Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
ORCID: 0009-0008-3263-2447
Preprint for priority. Revised 07 August 2026.
Deposited: Zenodo, 7 August 2026. DOI: 10.5281/zenodo.21832686.
Drawn from books and papers yet to be published.
(c) 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
Two established fields apply algebraic geometry to data. Singular learning theory characterizes the generalization behavior of statistical models whose Fisher information degenerates, with deep neural networks the motivating case, through invariants obtained by resolution of singularities. Algebraic statistics treats statistical models as algebraic varieties and performs estimation as geometry on those varieties. A third literature, performative prediction and its relatives, formalizes distributions that respond to the act of prediction.
This paper identifies a coordinate that connects all three and that none of them carries: coordination-dependence, the degree to which a measured quantity's data-generating structure is constituted by the ongoing coordination of the observers around it, quantified in prior work as the fraction b of variance attributable to the coordination-dependent component. Two claims are staked. First, a deposited proposition of The Geometry of Constraint Knowledge, that conservation laws in training data induce Fisher rank deficiency in trained models, places that program's empirical predictions at the entry point of singular learning theory, and a conjecture follows with a specifiable differential test: training data structured as attributed, timestamped, facet-decomposed disclosure shapes the singularity structure of the trained model measurably differently from unstructured text about the same domain, with restricted variants of the local learning coefficient the natural instrument and background architectural degeneracy the identification problem the design must difference out. Second, the fixed-variety assumption of algebraic statistics fails for high-b quantities in a way performative prediction describes dynamically but does not geometrize: the variety moves under the observation directed at it because the observers partly constitute the structure being estimated. A candidate formalization is stated, a family of varieties fibered over a coordination-state manifold whose primary structure is the closed elliptope of similarity matrices, in its Bures-Wasserstein geometry, with the similarity structure already load-bearing in the corpus's formal core, under which the fixed-variety case is recovered as the constant family and the fraction b acquires a candidate geometric referent.
The contribution is the coordinate and its connections. The underlying geometry belongs to the fields that built it, and the paper states the boundary between what is established, what is struck in the corpus, and what is open.
Keywords: singular learning theory, algebraic statistics, performative prediction, Fisher information, local learning coefficient, coordination-dependence, training data, moving variety, OCOT (Origin Chain of Thought)
This is a priority deposit drawing in part on a corpus not yet published. Published papers are cited in standard form with their deposits; unpublished works are not named, and the cover statement carries their attribution. Three terms of art recur. To strike a claim is to commit it to a dated, attributed record before the outcome is known. A facet-decomposed disclosure is a record in the corpus's six-part format (identity, input, method, output, scaling, transmit). The fraction b is defined counterfactually over variance components and is never computed by any runtime; a value traveling with a record is b-hat, an estimate struck by an identified originator with the method stated. OCOT, Origin Chain of Thought, names the field the corpus develops.
Algebraic geometry entered the study of data along two routes that have not needed each other. The first runs through the failure of classical statistical asymptotics. When a model's parameter-to-distribution map is not one-to-one, or its Fisher information matrix degenerates, the regularity behind the classical results fails, and Watanabe (2009) showed that the correct asymptotic theory for such singular models is governed by the real log canonical threshold, a birational invariant obtained through resolution of singularities, with the broader Bayesian treatment consolidated in Watanabe (2018) and explicit threshold computations for concrete model classes going back to Aoyagi and Watanabe (2005). The route would be a curiosity if singular models were rare. They are the norm: deep neural networks are singular models exhibiting complex degeneracies that flat-direction counting cannot capture (Wei et al., 2022), and a localized version of Watanabe's invariant, the local learning coefficient, has been made explicit and estimable at the scale of transformer language models (Lau et al., 2024), with refined variants that resolve the measure to model components now in use (Wang et al., 2024). The second route treats a statistical model as an algebraic variety and develops estimation as geometry on that object, from the lecture-course treatment of Drton, Sturmfels, and Sullivant (2009) through the textbook consolidation of Sullivant (2018).
