{
  "title": "Cambrian: theorem and research evidence",
  "reviewed": "2026-09-16",
  "scope": "Selected new formal completions in the institute research library, followed by documented internal research uses. Historical mathematical novelty here means an open library obligation or stronger generalization was proved; worldwide historical priority is not established. August results come from Cambrian’s AI research workflow, including third-party proof-authoring agents. September records identify generated intermediate proofs within larger arguments. Lean verification establishes stated mathematical conclusions from their premises; it is not experimental validation or a benchmark of autonomous AI. Public summaries omit proprietary methods and confidential projects.",
  "records": [
    {
      "id": "mesh-uniform-green-bound",
      "date": "2026-08-12",
      "date_basis": "Recorded library landing date (institute local date)",
      "theorem": "green_row_mass_uniform_bound",
      "result": "For a cube of side R > 0, the weighted absolute row mass of the specified discrete Dirichlet Green kernel is bounded by R²/8 at every positive mesh level.",
      "advance": "A mesh-dependent bound is strengthened to one constant chosen before the grid resolution.",
      "attribution": "Produced within Cambrian’s AI theorem-discovery workflow, including third-party AI proof-authoring agents and program coordination.",
      "verification": "Fresh Lean compilation of the frozen current sources and all 95 project dependency modules completed on 16 September 2026. A separate axiom audit of this theorem uses only Lean’s standard mathematical axioms. Existing external package caches were reused.",
      "limits": [
        "Mathematical result for the specified Dirichlet operator; no completed continuum-limit or simulation-convergence theorem is claimed. The page does not claim the constant is optimal."
      ],
      "record_sha256": {
        "verification_record": "35a670c3fd18db4c5b83f30d232f090a457c88b600a5e439d92c2ff50f3b1e51",
        "source_reviewed": "4a665f323903f625f88c6a3ac1409de3806387502808bc93d551f3e02e5dbb74",
        "fresh_verification_record": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
        "source_manifest": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5"
      },
      "verification_scope": "Current sources and compiled outputs are pinned by hashes in the institute’s verification record. This verifies the mathematical statement, not historical authorship, global novelty, or physical interpretation. Selected source excerpts are public; complete sources and dependencies remain private, with review access by agreement.",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ],
      "formal_problem": "R > 0, integer L >= 1, h = R/(L+1), L interior points per axis. A is the cubic Dirichlet Laplacian with diagonal 6/h^2, axial nearest-neighbor entries -1/h^2 and zero boundary values. Assuming AG = I/h^3, for every grid point x, sum_y |G(x,y)| h^3 <= R^2/8. The h^3 weight is grid-cell volume.",
      "supporting_theorem": "green_row_mass_le_Rsq_div_eight"
    },
    {
      "id": "geometric-kernel-span",
      "date": "2026-08-12",
      "date_basis": "Recorded library landing date (institute local date)",
      "theorem": "flatAngleJacobian_kernel_span",
      "result": "At the specified reference tetrahedron, the angle Jacobian annihilates exactly the uniform-scaling direction in squared-edge coordinates; its kernel is one-dimensional.",
      "advance": "The existing scaling direction is shown to be the only kernel direction.",
      "attribution": "Produced within Cambrian’s AI theorem-discovery workflow, including third-party AI proof-authoring agents and program coordination.",
      "verification": "Fresh Lean compilation of the frozen current sources and all 95 project dependency modules completed on 16 September 2026. A separate axiom audit of this theorem uses only Lean’s standard mathematical axioms. Existing external package caches were reused.",
      "limits": [
        "Specified reference geometry and infinitesimal angle map; not arbitrary-tetrahedron global rigidity."
