{ "schema": "recognitionphysics.cambrian-wins.v1", "updated": "2026-07-20", "scope": "Provenance-confirmed, AI-judge-accepted Cambrian theorems in canonical IndisputableMonolith domain modules", "wins": [ { "id": "CW-0001", "title": "The D=3 cube nullity law", "explanation": "Count the configurations of a three-dimensional M×N×P recognition block whose face records are closed. The answer is exactly 2^(M+N+P+1): an edge law, growing with the sides rather than the faces or the volume. The record-invisible share of the bulk vanishes even faster in three dimensions than in two.", "rs_significance": "Closes the named OPEN cube-nullity target in the Recognition Holography chapter (BP-4 spine). Together with the 2D perimeter law and the flat-cube collapse bridge, it completes the kernel-cardinality story across dimensions: what a boundary record cannot see is a vanishing sliver, in every dimension checked.", "statement": "∀ (M N P : ℕ), Nat.card {x : CubeCfg M N P // IsClosed M N P x} = 2 ^ (M + N + P + 1)", "theorem": "IndisputableMonolith.Holography.CubeKernelFalsifier.cube_nullity_card", "module": "IndisputableMonolith.Holography.CubeKernelFalsifier", "admitted": "2026-07-19", "proof": { "status": "THEOREM (shared-vertex lattice model)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.86, "receipt": "reality: Cambrian/campaign/novelty_admission_20260719/gate/runs/holography_card_v1/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine holography_card_v1.HC0001)", "held_out": "Selector-emitted HH0002: closed-sector count equals the free Kronecker parameter count, with HC0001 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "kronecker_kernel_card (hand-written compiled parent schema, permitted under the end-to-end authorship rule)" }, "claim_boundary": "THEOREM for the stated shared-vertex lattice model; inherits that named conditionality. No world-first or priority claim (PRIORITY_UNRESOLVED). The counting scaffolding parent was hand-built; the winning statement and its held-out consequence were authored by the ordinary runner." }, { "id": "CW-0002", "title": "The 2D affine-sector count", "explanation": "Count the configurations of a two-dimensional M×N recognition grid that are affine: built from whole-row and whole-column flips over a base value. The answer is exactly 2^(M+N+1), a perimeter law. These are precisely the configurations a boundary record cannot see, counted directly over their structural description rather than through the kernel of the record map.", "rs_significance": "Exposes the D=2 perimeter law in its structural form on the canonical Holography surface: the grid's record-invisible sector, described as affine configurations, has a count that grows with the perimeter and not the area. The held-out consequence ties this 2D count to the D=3 closed-sector count of the cube law (CW-0001) in one equality, connecting the dimension story the chapter tells.", "statement": "∀ (M N : ℕ), Nat.card {x : CornerCfg M N // IsAffine M N x} = 2 ^ (M + N + 1)", "theorem": "IndisputableMonolith.Holography.MulticellKernelFalsifier.affine_sector_card", "module": "IndisputableMonolith.Holography.MulticellKernelFalsifier", "admitted": "2026-07-19", "proof": { "status": "THEOREM (shared-vertex lattice model)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.78, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/grid_affine_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine grid_affine_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the 2D affine-sector count at matched perimeter equals the D=3 closed-sector count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "none (both parents were already canonical theorems)" }, "claim_boundary": "THEOREM for the stated shared-vertex lattice model; inherits that named conditionality. No world-first or priority claim (PRIORITY_UNRESOLVED). The judge framed this as canonical lemma utility, smaller in scope than CW-0001; the same statement family had appeared as unlanded quarantine evidence inside the CW-0001 package and was re-authored end to end for this win." }, { "id": "CW-0003", "title": "The cell's silent-move count", "explanation": "A move applied to a recognition cell is silent when it leaves the boundary record unchanged from every possible starting configuration. Exactly 16 of the cell's 256 moves are silent. The count is taken over the semantic property itself (invisible from every base), not over a pre-defined list of kernel elements.", "rs_significance": "Puts the whole-cell blindness law on the canonical Holography surface in its semantic form: what a boundary record can never see is a 16-element group of cell-global moves. The held-out consequence equates this silent-move