A third literature is closer to this paper's subject than either. Performative prediction formalizes settings in which the data distribution responds to the deployed predictor, defines performative stability and optimality, and proves convergence results for retraining dynamics (Perdomo et al., 2020), with strategic classification as an earlier special case (Hardt et al., 2016) and the performativity of financial models documented at book length in the sociology of markets (MacKenzie, 2006). That literature establishes the phenomenon this paper cares about, that estimation can move its own target. What it does not do is geometrize the target: its objects are distributions and equilibria of retraining maps, not varieties, and it models response to a deployed decision rule rather than constitution by ongoing observation. The distinction matters and Section 4 works it.
All three literatures share an unstated coordinate. Prior work in this corpus distinguishes observation-independent variance, arising from drivers that run whether or not measurement takes place, from coordination-dependent variance, arising from the ongoing coordination of parties around the measured quantity, and defines the fraction b of a quantity's variance carried by the second component, with disclosed estimators and an empirically testable identifying assumption (Beier, 2026, A Missing Coordinate in the Classification of Quantitative Data). A gold assay and a brand valuation are both data series; only one describes something that would persist through the withdrawal of every observer. The present paper carries that coordinate into the geometry. Section 3 places a struck proposition of the corpus at the entry of singular learning theory and states the conjecture the placement makes testable. Section 4 states what the coordinate does to the fixed-variety assumption, builds the base space from objects the corpus's formal core already carries, and locates the claim relative to performative prediction. Section 5 draws the property lines, and Section 6 specifies the experimental surface.
The corpus's existing geometric program operates at the linear special case, and it supplies more than precedent here: it supplies the raw material of the base space constructed in Section 4. The corpus's observation geometry, in a volume yet to be published, treats an observer as a linear subspace of a K-dimensional constraint space: a scalar reader occupies a point of Gr(1, K), a reader of aggregate-interaction structure spans a plane in Gr(2, K), and the blind set of an observation architecture is a Schubert variety whose collapse under an entering disclosure is computed by the Pieri rule. Grassmannians are projective varieties and Schubert calculus is their intersection theory, so the program is algebraic geometry in the strict sense, restricted to linear objects.
A configuration of observers in that geometry induces a coupling structure. Given N observer subspaces, the pairwise principal-angle cosines between them generate an N-by-N similarity matrix: observers reading overlapping subspaces are strongly coupled, observers reading orthogonal subspaces are not. The corpus's formal core takes exactly such a similarity matrix S among agents as a load-bearing object: the fit kernel L = diag(q) S diag(q), with q the vector of track-record qualities, whose subset determinants combine quality and diversity into one number and whose maximization forms the corpus's processing centers. The kernel form itself is public mathematics, the standard L-ensemble of determinantal point processes (Kulesza and Taskar, 2012), so the construction is checkable independently of the corpus application, which remains under the cover statement; and the aggregate correlation whose bracket, 1 + (N - 1) rho, governs both the accuracy and the stability of a coordinated population, with N over that bracket the effective number of independent judgments (corpus formal core, book yet to be published). One identity is worth recording, because it wires the corpus's master variable directly into the geometry that follows: the bracket 1 + (N - 1) rho is the leading eigenvalue of the equicorrelation similarity matrix, so the quantity the corpus already treats as the diagnostic of a population's coordination state is spectral data on the space of similarity matrices. The configuration of observers says who is in the room; the similarity matrix says how coupled the room is; the map from the first to the second is the principal-angle construction just stated.
The Geometry of Constraint Knowledge (Beier, 2026) made a claim with a specified test. Proposition CG-1 of that paper predicts that when a model's training distribution is generated by a process with conservation constraints, the Fisher information of the trained model vanishes along the conservation-violating direction, with that direction identified in weight space by the paper's Constraint Direction Probe, which supplies the pullback from the constraint in data space to the probed vector in parameter space. The paper also states the overparameterization caution in its own text: modern networks carry billions of near-zero Fisher eigenvalues unrelated to any physical constraint, so the operative test is the targeted projection along the probed direction, not the eigenvalue count. A companion result characterizes the mediation channel through which the constraint's fingerprint survives passage from process data to text about the process, with a computable attenuation bound governed by the density of conservation-relevant quantitative content and three sufficient conditions for preservation (Beier, 2026, Formal Extensions to the OCOT Corpus).