      ],
      "record_sha256": {
        "verification_record": "4a23a6ac70c05aa0882c214c7ced93b70507f12c33218f60c21fa7bac7566f9b",
        "source_reviewed": "4e5d346e47ada31ce07cdba1ffbce7b915aad77d63bb737f928d8bf372db43fe",
        "fresh_verification_record": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
        "source_manifest": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5"
      },
      "verification_scope": "Current sources and compiled outputs are pinned by hashes in the institute’s verification record. This verifies the mathematical statement, not historical authorship, global novelty, or physical interpretation. Selected source excerpts are public; complete sources and dependencies remain private, with review access by agreement.",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "quantum-rotation-characterization",
      "date": "2026-08-12",
      "date_basis": "Recorded library landing date (institute local date)",
      "theorem": "symm_invol_rotate_iff_anticomm",
      "result": "For real symmetric matrices M and N with M²=N²=I, every unit-circle combination cM+sN squares to I if and only if MN+NM=0.",
      "advance": "The missing converse completes the equivalence, and the new proof uses the earlier sufficiency theorem.",
      "attribution": "Produced within Cambrian’s AI theorem-discovery workflow, including third-party AI proof-authoring agents and program coordination.",
      "verification": "Fresh Lean compilation of the frozen current sources and all 95 project dependency modules completed on 16 September 2026. A separate axiom audit of this theorem uses only Lean’s standard mathematical axioms. Existing external package caches were reused.",
      "limits": [
        "Mathematical matrix model; no experimental claim or worldwide priority claim."
      ],
      "record_sha256": {
        "verification_record": "35a670c3fd18db4c5b83f30d232f090a457c88b600a5e439d92c2ff50f3b1e51",
        "source_reviewed": "9f9b5c54d73a004d8e401a7ba50a2009986890b905bc586ad521c9d090cdccfb",
        "fresh_verification_record": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
        "source_manifest": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5"
      },
      "verification_scope": "Current sources and compiled outputs are pinned by hashes in the institute’s verification record. This verifies the mathematical statement, not historical authorship, global novelty, or physical interpretation. Selected source excerpts are public; complete sources and dependencies remain private, with review access by agreement.",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ],
      "recorded_author_family": "Kimi-family AI proof-authoring agent within the Cambrian workflow."
    },
    {
      "id": "dimension-agreement",
      "date": "2026-08-12",
      "date_basis": "Recorded library landing date (institute local date)",
      "theorem": "k_fib_eq_k_int_iff_dim_three",
      "result": "The two defined dimension-dependent exponent formulas agree if and only if the natural-number dimension is three.",
      "advance": "Earlier coverage of dimensions one through four becomes a theorem over all natural-number dimensions.",
      "attribution": "Produced within Cambrian’s AI theorem-discovery workflow, including third-party AI proof-authoring agents and program coordination.",
      "verification": "Fresh Lean compilation of the frozen current sources and all 95 project dependency modules completed on 16 September 2026. A separate axiom audit of this theorem uses only Lean’s standard mathematical axioms. Existing external package caches were reused.",
      "limits": [
        "Uniqueness within the defined formulas; physical interpretation is separate."
      ],
      "record_sha256": {
        "verification_record": "35a670c3fd18db4c5b83f30d232f090a457c88b600a5e439d92c2ff50f3b1e51",
        "source_reviewed": "e678dfb4ad9907c8c4bdfb28b93c6bf9d90dd0febd6a0c57b69b5464fd9cea47",
        "fresh_verification_record": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
        "source_manifest": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5"
      },
      "verification_scope": "Current sources and compiled outputs are pinned by hashes in the institute’s verification record. This verifies the mathematical statement, not historical authorship, global novelty, or physical interpretation. Selected source excerpts are public; complete sources and dependencies remain private, with review access by agreement.",
      "axioms": [
        "propext",
        "Quot.sound"
      ],
      "recorded_author_family": "GPT-family AI proof-authoring agents within the Cambrian workflow."
