\\ldots,N\),0\]its signed diagonal eigenvalues are all distinct\. The zero set of a nonzero real polynomial has Lebesgue measure zero, and hence probability zero under every such law\. Forn=m=1n=m=1, the two factor rows are nonzero almost surely\. Joint rank one automatically givesC​J​C=λ​CCJC=\\lambda C; joint rank two automatically gives one positive and one negative signed spectral value\. The theorem then gives closure, in agreement with the known scalar case\[[1](https://arxiv.org/html/2610.27474#bib.bib1), Proposition 3\.10\]\. ∎

## Appendix CProofs of the geometric characterization

### C\.1Proof of Lemma[4\.1](https://arxiv.org/html/2610.27474#S4.Thmtheorem1)

###### Proof\.

Select finitely many bracket fieldsV1,…,VpV\_\{1\},\\ldots,V\_\{p\}independent atθ0\\theta\_\{0\}\. A nonzero frame minor stays nonzero on a smaller neighborhood; constant rank makes these fields a frame forℰ\\mathcal\{E\}there\. Their brackets belong pointwise toℰ\\mathcal\{E\}, and their frame coefficients are smooth, or analytic in the analytic case, by inversion of the fixed frame minor\. Leibniz’s rule then makes this subbundle involutive\. The corresponding smooth or analytic Frobenius theorem supplies coordinates\(x,h\)\(x,h\)in a product box, withℰ=ker⁡D​h\\mathcal\{E\}=\\ker Dh\. The components ofhhannihilate everygrad⁡ϕi\\grad\\phi\_\{i\}, so they are common conservation laws\. The seth=h⁡\(θ0\)h=h\(\\theta\_\{0\}\)in this box is a single connected plaque\. Ifp=Dp=D, one may takehhconstant\.

Becauseℋ⊂ℰ=T​L\\mathcal\{H\}\\subset\\mathcal\{E\}=TL, restriction ofD​ϕD\\phitoT​LTLhas image containing

image⁡\(D​ϕ​D​ϕ𝖳\)=image⁡D​ϕ\.\\operatorname\{image\}\(D\\phi D\\phi^\{\\mathsf\{T\}\}\)=\\operatorname\{image\}D\\phi\.The reverse containment is automatic\. Thus the restriction has the same rankqq\. No full\-output\-rank assumption is involved\. ∎

### C\.2Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)in the analytic case

###### Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)in the analytic case\.

The metric property implies ordinary closure by conservation, so consider the converse\. Ifq=0q=0,D​ϕ=0D\\phi=0locally,M=0M=0, and the zero global mobility and a constant conservation law suffice\. Assumeq\>0q\>0\. Use the connected analytic plaqueLLof Lemma[4\.1](https://arxiv.org/html/2610.27474#S4.Thmtheorem1)\. The clock endpoint construction in Lemma[2\.4](https://arxiv.org/html/2610.27474#S2.Thmtheorem4)gives a nonempty relatively open setO⊂LO\\subset Lon which

M⁡\(θ\)=K⁡\(ϕ⁡\(θ\)\),M\(\\theta\)=K\(\\phi\(\\theta\)\),with the same initially arbitraryKKas in ordinary closure\. Every vector inker⁡D⁡\(ϕ\|L\)\\ker D\(\\phi\|\_\{L\}\)at a point ofOOis tangent to an actual local fiber curve, because the restriction has constant rankqq\. Along such a curveϕ\\phiis constant, hence so isMM\. Therefore

ker⁡D⁡\(ϕ\|L\)⊂ker⁡D⁡\(M\|L\)on​O\.\\ker D\(\\phi\|\_\{L\}\)\\subset\\ker D\(M\|\_\{L\}\)\\quad\\hbox\{on \}O\.\(15\)This argument neither differentiatesKKnor assumes its continuity\.

RepresentMMby itsd2d^\{2\}entries\. The augmented differentialDL​\(ϕ,M\)D\_\{L\}\(\\phi,M\)has rankqqonOO\. In the connected analyticxx\-coordinate box ofLL, every\(q\+1\)\(q\+1\)\-minor of this differential is analytic and vanishes onOO\. The real\-analytic identity principle makes every such minor zero throughout that box\. Ifp=qp=q, the upper bound is automatic since there are onlyppcolumns\. SinceD⁡\(ϕ\|L\)D\(\\phi\|\_\{L\}\)has rankqqeverywhere, the augmented differential has rank exactlyqqeverywhere\. Thus the kernel inclusion above holds on the entire plaque, including nearθ0\\theta\_\{0\}\.

For completeness, this differential condition gives an equality on the actual local level, not just on its tangent\. Apply the analytic constant\-rank theorem toϕ\|L\\phi\|\_\{L\}nearθ0\\theta\_\{0\}\. In local source coordinates\(s,t\)∈ℝq×ℝp−q\(s,t\)\\in\\mathbb\{R\}^\{q\}\\times\\mathbb\{R\}^\{p\-q\}it has the formϕ⁡\(s,t\)=ι⁡\(s\)\\phi\(s,t\)=\\iota\(s\), whereι\\iotaparametrizes a locally embedded analytic imageSS\. Shrink to a product box so itstt\-fibers are connected\. The kernel inclusion says∂tM=0\\partial\_\{t\}M=0, henceM⁡\(s,t\)=m⁡\(s\)M\(s,t\)=m\(s\)\. Taking a local section, for examplet=0t=0, constructs an analytic mapkkonSSwithM=k∘ϕM=k\\circ\\phion this whole smaller plaque neighborhood\.

FlattenSSin analytic ambient target coordinates and extendkkconstantly in the normal coordinates, obtainingKlocK\_\{\\mathrm\{loc\}\}on an open target neighborhoodWWofϕ⁡\(θ0\)\\phi\(\\theta\_\{0\}\)\. Choose a smooth0≤χ≤10\\leq\\chi\\leq 1with compact support inWWand equal to one on a smaller neighborhood\. The map

K¯​\(z\)=χ⁡\(z\)​Kloc​\(z\)\+\(1−χ⁡\(z\)\)​Id\\overline\{K\}\(z\)=\\chi\(z\)K\_\{\\mathrm\{loc\}\}\(z\)\+\(1\-\\chi\(z\)\)I\_\{d\}is globally smooth, interpretingχ​Kloc\\chi K\_\{\\mathrm\{loc\}\}as zero outsideWW\. No analyticity of the global cutoff is asserted\. Finally shrink the original Frobenius product box so its entire initialhh\-level lies in the factorization neighborhood and its represented image lies whereχ=1\\chi=1\. This proves the exact local metric property\. Thehhare local conservation laws; they need not extend to a larger originally specified parameter domain\. ∎

### C\.3Proof of Lemma[4\.3](https://arxiv.org/html/2610.27474#S4.Thmtheorem3)

###### Proof\.

A self\-intersection ats≠ts\\neq twould require

a=−w⁡\(t\)−w⁡\(s\)t−s\.a=\-\\frac\{w\(t\)\-w\(s\)\}\{t\-s\}\.Cover the off\-diagonal square by countably many compact two\-dimensional pieces bounded away from the diagonal\. The secant map is smooth and Lipschitz on each piece\. Its image hasqq\-dimensional Lebesgue measure zero whenq\>2q\>2, by the elementary covering estimate for a Lipschitz image of a two\-dimensional set\. Failure of immersion requiresa=−w′​\(t\)a=\-w^\{\\prime\}\(t\); this one\-dimensional smooth image is also null\. Choose arbitrarily smallaja\_\{j\}outside their union\. Each resulting curve is injective and regular; compactness makes it an embedded compact arc\. TheC1C^\{1\}convergence is immediate, and compact containment gives the last claim\. Nothing in this proof asserts density of embedded planar arcs whenq=2q=2\. ∎

### C\.4Proof of Lemma[4\.4](https://arxiv.org/html/2610.27474#S4.Thmtheorem4)

###### Proof\.

ConstructKlocK\_\{\\mathrm\{loc\}\}first, and shrink into its domain and the ordinary neighborhood\. Constant rank gives one embedded represented imageSSand local coordinatesψ:S→ℝq\\psi:S\\to\\mathbb\{R\}^\{q\}\. Setπ=ψ∘ϕ\\pi=\\psi\\circ\\phi\. The kernels ofD​πD\\piandD​ϕD\\phiagree, so their Euclidean horizontal spaces agree\. The smooth right inverse

H⁡\(θ\)=D​π​\(θ\)𝖳​\[D​π​\(θ\)​D​π​\(θ\)𝖳\]−1H\(\\theta\)=D\\pi\(\\theta\)^\{\\mathsf\{T\}\}\[D\\pi\(\\theta\)D\\pi\(\\theta\)^\{\\mathsf\{T\}\}\]^\{\-1\}maps base velocities to their unique horizontal lifts\.