Remark AG-1 (Entry). At any parameter point where the conclusion of CG-1 holds, the trained model is a singular statistical model, since a degenerate Fisher information matrix is the defining condition of singularity.
The remark is definitional and is labeled accordingly. What it earns, and what it does not, both need stating. It does not by itself deliver Watanabe's asymptotics: those results carry conditions, analyticity of the model family among them, and the extension of the theory to non-analytic activations such as ReLU is an active research area rather than a settled inheritance, a gap the singular-networks literature itself discusses (Wei et al., 2022). Nor does it establish that the law-induced flat direction is among the degeneracies that govern the model's learning coefficient; a single flat direction can be asymptotically inert while other, deeper singularities control the invariant. What the remark earns is placement with instruments attached: the vanishing directional Fisher information CG-1 predicts is the visible edge of a degeneracy structure for which the singular learning literature has built a defined, estimable measure, the local learning coefficient (Lau et al., 2024), together with refined variants that resolve degeneracy to specified model components (Wang et al., 2024). Whether the law-induced degeneracy registers in that measure, and how strongly, is thereby an empirical question with an instrument, which is more than the proposition had before the placement and less than a theorem. The paper claims the placement and the question, not the inheritance.
The corpus's substrate thesis holds that training data structured as attributed, timestamped, facet-decomposed disclosure differs in kind from unstructured text as a learning substrate, and the mediation-channel results give the thesis information-theoretic form: structured disclosure raises the density of conservation-relevant quantitative content, satisfies the preservation conditions by construction, and yields strictly higher geometric fidelity along constrained directions, increasing in the coverage of the facet that carries aggregate-interaction structure. The mechanism runs through content statistics, quantitative density, temporal extent, and attributed uncertainty, not through surface formatting; the claim is not that a tokenizer treats a schema as an algebraic constraint, but that the statistics the schema enforces survive projection into the model by the same data-processing argument the attenuation bound rests on. Representational convergence at scale cannot equalize the arms by washing structure out, because the arms differ in what the text carries, not only in how it is arranged: the unstructured arm holds less conservation-relevant content by construction, and no invariance of the architecture can recover signal the corpus does not contain. What convergence can do is compress the margin that structure adds above matched density, which is why the conjecture predicts an ordering rather than a gap of stated size, and why the registration's power analysis carries the burden.
Conjecture AG-2 (Substrate-to-singularity). Hold architecture, scale, initialization scheme, and training procedure fixed. Train one model on a facet-structured disclosure corpus for a constraint domain and one on an unstructured text corpus about the same domain, matched on token count, vocabulary coverage, and domain scope, so that the manipulated variable is the conservation-relevant content density and its structure. Then the two models differ in the degeneracy induced along the domain's constrained directions, with the structured corpus inducing the deeper degeneracy, detectable as an ordering of refined local learning coefficients restricted to the relevant components.
Two features of the statement are load-bearing. The design is differential because the background problem is severe: every trained network of realistic size is massively degenerate for architectural and symmetry reasons, so absolute degeneracy identifies nothing; the source paper meets this with the directional projection along the probed constraint vector, and the corpus-level comparison here differences the same background out at the level of training substrates, the two controls composing rather than competing. Holding architecture fixed differences out the architectural degeneracy, and the comparison is between corpora, not against zero. The instrument is the restricted and refined family of local learning coefficients rather than the global scalar, because the global coefficient is not directionally resolved and cannot see "along the constrained directions" at all; the refined variants exist for exactly this purpose. The conjecture is specifiable rather than pre-registered: a registration must still fix the architecture and scale, the checkpoint or basin at which the coefficient is localized, the estimator settings whose sensitivity is a known limitation of current practice, the matching criteria in full, an effect size, and a power analysis, and until those are fixed, a null result is uninterpretable between falsification and underpowering. Section 6 lists the elements. Falsification, once the design is powered, is clean: indistinguishable degeneracy structure across matched corpora defeats the geometric form of the substrate thesis.