    },
    {
      "id": "fibonacci-family",
      "date": "2026-08-12",
      "date_basis": "Recorded library landing date (institute local date)",
      "theorem": "jcost_phi_pow_even_eq_fib",
      "result": "The institute’s cost function at every positive even golden-ratio power has an exact expression in neighboring Fibonacci numbers.",
      "advance": "Three exact examples and an existing rationality result are extended to an explicit formula for the infinite family.",
      "attribution": "Produced within Cambrian’s AI theorem-discovery workflow, including third-party AI proof-authoring agents and program coordination.",
      "verification": "Fresh Lean compilation of the frozen current sources and all 95 project dependency modules completed on 16 September 2026. A separate axiom audit of this theorem uses only Lean’s standard mathematical axioms. Existing external package caches were reused.",
      "limits": [
        "A formal library completion using established golden-ratio and Fibonacci mathematics, not a priority claim for classical identities."
      ],
      "record_sha256": {
        "verification_record": "0a974b9d675864a6859dc60913f8589de85a6edf08adf4ebeea238beb5e9afc8",
        "source_reviewed": "6e132b0a19e9eb06e8237f31a18b7aa2599949858303969ea3d05cdbf15b8967",
        "fresh_verification_record": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
        "source_manifest": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5"
      },
      "verification_scope": "Current sources and compiled outputs are pinned by hashes in the institute’s verification record. This verifies the mathematical statement, not historical authorship, global novelty, or physical interpretation. Selected source excerpts are public; complete sources and dependencies remain private, with review access by agreement.",
      "axioms": [
        "propext",
        "Classical.choice",
        "Quot.sound"
      ]
    },
    {
      "id": "atomic-calculation-support",
      "date": "2026-09-14",
      "result": "Under stated stability and boundary assumptions, the error of the specified polynomial calculation is bounded by the sum of its local equation defects. Cambrian generated the theorem proof within an assistant-developed package. The Atomic research agent accepted it for its next calculation.",
      "limits": [
        "Explicit mathematical assumptions; physical-model correspondence includes analytical arguments.",
        "No new atomic-energy prediction or experimental validation established by this contribution alone."
      ],
      "record_sha256": {
        "verification": "9d603b0abb68c961e75ddacee3ecf2edf789072f05bf60c29aa25438ac4331d3"
      },
      "recipient_status": "The Atomic research agent inspected and accepted the proof package and identified it for its next calculation. The supplied record does not establish a completed numerical calculation using this result.",
      "generated_contribution": {
        "theorem": "CambrianClenshawGenerated.clenshaw_full_defect_bound",
        "statement": "Under polynomial stability and two zero terminal values, the norm of the specified polynomial output error is at most the summed local equation-defect norms.",
        "scale": "One generated declaration. The proof applies the supplied defect_norm theorem to the supplied defect_identity theorem. The underlying identities, norm estimate and application are surrounding AI-assistant work.",
        "source_sha256": "22e789c219c78d89eae174b51e38b0250e346b4eefe35088ffecf6d38c88b7f9",
        "consumer": "The separately authored direct_output_error theorem invokes the generated declaration."
      }
    },
    {
      "id": "fundamental-physics-assumptions",
      "date": "2026-09-16",
      "question": "Which assumptions does the next step in a quantum calculation require?",
      "result": "An AI-assistant-developed package proves that a modeled quantum transformation preserves the identity operator (unitality), under explicit reversibility and model-structure assumptions. A machine-checked counterexample shows weaker norm-preserving assumptions insufficient. Cambrian generated a connecting proof used by the final formal argument. Another institute AI agent used the result as a premise for further stationary-state analysis.",
      "formal_check": "Saved successful verification of the complete fifteen-module argument, covering combined research work including Cambrian’s generated proof. A separate institute AI agent inspected source and saved receipts and used the result in further analysis; it did not perform a fresh build.",
      "attribution": "AI research assistants developed the surrounding mathematics, counterexample and physical interpretation. Cambrian generated a connecting proof used by the final formal argument; recipient agents work within the institute program.",
      "limits": [
        "Fifteen formal modules verified in the saved run. Recipient inspection was documentary, not a fresh build. Surrounding mathematics, controls and later recipient analysis are separate contributions. No experimental result, physical-constant prediction or autonomous discovery is established.",
        "Unitality alone does not establish existence of a stationary physical state. The subsequent analytical work belongs to the receiving AI agent."
