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"formalising", "formally", "formula", "formulas", "four", "fragment", "free", "fresh", "full", "fully", "funnelled", "g6", "gaps", "general", "generated", "generating", "generation", "generator", "genuinely", "geq", "giant", "gimbel", "gives", "gnu", "gpt", "graph", "graphs", "group", "groups", "grows", "guarantee", "half", "halves", "hand", "hard", "has", "hashes", "have", "headroom", "heart", "hermes", "homogeneous", "human", "hypothetical", "identified", "identifies", "identify", "implementation", "impossibility", "impossible", "include", "including", "independence", "independent", "independently", "index", "individual", "induces", "injected", "input", "inputs", "inspect", "instances", "intellectual", "interest", "into", "isabelle", "isolated", "isomorphic", "issue", "its", "itself", "john", "july", "keep", "kind", "knowledge", "known", "lacks", "ladder", "language", "layer", "lceil", "lean", "least", "lemmas", "leq", "level", "lie", "likely", "limitations", "line", "links", "linux", "lists", "literate", "local", "locally", "logical", "logically", "logs", "look", "lower", "lpr", "lrat", "lucky", "machine", "machinery", "macos", "maintain", "maintainers", "make", "makes", "manifest", "manifests", "manuscript", "many", "maps", "matches", "mathematical", "mathematician", "mathematicians", "mathematics", "mathrm", "max", "may", "mckay", "md", "mean", "mediation", "meet", "method", "might", "minimisation", "missing", "mix", "model", "models", "moderately", "most", "movement", "moving", "multiple", "must", "mutually", "named", "natural", "nauty", "necessary", "need", "needed", "needing", "needs", "negative", "network", "new", "next", "no", "nobody", "non", "nonzero", "not", "nous", "now", "number", "numbers", "objects", "obstructions", "obviously", "one", "only", "open", "orderly", "ordinary", "other", "our", "own", "package", "page", "pair", "pairings", "pairs", "paley", "paper", "particular", "partition", "partitioning", "parts", "passing", "pattern", "paul", "pdf", "peer", "pipeline", "pitch", "please", "plus", "points", "policy", "portable", "possible", "power", "precisely", "prediction", "print", "problem", "problems", "proceeds", "process", "produced", "program", "programs", "project", "projects", "proof", "proofs", "proposition", "propositional", "prose", "proved", "proximity", "public", "published", "push", "py", "python", "python3", "questions", "r44", "ramsey", "range", "rat", "rather", "raw", "rceil", "re", "readable", "readers", "readme", "reasoning", "recomputations", "recomputed", "recomputes", "recorded", "reduced", "reduces", "reduction", "refutations", "refuted", "regenerated", "regenerates", "regenerating", "rejected", "release", "released", "remain", "remaining", "remains", "replay", "report", "reports", "repository", "reproduce", "reproduces", "reproduction", "reproductions", "require", "rerun", "researchers", "residual", "resolving", "rest", "rests", "result", "results", "returning", "reusing", "reveal", "review", "roughly", "route", "routes", "rup", "said", "same", "sat", "satisfy", "scratch", "search", "second", "selection", "self", "semantic", "separate", "separately", "sequence", "set", "sets", "settle", "settles", "sh", "sha", "sha256", "sha256sum", "shasum", "ships", "short", "shortest", "should", "showing", "since", "single", "six", "size", "sized", "small", "so", "software", "sol", "solver", "solvers", "sometimes", "sort", "space", "specialist", "specialists", "split", "spot", "stages", "standard", "start", "states", "status", "step", "steps", "stops", "strong", "strongest", "structure", "study", "substitute", "success", "such", "suffice", "suites", "summary", "survive", "surviving", "symmetric", "system", "systems", "table", "tag", "take", "teaching", "team", "technical", "test", "tested", "testing", "text", "than", "their", "them", "then", "theorem", "theoretic", "theory", "these", "they", "third", "those", "three", "through", "time", "together", "tool", "tools", "tracker", "translated", "translation", "trim", "truncated", "trust", "trusted", "turning", "two", "uncertified", "unconnected", "under", "undergraduate", "unrefereed", "unsat", "unsatisfiability", "unsatisfiable", "unusually", "up", "upper", "upstream", "upwards", "usable", "use", "uses", "validated", "validation", "valuable", "value", "values", "variables", "verifiable", "verification", "verified", "verify", "vertex", "vertices", "very", "vr", "was", "week", "were", "where", "whether", "while", "who", "whose", "why", "will", "within", "worst", "would", "writes", "written", "you", "z20", "zenodo" ], "fingerprint": "45619e204c03524699c49c3cc38538b98b5593840cb033e079f9b8815fc7a889" } ], "cards": [ { "id": "finite-exact-certificate-triage", "kind": "research-move", "title": "Distinguish a finite convex formulation from exact certification", "summary": "The completion theorem supplies an exact real feasibility characterization. Rational failures admit rational violating posteriors, but rational primal completions and rational optimal coefficients are not guaranteed; floating-point boundary feasibility is not an exact certificate.", "releaseSlug": "finite-signal-quadratic-substitutes", "source": { "path": "papers/finite-signal-quadratic-substitutes/body.md", "locator": "## Evidence, assurance and limitations", "sha256": "5fe7e3e845805580847410bc9e028bdc6c4fa180d46c9009a0459887acd53543" }, "conditions": [ { "key": "finiteSignals", "value": true }, { "key": "scalarTarget", "value": true }, { "key": "squareIntegrableTarget", "value": true }, { "key": "quadraticValue", "value": true }, { "key": "positiveSourcePrior", "value": true }, { "key": "sourceOnlyRandomizedDisclosure", "value": true }, { "key": "exactModelTable", "value": true } ], "actions": [ { "label": "Choose the evidence object", "check": "For acceptance check all proposed row-loss inequalities exactly; for rejection check the strict posterior residual and admissible disclosure construction.", "stopWhen": "Stop an exact acceptance claim supported only by numerical solver status or rounded boundary coordinates." }, { "label": "Separate model estimation from theorem inputs", "check": "Record whether support and source means are exact or estimated, especially near coincident means.", "stopWhen": "Stop treating a plug-in empirical table as a statistical certificate without uncertainty analysis." } ], "limitations": [ "No general exact solver is supplied by the memory card or theorem statement.", "Rational failure-witness existence gives no stated polynomial certificate-length bound.", "This is a retrospective reusable checking move, not observed discovery chronology or demonstrated time saving." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "finite-scalar-knot-completion", "identification-target-invariance" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "1b5730eca7034c6a440e882a73e29e0a70d800a6626660fbff37f13ec62f12ec" }, { "id": "finite-full-support-collapse", "kind": "transformation", "title": "Collapse completions to directed pairs only under full support", "summary": "Under full rectangular support the completion anchored at x_j is forced to be row v_j. The sharp coefficient is the maximum directed weighted row difference divided by squared mean difference, after separately handling equal-mean incompatibility and the empty-maximum case.", "releaseSlug": "finite-signal-quadratic-substitutes", "source": { "path": "papers/finite-signal-quadratic-substitutes/body.md", "locator": "## Technical account", "sha256": "5fe7e3e845805580847410bc9e028bdc6c4fa180d46c9009a0459887acd53543" }, "conditions": [ { "key": "finiteSignals", "value": true }, { "key": "scalarTarget", "value": true }, { "key": "squareIntegrableTarget", "value": true }, { "key": "quadraticValue", "value": true }, { "key": "positiveSourcePrior", "value": true }, { "key": "sourceOnlyRandomizedDisclosure", "value": true }, { "key": "fullRectangularSupport", "value": true } ], "actions": [ { "label": "Check degeneracies before division", "check": "Compare equal-mean rows: incompatible conditional means imply infinite coefficient; all equal compatible means give zero.", "stopWhen": "Stop ratio division when x_i=x_j; use the degeneracy branch." }, { "label": "Preserve the normalized objective", "check": "Only under the stated nondegeneracy use inf G(T)/(V(A)-V(T))=1-R; retain the denominator and limiting-disclosure interpretation.", "stopWhen": "Stop if the requested result is a minimum absolute gap or general optimal disclosure." } ], "limitations": [ "Full support is needed for this forced-row formula, not for the scalar completion theorem.", "An infimum can occur only as a limit; it is not automatically attained by a disclosure.", "A target-dependent coefficient is not a channel-only SDPI coefficient." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "finite-scalar-knot-completion", "finite-vector-pairwise-boundary" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "6736f90cb4d74694d401facc7c12aa6f4b8210a1fc1ac28344b19ec74d73cab6" }, { "id": "finite-scalar-knot-completion", "kind": "lemma", "title": "Replace universal scalar disclosure tests with simultaneous knot completions", "summary": "For finite scalar quadratic signal tables, including structural zeros, N(q) <= C D(q) for every posterior is equivalent to a common completion z at every distinct source mean t satisfying each supported row loss L_i(z) <= C(x_i-t)^2. C=1 tests the specified disclosure orientation.", "releaseSlug": "finite-signal-quadratic-substitutes", "source": { "path": "papers/finite-signal-quadratic-substitutes/body.md", "locator": "## Summary for specialists", "sha256": "5fe7e3e845805580847410bc9e028bdc6c4fa180d46c9009a0459887acd53543" }, "conditions": [ { "key": "finiteSignals", "value": true }, { "key": "scalarTarget", "value": true }, { "key": "squareIntegrableTarget", "value": true }, { "key": "quadraticValue", "value": true }, { "key": "positiveSourcePrior", "value": true }, { "key": "sourceOnlyRandomizedDisclosure", "value": true }, { "key": "finiteNonnegativeThreshold", "value": true } ], "actions": [ { "label": "Build supported row losses", "check": "Check stochastic rows, supported cell means, x_i=sum_b r_ib v_ib, and one shared completion vector per distinct mean. Do not fill impossible cells with assumed conditional expectations.", "stopWhen": "Stop theorem application if target scalarity, admissibility, positive prior or supported-data semantics is unverified." }, { "label": "Separate orientations", "check": "Use C=1 for this orientation; reconstruct the reverse conditional table before making a two-sided statement.", "stopWhen": "Stop a two-sided claim after only one orientation has been checked." } ], "limitations": [ "The cited source is a public summary of an unrefereed analytic candidate, not a proof-assistant certificate.", "Finite convex representation is not an implemented general exact solver or polynomial-time bit-complexity theorem.", "No extension to arbitrary pairs of partially disclosed signals or other scoring rules." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "finite-full-support-collapse", "finite-sparse-joint-obstruction", "finite-exact-certificate-triage" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "f5a62c4a13750a8c193ed62299e0a6934e72b1fd19499efb39e2250554bd0107" }, { "id": "finite-sparse-joint-obstruction", "kind": "obstruction", "title": "Pairwise-passing sparse rows can fail jointly", "summary": "In the displayed three-row sparse family, C*=4a^2/(a+b)^2. At (a,b)=(1,1/2), all two-state posterior restrictions pass in one orientation but C*=16/9 and the uniform no-disclosure gap is -1/24. At (1/2,1), C*=4/9 and the normalized margin is positive.", "releaseSlug": "finite-signal-quadratic-substitutes", "source": { "path": "papers/finite-signal-quadratic-substitutes/body.md", "locator": "### One sparse family, two different answers", "sha256": "5fe7e3e845805580847410bc9e028bdc6c4fa180d46c9009a0459887acd53543" }, "conditions": [ { "key": "finiteSignals", "value": true }, { "key": "scalarTarget", "value": true }, { "key": "squareIntegrableTarget", "value": true }, { "key": "quadraticValue", "value": true }, { "key": "positiveSourcePrior", "value": true }, { "key": "sourceOnlyRandomizedDisclosure", "value": true }, { "key": "fullRectangularSupport", "value": false }, { "key": "displayedSparseFamily", "value": true } ], "actions": [ { "label": "Reconstruct the exact family", "check": "Use channel rows (1/2,1/2,0), (0,1/2,1/2), (1/2,0,1/2) and exactly the source's supported cell means and parameter bounds.", "stopWhen": "Stop transferring 16/9, 4/9 or -1/24 to another table." }, { "label": "Test joint completion", "check": "Check the simultaneous prescriptions from equal-mean rows rather than separate binary restrictions.", "stopWhen": "Stop the claim that all pairwise checks certify arbitrary sparse scalar tables." } ], "limitations": [ "This is a specified family and disclosure orientation, not an all-sparse rejection theorem.", "The same sparse support has positive and negative cases; zeros alone do not imply failure.", "No inference about estimated real-world forecasts." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "finite-scalar-knot-completion", "finite-vector-pairwise-boundary" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "24c7684392a05bae4fbab88b9f171ac44c8adfa9703526e986825eb631ec4b2e" }, { "id": "finite-vector-pairwise-boundary", "kind": "obstruction", "title": "Full support does not rescue pairwise tests for vector targets", "summary": "The public source records a full-support vector example with six positive directed pairwise margins and a negative three-state residual. A strictly positive four-category probability embedding preserves the failure under quadratic probability scoring.", "releaseSlug": "finite-signal-quadratic-substitutes", "source": { "path": "papers/finite-signal-quadratic-substitutes/body.md", "locator": "## Evidence, assurance and limitations", "sha256": "5fe7e3e845805580847410bc9e028bdc6c4fa180d46c9009a0459887acd53543" }, "conditions": [ { "key": "finiteSignals", "value": true }, { "key": "quadraticValue", "value": true }, { "key": "vectorTarget", "value": true }, { "key": "fullRectangularSupport", "value": true } ], "actions": [ { "label": "Use the example as a transfer test", "check": "Inspect the archived vector table and recompute its global residual before treating scalar pairwise sufficiency as a vector theorem.", "stopWhen": "Stop the unrestricted vector pairwise-sufficiency extension when the exact example is verified." }, { "label": "Separate failed proof mechanism from impossibility", "check": "Record that scalar ordering underlies the interpolation step; search for another finite vector characterization if needed.", "stopWhen": "Stop short of claiming that no finite vector criterion can exist." } ], "limitations": [ "One-orientation counterexample, not a classification of vector targets.", "Four categories are a realization, not a minimality theorem.", "The source summary points to vector_exact.py; this card does not reproduce the table or certify it." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "finite-full-support-collapse", "finite-sparse-joint-obstruction" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "4a958545cfc90a09d942f1a383bdc854e346b42cc14361f98f4e968f9feea24e" }, { "id": "graph-budget-versus-restricted-unsat", "kind": "research-move", "title": "Separate budget exhaustion, restricted infeasibility and global impossibility", "summary": "The workbench requires an exhausted assignment budget to be reported as inconclusive, not UNSAT. Even complete failure to find a transversal of a supplied matching does not exclude an ordinary dominating set: its synthetic 10-vertex control has no such transversal but does have a 3-vertex dominating set.", "releaseSlug": "txgraffiti-order48-successor", "source": { "path": "docs/ATLAS_RESEARCH_WORKBENCH.md", "locator": "## Tests that prevent a misleading success", "sha256": "7208f89886e791f2116894bfb70695193a89819ebf41cae63150c978998454f5" }, "conditions": [ { "key": "finiteSimpleGraph", "value": true }, { "key": "endpointTransversalSearch", "value": true } ], "actions": [ { "label": "Audit search coverage", "check": "Record tested versus total assignments and whether cancellation or a budget ended the run.", "stopWhen": "Stop any UNSAT claim when the relevant finite space was not exhausted or otherwise certified." }, { "label": "Audit the quantified space", "check": "Even after exhaustive transversal failure, report only no transversal for this matching; seek a separate global reduction before excluding all dominating sets.", "stopWhen": "Stop the inference from restricted UNSAT to ordinary-domination impossibility." } ], "limitations": [ "The 50-vertex route control also concerns a supplied matching, not a replay of a global domination lower bound.", "Budget exhaustion is missing search evidence, not a mathematical counterexample.", "This extraction does not claim that memory would have prevented a past error." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "graph-matching-sat-bridge", "graph48-witness-stops-strengthening" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "4c5a18da686abbebacb2d00e438a7e0b550c306ea35c1d6f58ea513f58b3975b" }, { "id": "graph-matching-sat-bridge", "kind": "transformation", "title": "Encode dominating endpoint transversals without changing the graph claim", "summary": "For a finite simple graph with a maximal matching, choosing one endpoint per matching edge yields Boolean variables. Clauses require domination of unmatched vertices; maximality makes unmatched vertices independent, and each matched vertex is dominated by its edge's chosen endpoint. Satisfying assignments are exactly dominating transversals of that matching.", "releaseSlug": "txgraffiti-order48-successor", "source": { "path": "docs/ATLAS_RESEARCH_WORKBENCH.md", "locator": "The external bridge is Gupta's Definition 2 and Proposition 6.", "sha256": "7208f89886e791f2116894bfb70695193a89819ebf41cae63150c978998454f5" }, "conditions": [ { "key": "finiteSimpleGraph", "value": true }, { "key": "matchingMaximal", "value": true } ], "actions": [ { "label": "Recompute the bridge hypotheses", "check": "Verify the graph and matching directly from edges, then derive clauses from unmatched vertices, retaining tautologies correctly.", "stopWhen": "Stop if the supplied matching is not maximal or stored metadata disagrees with edges." }, { "label": "Check the original representation", "check": "Reconstruct the selected endpoints and directly test domination of every original vertex.", "stopWhen": "Stop promoting a SAT assignment that fails graph-level witness checking." } ], "limitations": [ "The bridge does not need cubic regularity, but conclusions about minimum maximal matching may.", "It searches transversals of one supplied matching, not all dominating sets.", "Implementation-diverse internal checks are not unaffiliated reproduction." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "graph48-witness-stops-strengthening", "graph-budget-versus-restricted-unsat" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "4137e4cfc051e8912b29f05954a9114d5f4784cfd6cb0bd73a8d853e790aefe5" }, { "id": "graph48-witness-stops-strengthening", "kind": "obstruction", "title": "An independent-domination counterexample need not refute ordinary domination", "summary": "The recorded pilot finds 317 endpoint-transversal assignments for the tagged 48-vertex graph and an ordinary dominating set of size 15. With 72 edges and maximum degree 3, the matching lower bound ceil(72/5)=15 and checked matching give gamma <= 15 = gamma_e. The source release separately records independent domination i=16.", "releaseSlug": "txgraffiti-order48-successor", "source": { "path": "docs/ATLAS_RESEARCH_WORKBENCH.md", "locator": "For the tagged 48-vertex graph the program finds **317** satisfying assignments", "sha256": "7208f89886e791f2116894bfb70695193a89819ebf41cae63150c978998454f5" }, "conditions": [ { "key": "finiteSimpleGraph", "value": true }, { "key": "canonicalGraph48", "value": true } ], "actions": [ { "label": "Bind the witness to its graph", "check": "Verify the set {0,2,4,7,8,10,13,14,16,18,21,22,24,26,28} against the canonical edges, not merely the order or title.", "stopWhen": "Stop reuse if input identity or direct domination fails." }, { "label": "Keep optimization targets distinct", "check": "Use the size-15 set as an ordinary-domination upper bound; keep the independent-domination lower bound in its own source scope.", "stopWhen": "Stop the strengthening route from i>=16 to gamma>=16 on this graph." } ], "limitations": [ "No exact ordinary domination number is established.", "The pilot does not recompute the release's independent-domination exclusion certificate.", "317 became a regression expectation after execution; it was not the prospective success criterion.", "Reduced candidate count is not measured researcher or solver acceleration." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "graph-matching-sat-bridge", "graph-budget-versus-restricted-unsat" ], "releaseVersion": "0.1.0-candidate", "fingerprint": "4be7ad778536f9bd34164708dc4a3c077c9e011142edab0e7becbff27812c5fe" }, { "id": "identification-second-condition", "kind": "research-move", "title": "Choose a measurement that distinguishes unresolved responses", "summary": "The public source attributes to Thommen a reaction-network example: two models have identical internal-state path laws at one environmental condition but different stationary responses when that condition changes. The associated benchmark separates a discriminating second condition from condition-specific parameter refitting.", "releaseSlug": "unique-answer-not-identified", "source": { "path": "papers/unique-answer-not-identified/body.md", "locator": "## Subsequent related work", "sha256": "c82a01cc08ad6d69e4c9b43ba95a8d1268bde3a60e3d5ba1339abed9a1e772a9" }, "conditions": [ { "key": "singleEnvironmentalCondition", "value": true }, { "key": "responseToChangedConditionTarget", "value": true } ], "actions": [ { "label": "Freeze the comparison before adding data", "check": "Specify which parameters and channel/environment dependence are held fixed across conditions; compare response predictions under those same rules.", "stopWhen": "Stop claiming discrimination if each model is refitted independently by condition to erase the contrast." }, { "label": "Test the new observation direction", "check": "Check whether the proposed second condition changes predictions along the unresolved direction; include a redundant observation as a negative control.", "stopWhen": "Stop calling a measurement rank-restoring solely because it adds precision at the original condition." } ], "limitations": [ "Attributed source example, not a new reaction-network theorem derived by this memory.", "The public body provides a conceptual counterexample and benchmark pointer; full algebra must be inspected before applying it.", "Exact observation-law arguments do not establish identifiability from finite noisy samples.", "No empirical effectiveness claim for the Selector Ledger or Atlas." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "identification-target-invariance" ], "releaseVersion": "1.0.0-candidate", "fingerprint": "d3d4bbafd43ddd1879ee45902d550f8dcddb61232a19bbeac5188e2ad55fa99a" }, { "id": "identification-target-invariance", "kind": "obstruction", "title": "A unique numerical answer need not be observation-identified", "summary": "Under a fixed observation regime, two structural states with identical observable laws cannot be distinguished by more sampling from that unchanged regime. If a target differs across those states, a unique output produced by a prior, constraint or tie-breaker does not identify that target from those observations.", "releaseSlug": "unique-answer-not-identified", "source": { "path": "papers/unique-answer-not-identified/body.md", "locator": "## The central claim or finding", "sha256": "c82a01cc08ad6d69e4c9b43ba95a8d1268bde3a60e3d5ba1339abed9a1e772a9" }, "conditions": [ { "key": "fixedObservationRegime", "value": true }, { "key": "observationallyEquivalentAlternatives", "value": true }, { "key": "targetDiffersAcrossAlternatives", "value": true } ], "actions": [ { "label": "Exhibit the semantic counterexample", "check": "Write the forward observation map and two admitted states with equal observation laws but different target values; keep any selector explicit.", "stopWhen": "Stop describing the selected target as identified by the focal observations alone." }, { "label": "Check the target, not only parameters", "check": "Ask whether the decision-relevant target remains invariant even when the full parameter does not.", "stopWhen": "Stop a blanket claim that every target or decision is unidentified merely because the parameter map is noninjective." } ], "limitations": [ "Classical identification principle, not a new theorem claimed by the synthesis.", "Exact equivalence must not be replaced silently by a weak or practical ridge.", "Nonidentification is not a zero effect and need not preclude a robust decision." ], "provenance": { "basis": "retrospective-source-extraction", "reviewedAt": "2026-10-15T18:36:44Z" }, "relatedCards": [ "identification-second-condition", "finite-exact-certificate-triage" ], "releaseVersion": "1.0.0-candidate", "fingerprint": "3a1efe732f733c50ecba767a584d2c7c94c0d620aa0b7261a3b7632d1013e415" }, { "id": "tensor-literal-normalization-obstruction", "kind": "obstruction", "title": "Stress-test literal tensorization on a pure tensor", "summary": "Under base-2 analytic rank and fixed-order Kronecker powers, a nonzero pure order-three tensor has constant rank a=log_2(4/3). The side-two identity has rank 2a<1, so the literal scalar comparator (2ac)^m tends to zero for 0