Algebraic statistics performs estimation on a variety that does not respond to being estimated. For quantities dominated by observation-independent drivers the assumption is correct, and the field's exactness where it applies is the benchmark any extension must preserve. Performative prediction studies the neighboring situation in which the assumption fails dynamically: a deployed predictor shifts the distribution, and the theory characterizes the equilibria of the resulting retraining map (Perdomo et al., 2020). The failure this paper's coordinate names is related but not the same. In the performative setting the distribution responds to a decision rule, and absent deployment there is a well-defined resting distribution. For a high-b quantity there is no resting structure to return to: the coordination of the observers, including the act of sustained estimation itself, is part of what constitutes the quantity, which is the counterfactual content of the fraction's definition, the share of value that would begin drifting toward incoherence under full observation withdrawal. Performativity is a force applied to a standing object; constitution means the object is partly made of the force. The cheap collapse, that this is performative prediction with a richer index set, fails on the framework's own primitives: performative prediction's objects are distribution maps indexed by a deployed decision rule, with stability defined as a fixed point of the retraining dynamic, and in the constitution setting there is no privileged decision rule to index by and no null deployment defining a resting point, so performative stability is not unattained here but undefined, and what replaces the equilibrium concept is a geometric object, the family of Section 4.2, which the performative framework does not carry. The stateful extension of that framework, in which the distribution depends on accumulated state rather than the current rule alone (Brown, Hod, and Kalemaj, 2022), is the strongest remaining version of the collapse, and it does not close the gap either: even stateful maps are indexed by deployment history, and the constitution setting has no deployment to accumulate. The financial cases that motivate the fraction are the cases MacKenzie (2006) documents sociologically, and the geometric claim is that they deserve an object, not only an equilibrium concept.
Definition AG-3 (Coordination-state manifold and the moving variety). Let C denote the coordination-state manifold: the space whose points c specify the coordination directed at a quantity, carrying as primary structure the similarity matrix S(c) among the N observers. Since principal-angle cosines yield unit diagonal and only positive semidefiniteness, S(c) lives on the elliptope, the compact convex body of correlation matrices, whose interior is the positive-definite stratum and whose boundary strata are the singular correlation matrices; C maps to the closed elliptope, and the metric geometry adopted is the Bures-Wasserstein metric, which extends continuously to the semidefinite closure and keeps the boundary at finite distance. The affine-invariant metric on the open SPD cone is rejected for exactly the reason the conjecture below requires: it places the rank-deficient boundary at infinite distance, and the conjecture lives on that boundary. The observer-configuration data of Section 2 is the refinement generating S(c) through the principal-angle construction. A moving variety is then the following object. Fix m and a degree bound d. Each generator f_i(x, c) is a polynomial in the model variables x in R^m of degree at most d, whose coefficient functions c maps to a_i(c) are continuous on the closed elliptope and smooth on its interior; the fiber V_c is the real zero set in R^m of the ideal I_c = < f_1(x, c), ..., f_k(x, c) >, and the family is the total space of pairs (x, c) with all f_i(x, c) = 0, together with its projection to C. The family is constant when V_c is one variety for all c. Whether the family is flat over C, and over which strata, is not asserted; it is the first structural question of the formalization program. One objection dissolves at the definition and is worth dissolving in print: the corpus's own dynamics include sudden moves, stampedes and coalition collapses, and these do not tear the family, because continuity is required of the assignment from state to fiber, not of the trajectory a society traces through the states. A stampede is a fast path or a jump in the base; the fiber over each state is what it is. Where the assignment itself changes character, the elliptope's stratification is the natural home, with phase transitions living as stratum crossings, which is exactly where the degeneration conjecture already points.
