| 2026-10-01 | Theorem 9's method applied to the box lemma R_b: the integral family, the branch certificates, and why no closed form in the branches 2026-10-01-opus-rbox.md | closed-r-then-monodromy |
| 2026-10-01 | Agent pubsite: the public notes repository, its sync, validator and site, and draft manifests for nine notes 2026-10-01-opus-pubsite.md | |
| 2026-10-01 | Hostile check of Lemma L, Lemma A, Theorem P through P1, Lemma 21 and the chain H + P + A to periodicity, with independent programs: four confirmed, and the chain corrected on thin cells, where a frozen zero-flux state with no translation symmetry exists 2026-10-01-opus-pacheck.md | lemma-21, periodicity, thin-cells-aperiodic |
| 2026-10-01 | Hostile check of the two linear-algebra steps of P1 (completeness of the chain modes, and ker L = S + G with dim H_A = 4(a + b + c) - 2), with an independent model and an independent stratification: both confirmed as proofs for every cell, with three expository corrections 2026-10-01-opus-p1check.md | parametrisation-c, periodicity |
| 2026-10-01 | Checker kucheck on kucone's construction: the first step of K_u given orientation is the sum of two Theorem 9 identities 2026-10-01-opus-kucheck.md | ku-first-step-hand |
| 2026-10-01 | Falsify: a whole-torus SAT search for a counterexample to Theorem U on (3,3,9), (4,4,8), (7,7,7), (8,8,8), (6,6,9) and (3,3,12); none exists, and Theorem U′ is certified on all six 2026-10-01-opus-falsify.md | u-prime-certificates-2026-10-01 |
| 2026-10-01 | Agent kucone (Opus, 30 September 2026), stopped before it reported: what its scripts and logs show. The first step of K_u given orientation is the sum of two Theorem 9 certificates, checked exactly on the infinite lattice for d = 2 to 30 in both slabs. Computed, no checker, not a result of the note 2026-10-01-kucone-partial.md | ku-first-step-hand |
| 2026-10-01 | (V) given Statement R: the reach drops by exactly the literal X(v_2) = 1 and no more, the models short of threshold are many and are not label classes, both cut cylinders still refute (V), and on the B grid x = 1 the parametrisation (C) has no linear content at all 2026-10-01-fable-screw.md | closed-r-then-monodromy |
| 2026-10-01 | The kink vectors of the x-chain classes: a wave equation along the rod axis (Law C per class, every cell), a defect between the two planes of a chain that is one constant on the torus and is forced flat, then non-negative, then zero by the pattern and the negation on windows of 3, 2g-2 and 2g-1 grids; and why the counts still admit the negation 2026-10-01-fable-labels.md | kink-counts-per-class, closed-kink-counts |
| 2026-09-30 | The first step of K_u beside a wall: is it a window lemma? 2026-09-30-opus-kuwall.md | |
| 2026-09-30 | Check of the zeroflux report: Theorem U implies Theorem H on every cell without K_u. Verdict CONFIRMED, with wording corrections; every step is within the hypotheses of the note's theorems, the corpus checks re-run by independent code with 0 exceptions, and the solver finds no oriented frozen state with six directions on both sublattices at (3,3,3) or (2,2,4) 2026-09-30-opus-check10.md | thm-1 |
| 2026-09-30 | Z(L) given Theorem U, and more: Theorem U implies Theorem H on every cell without K_u. The stripe structure of every two-way axis on both sublattices (Theorem 6 with Lemma 4, all three cases of Theorem 7), the profiles of Theorem 15 that it kills, and the diagonal law on full lines give the contradiction over the three axes; neither W = 0, the walls, the uniform chains nor per-grid balance is needed. Every ingredient checked with 0 exceptions on the 1,357 odd cubic states, 48 oriented odd states of four other cells and 11 odd W = 0 states 2026-09-30-fable-zeroflux.md | thm-1 |
| 2026-09-30 | Statement R (the whole line of the rods): its reach, the height language, and what is proved by hand 2026-09-30-fable-rodline.md | statement-r-box-lemma, weights-integrality-correction |
| 2026-09-29 | Probe of a phase quantity Q on the solutions of the negation of (V′): none of the listed candidates is defined or constrained on the dense solutions; the integer solutions of a window span a much smaller affine space than the real solutions, cut out by a backbone and two-variable equations 2026-09-29-opus-moment.md | closed-tilted-sheet |
| 2026-09-28 | Package D (agent twochain): on every cell (2a', 2b', 2c') with a', b', c' pairwise coprime, (V) is proved by hand — in a type-model solution containing the unoriented cage, sublattice B has no vertex of type x and the two planes of bridgeheads are entirely of types y and z. The proof is the profile description (nine functions of one coordinate on each sublattice, the B data a shift of the A data), one linear identity from the four bridgeheads, and a finite parity case analysis. With Theorem M′ this is Theorem U on the whole family, and with Theorem 1′ and K_u it gives Theorem H on the members with every extent at least 4; the thin members are not covered by the recorded scope of K_u 2026-09-28-opus-twochain.md | thm-22 |