For a smooth horizontal pathθ⁡\(t\)\\theta\(t\), putw⁡\(t\)=π⁡\(θ⁡\(t\)\)w\(t\)=\\pi\(\\theta\(t\)\)\. It solvesθ′=H⁡\(θ\)​w′\\theta^\{\\prime\}=H\(\\theta\)w^\{\\prime\}\. Apply Lemma[4\.3](https://arxiv.org/html/2610.27474#S4.Thmtheorem3)and liftwjw\_\{j\}from exactlyθ0\\theta\_\{0\}byθj′=H⁡\(θj\)​wj′\\theta\_\{j\}^\{\\prime\}=H\(\\theta\_\{j\}\)w\_\{j\}^\{\\prime\}\. Differentiation givesπ⁡\(θj​\(t\)\)=wj​\(t\)\\pi\(\\theta\_\{j\}\(t\)\)=w\_\{j\}\(t\)\. Standard ODE stability on a compact state neighborhood gives uniform convergence of lifts on\[0,T\]\[0,T\]\. One explicit bound, while both lifts remain there, is

supt≤T‖θj​\(t\)−θ⁡\(t\)‖≤C​‖wj′−w′‖L1​exp⁡\(C​∫0T‖wj′​\(t\)‖​𝑑t\)\.\\sup\_\{t\\leq T\}\\\|\\theta\_\{j\}\(t\)\-\\theta\(t\)\\\|\\leq C\\\|w\_\{j\}^\{\\prime\}\-w^\{\\prime\}\\\|\_\{L^\{1\}\}\\exp\\\!\\left\(C\\int\_\{0\}^\{T\}\\\|w\_\{j\}^\{\\prime\}\(t\)\\\|\\,dt\\right\)\.A slightly larger compact neighborhood and continuation show that all large\-jjlifts exist on the full interval and remain in the required open set\. Their represented arcszj=ϕ⁡\(θj\)=ψ−1​\(wj\)z\_\{j\}=\\phi\(\\theta\_\{j\}\)=\\psi^\{\-1\}\(w\_\{j\}\)are embedded and regular\.

Here is the exact autonomous realization used for each arc\. The constant\-rank Moore–Penrose inverseM†M^\{\\dagger\}is smooth\. Putgj=−M​\(θj\)†​zj′g\_\{j\}=\-M\(\\theta\_\{j\}\)^\{\\dagger\}z\_\{j\}^\{\\prime\}\. Horizontality gives−D​ϕ𝖳​gj=θj′\-D\\phi^\{\\mathsf\{T\}\}g\_\{j\}=\\theta\_\{j\}^\{\\prime\}and

⟨gj,zj′⟩=−‖θj′‖2<0\.\\langle g\_\{j\},z\_\{j\}^\{\\prime\}\\rangle=\-\\\|\\theta\_\{j\}^\{\\prime\}\\\|^\{2\}<0\.Slightly extend the regular embedded arc at its two ends\. In tubular coordinateszj​\(t\)\+∑byb​eb​\(t\)z\_\{j\}\(t\)\+\\sum\_\{b\}y\_\{b\}e\_\{b\}\(t\), where theebe\_\{b\}form a smooth normal frame, prescribe

A′​\(t\)=⟨gj​\(t\),zj′​\(t\)⟩,floc​\(t,y\)=A⁡\(t\)\+∑byb​⟨gj​\(t\),eb​\(t\)⟩\.A^\{\\prime\}\(t\)=\\langle g\_\{j\}\(t\),z\_\{j\}^\{\\prime\}\(t\)\\rangle,\\qquad f\_\{\\mathrm\{loc\}\}\(t,y\)=A\(t\)\+\\sum\_\{b\}y\_\{b\}\\langle g\_\{j\}\(t\),e\_\{b\}\(t\)\\rangle\.Its gradient on the arc is exactlygjg\_\{j\}: the primitive supplies the tangential derivative, and the linear terms supply all normal derivatives\. A cutoff equal to one around the original compact arc and supported in this slightly extended tube yields one global smooth represented loss\. ODE uniqueness identifies its actual parameter flow fromθ0\\theta\_\{0\}withθj\\theta\_\{j\}\. The construction requires no uniform derivative bound overjj\.

Ordinary closure therefore givesM⁡\(θj​\(t\)\)=K⁡\(ϕ⁡\(θj​\(t\)\)\)=Kloc​\(ϕ⁡\(θj​\(t\)\)\)M\(\\theta\_\{j\}\(t\)\)=K\(\\phi\(\\theta\_\{j\}\(t\)\)\)=K\_\{\\mathrm\{loc\}\}\(\\phi\(\\theta\_\{j\}\(t\)\)\)\. Pass to the limit through the separately forced smoothKlocK\_\{\\mathrm\{loc\}\}\. This proves the asserted identity on the original horizontal path without assuming continuity of the originalKK\. ∎

### C\.5Proof of Lemma[4\.5](https://arxiv.org/html/2610.27474#S4.Thmtheorem5)

###### Proof\.

Use positive durations and the finite symmetric field family\{grad⁡ϕi,−grad⁡ϕi\}\\\{\\grad\\phi\_\{i\},\-\\grad\\phi\_\{i\}\\\}\. A uniform speed bound on a compact chart gives a sufficiently short total horizon for all words under consideration\. Maximize the differential rank of their endpoint maps over all finite words and strictly positive duration vectors\. The maximumQQis attained, since ranks are integers bounded bypp\. Around a maximizing vector, a nonzeroQQ\-minor persists and no higher rank is possible\. The constant\-rank theorem gives an embedded image patchPPof dimensionQQwith a strict remaining horizon margin\.

Append any one of the allowed fields for an additional positive duration\. Maximality bounds the augmented rank byQQ\. Taking that duration to zero makes its extra derivative tangent toPP; the originalQQ\-minor still persists\. Thus all the generators, and hence their brackets, are tangent toPP\. Their Lie rank ispp, soQ=pQ=pandPPis open inLL\.

Let the fixed maximizing word end atθ∗\\theta\_\{\*\}\. Its inverse word is a local diffeomorphism sendingθ∗\\theta\_\{\*\}toθ0\\theta\_\{0\}and a smaller open patchPPto a neighborhood ofθ0\\theta\_\{0\}inLL\. Every point of that neighborhood is obtained by a nearby forward word followed by this fixed inverse word\. Finite\-path continuous dependence ensures that all these paths stay in the larger chart after shrinkingPP\. This uses control words only as horizontal paths; it does not call their concatenation a single autonomous loss\.

For each segment, replace its time parameter by a smooth increasing function with the same endpoints and all derivatives zero at both ends\. The resulting joined path is smooth and horizontal, including at the joins, and has exactly the same image and endpoint\. ∎

### C\.6Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq≥3q\\geq 3and the elementary low\-rank cases

###### Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq≥3q\\geq 3and the elementary low\-rank cases\.

Forq≥3q\\geq 3, Lemmas[4\.4](https://arxiv.org/html/2610.27474#S4.Thmtheorem4)and[4\.5](https://arxiv.org/html/2610.27474#S4.Thmtheorem5)give the mobility identity on an entire neighborhood ofθ0\\theta\_\{0\}in its initial plaque\. Shrink the original Frobenius product box so its actual initial level lies there\. A target cutoff as in the analytic proof makesKlocK\_\{\\mathrm\{loc\}\}globally smooth and proves the metric property\.

Ifq=0q=0, the stationary argument already applies\. Ifq=1q=1, the regular horizontal line bundle is involutive, sop=1p=1\. More generally, wheneverp=qp=q,ϕ\\phiis locally injective on the initial plaque by the constant\-rank theorem\. A local section factorsMMsmoothly through its represented image, and target extension and cutoff give the metric property without assuming ordinary closure\. These are the classical local recoverability cases\. It remains to prove the smoothq=2,p=3q=2,p=3case\. ∎

### C\.7Proof of Lemma[4\.6](https://arxiv.org/html/2610.27474#S4.Thmtheorem6)

###### Proof\.

The coordinate\-gradient fields andH1,H2H\_\{1\},H\_\{2\}are smooth frame changes of the same horizontal bundle\. Their generated Lie modules have the same evaluated distribution, namelyT​LTL\.