The base space is not invented for the occasion. Its primary structure is the similarity matrix the corpus's fit kernel already operates on, its metric geometry is the Bures-Wasserstein geometry of the closed elliptope stated in the definition, and its spectral coordinates carry the corpus's master variable through the bracket identity of Section 2. A working ansatz gives the parameterized ideal the shape the Mixing decomposition wants: each generator splitting as an invariant part f_ind(x), the observation-independent constraints, plus a coordination-directed perturbation f_coord(x, lambda(S(c))) driven by the spectrum of the similarity matrix, so that rising aggregate correlation amplifies the perturbing term and deforms the variety away from its invariant constraints. The ansatz is a candidate form, adopted because it makes the decomposition legible, not a derived necessity.
Proposition AG-4 (Anchoring, conditional). Assume the identification the bridging program is to establish: that the quantity's data-generating structure at coordination state c is the fiber V_c. Under that identification, if the family is constant, then b = 0; equivalently, b > 0 implies the family is non-constant. The converse fails in general even under the identification: a family can move along directions that contribute nothing to the variance decomposition, so b = 0 does not imply constancy.
Proof sketch. Suppose b > 0. Then a strictly positive share of the quantity's variance is attributable to the coordination-dependent component, so there exist coordination states across which the data-generating structure differs, which under the assumed identification is non-constancy of the family; the contrapositive gives the first statement. For the failure of the converse, let the family vary only in generators whose zero sets coincide on the support of the data distribution, or vary along directions orthogonal to the observed functionals entering the variance decomposition; then the family moves while the decomposition registers nothing, so b = 0 with a non-constant family. End sketch.
The proposition earns two things. It fixes the inclusion, that the classical fixed-variety setting embeds as the constant family and the extension costs nothing where the classical assumption holds, and it does so conditionally: the proposition inherits exactly the structure-to-fiber identification the bridging theorem must supply, which is stated in its hypothesis rather than smuggled through the sketch. And its second clause, the failure of the converse, is itself informative: b is blind to motion that never reaches the variance decomposition, so the fraction is a lower witness of the family's motion, not a measure of it, and any theorem giving b a full geometric meaning must state which motions it sees. That lossiness is a property, not a defect, of the diagnostic role: b is the protocol's disclosed coordinate, struck by originators about the quantities they carry, and the bridging theorem's job is to characterize the cone of motions the coordinate witnesses, not to promote it to a complete invariant it was never defined to be. That bridging theorem, from the variance decomposition to a deformation decomposition on the family with b as the weight of the base-directed part, is the program's central open problem, now well-posed by the definition but not solved by it. A further conjecture gives the program its singular-theory face and is recorded as a conjecture: as S(c) moves along a Bures-Wasserstein geodesic toward the all-ones matrix, the rank-one stratum of the elliptope boundary that the chosen metric reaches at finite distance, and the regime the corpus's correlation analysis names the correlation trap (book yet to be published), the parameterized ideal's Jacobian should lose rank as a function of the falling effective rank of S(c), so that the blindness of a maximally correlated population is mirrored by a deepening singularity of the variety a model learns from its output. Nothing in this paper proves that mirror; it is stated because it is falsifiable and because it joins Section 3's instrument to Section 4's object.
One notational discipline is recorded for the corpus: S names the similarity matrix, as the formal core already uses it, and the base space is C. The symbols are scoped and do not overload.
The division of intellectual property is meant to be checkable line by line. Singular learning theory, the real log canonical threshold, and the local learning coefficient belong to Watanabe (2009, 2018), Aoyagi and Watanabe (2005), and Lau et al. (2024) with the refined-variant literature; the fixed-variety apparatus belongs to algebraic statistics (Drton, Sturmfels, and Sullivant, 2009; Sullivant, 2018); the singularity of deep networks is that literature's own observation (Wei et al., 2022); the responsiveness of distributions to prediction belongs to performative prediction and its relatives (Perdomo et al., 2020; Hardt et al., 2016), and the sociology of the financial cases to MacKenzie (2006). The constraint-geometry propositions (CG-1 through CG-3 of The Geometry of Constraint Knowledge), the mediation-channel results, the Grassmannian observation program, the fit kernel and the bracket, and the coordination-dependence fraction with its estimators are struck in the corpus, the published items cited above and the unpublished remainder covered by the cover statement. What this paper claims as new is the coordinate and its two connections: the placement of CG-1 at the entry of singular learning theory together with the differential substrate conjecture that placement makes specifiable, and the identification of constitution, as distinct from performative response, as a failure mode of the fixed-variety assumption, with the coordination-state manifold built from the corpus's own similarity structure as the candidate base and the anchoring proposition, in its corrected one-directional form, as the inclusion result. The formalization program those connections open is listed as open.
