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| { | |
| "schema_version": "1.0.0", | |
| "title": "AELYRAITH III: The Nonlinear Continuation Theorem", | |
| "index_status": "Navigation derived from the supplied release; not an independent proof or novelty certification.", | |
| "authoritative_source": "manuscript/aelyraith3.tex", | |
| "pdf": "manuscript/AELYRAITH_III_Nonlinear_Continuation_Theorem.pdf", | |
| "implementation": "src/continuation.py", | |
| "global_hypotheses": [ | |
| "States range over the full probability simplex in R^n; n >= 2.", | |
| "A finite known library contains strictly positive real n by n branch matrices.", | |
| "Branches act by positive normalization T*x/(e*T*x).", | |
| "Protected readouts are beta*x/(alpha*x), with every alpha entry strictly positive.", | |
| "The preserved contract is conditional readout equality for every finite supplied branch word, including the empty word.", | |
| "Stochastic normalization, when required, is an extra condition per action." | |
| ], | |
| "word_convention": { | |
| "branch_indices": "zero-based", | |
| "words": "chronological", | |
| "state_product": "T_aL * ... * T_a1", | |
| "row_evaluation": "multiply row by matrices in reverse chronological order", | |
| "explicit_family_labels": { | |
| "0": "A", | |
| "1": "B" | |
| } | |
| }, | |
| "results": [ | |
| { | |
| "id": "thm:exterior", | |
| "title": "Finite nonlinear continuation theorem", | |
| "kind": "theorem", | |
| "section": "2", | |
| "number": "2.1", | |
| "tex_label": "thm:exterior", | |
| "equation_labels": [ | |
| "eq:wedge", | |
| "eq:compound", | |
| "eq:closure", | |
| "eq:signature" | |
| ], | |
| "conclusion": "Finite exterior orbit closure exactly characterizes complete conditional behavior. The selected ratio signature defines a coarsest sufficient information partition and admits representative-independent branch updates.", | |
| "additional_hypotheses": [], | |
| "implementation_entrypoints": [ | |
| "compile_exterior", | |
| "compile_ratios", | |
| "ExteriorCertificate.verify", | |
| "ExteriorCertificate.signature_exact" | |
| ], | |
| "related_tests": [ | |
| "AlgebraTests", | |
| "CertificateIntegrityTests", | |
| "FamilyTests.test_all_words_six" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "The signature coordinate count is not its intrinsic dimension or a bit count.", | |
| "Certificate completeness requires seed coverage, genuine provenance, and transport, not transport alone.", | |
| "No equality of label distributions is implied." | |
| ] | |
| }, | |
| { | |
| "id": "prop:fiber", | |
| "title": "Exact fiber geometry", | |
| "kind": "proposition", | |
| "section": "3", | |
| "number": "3.1", | |
| "tex_label": "prop:fiber", | |
| "equation_labels": [ | |
| "eq:L", | |
| "eq:fiber" | |
| ], | |
| "conclusion": "Every feasible signature fiber equals the simplex intersected with the affine equations L(z)*x=0 and is compact and convex; any representative gives the same exact signature update.", | |
| "additional_hypotheses": [], | |
| "implementation_entrypoints": [ | |
| "ExteriorCertificate.fiber_matrix", | |
| "ExteriorCertificate.reconstruct_fiber_float", | |
| "ExteriorCertificate.update_float" | |
| ], | |
| "related_tests": [ | |
| "FamilyTests.test_lp_representative_update" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "The delivered representative solver is floating-point linear programming, not an exact feasibility certificate.", | |
| "A representative need not be the true hidden state.", | |
| "Temporary workspace can remain n-dimensional." | |
| ] | |
| }, | |
| { | |
| "id": "thm:dimension", | |
| "title": "Exact local continuous dimension", | |
| "kind": "theorem", | |
| "section": "3", | |
| "number": "3.2", | |
| "tex_label": "thm:dimension", | |
| "equation_labels": [ | |
| "eq:rank" | |
| ], | |
| "conclusion": "At each interior regular point, generic signature rank r equals rank L(sigma(x)); r smooth coordinates suffice locally and every continuous exact encoder needs at least r local coordinates.", | |