Letℐ\\mathcal\{I\}be the smooth ideal generated by all orderedHiH\_\{i\}\-derivatives ofcc, includingcc\. It is stable underH1,H2H\_\{1\},H\_\{2\}by Leibniz’s rule\. The module

𝒩=spanC∞\{H1,H2\}\+ℐ∂η\\mathcal\{N\}=\\Span\_\{C^\{\\infty\}\}\\\{H\_\{1\},H\_\{2\}\\\}\+\\mathcal\{I\}\\partial\_\{\\eta\}is closed under brackets\. In fact, withA1=A,A2=BA\_\{1\}=A,A\_\{2\}=B,

\[Hi,a∂η\]=\(Hia−a∂ηAi\)∂η,\[a∂η,b∂η\]=\(a∂ηb−b∂ηa\)∂η\.\[H\_\{i\},a\\partial\_\{\\eta\}\]=\(H\_\{i\}a\-a\\partial\_\{\\eta\}A\_\{i\}\)\\partial\_\{\\eta\},\\quad\[a\\partial\_\{\\eta\},b\\partial\_\{\\eta\}\]=\(a\\partial\_\{\\eta\}b\-b\\partial\_\{\\eta\}a\)\\partial\_\{\\eta\}\.Both coefficients belong toℐ\\mathcal\{I\}whena,b∈ℐa,b\\in\\mathcal\{I\}; stability under∂η\\partial\_\{\\eta\}is not needed\. Horizontal brackets are included byc∈ℐc\\in\\mathcal\{I\}, and smooth coefficients cause only the displayed terms and horizontal multiples\. If all generators ofℐ\\mathcal\{I\}vanished atθ0\\theta\_\{0\},𝒩⁡\(θ0\)\\mathcal\{N\}\(\\theta\_\{0\}\)would have dimension at most two, contradicting Lie rank three\. Thus a finite least ordermmexists\.

Form≥2m\\geq 2, interchange adjacent horizontal operators in anmm\-word applied tocc\. Its change is an outer horizontal word applied to±c∂η\(Hinnerc\)\\pm c\\partial\_\{\\eta\}\(H\_\{\\mathrm\{inner\}\}c\)\. The outer word has length at mostm−2m\-2\. Every Leibniz term contains a firstccfactor acted on by at mostm−2m\-2horizontal derivatives, which vanishes atθ0\\theta\_\{0\}by minimality\. All adjacent interchanges therefore preserve the evaluation\. Orders zero and one are immediate\. The nonzero leading tensor is symmetric; polarization then shows that its diagonal polynomialPmP\_\{m\}is nonzero\. ∎

### C\.8Proof of Lemma[4\.7](https://arxiv.org/html/2610.27474#S4.Thmtheorem7)

###### Proof\.

Choose independent unit vectorse\(1\),e\(2\)e^\{\(1\)\},e^\{\(2\)\}avoiding the zeros ofPmP\_\{m\}\. Their negatives also avoid these zeros by homogeneity\. Denote the four directions by𝒟\\mathcal\{D\}and lete⟂e^\{\\perp\}be the counterclockwise unit rotation ofee\. Sincedet\(e,e⟂\)=1\\det\(e,e^\{\\perp\}\)=1,\[He,He⟂\]=c∂η\[H\_\{e\},H\_\{e^\{\\perp\}\}\]=c\\partial\_\{\\eta\}\.

LetΦXt\\Phi\_\{X\}^\{t\}denote the local flow ofXX\. The lift of the base rectangle

0⟶ε​e⟶ε​e\+δ​e⟂⟶δ​e⟂⟶00\\longrightarrow\\varepsilon e\\longrightarrow\\varepsilon e\+\\delta e^\{\\perp\}\\longrightarrow\\delta e^\{\\perp\}\\longrightarrow 0has endpoint

he​\(δ\)=ΦHe⟂−δ​ΦHe−ε​ΦHe⟂δ​ΦHeε​\(θ0\)\.h\_\{e\}\(\\delta\)=\\Phi\_\{H\_\{e^\{\\perp\}\}\}^\{\-\\delta\}\\Phi\_\{H\_\{e\}\}^\{\-\\varepsilon\}\\Phi\_\{H\_\{e^\{\\perp\}\}\}^\{\\delta\}\\Phi\_\{H\_\{e\}\}^\{\\varepsilon\}\(\\theta\_\{0\}\)\.It lies in the base\-zero fiber, is smooth in signedδ\\delta, andhe​\(0\)=θ0h\_\{e\}\(0\)=\\theta\_\{0\}\. Differentiating at zero gives the vertical vector

Δe​\(ε\)=\(ΦHe−ε\)∗​He⟂​\(ΦHeε​\(θ0\)\)−He⟂​\(θ0\)\.\\Delta\_\{e\}\(\\varepsilon\)=\(\\Phi\_\{H\_\{e\}\}^\{\-\\varepsilon\}\)\_\{\*\}H\_\{e^\{\\perp\}\}\(\\Phi\_\{H\_\{e\}\}^\{\\varepsilon\}\(\\theta\_\{0\}\)\)\-H\_\{e^\{\\perp\}\}\(\\theta\_\{0\}\)\.For the iterated bracketadHej​He⟂\\operatorname\{ad\}\_\{H\_\{e\}\}^\{j\}H\_\{e^\{\\perp\}\}, write its vertical coefficient asfjf\_\{j\}\. Thenf1=cf\_\{1\}=candfj\+1=He​fj−fj​∂ηAef\_\{j\+1\}=H\_\{e\}f\_\{j\}\-f\_\{j\}\\partial\_\{\\eta\}A\_\{e\}, whereAeA\_\{e\}is the vertical coefficient ofHeH\_\{e\}\. Induction yields

fj=Hej−1​c\+∑ℓ=0j−2aj​ℓ​Heℓ​cf\_\{j\}=H\_\{e\}^\{j\-1\}c\+\\sum\_\{\\ell=0\}^\{j\-2\}a\_\{j\\ell\}H\_\{e\}^\{\\ell\}cfor smooth coefficientsaj​ℓa\_\{j\\ell\}\. Minimality ofmmkills all lower terms atθ0\\theta\_\{0\}\. The finite Taylor formula for the smooth flow pullback therefore gives

Δe\(ε\)=εm\+1\(m\+1\)\!Pm\(e\)∂η\+o\(εm\+1\)\.\\Delta\_\{e\}\(\\varepsilon\)=\\frac\{\\varepsilon^\{m\+1\}\}\{\(m\+1\)\!\}P\_\{m\}\(e\)\\partial\_\{\\eta\}\+o\(\\varepsilon^\{m\+1\}\)\.This uses a finite Taylor remainder, not convergence of a smooth Taylor series\. One sufficiently small positiveε\\varepsilonmakes all four derivatives nonzero\. The one\-dimensional inverse function theorem makes each signedδ\\deltafamily cover an open fiber interval around zero\. Their finite intersection contains a common open intervalII\. Shrink theδ\\deltaranges,II, and all state boxes so the finitely many loop families stay inside the ordinary andKlocK\_\{\\mathrm\{loc\}\}domains\. ∎

### C\.9Proof of Lemma[4\.8](https://arxiv.org/html/2610.27474#S4.Thmtheorem8)

###### Proof\.

For eachb≠0b\\neq 0, at least onee∈𝒟e\\in\\mathcal\{D\}satisfiese⋅b<0e\\cdot b<0, because the two chosen axes are independent\. The rectangle lies ine⋅w≥0e\\cdot w\\geq 0, for either sign ofδ\\delta; the tail lies ine⋅w<0e\\cdot w<0except at zero\. For0<ρ<10<\\rho<1, follow its first three edges and the returning fourth edge only toρ​δ​e⟂\\rho\\delta e^\{\\perp\}, join that point toρ​b\\rho bby a segment, then follow the remaining tail tobb\. The connector has negativeee\-projection after its first point, so it cannot meet the rectangle\. It meets the tail only atρ​b\\rho b, sincebbis not parallel toe⟂e^\{\\perp\}\. The full polygon is consequently injective and still starts exactly at zero\.

For an explicit velocity estimate, assign each original rectangle edge and the tail one unit of time\. Cut the return at time4−ρ4\-\\rho, traverse the connector until time4\+ρ4\+\\rho, and then resume the original tail\. The connector velocity is\(b−δ​e⟂\)/2\(b\-\\delta e^\{\\perp\}\)/2\. All changes occur on an interval of length2​ρ2\\rho, givingL1L^\{1\}error at mostC​ρ​\(‖b‖\+\|δ\|\)C\\rho\(\\\|b\\\|\+\|\\delta\|\)\. Round finitely many corners in mutually disjoint shrinking disks\. Elementary smooth rounding of these nondegenerate polygonal corners preserves injectivity, endpoints, and regularity, and adds a vanishingL1L^\{1\}velocity error\. The disks may depend on the fixedb,δb,\\delta; no uniform radius over these parameters is required\. The total variations are bounded\.

Ifb=0b=0, truncate the final return atρ​δ​e⟂\\rho\\delta e^\{\\perp\}and round its corners\. On a fixed last time interval one can reduce the return velocity by the factor1−ρ1\-\\rho; the sameL1L^\{1\}conclusion holds and the terminal base point tends to zero\. The caseδ=0\\delta=0is obtained later by continuity inδ\\delta, without approximating a degenerate out\-and\-back path by embedded arcs\.