Three tests bear on the claims, in ascending order of difficulty. The first is the differential substrate experiment of Conjecture AG-2. A registration must fix: architecture and scale, held identical across arms; the corpus-matching criteria (token count, vocabulary coverage, domain scope) with conservation-relevant content density as the manipulated variable; the checkpoint and localization at which coefficients are estimated; the restricted-coefficient components corresponding to the domain's constrained directions; estimator settings, with sensitivity analysis across them, since estimator variance is a known limitation; and an effect size with a power analysis, without which a null is uninterpretable. The prediction is the ordering stated in the conjecture. The second test targets the anchoring correspondence at panel scale: across quantities with independently struck b-hat values, the measured motion of estimated structure across coordination settings, operationalized through the meta-regression residual framework of the fraction's estimator, should order the panel consistently with the disclosed b-hat ordering on the motions the decomposition can see, the qualification Proposition AG-4's second clause makes mandatory. The third is the formalization program: a definition of the family under which the Mixing decomposition is a theorem about its variation, the fraction is the weight of the base-directed part on the visible motions, and the degeneration conjecture is a statement about ideals over a Bures-Wasserstein geodesic in the closed elliptope reaching the rank-one stratum at finite distance. The first test is specifiable now; the second requires the disclosure substrate at panel scale; the third requires only the mathematics.
V2.0, 7 August 2026. Deposited. The Zenodo record is live at DOI 10.5281/zenodo.21832686 and is now the immutable published form; this file is the source of record for the lineage. The deposited PDF carries the header date 06 August, accepted as printed because the paper's content was finalized on the 6th and the 7 August deltas touched only the version record, which the deposit rightly omits. The version rolls from 1.9 to 2.0 under the versioning rule; any future revision is a new Zenodo version citing this DOI. The deposit ledger for the year now reads: The Geometry of Constraint Knowledge, 19 March (10.5281/zenodo.19439254); A Missing Coordinate in the Classification of Quantitative Data, 3 July (10.5281/zenodo.21154566); the Machine-Origination Boundary, 20 July; Formal Extensions to the OCOT Corpus (10.5281/zenodo.21198572); Singular Models and Moving Varieties, 7 August (10.5281/zenodo.21832686).
V1.9, 7 August 2026. Header revision date aligned to the record (07 August; the 06 August header predated the voice pass). Deposit-final. Note for the record: this file sits at V1.9, so any post-deposit revision rolls to V2.0 as a new Zenodo version citing this one.
V1.8, 7 August 2026. Voice conformance on my ruling of 7 August, enforced at gate v2.1 and recorded in Conformance Contract V1.5: three third-person self-references in the version record rewritten in the first person. The paper's body already spoke correctly; a deposit under my byline has no narrator. No content change.
V1.7, 6 August 2026. One repair: the Brown, Hod, and Kalemaj (2022) entry added to the references, closing the dangling citation the V1.6 stateful-performativity clause introduced. Every citation now resolves to a public record or sits under the cover statement; the geometry is uniform across abstract, definition, conjecture, and experimental surface; every claim carries its label. The deposit is internally consistent end to end, and the next version number that matters is the one attached to an AG-2 run, the bridging theorem, or the foundations rewrite.