      ],
      "record_sha256": {
        "formal_verification": "bb30e96e1715e9ce1b46a0b095f9deecd5394f90842c3b46a592e142e84b24e0",
        "recipient_record": "ed632ed014ab227bce7bf6c1020b3b7e5da52c5d90dcdb57d22fdf43dec4f0f0"
      },
      "displayed_generated_contribution": {
        "theorem": "CambrianLoadedUnitalGenerated.reverse_inner",
        "statement": "Under the supplied inner-product and support assumptions, the reverse recorded inner product equals the input inner product.",
        "scale": "One generated declaration; three proof-body lines applying equality transitivity to two supplied lemmas. Lines are descriptive, not a complexity or performance measure.",
        "source_sha256": "8d3e26cf8ff2867a03f8c24864f14d5af8c2535565387ef149848834c06c41a5",
        "consumer": "The separately authored reverse_inner_product theorem uses this generated lemma in the combined argument."
      }
    }
  ],
  "record_identifiers": "Hashes identify the private verification records and the exact public excerpts. Complete sources and dependencies require an agreed review scope.",
  "access_contact": "jon@recognitionphysics.org",
  "theoretical_discovery_example": {
    "title": "The Four Indices",
    "date": "2026-09-06",
    "attribution": "The human-directed Cambrian research workflow using AI agents; not attributed to autonomous operation of the native proof generator.",
    "result": "Within the established posting geometry, a finite coordinate index represents its covectors faithfully and completely exactly when its size is the spatial dimension plus one. At the model’s spatial dimension three, the count is four. Every larger coordinate index has a nonzero covector that reads zero on every realized move.",
    "scientific_contribution": "Constructing the coordinate carrier, identifying an insufficient completeness-only criterion, and proving the corrected characterization.",
    "scope": "Theoretical discovery within the specified geometry. The identification of physical field equations with this geometry is part of the model’s interpretation. No experimental result or worldwide priority claim is made.",
    "verification": "Primary source and paper argument inspected on 16 September 2026. The owning research ledger records a successful Lean build and standard-axiom audit from 6 September. This example was not included in the current five-theorem rebuild.",
    "record_sha256": {
      "source_reviewed": "5aac4f09821dd52867287019357d96f12c9f93c7162b1451aed12b7f4a43a61c",
      "paper_reviewed": "8e6dff5cebd6648bc49c32afb59f701a1dd712d9c087a4f03c2f6e1bd857238e"
    },
    "historical_ledger_record": "12c302d86359"
  },
  "fresh_theorem_verification": {
    "status": "PASS",
    "date": "2026-09-16",
    "project_modules_freshly_compiled": 95,
    "toolchain": "Lean (version 4.27.0-rc1, x86_64-unknown-linux-gnu, commit 2fcce7258eeb6e324366bc25f9058293b04b7547, Release)",
    "source_manifest_sha256": "0810794591dfdc24fad0fcf233456c80c10355c78141355c88304235f501e7d5",
    "package_manifest_sha256": "1001974f24ccd456face62c6f734c26160ac2d9217ee739dd0206723c1ac00d9",
    "axiom_output_sha256": "97213bfdf4b06d8052cc5973c60c7e163753d82e6c6b60d3f5260b62c4d2f643",
    "verification_record_sha256": "52281ac29cabbae8c35118020c1533f48fddfd0c67449b693b56d1a4d6a4a475",
    "package_cache_reused": true,
    "project_cached_oleans_used": false
  },
  "source_excerpts": [
    {
      "id": "quantum-proof",
      "source_file": "RotationConverse.lean",
      "source_sha256": "9f9b5c54d73a004d8e401a7ba50a2009986890b905bc586ad521c9d090cdccfb",
      "line_start": 70,
      "line_end": 74,
      "excerpt_kind": "Proof body",
      "text": "  constructor\n  · intro hrot\n    exact anticomm_of_symm_invol_rotate M N hMsym hNsym hMinv hNinv hrot\n  · intro hanti c s hcs\n    exact (symm_invol_rotate M N hanti hNinv hMinv hNsym hMsym hcs).2",
      "excerpt_sha256": "200f4fe936d470364c42b92d1ed43559afe7d7aa7826d2247151d104cf122b01",
      "related_theorem": "update-quantum",
      "verification": "Exact excerpt from the frozen source freshly checked with its project dependencies on 16 September 2026. Snippets rely on surrounding declarations and imports; they are not standalone build files."