For the lift equationθ′=H1​\(θ\)​u′\+H2​\(θ\)​v′\\theta^\{\\prime\}=H\_\{1\}\(\\theta\)u^\{\\prime\}\+H\_\{2\}\(\\theta\)v^\{\\prime\}, bounded state derivatives on a compact chart give

supt≤T‖θj​\(t\)−θ⁡\(t\)‖≤C​‖wj′−w′‖L1​exp⁡\(C​∫0T‖wj′​\(t\)‖​𝑑t\)\.\\sup\_\{t\\leq T\}\\\|\\theta\_\{j\}\(t\)\-\\theta\(t\)\\\|\\leq C\\\|w\_\{j\}^\{\\prime\}\-w^\{\\prime\}\\\|\_\{L^\{1\}\}\\exp\\\!\\left\(C\\int\_\{0\}^\{T\}\\\|w\_\{j\}^\{\\prime\}\(t\)\\\|\\,dt\\right\)\.The bounded variations and vanishingL1L^\{1\}error thus give uniform convergence of lifts from the sameθ0\\theta\_\{0\}, and continuation keeps the fine approximants inside the ordinary neighborhood\. Each embedded regular represented lift is realized by one global smooth loss using the full compatible jet construction in Lemma[2\.3](https://arxiv.org/html/2610.27474#S2.Thmtheorem3)\. Ordinary closure holds at its endpoint\. Passing to the endpoint limit through the already smoothKlocK\_\{\\mathrm\{loc\}\}proves the required equality at every rectangle and tail endpoint, including closed\-loop endpoints\. Those limiting closed loops are not asserted to be nonstationary autonomous loss trajectories themselves\. Continuity ofhe​\(δ\)h\_\{e\}\(\\delta\)and of its straight\-tail transport coversδ=0\\delta=0\. ∎

### C\.10Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq=2,p=3q=2,p=3

###### Proof of Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)forq=2,p=3q=2,p=3\.

LetPbP\_\{b\}be horizontal transport along the straight base segment from zero tobb, starting at an arbitrary initial fiber coordinateη∈I\\eta\\in I\. For everyb≠0b\\neq 0select an exterior directioneeand a signedδ\\deltasuch thathe​\(δ\)h\_\{e\}\(\\delta\)is precisely that fiber point\. Lemma[4\.8](https://arxiv.org/html/2610.27474#S4.Thmtheorem8)gives

M⁡\(Pb​\(η\)\)=Kloc​\(ϕ⁡\(Pb​\(η\)\)\)\.M\(P\_\{b\}\(\\eta\)\)=K\_\{\\mathrm\{loc\}\}\(\\phi\(P\_\{b\}\(\\eta\)\)\)\.Forb=0b=0the same follows from the truncated\-return limit\. The mapΨ⁡\(b,η\)=Pb​\(η\)\\Psi\(b,\\eta\)=P\_\{b\}\(\\eta\)is smooth, has base projectionbb, and has an invertible fiber derivative at\(0,0\)\(0,0\)\. The inverse function theorem therefore makes it a local diffeomorphism from a smaller base disk times fiber interval onto a plaque neighborhood ofθ0\\theta\_\{0\}\. The equality holds at every such point\. Discontinuous choice of a witnessing directioneeis harmless:Ψ\\Psiitself is smooth, and the equality is pointwise for every\(b,η\)\(b,\\eta\)\.

Finally shrink the original ambient Frobenius product box so its entire initial level lies in this neighborhood\. ExtendKlocK\_\{\\mathrm\{loc\}\}globally by a target cutoff equal to one nearϕ⁡\(θ0\)\\phi\(\\theta\_\{0\}\), and shrink once more into that equality region\. This proves the metric property on the actual local invariant level and completes Theorem[4\.2](https://arxiv.org/html/2610.27474#S4.Thmtheorem2)\. ∎

### C\.11Proof of Proposition[4\.9](https://arxiv.org/html/2610.27474#S4.Thmtheorem9)

###### Proof\.

The stationary case is immediate\. Otherwise fix one open clock ballBBand a state box inside the domain ofKlocK\_\{\\mathrm\{loc\}\}\. For any fixed finite sequenceg1,…,gk∈Bg\_\{1\},\\ldots,g\_\{k\}\\in B, letE⁡\(t\)E\(t\)be the flow\-word endpoint at durationst1,…,tkt\_\{1\},\\ldots,t\_\{k\}\. All sufficiently small positive duration vectors have a strict represented clock\. Smooth control ramps and the compatible autonomous jet construction give genuine single\-global\-loss endpoints tending toE⁡\(t\)E\(t\)\. The smooth representative permits the limit, soR⁡\(E⁡\(t\)\)=0R\(E\(t\)\)=0throughout the positive open duration orthant near zero\.

The finite endpoint map extends smoothly to a full neighborhood of zero durations\. Every mixed derivative ofR∘ER\\circ Eon the positive orthant is zero and remains zero at its corner by continuity\. Differentiating once in each duration yields

Yg1⋯YgkR\(θ0\)=0,Yg=−Dϕ𝖳g,Y\_\{g\_\{1\}\}\\cdots Y\_\{g\_\{k\}\}R\(\\theta\_\{0\}\)=0,\\qquad Y\_\{g\}=\-D\\phi^\{\\mathsf\{T\}\}g,with the naming order fixed by pullback convention; every sequence order is available\. The zero\-order identity follows at the initial time\. Negative durations are used only to define derivatives of a smooth map, not as assumed test trajectories\.

For each fixedkkthe displayed evaluation is multilinear in its constant controls\. Vanishing onBkB^\{k\}extends it to all controls, and therefore to all ordered coordinate\-gradient words\. Every bracket field is a finite signed combination of such differential operator words, and products of bracket fields retain this property\. Choose a finite local bracket frameV1,…,VpV\_\{1\},\\ldots,V\_\{p\}forℰ\\mathcal\{E\}\. All its ordered products annihilateRRatθ0\\theta\_\{0\}\. Each plaque\-coordinate vector field is a smooth linear combination of this frame\. Repeated Leibniz expansion of any coordinate derivative consequently consists of smooth coefficients times ordered frame derivatives ofRR, all zero atθ0\\theta\_\{0\}\. This proves flatness in every plaque chart\. The argument does not identify a smooth flat function with zero on a neighborhood\. ∎

### C\.12Proof of Proposition[4\.10](https://arxiv.org/html/2610.27474#S4.Thmtheorem10)

###### Proof\.

Sinceq=2q=2,X,YX,Ysmoothly frameℋ\\mathcal\{H\}\. Putλ=⟨X,Y⟩=a𝖳​M​b\\lambda=\\langle X,Y\\rangle=a^\{\\mathsf\{T\}\}Mb\. Euclidean differentiation gives the exact identity

\[X,Y\]=grad⁡λ−2​D2​u​Y\.\[X,Y\]=\\grad\\lambda\-2D^\{2\}u\\,Y\.The bracket lies inℰ\\mathcal\{E\}and the last term is horizontal, so the full ambientgrad⁡λ\\grad\\lambdabelongs toℰ\\mathcal\{E\}\. If ordinary closure holds, Lemma[2\.4](https://arxiv.org/html/2610.27474#S2.Thmtheorem4)gives a nonempty plaque\-open patchOOwith the common full\-matrix factorization\. Actual constant\-rank fibers ofϕ\|L\\phi\|\_\{L\}makeλ\\lambdaconstant along them\. Its gradient is therefore orthogonal toℰ∩ker⁡D​ϕ\\mathcal\{E\}\\cap\\ker D\\phi\. Because it also lies inℰ\\mathcal\{E\}, it belongs toℋ\\mathcal\{H\}, the orthogonal complement of that vertical subspace insideℰ\\mathcal\{E\}\. The identity above then makes\[X,Y\]\[X,Y\]horizontal onOO\.