| "additional_hypotheses": [ | |
| "Interior regular point of the rational signature; r is its maximum interior rank." | |
| ], | |
| "implementation_entrypoints": [ | |
| "jacobian_rank", | |
| "ExteriorCertificate.fiber_matrix" | |
| ], | |
| "related_tests": [ | |
| "FamilyTests.test_local_rank_two", | |
| "RewriteTests.test_new_local_rank_full" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "The generic theorem is local; a globally minimal chart is not asserted.", | |
| "jacobian_rank evaluates one state and does not by itself prove regularity or generic maximal rank.", | |
| "Discontinuous real encodings are not bounded by this continuous dimension statement." | |
| ] | |
| }, | |
| { | |
| "id": "lem:ratio", | |
| "title": "Affine information required by a nonconstant ratio", | |
| "kind": "lemma", | |
| "section": "4", | |
| "number": "4.1", | |
| "tex_label": "lem:ratio", | |
| "equation_labels": [], | |
| "conclusion": "Any fixed affine row space containing e and permitting arbitrary decoding of a nonconstant positive-denominator ratio must contain both its numerator and denominator.", | |
| "additional_hypotheses": [ | |
| "Nonconstant ratio on the full simplex; affine encoding row span contains e." | |
| ], | |
| "implementation_entrypoints": [], | |
| "related_tests": [], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "An initialization statement about fixed affine information; it does not impose recurrence." | |
| ] | |
| }, | |
| { | |
| "id": "thm:two", | |
| "title": "Sharp affine-snapshot law", | |
| "kind": "theorem", | |
| "section": "4", | |
| "number": "4.2", | |
| "tex_label": "thm:two", | |
| "equation_labels": [ | |
| "eq:S", | |
| "eq:factor2" | |
| ], | |
| "conclusion": "The minimum affine initial-snapshot dimension is s = dim span(e,a_i,b_i)-1. With generic nonlinear dimension r, r <= s <= 2*r, and both endpoints are attained.", | |
| "additional_hypotheses": [], | |
| "implementation_entrypoints": [ | |
| "snapshot_space", | |
| "factor_two_fixture" | |
| ], | |
| "related_tests": [ | |
| "SnapshotTests.test_factor_two_sharp", | |
| "SnapshotTests.test_family_snapshot_three", | |
| "SnapshotTests.test_snapshot_initializes_compact" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "Snapshot coordinates need not update from themselves.", | |
| "The theorem rules out an unbounded nonlinear scalar advantage for affine initialization under this contract." | |
| ] | |
| }, | |
| { | |
| "id": "thm:family", | |
| "title": "Unbounded recurrent affine gap", | |
| "kind": "theorem", | |
| "section": "5", | |
| "number": "5.1", | |
| "tex_label": "thm:family", | |
| "equation_labels": [ | |
| "eq:params", | |
| "eq:LA", | |
| "eq:LB", | |
| "eq:physical", | |
| "eq:compact", | |
| "eq:updates" | |
| ], | |
| "conclusion": "An explicit strictly positive instrument has branch probabilities in [3/8,5/8], minimum continuous nonlinear memory 2, minimum affine snapshot 3, and minimum fixed affine recurrent memory n-1.", | |
| "additional_hypotheses": [ | |
| "n is a power of two and n >= 4.", | |
| "Explicit Hadamard construction; c=1/2, tau=1/4, gamma=kappa=1/8, lambda=1/(8*n); delta=0." | |
| ], | |
| "implementation_entrypoints": [ | |
| "hadamard_exact", | |
| "separation_family", | |
| "Family.signature", | |
| "Family.compact_step", | |
| "affine_closure" | |
| ], | |
| "related_tests": [ | |
| "FamilyTests", | |
| "CertificateIntegrityTests.test_mode_aware_same_evidence_control" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "The affine recurrence lower bound applies to a fixed affine encoder on the entire state base.", | |
| "A mode-aware conventional controller matches outputs with three initial affine values and subsequently at most one scalar plus a mode flag.", | |
| "No superiority over all conventional controllers is established." | |
| ] | |
| }, | |
| { | |
| "id": "thm:rewrite", | |
| "title": "Nonlinear memory dimension transition", | |
| "kind": "theorem", | |
| "section": "6", | |
| "number": "6.1", | |