    },
    {
      "id": "geometry-proof",
      "source_file": "ReggeTTJacobianKernelSpan.lean",
      "source_sha256": "4e5d346e47ada31ce07cdba1ffbce7b915aad77d63bb737f928d8bf372db43fe",
      "line_start": 195,
      "line_end": 198,
      "excerpt_kind": "Complete declaration",
      "text": "theorem flatAngleJacobian_kernel_span (v : Fin 6 → ℝ)\n    (hv : ∀ f : Fin 6, ∑ g : Fin 6, flatAngleJacobian f g * v g = 0) :\n    ∃ c : ℝ, v = fun g => c * freudenthalTetSqEdges g :=\n  rawJacobianCoefficient_kernel_span v ((kernel_flat_iff_kernel_raw v).mp hv)",
      "excerpt_sha256": "e9e4480611e6fe96de196f2fc8501dcfbb15b7ab6816bc87e37f0590c4040c23",
      "related_theorem": "update-geometry",
      "verification": "Exact excerpt from the frozen source freshly checked with its project dependencies on 16 September 2026. Snippets rely on surrounding declarations and imports; they are not standalone build files."
    },
    {
      "id": "native-proof",
      "file": "GeneratedUnitaryRows.lean",
      "source_sha256": "8d3e26cf8ff2867a03f8c24864f14d5af8c2535565387ef149848834c06c41a5",
      "lines": [
        8,
        10
      ],
      "kind": "Proof body of one in-house-generated theorem",
      "text": "by\n  intro x0 x1 x2 x3 x4 x5 x6 x7 x8 x9\n  exact Eq.trans (@CambrianLoadedUnital.reverse_loaded_to_paths.{u0} x0 x1 x2 x3 x4 x5 x6) (@CambrianLoadedUnital.reverse_paths_to_input.{u0} x0 x1 x2 x3 x4 x7 x8 x9 x5 x6)",
      "excerpt_sha256": "4cb974632f700909572d3a07a8d7b8cb7122b30e76413faa7b4b65284ddf2f29",
      "related_theorem": "update-physics",
      "attribution": "Cambrian in-house proof generator; two input lemmas and surrounding research supplied by other AI research agents.",
      "verification": "Saved successful check, separately from the five August theorem sources rebuilt on 16 September."
    }
  ],
  "program_identity": "Cambrian is an AI for mathematical discovery: a human-directed system combining its native theorem generator, external AI research agents and independent Lean proof checking. Authorship and the native generator’s specific contribution are attributed for each example.",
  "evaluation_status": {
    "selection": "Five selected successful completions; no complete attempt count or comparable effort log is supplied here.",
    "proposal": "Frozen new problems, every attempt reported; compare complete workflow with the same third-party models without the in-house generator, measuring checked completions, total time, cost and human interventions.",
    "continuation": "Expand if the generator adds checked completions or reduces total effort; revise if neither improves.",
    "commitments": "No outside reviewer, date, result or performance advantage is committed or claimed.",
    "historical_attempt_records": "Partial campaign logs record attempt counts, duplicate proposals and failed first attempts. They do not supply a complete harmonized attempt/effort census across the five selected results."