Smooth frame coefficients imply that the horizontal bundle is involutive on this leaf\-open patch\. Restriction of ambient tangent fields toLLcommutes with brackets\. All coordinate gradients are smooth combinations ofX,YX,Y, so their generated distribution onOOhas rank two\. This contradicts constantp\>2p\>2; an ambient\-open involutivity patch is not needed\. Conversely, whenp=q=2p=q=2, the local recoverability argument already proves the metric property and hence ordinary closure\. FinallyD2​u=0D^\{2\}u=0in the affine case andD2​u=α​ID^\{2\}u=\\alpha Iin the isotropic quadratic case, giving the stated specializations without changing the Euclidean parameter metric\. ∎

## Appendix DProofs of the autonomous\-loss controls

### D\.1Proof of Proposition[5\.1](https://arxiv.org/html/2610.27474#S5.Thmtheorem1)

###### Proof\.

WriteF⁡\(x\)=xkF\(x\)=x^\{k\}\. The coordinate gradients are

X=∂x,Y=\(F′\(x\)z,1,F\(x\)\)\.X=\\partial\_\{x\},\\qquad Y=\(F^\{\\prime\}\(x\)z,1,F\(x\)\)\.They are independent everywhere andadXkY=k\!∂z\\operatorname\{ad\}\_\{X\}^\{k\}Y=k\!\\partial\_\{z\}, so they generate a rank\-three distribution everywhere\. At the originY−∂yY\-\\partial\_\{y\}has polynomial coefficients homogeneous of degreekk, independent ofyy\. Every nonzero bracket of lengthℓ≥2\\ell\\geq 2contains at least one such nonconstant factor and at mostℓ−1\\ell\-1coefficient differentiations, hence vanishes at the origin whenℓ≤k\\ell\\leq k\. Thus the stated delay is exact\. The mobility is

M=\[1F′​zF′​z1\+F2\+\(F′​z\)2\]\.M=\\begin\{bmatrix\}1&F^\{\\prime\}z\\\\ F^\{\\prime\}z&1\+F^\{2\}\+\(F^\{\\prime\}z\)^\{2\}\\end\{bmatrix\}\.Its first parameter derivative is zero at the origin fork≥2k\\geq 2\. Consequentlyrank⁡D⁡\(ϕk,M\)​\(0\)=2\\rank D\(\\phi\_\{k\},M\)\(0\)=2, so this necessary first\-order test alone does not detect the failure below\.

Fixa\>0a\>0andϵ\>0\\epsilon\>0, and set, for all realuu,

h⁡\(u\)\\displaystyle h\(u\)=ϵ​u​\(a−u\),\\displaystyle=\\epsilon u\(a\-u\),ζ⁡\(u\)\\displaystyle\\zeta\(u\)=11\+F​\(u\)2​∫0uF⁡\(s\)​h′​\(s\)1\+F​\(s\)2​𝑑s,\\displaystyle=\\frac\{1\}\{\\sqrt\{1\+F\(u\)^\{2\}\}\}\\int\_\{0\}^\{u\}\\frac\{F\(s\)h^\{\\prime\}\(s\)\}\{\\sqrt\{1\+F\(s\)^\{2\}\}\}\\,ds,B⁡\(u\)\\displaystyle B\(u\)=F′​\(u\)​ζ​\(u\)−h′​\(u\)1\+F​\(u\)2,\\displaystyle=\\frac\{F^\{\\prime\}\(u\)\\zeta\(u\)\-h^\{\\prime\}\(u\)\}\{1\+F\(u\)^\{2\}\},A⁡\(u\)\\displaystyle A\(u\)=∫0u\[−1−\(h′​\(s\)−F′​\(s\)​ζ​\(s\)\)21\+F​\(s\)2\]​𝑑s\.\\displaystyle=\\int\_\{0\}^\{u\}\\left\[\-1\-\\frac\{\(h^\{\\prime\}\(s\)\-F^\{\\prime\}\(s\)\\zeta\(s\)\)^\{2\}\}\{1\+F\(s\)^\{2\}\}\\right\]ds\.All these functions are globally smooth, since the denominators are strictly positive\. Take

f1​\(u,v\)=−u,f2​\(u,v\)=A⁡\(u\)\+B⁡\(u\)​\(v−h⁡\(u\)\)\.f\_\{1\}\(u,v\)=\-u,\\qquad f\_\{2\}\(u,v\)=A\(u\)\+B\(u\)\(v\-h\(u\)\)\.The first trajectory from the origin is\(x,y,z\)=\(t,0,0\)\(x,y,z\)=\(t,0,0\)\. The second is

x=t,z=ζ⁡\(t\),y=h⁡\(t\)−F⁡\(t\)​ζ​\(t\),ϕk=\(t,h⁡\(t\)\)\.x=t,\\qquad z=\\zeta\(t\),\\qquad y=h\(t\)\-F\(t\)\\zeta\(t\),\\qquad\\phi\_\{k\}=\(t,h\(t\)\)\.Indeed, differentiating the defining integral givesζ′=F⁡\(h′−F′​ζ\)/\(1\+F2\)=−F​B\\zeta^\{\\prime\}=F\(h^\{\\prime\}\-F^\{\\prime\}\\zeta\)/\(1\+F^\{2\}\)=\-FB\. Along the displayed graphgrad⁡f2=\(A′−B​h′,B\)\\grad f\_\{2\}=\(A^\{\\prime\}\-Bh^\{\\prime\},B\), and the factor equations are

x˙=−A′\+B⁡\(h′−F′​ζ\)=1,z˙=−F​B=ζ′,y˙=−B\.\\dot\{x\}=\-A^\{\\prime\}\+B\(h^\{\\prime\}\-F^\{\\prime\}\\zeta\)=1,\\qquad\\dot\{z\}=\-FB=\\zeta^\{\\prime\},\\qquad\\dot\{y\}=\-B\.The asserted derivative ofyyequals\(h′−F′​ζ\)/\(1\+F2\)=−B\(h^\{\\prime\}\-F^\{\\prime\}\\zeta\)/\(1\+F^\{2\}\)=\-Bas well\. Thus uniqueness verifies the actual autonomous trajectories, not merely a controlled proxy\.

At timeaaboth outputs equal\(a,0\)\(a,0\)\. Integration by parts gives

ζ\(a\)1\+F​\(a\)2=−∫0ah⁡\(s\)​F′​\(s\)\(1\+F​\(s\)2\)3/2ds<0\.\\zeta\(a\)\\sqrt\{1\+F\(a\)^\{2\}\}=\-\\int\_\{0\}^\{a\}\\frac\{h\(s\)F^\{\\prime\}\(s\)\}\{\(1\+F\(s\)^\{2\}\)^\{3/2\}\}\\,ds<0\.The first trajectory hasM12=0M\_\{12\}=0and the second hasM12=F′​\(a\)​ζ​\(a\)<0M\_\{12\}=F^\{\\prime\}\(a\)\\zeta\(a\)<0\. They cannot share one entire mobilityKKat the common output\. Asa↓0a\\downarrow 0withϵ\\epsilonfixed, both parameter paths on\[0,a\]\[0,a\]converge uniformly to the origin\. Hence the witness lies inside every proposed ordinary neighborhood for a small enoughaa\. This proves failure of \(O\) with the exact required quantifiers\. ∎

### D\.2Proof of Proposition[5\.2](https://arxiv.org/html/2610.27474#S5.Thmtheorem2)

###### Proof\.

HereX=\(1,0,0,0\)X=\(1,0,0,0\)andY=\(z,1,x\+w,z\)Y=\(z,1,x\+w,z\)are independent;\[X,Y\]=∂z\[X,Y\]=\\partial\_\{z\}and\[∂z,Y\]=∂x\+∂w\[\\partial\_\{z\},Y\]=\\partial\_\{x\}\+\\partial\_\{w\}\. These four vectors spanℝ4\\mathbb\{R\}^\{4\}everywhere\. The first output is affine, so Proposition[4\.10](https://arxiv.org/html/2610.27474#S4.Thmtheorem10)excludes ordinary closure\. Directly,

M=\[1zz1\+\(x\+w\)2\+2​z2\],v=\(0,−\(x\+w\),1,0\)∈ker⁡D​ϕM=\\begin\{bmatrix\}1&z\\\\ z&1\+\(x\+w\)^\{2\}\+2z^\{2\}\\end\{bmatrix\},\\qquad v=\(0,\-\(x\+w\),1,0\)\\in\\ker D\\phihasD​M12​\[v\]=1DM\_\{12\}\[v\]=1, violating the initial\-plaque flatness consequence\. This example satisfies the rank hypotheses but not the universal ordinary premise; it is not a counterexample to the equivalence in its proved regimes\. ∎

## Appendix EProofs for restricted loss families

### E\.1Proof of Proposition[6\.1](https://arxiv.org/html/2610.27474#S6.Thmtheorem1)

###### Proof\.

LetE⁡\(v\)E\(v\)be the time\-one endpoint fromθ0\\theta\_\{0\}ofθ˙=−D​ϕ​\(θ\)T​v\\dot\{\\theta\}=\-D\\phi\(\\theta\)^\{T\}v\. Forvvin a sufficiently small ball, small\-field local ODE theory gives existence on\[0,1\]\[0,1\]and smooth dependence onvv\. Atv=0v=0,