count with the posted-record image count, exhibiting the rank-nullity balance of the cell record map as a single cardinality equation.", "statement": "Nat.card {d : CellCfg // ∀ c : CellCfg, faceRecord (xorCfg c d) = faceRecord c} = 16", "theorem": "IndisputableMonolith.Holography.CellSilentMoves.silent_moves_card", "module": "IndisputableMonolith.Holography.CellSilentMoves", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.86, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/cell_silent_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine cell_silent_card_v1.MINT0)", "held_out": "Selector-emitted HX0001: the silent-move count equals the posted-record image count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "recordKernel_nat_card (hand-written compiled schema parent, permitted and counted under the end-to-end authorship rule)" }, "claim_boundary": "THEOREM over the finite cell model; the bridge from universal invisibility to the kernel is the exhaustively checked invisible_iff_kernel. No world-first or priority claim (PRIORITY_UNRESOLVED). A sibling candidate from the same session (the glued-pair kernel count) was denied by the same judge as thin packaging and stays quarantined; this entry cleared that bar because the counted predicate is not the kernel's definition." }, { "id": "CW-0004", "title": "Sync-minimization selects exactly one dimension", "explanation": "Among all possible spatial dimension counts, ask which ones are admissible (at least three) and minimize the recognition synchronization period. The answer set has exactly one element. The count is taken over the semantic minimization property across ALL naturals, not over a pre-listed answer.", "rs_significance": "Packages dimensional rigidity as a single counting law on the canonical surface: the (S) synchronization constraint of the dimensional-rigidity paper does not merely imply D = 3, its solution set is literally a one-element set. This is the counting form of why reality has three spatial dimensions, sitting beside the T8 guidepost on the Skeleton spine.", "statement": "Nat.card {D : ℕ // ConstraintS D} = 1", "theorem": "IndisputableMonolith.Papers.DraftV1.constraintS_card", "module": "IndisputableMonolith.Papers.DraftV1", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.91, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/constraintS_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine constraintS_card_v1.MINT0)", "held_out": "Selector-emitted: the ConstraintS solution count equals the literal D = 3 fiber count, with MINT0 load-bearing; an earlier vacuous continuation (hypothesis-laden) was rejected by the runner's hypothesis-freedom filter before judging", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "nat_eq_three_card (hand-written compiled schema parent, permitted and counted under the end-to-end authorship rule)" }, "claim_boundary": "THEOREM over the formalized (S) constraint (admissibility plus syncPeriod minimization as defined in the Lean paper surface). No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0005", "title": "The self-dual coupling dimension is unique", "explanation": "The coupling-dimension duality swaps the two outer dimensions and fixes the middle one. Counting its fixed points over the fixed-point property itself gives exactly one. This uniqueness is the mechanism that forces the colorless lepton sector to the loop dimension in the sector-assignment derivation.", "rs_significance": "Puts the fixed-point uniqueness of the coupling-dimension duality on the canonical Masses surface in counting form. The sector-dimension derivation forces the lepton to the self-dual dimension precisely because an equivariant bijection must carry the unique conjugation fixed point to the unique duality fixed point; this theorem is that uniqueness as a cardinality law.", "statement": "Nat.card {d : Fin 3 // dimDual d = d} = 1", "theorem": "IndisputableMonolith.Masses.SectorDualityFixedCard.dimDual_fixed_card", "module": "IndisputableMonolith.Masses.SectorDualityFixedCard", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.74, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/dimDual_fixed_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine dimDual_fixed_card_v1.MINT0)", "held_out": "Selector-emitted: the fixed-point count equals the literal middle-index fiber count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "fin3_eq_one_card (hand-written compiled schema parent, permitted and counted under the end-to-end authorship rule)" }, "claim_boundary": "THEOREM over the finite coupling-dimension model (Fin 3). The judge noted the small ambient type lowers novelty strength and the held-out leans on the mint's own scaffold; win credit was granted at 0.74 confidence because the counted predicate (involution fixed point) is semantically distinct from the literal index. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0006", "title": "The cell's gauge-fiber count", "explanation": "Fix any configuration of the recognition cell and count the configurations that carry the same boundary record. The answer is exactly 16, for every starting configuration. Gauge classes are cosets of the 16-element record kernel, so every fiber has the kernel's size: what the boundary cannot distinguish is the same 16-fold blindness everywhere in the state space.", "rs_significance": "Completes the cell's gauge story on the canonical Holography surface: CW-0003 counted the moves invisible from every base; this counts, for each base, the states the record confuses with it. It is the cardinality form of gauge-classes-are-kernel-cosets, the structure the fork selector uses to define physical states. It is also the first theorem minted by the fiber-transport operator, built to close the typed wall this exact statement raised one cycle earlier.", "statement": "∀ (c : CellCfg), Nat.card {c' : CellCfg // gaugeRel c c'} = 16", "theorem": "IndisputableMonolith.Holography.CellGaugeFiber.gauge_fiber_card", "module": "IndisputableMonolith.Holography.CellGaugeFiber", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.86, "receipt": "reality: Cambrian/campaign/operator_expansion_20260719/runs/cell_gauge_fiber_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian fiber-transport mint (quarantine cell_gauge_fiber_card_v1.MINT0), the first product of the translation-fiber operator whose frozen positive/negative controls live at Cambrian/campaign/operator_expansion_20260719/gate/OPERATOR_GATE.json", "held_out": "Runner-emitted HX0000: every gauge-fiber count equals the record-kernel count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "recordKernel_nat_card (prior-session compiled schema parent, already counted); zero new scaffolding units for this target" }, "claim_boundary": "THEOREM over the finite cell model; the coset characterization gauge_iff_kernel is exhaustively checked. No world-first or priority claim (PRIORITY_UNRESOLVED). This pair was a typed wall (MISSING_OPERATOR:translation-fiber-transport) for the previous operator set; the statement became reachable with zero per-target hand work once the generic operator landed." }, { "id": "CW-0007", "title": "The 8-tick equation has exactly one solution", "explanation": "Ask which spatial dimension counts D satisfy the 8-tick equation, two to the power D equals eight. Over all naturals the solution set has exactly one element. The count is taken over the equation itself, not over a pre-listed answer.", "rs_significance": "Packages the arithmetic kernel of the 8-tick to three-dimensions hinge (registry item F-003) as a counting law on the canonical Unification surface, beside the T7/T8 guideposts. Together with CW-0004 it gives the dimension-selection story two independent counting forms: one from synchronization minimization, one from the 8-tick equation, both landing on the same singleton.", "statement": "Nat.card {D : ℕ // 2 ^ D = 8} = 1", "theorem": "IndisputableMonolith.Unification.DimensionThreeConsequences.two_pow_eq_eight_card", "module": "IndisputableMonolith.Unification.DimensionThreeConsequences", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.78, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/two_pow_eq_eight_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine two_pow_eq_eight_card_v1.MINT0)", "held_out": "Selector-emitted: the equation's solution count equals the literal D = 3 fiber count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "none new: reuses the shared D3 singleton schema parent already counted for CW-0004 (first scaffold-level compounding of the campaign)" }, "claim_boundary": "THEOREM over the naturals. The judge noted the bridge to D = 3 is elementary arithmetic, weaker than CW-0004's minimization principle, and granted win credit at 0.78. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0008", "title": "The octave unit group has four elements", "explanation": "Among the eight residues of the octave clock, exactly four are invertible: the odd classes 1, 3, 5, and 7. The theorem counts the semantic unit property itself. The explicit odd list is the checked bridge, not the definition of the counted set.", "rs_significance": "Puts the unit-group count of the octave algebra on the canonical Foundation spine. The four invertible phases form the Klein four-group already used in the chapter, so the count connects the eight-tick clock to its full automorphism structure in one reusable cardinal law.", "statement": "Nat.card {x : ZMod 8 // IsUnit x} = 4", "theorem": "IndisputableMonolith.Algebra.OctaveAlgebra.zmod8_unit_card", "module": "IndisputableMonolith.Algebra.OctaveUnitCard", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.82, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/zmod8_unit_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine zmod8_unit_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the unit count is substituted into the octave WToken arithmetic identity, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "odd_residues_card (hand-written compiled schema parent, permitted and already counted when the target first reached its typed wall)" }, "claim_boundary": "THEOREM over the finite ring ZMod 8. The judge found the unit predicate semantically distinct from the explicit odd-residue schema. The held-out consequence is valid but only moderately independent, which lowered confidence to 0.82. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0009", "title": "The 8-by-45 synchronization equation has one solution", "explanation": "Ask which natural dimensions make the least common multiple of 2^D and 45 equal 360. The solution set has exactly one element. The theorem counts the synchronization equation itself rather than a pre-listed answer.", "rs_significance": "Adds the 8-to-45 hinge to the canonical dimension-forcing spine as a counting law. CW-0007 says the eight-tick equation has one solution. This theorem shows the same uniqueness survives when the gap-45 cycle is folded into the least-common-multiple condition, and the runner's held-out connects the two counts directly.", "statement": "Nat.card {D : ℕ // Nat.lcm (2 ^ D) 45 = 360} = 1", "theorem": "IndisputableMonolith.RecogSpec.lcm_pow2_45_card", "module": "IndisputableMonolith.RecogSpec.LCMPow2Card", "admitted": "2026-07-19", "proof": { "status": "THEOREM", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.86, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/lcm_pow2_45_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine lcm_pow2_45_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the LCM-solution count equals the independently admitted 2^D = 8 solution count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "none new: reuses the D = 3 singleton schema parent already counted for CW-0004" }, "claim_boundary": "THEOREM over the naturals. The equation is formalized exactly as Nat.lcm (2^D) 45 = 360. The result carries no world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0010", "title": "Kepler non-precession selects one dimension", "explanation": "Ask which natural dimensions make the closed-form apsidal angle equal one full turn, so an orbit returns without precession. The answer set has exactly one element. The theorem counts the Kepler condition itself rather than a pre-listed dimension.", "rs_significance": "Adds the Kepler side of dimensional rigidity to the canonical Foundation spine as a counting law. CW-0004 counted the synchronization-minimizing dimensions. This theorem counts the non-precessing dimensions, and the runner's held-out proves those two solution sets have the same size.", "statement": "Nat.card {D : ℕ // apsidalAngle D = 2 * Real.pi} = 1", "theorem": "IndisputableMonolith.Papers.DraftV1.constraintK_card", "module": "IndisputableMonolith.Papers.DraftV1", "admitted": "2026-07-19", "proof": { "status": "THEOREM (closed-form Kepler specialization)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.88, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/constraintK_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine constraintK_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the Kepler solution count equals the admitted synchronization-minimization solution count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "none new: reuses the D = 3 singleton schema parent already counted for CW-0004" }, "claim_boundary": "THEOREM for the closed-form apsidal-angle specialization formalized in DraftV1. It does not add the classical-mechanics derivation that precedes that specialization. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0011", "title": "The golden return-map family has one golden parameter", "explanation": "Ask which integer linking parameters make the algebraic return map satisfy the golden relation. The solution set has exactly one element. The theorem counts the golden predicate over the whole integer family rather than a pre-listed parameter.", "rs_significance": "Adds a direct cardinality interface to the golden monodromy front-end. Downstream proofs can now use the uniqueness of the golden return-map fiber without rebuilding the subtype transport from the characterization theorem.", "statement": "Nat.card {k : ℤ // GoldenRelation (returnMap k)} = 1", "theorem": "IndisputableMonolith.Masses.GoldenMonodromyReturn.returnMap_golden_card", "module": "IndisputableMonolith.Masses.GoldenMonodromyReturn", "admitted": "2026-07-19", "proof": { "status": "THEOREM (algebraic return-map family)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.84, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/returnMap_golden_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine returnMap_golden_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the golden return-map count equals the integer singleton count, with MINT0 load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "one new compiled parent: Verification.IntegerCardSchema.int_eq_one_card" }, "claim_boundary": "THEOREM only for the algebraic family k ↦ returnMap k: exactly one integer parameter satisfies GoldenRelation. It does not derive the physical linking number from the kernel or construct the underlying topology. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0012", "title": "One Wang matrix inverts against its return map", "explanation": "Ask which integer parameters make the Wang matrix multiply its return map to the identity. The solution set has exactly one element. The theorem counts the inverse condition itself across the full integer family.", "rs_significance": "This is the first direct machine-to-machine compound in the public sequence. Its held-out continuation uses CW-0011 as a load-bearing parent and proves that the Wang-inverse fiber and golden-return fiber have the same size, with no new scaffolding.", "statement": "Nat.card {k : ℤ // wangMatrix k * returnMap k = 1} = 1", "theorem": "IndisputableMonolith.Masses.GoldenTwistedH1.wang_inv_golden_card", "module": "IndisputableMonolith.Masses.GoldenTwistedH1", "admitted": "2026-07-19", "proof": { "status": "THEOREM (algebraic Wang-return family)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.82, "receipt": "reality: Cambrian/campaign/compounding_loop_20260719/runs/wang_inv_golden_card/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian card-transport mint (quarantine wang_inv_golden_card_v1.MINT0)", "held_out": "Selector-emitted HX0000: the Wang-inverse count equals CW-0011's golden return-map count, with both machine theorems load-bearing", "causal_arms": "full accepts; omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "none new: reuses Verification.IntegerCardSchema.int_eq_one_card from CW-0011" }, "claim_boundary": "THEOREM only for the banked algebraic Wang-matrix and return-map family. It does not derive the physical linking number, construct the topological carrier, or add semantics beyond the existing Wang inverse characterization. No world-first or priority claim (PRIORITY_UNRESOLVED)." }, { "id": "CW-0013", "title": "Later radiation fits inside earlier capacity", "explanation": "Choose any two channel ticks with the first no later than the second. The radiation entropy at the later tick cannot exceed the channel capacity that remained at the earlier tick.", "rs_significance": "Adds a reusable cross-time Page-curve bound to the canonical Holography chapter. It is also the first public theorem minted by Cambrian after learning a new inequality-composition move from its own typed wall receipts.", "statement": "∀ (ch : FiniteChannel) (t2 t1 : ℕ), t1 ≤ t2 → ch.radiationEntropy t2 ≤ ch.remainingCapacity t1", "theorem": "IndisputableMonolith.Holography.PageCurve.FiniteChannel.radiationEntropy_le_remaining_of_le", "module": "IndisputableMonolith.Holography.PageCurve", "admitted": "2026-07-19", "proof": { "status": "THEOREM (finite-channel Page curve)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Codex (cross-family)", "confidence": 0.78, "receipt": "reality: Cambrian/campaign/move_learning_20260719/runs_remote/page_rad_remaining_bound/AI_JUDGE.json" }, "provenance": { "author": "Ordinary Cambrian ineq-transport mint (quarantine page_rad_remaining_bound_v1.MINT0)", "held_out": "Selector-emitted HX0000: a second cross-time remaining-capacity transport with MINT0 load-bearing", "causal_arms": "full accepts; both omit-parent arms and typed false decoy reject; replay import clean", "scaffolding": "one learned operator organ, emitted by scripted synthesis and admitted by a frozen gate; zero hand proof in the mint" }, "claim_boundary": "THEOREM for the finite-channel PageCurve model: later radiation entropy is bounded by earlier remaining capacity whenever the ticks are ordered." }, { "id": "CW-0014", "title": "The exact best contraction rate on a cost budget", "explanation": "Put phantom coupling on a J-cost budget and ask for the best uniform contraction