  },
  "discovery_case": {
    "title": "From special-case quantum calculations to a general law and its converse",
    "dates": [
      "2026-08-11",
      "2026-08-12"
    ],
    "question": "The existing quantum-correlation construction discharged a matrix property by checking entries of specific two-by-two matrices. What general statement could replace that step?",
    "result": "Cambrian’s research workflow produced symm_invol_rotate for every finite matrix size. The later equivalence combines that sufficiency theorem with the separately proved necessity result anticomm_of_symm_invol_rotate; the necessity proof itself does not invoke symm_invol_rotate.",
    "use": "The first theorem is used in the formal quantum-correlation saturation construction.",
    "attribution": "Research tools identified the generalization opportunity; the workflow developed the statement and an external AI prover supplied its proof. The converse has recorded Kimi-family authorship. This is not wholly autonomous native-engine discovery.",
    "scope": "New general statements in this library; no worldwide mathematical priority or empirical discovery claimed.",
    "evidence": "Source, discovery paper and saved axiom audit inspected on 16 September; the converse and its dependency closure are covered by the page’s current 95-module rebuild.",
    "record_sha256": {
      "discovery_paper": "1aa3b52f477b35bf6ed8e898db4a89f9fc41e23031477a47662945424a6db793",
      "rotation_source": "583834b50ac064ab8f606e073b54799f0acf21860220e86578ec10c69bd97cb5"
    }
  },
  "cumulative_proof_audit": {
    "date": "2026-08-14",
    "machine_authored_lemmas_checked": 16,
    "proof_dependencies_between_them": 19,
    "longest_chain_levels": 4,
    "meaning": "A saved proof-term audit found later machine-authored lemmas invoking earlier ones; this is actual proof dependence, not merely similarity or citations in prose.",
    "scope": "One audited collection in the broader AI research workflow; not a growth-rate measurement or evidence of native autonomous self-improvement.",
    "verification": "Saved audit inspected and source dependency edges checked on 16 September. This collection was not rebuilt for the page.",
    "record_sha256": "d318ef2a30c3980f07162aae5b4a39eabb4a6627324c6e97e35ef0e775befe5e"
  },
  "infinite_series_application": {
    "date": "2026-09-16",
    "result": "Under coefficient-mass and uniform approximation hypotheses, the actual integral error series is summable with an explicit total norm bound; the total tends to zero when the uniform error parameter tends to zero.",
    "native_contribution": "One generated theorem connects a supplied series certificate to a supplied general norm comparison. The actual integral consumer calls it directly, using unchanged generated source.",
    "surrounding_work": "Other AI research agents supplied the scientific formulation, selected parent theorems, ordinary parent proofs, integral application and counterexample.",
    "verification": "Saved successful remote build of five final targets and 38 standard-axiom reports inspected. Source and evidence hashes match the canonical record. No fresh rebuild was run for this page addition.",
    "limits": "Selected-parent composition; no corpus-wide novelty, autonomous physical discovery, complete quantum theory or experimental validation claim. The formal consumer is evidence of use within the proof package, not a completed downstream physics experiment.",
    "record_sha256": {
      "GeneratedFactorialBound.lean": "7f5b698fd39ce233f0fa0b0327ec8eec096d104baef8d00d50abd4a363a08843",
      "VacuumErrorConsumer.lean": "0909cdc5a4b9c713c0e80d949a16207728e2d58711532ef6ed3bc80da86b84e2",
      "VERIFICATION.json": "9640b87329f45823b0cbd38fa51d5f7548478c2d0469539f5dd1bcb3ac157e99"
    },
    "run_accounting": {
      "generated_theorems": 1,
      "deliberate_rejection_controls": 4,
      "interpretation": "One positive composition plus four deliberate rejection controls, not a search success rate. Earlier failed attempts and supplied-input revisions are retained in the development record.",
      "record_sha256": "fcfb9e87b3c71f3245d15e74d3d45df99a48c4cf8e4769ccbb20269589432650"
    }
  },
  "native_engine_capability": {
    "scope": "Within supplied mathematical inputs, constructs statements with proofs, excludes matching statement shapes in its index, and supports further deductions using newly generated results.",
    "source_basis": "Actual engine functions and accepted output inspected; bounded capabilities, not autonomous research or a measured improvement over external AI provers."
  }
}