E⁡\(0\)=θ0,D​E​\(0\)=−D​ϕ​\(θ0\)T\.E\(0\)=\\theta\_\{0\},\\qquad DE\(0\)=\-D\\phi\(\\theta\_\{0\}\)^\{T\}\.Putz0=ϕ⁡\(θ0\)z\_\{0\}=\\phi\(\\theta\_\{0\}\)andΨ⁡\(v\)=ϕ⁡\(E⁡\(v\)\)\\Psi\(v\)=\\phi\(E\(v\)\)\. ThenD​Ψ​\(0\)=−M0D\\Psi\(0\)=\-M\_\{0\}is invertible\. Shrinking to a ballBδ​\(0\)B\_\{\\delta\}\(0\), the inverse function theorem makesΨ\\Psia diffeomorphism onto an open target neighborhoodVV\. Define

Kloc​\(z\)=M⁡\(E⁡\(Ψ−1​\(z\)\)\),z∈V\.K\_\{\\rm loc\}\(z\)=M\\bigl\(E\(\\Psi^\{\-1\}\(z\)\)\\bigr\),\\qquad z\\in V\.Full row rank persists on a parameter neighborhoodU0U\_\{0\}, on whichM≥m​IM\\geq mIfor somem\>0m\>0\. Choose a smallerU∋θ0U\\ni\\theta\_\{0\}with closure inU0U\_\{0\}, withϕ⁡\(U\)\\phi\(U\)contained in a target neighborhoodV1V\_\{1\}whose closure is compact inVV, and with\|ϕ⁡\(θ\)−z0\|<r\|\\phi\(\\theta\)\-z\_\{0\}\|<ronUU, wherer/m<δr/m<\\delta\.

Forg≠0g\\neq 0letn=g/\|g\|n=g/\|g\|\. As long as its actual flow remains inUU,

dd​t​\[nT​\(ϕ⁡\(θ⁡\(t\)\)−z0\)\]=−\|g\|​nT​M​\(θ⁡\(t\)\)​n≤−m​\|g\|\.\\frac\{d\}\{dt\}\\bigl\[n^\{T\}\(\\phi\(\\theta\(t\)\)\-z\_\{0\}\)\\bigr\]=\-\|g\|\\,n^\{T\}M\(\\theta\(t\)\)n\\leq\-m\|g\|\.Integration and the target\-radius bound implyt​\|g\|<r/m<δt\|g\|<r/m<\\delta\. Time rescaling of the constant\-control vector field givesθ⁡\(t\)=E⁡\(t​g\)\\theta\(t\)=E\(tg\)\. Thust​gtgbelongs to the inverse chart and

t​g=Ψ−1​\(ϕ⁡\(θ⁡\(t\)\)\),M⁡\(θ⁡\(t\)\)=Kloc​\(ϕ⁡\(θ⁡\(t\)\)\)\.tg=\\Psi^\{\-1\}\(\\phi\(\\theta\(t\)\)\),\\qquad M\(\\theta\(t\)\)=K\_\{\\rm loc\}\(\\phi\(\\theta\(t\)\)\)\.This argument covers the whole interval before exit, including arbitrarily largegg; it does not assume thatt​gtgremains small without proving it\. Forg=0g=0the solution is stationary\. A smooth target cutoff equal to one nearV¯1\\overline\{V\}\_\{1\}and supported inVVextendsKlocK\_\{\\rm loc\}by zero to a global smoothKK\. No positivity outside the tested image is required\. ∎

### E\.2Proof of Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3)

###### Proof\.

Writev=y\+xk​zv=y\+x^\{k\}z\. The exact parameter gradient equations are

x˙\\displaystyle\\dot\{x\}=b1−x−γ​v\+k​xk−1​z​\(b2−γ​x−v\),\\displaystyle=b\_\{1\}\-x\-\\gamma v\+kx^\{k\-1\}z\(b\_\{2\}\-\\gamma x\-v\),y˙\\displaystyle\\dot\{y\}=b2−γ​x−v,z˙=xk​\(b2−γ​x−v\)\.\\displaystyle=b\_\{2\}\-\\gamma x\-v,\\qquad\\dot\{z\}=x^\{k\}\(b\_\{2\}\-\\gamma x\-v\)\.For one fixedc\>0c\>0, introduce

s=tc​a,x=a​X,y=a2​Y,z=ak\+2​Z,b1=B/c,b2=a​C\.s=\\frac\{t\}\{ca\},\\quad x=aX,\\quad y=a^\{2\}Y,\\quad z=a^\{k\+2\}Z,\\qquad b\_\{1\}=B/c,\\quad b\_\{2\}=aC\.PutV=Y\+a2​k​Xk​ZV=Y\+a^\{2k\}X^\{k\}Z, so thatv=a2​Vv=a^\{2\}V\. On0≤s≤10\\leq s\\leq 1the exact scaled initial\-value problem, starting at zero, is

X′\\displaystyle X^\{\\prime\}=B−c​a​X−c​γ​a2​V\+c​k​a2​k\+2​Xk−1​Z​\(C−γ​X−a​V\),\\displaystyle=B\-caX\-c\\gamma a^\{2\}V\+cka^\{2k\+2\}X^\{k\-1\}Z\(C\-\\gamma X\-aV\),\(16\)Y′\\displaystyle Y^\{\\prime\}=c⁡\(C−γ​X−a​V\),\\displaystyle=c\(C\-\\gamma X\-aV\),\(17\)Z′\\displaystyle Z^\{\\prime\}=c​Xk​\(C−γ​X−a​V\)\.\\displaystyle=cX^\{k\}\(C\-\\gamma X\-aV\)\.\(18\)These equations are polynomial in states and parametersa,B,Ca,B,C\. Ata=0a=0the solutions on\[0,1\]\[0,1\]are bounded polynomials\. Local ODE dependence therefore gives existence and analytic dependence on the whole interval for parameters in a small neighborhood of the reference values\. Concretely, choose a compact tube around the limiting trajectory; the integral equation and a Gronwall estimate prevent exit for nearby parameters\. No global completeness of polynomial vector fields is asserted\.

LetEc​\(a,B,C\)=\(X⁡\(1\),V⁡\(1\)\)E\_\{c\}\(a,B,C\)=\(X\(1\),V\(1\)\)\. Ata=0a=0,

Ec​\(0,B,C\)=\(B,c⁡\(C−γ​B/2\)\)\.E\_\{c\}\(0,B,C\)=\\bigl\(B,c\(C\-\\gamma B/2\)\\bigr\)\.At\(B,C\)=\(1,γ/2\)\(B,C\)=\(1,\\gamma/2\)the endpoint is\(1,0\)\(1,0\), and the shooting Jacobian is

D\(B,C\)​Ec​\(0,1,γ/2\)=\(10−cγ/2c\),detD\(B,C\)​Ec=c\>0\.D\_\{\(B,C\)\}E\_\{c\}\(0,1,\\gamma/2\)=\\begin\{pmatrix\}1&0\\\\ \-c\\gamma/2&c\\end\{pmatrix\},\\qquad\\det D\_\{\(B,C\)\}E\_\{c\}=c\>0\.The analytic implicit function theorem suppliesBc​\(a\),Cc​\(a\)B\_\{c\}\(a\),C\_\{c\}\(a\)withBc​\(0\)=1B\_\{c\}\(0\)=1,Cc​\(0\)=γ/2C\_\{c\}\(0\)=\\gamma/2andEc​\(a,Bc​\(a\),Cc​\(a\)\)=\(1,0\)E\_\{c\}\(a,B\_\{c\}\(a\),C\_\{c\}\(a\)\)=\(1,0\)exactly\. Thusbc​\(a\)=\(Bc​\(a\)/c,a​Cc​\(a\)\)b\_\{c\}\(a\)=\(B\_\{c\}\(a\)/c,aC\_\{c\}\(a\)\)gives the exact represented endpoint\(a,0\)\(a,0\)\. Use a common sufficiently smalla∗a\_\{\*\}forc1,c2c\_\{1\},c\_\{2\}\. The equality is an actual endpoint identity, not a formal\-series substitute for a shooting solution\.