| "tex_label": "thm:rewrite", | |
| "equation_labels": [ | |
| "eq:perturb", | |
| "eq:inverse", | |
| "eq:recover" | |
| ], | |
| "conclusion": "An arbitrarily small positivity-preserving rank-one rewrite keeps branch probability functions and individual/aggregate characteristic polynomials unchanged, but makes complete conditional behavior injective and raises minimum continuous exact memory to n-1.", | |
| "additional_hypotheses": [ | |
| "Explicit Hadamard family with 0 < delta < 1/8." | |
| ], | |
| "implementation_entrypoints": [ | |
| "separation_family", | |
| "reconstruct_hidden_from_B_readouts" | |
| ], | |
| "related_tests": [ | |
| "RewriteTests.test_rank_one_and_equal_spectra", | |
| "RewriteTests.test_uniform_stationary", | |
| "RewriteTests.test_reconstruction_inversion", | |
| "RewriteTests.test_rewrite_distinguishes_old_memory", | |
| "RewriteTests.test_new_local_rank_full" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "Exact inverse recovery can be ill-conditioned for small delta and deep coordinates.", | |
| "The result does not imply a large error at a fixed practical tolerance.", | |
| "The isospectral statement concerns the explicit family." | |
| ] | |
| }, | |
| { | |
| "id": "thm:approxjump", | |
| "title": "Uniform approximation across the dimension jump", | |
| "kind": "theorem", | |
| "section": "7", | |
| "number": "7.1", | |
| "tex_label": "thm:approxjump", | |
| "equation_labels": [ | |
| "eq:uniformdelta" | |
| ], | |
| "conclusion": "The old compact updates, initialized with the true old signature, retain output error at most 8*delta/3 uniformly over states, supplied label words, time, and dimension.", | |
| "additional_hypotheses": [ | |
| "Explicit perturbed Hadamard family; true old signature at initialization." | |
| ], | |
| "implementation_entrypoints": [ | |
| "Family.signature", | |
| "Family.compact_step", | |
| "separation_family" | |
| ], | |
| "related_tests": [ | |
| "RewriteTests.test_perturbation_output_bound" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "Approximate protected-output guarantee, not exact compatibility or full-state recovery.", | |
| "This family-specific bound is not a generic robustness theorem." | |
| ] | |
| }, | |
| { | |
| "id": "finite-bit-implementation", | |
| "title": "Finite-bit recurrent implementation", | |
| "kind": "derived bound", | |
| "section": "7", | |
| "number": null, | |
| "tex_label": null, | |
| "equation_labels": [ | |
| "eq:quant" | |
| ], | |
| "conclusion": "For the explicit contractive compact maps, rounding on spacing Delta yields uniform coordinate error at most Delta/(2*(1-kappa)); deterministic worst-state state-bit cost has leading term 2*log2(1/epsilon).", | |
| "additional_hypotheses": [ | |
| "Unperturbed explicit family; kappa=1/8; specified rounding and initialization error.", | |
| "Matching packing lower bound uses current output and next-A output accuracy." | |
| ], | |
| "implementation_entrypoints": [ | |
| "Family.compact_step" | |
| ], | |
| "related_tests": [], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "The argument is in an unnumbered subsection, not a separately numbered theorem.", | |
| "Code, model, acquisition, and metadata costs are excluded.", | |
| "No claim for a different average-error or randomized encoding contract." | |
| ] | |
| }, | |
| { | |
| "id": "thm:admit", | |
| "title": "Exact nonlinear rewrite admission", | |
| "kind": "theorem", | |
| "section": "8", | |
| "number": "8.1", | |
| "tex_label": "thm:admit", | |
| "equation_labels": [ | |
| "eq:QE", | |
| "eq:suff" | |
| ], | |
| "conclusion": "A proposed positive branch admits updates from the old signature exactly when every transported exterior row annihilates all feasible equivalent simplex pairs. An analogous pair test governs new readouts.", | |
| "additional_hypotheses": [ | |
| "Proposed branch is strictly positive." | |
| ], | |
| "implementation_entrypoints": [], | |
| "related_tests": [ | |