V1.6, 6 August 2026. Propagation and hardening pass on the third referee cycle. The elliptope and Bures-Wasserstein correction, which V1.4 landed only in the definition and the conjecture, now runs the full document: the abstract, the paragraph adjacent to AG-3, and the Section 6 restatement all carry the corrected geometry. The abstract's leftover program designation replaced with the deposited paper's title; the Section 3 citation of the formal extensions aligned with its live DOI; the reference list alphabetized. Three hardenings from the red team: the latent-washing attack on AG-2 answered on the data-processing ground, that convergence cannot recover content the corpus does not contain, with the ordering-not-gap prediction and the power burden stated; the continuity objection answered at the definition, continuity of the assignment versus continuity of the trajectory, with stratum crossings as the home of phase transitions; the stateful performative extension acknowledged and distinguished, deployment history being still a deployment. The lower-witness reading of b stated as the diagnostic's designed role. Genre verdict recorded as the referee keeps it: this is the deposit; the three exit paths are the game.
V1.5, 6 August 2026. The two Zenodo DOI slots filled from my records: 10.5281/zenodo.19439254 for The Geometry of Constraint Knowledge and 10.5281/zenodo.21198572 for Formal Extensions to the OCOT Corpus, whose title is corrected to the deposited form, Extensions to rather than Extensions of. Every reference now resolves or is covered by the cover statement. No other change; the deposit is complete.
V1.4, 6 August 2026. Second referee cycle, six repairs. AG-4 made explicitly conditional on the structure-to-fiber identification, in its hypothesis rather than its sketch. The base geometry corrected: similarity matrices live on the elliptope, not the open SPD cone, and the metric is Bures-Wasserstein, which reaches the rank-deficient boundary at finite distance; the affine-invariant metric is rejected in text for placing the degeneration conjecture's locus at infinity, and the conjecture is restated along a finite geodesic to the rank-one stratum. AG-3 made computable: fixed ambient dimension and degree bound, polynomial generators with coefficients continuous on the closed elliptope and smooth on its interior, fibers as real zero sets, the total-space construction stated, flatness posed as the program's first structural question. The fit-kernel construction grounded in public mathematics through the determinantal point process literature, with the corpus application remaining under the cover statement, and the one uncovered insider term removed. The refined-coefficient citation completed with its authors, and DOI slots opened for the two Zenodo deposits pending the strings. The performativity collapse move pre-empted in Section 4.1 on the framework's own primitives. Genre unchanged: this is the deposit, and the three exit paths toward a submission stand as the referee left them.
V1.3, 6 August 2026. Published-record corrections on my ruling: the Missing Coordinate paper cited as the live Zenodo deposit with its DOI; the constraint-geometry source cited under its deposited title, The Geometry of Constraint Knowledge, with its own proposition labels (CG-1 replacing the corpus-internal NR-1 designation) and its own Constraint Direction Probe and overparameterization clarification credited in Section 3; the unpublished-corpus collective reference removed entirely, the cover statement carrying all unpublished attribution. No unpublished book or article is named anywhere in the text or references.
V1.2, 6 August 2026. Unpublished book titles removed from the text and references on my ruling; the cover statement, drawn from books and papers yet to be published, carries the attribution, and one collective reference entry points to it. No content change.
V1.1, 6 August 2026. Revision on the full QA cycle of the same day: one cold-read referee report, one red-team pass, and the corpus verification that followed. The changes: AG-4 restated in its correct direction (constant implies b = 0), with the failure of the converse recorded and put to work; AG-1 demoted from proposition to the definitional remark it is, with the conditions separating entry from inheritance stated and the trivial-degeneracy caution added; AG-2 rebuilt as a differential design on refined and restricted local learning coefficients, with the background-degeneracy identification problem, the estimator-sensitivity caveat, and the registration elements named, and "pre-registrable" corrected to "specifiable"; the base space rebuilt from canon as the coordination-state manifold over the SPD manifold of similarity matrices, citing the fit kernel, the bracket, and the correlation trap, with the principal-angle construction joining the Grassmannian configuration to the similarity structure and the S-versus-C symbol scoping fixed; a prior-art treatment of performative prediction, strategic classification, and performativity added, with the novelty claim narrowed to constitution versus response and the geometric object; the reference base roughly doubled; the parameterized-ideal ansatz and the degeneration conjecture entered with status marked; the vocabulary note added for readers outside the corpus. V1.0 of the same date is superseded. The volume sibling is Part 10 of the Beierspace theory volume, V2.7 lineage, which will want a conforming pass at its next bump.