Ata=0a=0and these limiting shooting parameters,X⁡\(s\)=sX\(s\)=sand

Z⁡\(1\)=c​γ​∫01sk​\(1/2−s\)​𝑑s=−c​γ​k2​\(k\+1\)​\(k\+2\)\.Z\(1\)=c\\gamma\\int\_\{0\}^\{1\}s^\{k\}\(1/2\-s\)\\,ds=\-\\frac\{c\\gamma k\}\{2\(k\+1\)\(k\+2\)\}\.Smooth dependence in \([18](https://arxiv.org/html/2610.27474#A5.E18)\) yields

z⁡\(c​a\)=−c​γ​k2​\(k\+1\)​\(k\+2\)​ak\+2\+O⁡\(ak\+3\)\.z\(ca\)=\-\\frac\{c\\gamma k\}\{2\(k\+1\)\(k\+2\)\}a^\{k\+2\}\+O\(a^\{k\+3\}\)\.The full mobility is

M⁡\(x,y,z\)=\(1k​xk−1​zk​xk−1​z1\+x2​k\+\(k​xk−1​z\)2\)\.M\(x,y,z\)=\\begin\{pmatrix\}1&kx^\{k\-1\}z\\\\ kx^\{k\-1\}z&1\+x^\{2k\}\+\(kx^\{k\-1\}z\)^\{2\}\\end\{pmatrix\}\.At each exact endpointx=ax=a, multiplication byk​ak−1ka^\{k\-1\}and subtraction give \([11](https://arxiv.org/html/2610.27474#S6.E11)\)\. The constants implicit in the remainders depend on the fixedk,γ,c1,c2k,\\gamma,c\_\{1\},c\_\{2\}\. A nonzero leading coefficient implies inequality for all sufficiently small positiveaa\.

The scaled states stay uniformly bounded on\[0,1\]\[0,1\]asa→0a\\to 0\. Consequentlyx=O⁡\(a\)x=O\(a\),y=O⁡\(a2\)y=O\(a^\{2\}\)andz=O⁡\(ak\+2\)z=O\(a^\{k\+2\}\)uniformly on the whole actual trajectories\. They therefore fit inside every proposed common neighborhood for small enoughaa\. A singleKKcannot take both unequal mobility values at the same output\(a,0\)\(a,0\)\. This is precisely the required failure of the restricted ordinary\-closure property\. Its quantifier permits different horizons\. ∎

### E\.3Proof of Corollary[6\.4](https://arxiv.org/html/2610.27474#S6.Thmtheorem4)

###### Proof\.

Forτ=W−1​b\\tau=W^\{\-1\}b,

ℓτ​\(ψk​\(θ\)\)=fb​\(ϕk​\(θ\)\)\+12​bT​H−1​b\.\\ell\_\{\\tau\}\(\\psi\_\{k\}\(\\theta\)\)=f\_\{b\}\(\\phi\_\{k\}\(\\theta\)\)\+\\tfrac\{1\}\{2\}b^\{T\}H^\{\-1\}b\.The additive constant changes no Euclidean parameter gradient field\. Thus the two actual paths in Theorem[6\.3](https://arxiv.org/html/2610.27474#S6.Thmtheorem3)are also paths for ordinary squared error on this one fixed representation, and their represented endpoints both equalW​\(a,0\)TW\(a,0\)^\{T\}\. Moreover

D​ψk=W​D​ϕk,Mψk=W​Mϕk​WT\.D\\psi\_\{k\}=WD\\phi\_\{k\},\\qquad M\_\{\\psi\_\{k\}\}=WM\_\{\\phi\_\{k\}\}W^\{T\}\.Invertibility ofWWpreserves the nonzero full mobility difference; a particular matrix entry need not be preserved\. The coordinate gradients are invertible constant linear combinations of those ofϕk\\phi\_\{k\}, so their generated Lie distributions are the same\. The ranks follow from Remark[6\.2](https://arxiv.org/html/2610.27474#S6.Thmtheorem2)\. The formula forMϕkM\_\{\\phi\_\{k\}\}givesD​Mϕk​\(0\)=0DM\_\{\\phi\_\{k\}\}\(0\)=0fork≥2k\\geq 2; constant congruence gives the same property forMψkM\_\{\\psi\_\{k\}\}\. ∎

### E\.4Proof of Proposition[6\.6](https://arxiv.org/html/2610.27474#S6.Thmtheorem6)

###### Proof\.

Atb=0b=0the parameter flow fromθ0\\theta\_\{0\}is stationary\. LetJ0=D​ϕ​\(θ0\)J\_\{0\}=D\\phi\(\\theta\_\{0\}\)andM0=J0​J0TM\_\{0\}=J\_\{0\}J\_\{0\}^\{T\}\. At one fixed sufficiently short timeτ\>0\\tau\>0, smooth flow dependence defines the endpoint for all sufficiently smallbb, with the entire path in the ordinary neighborhood\. IfP⁡\(t\)P\(t\)is the derivative of the parameter state with respect tobbat zero, variation of the vector field gives

P˙=−J0T​J0​P−J0T,P⁡\(0\)=0\.\\dot\{P\}=\-J\_\{0\}^\{T\}J\_\{0\}P\-J\_\{0\}^\{T\},\\qquad P\(0\)=0\.ConsequentlyQ⁡\(t\)=J0​P​\(t\)Q\(t\)=J\_\{0\}P\(t\)satisfiesQ˙=−M0​Q−M0\\dot\{Q\}=\-M\_\{0\}Q\-M\_\{0\},Q⁡\(0\)=0Q\(0\)=0, so

Db​ϕ​\(θb​\(τ\)\)\|b=0=−\(I−e−τ​M0\)\.D\_\{b\}\\phi\(\\theta\_\{b\}\(\\tau\)\)\\big\|\_\{b=0\}=\-\(I\-e^\{\-\\tau M\_\{0\}\}\)\.\(19\)This identity does not requireM0M\_\{0\}to be invertible\. OnS=im⁡M0S=\\operatorname\{im\}M\_\{0\}, its eigenvalues are−\(1−e−τ​λ\)\-\(1\-e^\{\-\\tau\\lambda\}\)forλ\>0\\lambda\>0, hence nonzero\. Restrictbbto theqq\-dimensional spaceSS\. The constant\-rank theorem supplies a local embedded image ofϕ\\phiwith tangentSSatz0z\_\{0\}, after shrinking the parameter neighborhood\. In a chart of that image, \([19](https://arxiv.org/html/2610.27474#A5.E19)\) is an invertibleqq\-dimensional endpoint derivative\. The inverse function theorem produces a sectionσ\\sigmasatisfyingϕ⁡\(σ⁡\(z\)\)=z\\phi\(\\sigma\(z\)\)=zon a represented neighborhood ofz0z\_\{0\}, where eachσ⁡\(z\)\\sigma\(z\)is an actual endpoint of one of the tested flows\. Thus

K⁡\(z\)=M⁡\(σ⁡\(z\)\)K\(z\)=M\(\\sigma\(z\)\)there, by the common\-closure hypothesis\. The right side is smooth\. A local target chart flattening the embedded image extends it normally; a target cutoff then gives a global smooth extension agreeing on a smaller image\. In the analytic case the local ODE, image charts and inverse theorem give an analytic section and a local analytic extension; no global analytic cutoff is asserted\. Forq<dq<dthe inverse theorem has only been used on the image, not on all ofℝd\\mathbb\{R\}^\{d\}\. ∎

## Appendix FFinite checks and reproducibility

The universal statements above rest on the included proofs\. Existing exact algebraic checks are finite implementation evidence and were not rerun to produce this manuscript\. The latest boundary validation used rational arithmetic on 39 fixtures: 24 effective scalar\-imbalance and 15 nonscalar cases, including four zero\-scalar cases\. It checked 259 hidden\-rotation and tangent identities and reported zero mismatches\. These were bounded algebraic checks, not model training\. These finite checks neither prove the universal classification nor establish task accuracy, statistical uncertainty, approximate\-closure error or a practical speedup\.

The local provenance audit preserves the available inputs, proof versions, automated review records, primary source texts, and their hashes\. Complete local source custody is distinguished from selected theorem\-and\-proof reading\. The two closest source versions are arXiv:2508\.07370v2 \(12 March 2026, ICLR 2026 rendition\) and arXiv:2307\.00144v2 \(10 July 2024 revision of the NeurIPS 2023 work\)\. The following numerical record is copied from a frozen, actually executed experiment\. No new scientific run was used to assemble this manuscript\. The algebraic and autonomous\-loss records answer different finite questions\.

## Appendix GA finite autonomous\-loss implementation check

We executed one deterministic CPU sentinel for the engineered smooth\-loss construction in Proposition[5\.1](https://arxiv.org/html/2610.27474#S5.Thmtheorem1)\. This tests an implementation of a proved mechanism, rather than a learning algorithm\. The fixed\-Hessian quadratic and ordinary squared\-loss extension in this manuscript has*not*been executed: its numericala∗a\_\{\*\}and any executed quadratic\-loss grid remain unknown\. The measurements below must not be transferred to that new loss family\.