factor that holds across the whole budget. The theorem names the exact factor, one minus kappa over twice the budget's upper endpoint, proves it works everywhere on the budget, and proves nothing smaller can: the rate is attained at the budget's edge.", "rs_significance": "Generation one of the breeder's sustained pass. The parent contraction theorem said the budget factor is admissible; this theorem says it is the least admissible factor, an exact optimum rather than a bound. That exactness is the surface generation two stands on.", "statement": "IsLeast {q : ℝ | ∀ r : ℝ, 1 / 2 ≤ r → Cost.Jcost r ≤ c → |phiCouplingNewRatio kappa r - 1| ≤ q * |r - 1|} (1 - kappa / (2 * Cost.jcostSublevelHi c))", "theorem": "IndisputableMonolith.Intelligence.PhantomCouplingBridge.phi_coupling_sublevel_optimal_contraction_rate", "module": "IndisputableMonolith.Intelligence.PhantomCouplingOptimalRate", "admitted": "2026-07-20", "proof": { "status": "THEOREM (exact least rate on the positive J-cost sublevel)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Dual cross-family: Codex and Claude, both ACCEPT_CANONICAL", "receipt": "reality: Cambrian/campaign/autonomous_breeder_20260720/runs/g1_G1_phi_coupling_sublevel_Jcost_contraction_correction1_20260720T184212Z/" }, "provenance": { "author": "Autonomous Cambrian breeder, generation one (one disclosed correction pass, wrapper-only)", "held_out": "Causal arms on the frozen protocol: full proof accepts; bare and typed false decoy reject; statement absent from the frozen baseline", "causal_arms": "full accepts under allowed axioms; omit-parent, bare, and typed false decoy all reject", "scaffolding": "none beyond catalogued parents; correction pass disclosed as wrapper-only" }, "claim_boundary": "THEOREM for the formalized phantom-coupling model over the positive J-cost sublevel; sharpness is attained at the upper sublevel endpoint. One correction pass was used and disclosed. No world-first or priority claim." }, { "id": "CW-0015", "title": "The golden fold has the strictly smallest exact rate", "explanation": "Recognition growth comes in folds, and every integer fold sets its own J-cost budget. Compare exact least contraction rates across those budgets. The golden fold's rate is strictly smallest: any uniform factor that survives a higher integer-fold budget is strictly larger than the golden fold's exact least rate.", "rs_significance": "The first Cambrian theorem that stands on another Cambrian theorem as a load-bearing parent under the frozen breeder protocol. Remove the generation-one exact-rate law and this proof dies; the omission arm checks exactly that. It turns the mass-fold cost ordering into a strict rate law and lands the phantom-coupling story on the golden ratio.", "statement": "IsLeast {q : ℝ | ∀ r : ℝ, 1 / 2 ≤ r → Cost.Jcost r ≤ foldCost 1 → |phiCouplingNewRatio kappa r - 1| ≤ q * |r - 1|} (1 - kappa / (2 * Cost.jcostSublevelHi (foldCost 1))) ∧ ∀ q : ℝ, (∀ r : ℝ, 1 / 2 ≤ r → Cost.Jcost r ≤ foldCost k → |phiCouplingNewRatio kappa r - 1| ≤ q * |r - 1|) → 1 - kappa / (2 * Cost.jcostSublevelHi (foldCost 1)) < q", "theorem": "IndisputableMonolith.Masses.GoldenFoldPhantomCouplingRate.golden_fold_minimizes_least_phantom_coupling_rate", "module": "IndisputableMonolith.Masses.GoldenFoldPhantomCouplingRate", "admitted": "2026-07-20", "proof": { "status": "THEOREM (fold-budget model, generation-one parent load-bearing)", "axioms": [ "propext", "Classical.choice", "Quot.sound" ] }, "judge": { "verdict": "ACCEPT_CANONICAL", "win_credit": "QUALIFIES_AS_CAMBRIAN_WIN", "model": "Dual cross-family: Codex and Claude, both ACCEPT_CANONICAL", "receipt": "reality: Cambrian/campaign/autonomous_breeder_20260720/runs/g2_G2_Q03_golden_fold_minimizes_least_coupling_rate_correction1_20260720T205637Z/" }, "provenance": { "author": "Autonomous Cambrian breeder, generation two, with CW-0014 load-bearing (one disclosed correction pass)", "held_out": "Causal arms on the frozen protocol: full proof accepts; omit-parent arm fails without the generation-one theorem; bare and typed false decoy reject; statement absent from the frozen post-landing baseline", "causal_arms": "full accepts under allowed axioms; omit-parent, bare, and typed false decoy all reject; generation-one parent verified load-bearing", "scaffolding": "none beyond catalogued parents; two sibling generation-two candidates were denied by the cross-family judges as equivalence transports and did not land" }, "claim_boundary": "THEOREM for the formalized fold-budget model with kappa in (0,1] and integer folds k ≥ 2. Generation one is load-bearing: the omission arm fails without it. One correction pass was used and disclosed. No world-first or priority claim." } ] }