| "RewriteTests.test_span_admission_not_necessary" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "General real quantifier elimination is a theoretical decision route and is not implemented.", | |
| "Linear exterior transport invariance is sufficient but not necessary.", | |
| "A failed span test alone is not a valid impossibility certificate." | |
| ] | |
| }, | |
| { | |
| "id": "prop:all", | |
| "title": "No exact compression under arbitrary positive rewrites", | |
| "kind": "proposition", | |
| "section": "8", | |
| "number": "8.2", | |
| "tex_label": "prop:all", | |
| "equation_labels": [], | |
| "conclusion": "Preserving a nonconstant affine readout under every strictly positive stochastic rewrite forces injectivity on the simplex; continuous encoders need at least n-1 real coordinates.", | |
| "additional_hypotheses": [ | |
| "Every strictly positive stochastic rewrite is permitted.", | |
| "At least one protected affine readout is nonconstant." | |
| ], | |
| "implementation_entrypoints": [], | |
| "related_tests": [], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "Restricted rewrite libraries and controlled approximate memory are not ruled out." | |
| ] | |
| }, | |
| { | |
| "id": "thm:robust", | |
| "title": "Finite continuation-error certificate", | |
| "kind": "theorem", | |
| "section": "9", | |
| "number": "9.1", | |
| "tex_label": "thm:robust", | |
| "equation_labels": [ | |
| "eq:residual", | |
| "eq:boundC", | |
| "eq:robust" | |
| ], | |
| "conclusion": "Valid transport residual and denominator bounds give a componentwise one-step signature error bound; a common weighted contraction gives a geometric all-horizon bound with implementation error charges.", | |
| "additional_hypotheses": [ | |
| "Declared feasible compact state set and valid positive denominator lower bounds.", | |
| "For the lifetime claim, common positive vector v, q<1, and C_a*v <= q*v.", | |
| "Represented-state updates have genuine weighted signature error charges." | |
| ], | |
| "implementation_entrypoints": [], | |
| "related_tests": [], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "No generic robust certificate solver is delivered.", | |
| "Ambient box quantization is not automatically feasible for a general signature image.", | |
| "Small residual alone does not establish all-horizon stability." | |
| ] | |
| }, | |
| { | |
| "id": "successor-gain-gate", | |
| "title": "Successor decisions and a sound recursive deployment rule", | |
| "kind": "protocol and counterexample", | |
| "section": "10", | |
| "number": null, | |
| "tex_label": null, | |
| "equation_labels": [ | |
| "eq:gate" | |
| ], | |
| "conclusion": "A positive full-set gain lower bound with valid uniform model errors, common anchor, charged costs, compatible memory, and separate operational requirements supports the stated bounded promotion rule. Vertex-only evaluation can be unsound.", | |
| "additional_hypotheses": [ | |
| "Valid common uncertainty event and transition model.", | |
| "Actual infimum bound over the full feasible uncertainty set.", | |
| "Uniform error guarantees and actual information acquisition when needed." | |
| ], | |
| "implementation_entrypoints": [], | |
| "related_tests": [ | |
| "RewriteTests.test_vertex_promotion_failure" | |
| ], | |
| "status": "written argument in supplied manuscript; see implementation fields for executable support", | |
| "boundaries": [ | |
| "No general nonlinear gain minimizer is implemented.", | |
| "No guarantee of useful proposals, recursive acceleration, or inclusion of every human requirement.", | |
| "The protocol is not a separately numbered theorem." | |
| ] | |
| } | |
| ], | |
| "unestablished_claims": [ | |
| "Unknown-model or arbitrary neural-agent faithful realization.", | |
| "Measured frontier-agent performance or research-productivity advantage.", | |
| "Universal superiority over known methods.", | |
| "Exhaustively established publication priority.", | |
| "Proof-assistant formalization or external expert certification.", | |
| "A justified T9 significance classification." | |
| ] | |
| } | |