#### Actual objective and parameter flow\.

SetF⁡\(u\)=ukF\(u\)=u^\{k\},h⁡\(u\)=u⁡\(a−u\)h\(u\)=u\(a\-u\)andD⁡\(u\)=1\+F​\(u\)2D\(u\)=1\+F\(u\)^\{2\}\. The implemented global objective was

f2​\(u,v\)=A⁡\(u\)\+B⁡\(u\)​\(v−h⁡\(u\)\),B=F′​ζ−h′D,A′=−1−\(h′−F′​ζ\)2D,A⁡\(0\)=0,f\_\{2\}\(u,v\)=A\(u\)\+B\(u\)\(v\-h\(u\)\),\\qquad B=\\frac\{F^\{\\prime\}\\zeta\-h^\{\\prime\}\}\{D\},\\qquad A^\{\\prime\}=\-1\-\\frac\{\(h^\{\\prime\}\-F^\{\\prime\}\\zeta\)^\{2\}\}\{D\},\\quad A\(0\)=0,where

ζ\(u\)=D\(u\)−1/2∫0uF\(s\)h′\(s\)D\(s\)−1/2ds\.\\zeta\(u\)=D\(u\)^\{\-1/2\}\\int\_\{0\}^\{u\}F\(s\)h^\{\\prime\}\(s\)D\(s\)^\{\-1/2\}\\,ds\.Writingv=y\+F⁡\(x\)​zv=y\+F\(x\)z, its full output gradient and actual parameter ODE were

gu\\displaystyle g\_\{u\}=A′\+B′​\(v−h\)−B​h′,\\displaystyle=A^\{\\prime\}\+B^\{\\prime\}\(v\-h\)\-Bh^\{\\prime\},gv\\displaystyle g\_\{v\}=B,\\displaystyle=B,\(x˙,y˙,z˙\)\\displaystyle\(\\dot\{x\},\\dot\{y\},\\dot\{z\}\)=−\(gu\+F′​\(x\)​z​B,B,F⁡\(x\)​B\)\.\\displaystyle=\-\(g\_\{u\}\+F^\{\\prime\}\(x\)zB,\\ B,\\ F\(x\)B\)\.In particular the off\-graph termB′​\(v−h\)B^\{\\prime\}\(v\-h\)was included; substituting the analytic graph for this ODE would not qualify the implementation\. The analytic comparison path is\(t,h⁡\(t\)−F⁡\(t\)​ζ​\(t\),ζ⁡\(t\)\)\(t,h\(t\)\-F\(t\)\\zeta\(t\),\\zeta\(t\)\)\. The comparison lossf1​\(u,v\)=−uf\_\{1\}\(u,v\)=\-uwas also integrated, with analytic path\(t,0,0\)\(t,0,0\)\.

#### Frozen protocol and discriminating controls\.

The twelve witness cases usedk∈\{2,3,4\}k\\in\\\{2,3,4\\\}anda∈\{0\.5,0\.35,0\.25,0\.175\}a\\in\\\{0\.5,0\.35,0\.25,0\.175\\\}, withϵ=1\\epsilon=1\. Four matched controls usedF≡0F\\equiv 0at the sameaavalues; these havep=q=2p=q=2, mobilityI2I\_\{2\}and zero gap\. We used DOP853 with relative tolerance2×10−122\\times 10^\{\-12\}, absolute tolerance10−1410^\{\-14\}and 21 stored times per trajectory\. Working scalar quadratures used absolute and relative tolerances10−1310^\{\-13\}and at most 100 subdivisions\. Independently, a 50\-decimal\-digit quadrature evaluated the cancellation\-avoiding reference

Δref=−k2​a2​k\+11\+a2​k∫01tk​\(1−t\)\(1\+a2​k​t2​k\)3/2dt\.\\Delta\_\{\\rm ref\}=\-\\frac\{k^\{2\}a^\{2k\+1\}\}\{\\sqrt\{1\+a^\{2k\}\}\}\\int\_\{0\}^\{1\}\\frac\{t^\{k\}\(1\-t\)\}\{\(1\+a^\{2k\}t^\{2k\}\)^\{3/2\}\}\\,dt\.\(20\)This is a high\-precision estimate, not a certified quadrature interval\.

Five\-point central differences, step2×10−42\\times 10^\{\-4\}, differentiated the globally defined scalar loss composed withϕk\\phi\_\{k\}in all three parameter coordinates\. Both probes were off graph:\(x,y,z\)=\(0,0\.1,0\.03\)\(x,y,z\)=\(0,0\.1,0\.03\), andx=0\.3​ax=0\.3a,z=0\.02​az=0\.02a,v=h⁡\(x\)\+0\.05​av=h\(x\)\+0\.05a, withy=v−F⁡\(x\)​zy=v\-F\(x\)z\. Each coordinate used tolerance5×10−8\+5×10−6​\|gFD\|5\\times 10^\{\-8\}\+5\\times 10^\{\-6\}\|g\_\{\\rm FD\}\|\. The deliberately incorrect gradient omittingB′​\(v−h\)B^\{\\prime\}\(v\-h\)had to fail the same tolerance at both probes\. Its first\-probe discrepancy had to exceed10−310^\{\-3\}\. This matters because the omitted term vanishes on the analytic graph\.

Before execution we fixed state error≤3×10−10\\leq 3\\times 10^\{\-10\}, represented endpoint error≤10−10\\leq 10^\{\-10\}, nonzero\-gap relative error≤5×10−6\\leq 5\\times 10^\{\-6\}, and a zero\-control gap threshold10−1210^\{\-12\}\. ForL=k2​a2​k\+1/\(\(k\+1\)​\(k\+2\)\)L=k^\{2\}a^\{2k\+1\}/\(\(k\+1\)\(k\+2\)\), the measured−Δ/L\-\\Delta/Lwas compared with\[\(1\+a2​k\)−2,\(1\+a2​k\)−1/2\]\[\(1\+a^\{2k\}\)^\{\-2\},\(1\+a^\{2k\}\)^\{\-1/2\}\]using slack5×10−65\\times 10^\{\-6\}\. Nonfinite outputs could not pass\. The first completed failed case or exception would stop the sole attempt; no grid or tolerance was selected after observing results\. There are no random seeds or statistical confidence intervals\.

Table 1:Measured endpoint mobility gapΔ=M12​\(f2,a\)−M12​\(f1,a\)\\Delta=M\_\{12\}\(f\_\{2\};a\)\-M\_\{12\}\(f\_\{1\};a\)and relative discrepancy\|Δ−Δref\|/\|Δref\|\|\\Delta\-\\Delta\_\{\\rm ref\}\|/\|\\Delta\_\{\\rm ref\}\|\. All twelve frozen engineered\-loss witnesses are shown\.
#### Observed results and limits\.

All twelve witnesses and four matched controls passed the frozen criteria\. The maximum state discrepancy over the 21 stored times was2\.015×10−122\.015\\times 10^\{\-12\}and represented endpoint discrepancy1\.859×10−131\.859\\times 10^\{\-13\}; all fourF=0F=0gaps were stored as zero and their identity\-mobility checks passed\. All 32 off\-graph probes passed, with maximum coordinate FD discrepancy1\.043×10−131\.043\\times 10^\{\-13\}\. Every omitted\-term mutant was rejected; the minimum across probes of its largest coordinate error was0\.01750\.0175\. Thek=4,a=0\.175k=4,a=0\.175ratio band has width1\.3195×10−61\.3195\\times 10^\{\-6\}, smaller than the allowed slack, and is explicitly tolerance dominated\. Passing that band does not demonstrate sharp\-band or next\-order asymptotic resolution\. The integrator tolerances are not global error certificates, and working quadrature error estimates are heuristic\.

The historical implementation used 3122f2f\_\{2\}right\-hand\-side evaluations and 9057 quadrature calls\. No GPU computation or model training was used\. Raw trajectories, gradient probes, numerical references, code/input hashes are preserved in the local historical record\. The supplied numerical payload can be run independently with the listed dependencies; two original external resource\-wrapper files are unavailable, so the historical execution environment and resource measurements are not fully reproducible from this package\. The public historical data consist of the exported summary and trajectory CSVs; the original raw probe/reference JSON files and source\-custody receipts are not distributed in this package\. These finite observations support implementation and mechanism agreement; the exact endpoint equality and universal quantifiers rest on the proofs\. They establish no learning improvement, algorithmic speedup, numerical validation of the new quadratic theorem, or general asymptotic law\.

#### Separate pre\-submission replay\.

A separate pre\-submission replay under Python 3\.12\.14, NumPy 2\.3\.5, SciPy 1\.17\.0, and mpmath 1\.3\.0 passed all 16 cases and 32 off\-graph probes, rejecting every omitted\-term mutant\. It used 3116f2f\_\{2\}right\-hand\-side evaluations and 7035 quadrature calls\. Source copies, outputs, environment, and comparisons are inanc/audit\_replay/\. These new results are separate from the historical measurements; they do not reproduce the historical environment or resource wrappers, and do not test the fixed\-Hessian quadratic theorem\.