| 2026-09-28 | The near-miss of (V′) at the one-sided threshold is one sheet on the labels of the x-chains, given by one formula at every g; opening the transverse torus does not remove g but replaces it by the number of chain labels of the surface that is left, and on the plane (V′) is false 2026-09-28-opus-sheet.md | closed-tilted-sheet |
| 2026-09-28 | Signed completeness, proved for every cell: the kernel of law P^k is the span of the chains of the two other axes (dimension 4b·gcd(a,c) + 4c·gcd(a,b) − 4·gcd(a,b,c) + 2 for k = x); the joint system of the three laws and the bond rule has as solutions exactly the sums over chains of a weight times the chain's own direction, n = ½ Σ g(C) u_C 1_C; the six direction indicators of a state with every hole oriented are sums over chains in which each chain carries two weights, each coupling two directions whose difference is parallel to the chain; the signed line fractions have six family profiles and do not add to a constant 2026-09-28-opus-scomp.md | thm-23 |
| 2026-09-28 | Package dfour: Theorem H proved by hand on every cell (2a', 2b', 2c') with a', b', c' pairwise coprime, thin members included, without K_u. In an odd frozen state with three types both sublattices carry the same one of six profile cases, (i)_k (the profile of coordinate k constant, the other two of spread 1) or (ii)_k (every k-line alternates), and axis k is one-way. The proof uses Theorem U on the family, Theorem M′, the profile description of twochain, the current description of the signed moments (Theorems 17 and 19 of the note) and Lemma 4. The case (i) and the equal case (ii) are settled by the signed line fractions, the mixed case (ii) by a parity argument on the E-data of the types. Theorem M′ is solver-UNSAT at extent 2 on (2,2,2), (2,2,6), (2,4,6), 18 of 18 with controls 2026-09-28-opus-dfour.md | thm-23 |
| 2026-09-28 | Package E (agent cubethree): at three chains the profile description is proved on every member (3a', 3b', 3c'), with Eisenstein line data, integer profiles with budget 3, and a colour rule modulo (1 - omega) in place of the parity rule. The proof of Theorem V goes over in Steps 2 to 4, with a three-plane sum Sigma(0) in place of the difference E(0,-) - E(4,-). It stops at two statements. Lemma R: the rods and two bridgeheads complete the rod line. Lemma S: once the rod line is complete, Sigma(0) = 0. Both hold by solver on six members and per cell by integrality of a few lines plus linear programming, but neither is proved. (V') is therefore reduced to R and S, not proved 2026-09-28-opus-cubethree.md | |
| 2026-09-28 | Independent check of the signed moments (Lemmas 11 and 12, Theorems 16 to 19 of the note) and of package dfour (Theorem H on the family (2a', 2b', 2c') without K_u) 2026-09-28-opus-check9.md | thm-23 |
| 2026-09-28 | Independent check of package D (agent twochain): Theorem P, identity (★), Theorem V, the symmetry reduction, and Theorems U and H on the family (2a′, 2b′, 2c′) 2026-09-28-opus-check8.md | thm-22 |
| 2026-09-28 | Independent check of Theorem S, Theorem S2, Corollary H, Lemma J, Lemma T′, the repair of Theorem R1 and the law of line fractions (attack proposal of 28 September) 2026-09-28-opus-check7.md | thm-14 |
| 2026-09-28 | Attack proposal of 28 September: Theorem H is proved by hand on every cell in which two of the three extents are coprime, by a lemma on the types of one sublattice; what is left is the cells in which every two extents share a factor, where odd states with all three types exist and the signs are needed; a proposal for that case 2026-09-28-attack-proposal.md | thm-14 |
| 2026-09-27 | Synthesis of the first Opus round: Theorem H is reduced to three sign laws, H ⇐ K_u + Lemma O + (R⁺) + (S≥1), of which only Lemma O (Theorem U, hand residue (V)) needs oddness; audited by the main loop 2026-09-27-synthesis.md | |
| 2026-09-27 | (V) through the signed cage defect (agent vsign): the defect of the signed cage law is the hole's charge c = #away − #toward, the same for all three axes (proved); the signed line counts obey a Poisson equation whose source is a divergence of cross counts on the other sublattice (proved), so the charges are neutral on every plane (the flux identity) and in every rectangle with empty corner lines; but no hole's orientation is forced by an unoriented hole, the sign field shortens no window, the signed LP stays positive, and (V) is not proved. What is new and quantitative: the gap-1 line-sign law on x-lines (or (R⁺) at x-separation 5) cuts the x-reach of Theorem U from 2L − 2 to 2⌈L/3⌉ + 1 (solver, L = 4 … 10), while the transverse torus stays fully needed; integrality of the hole charges alone closes the signed LP at L = 3 2026-09-27-opus-vsign.md | |
| 2026-09-27 | The turn beside a free plane (agent uwall): the cage at a turn and the corner-pattern lemma proved by hand; the corner case is exactly the exit of a length-2 excursion (proved) and is refuted with the plane hypothesis on ONE line of the free plane, needs integrality on ONE line of 2L indicators (branching on 1 variable at L = 4, 2 at L = 5), and gains nothing from the sign field; 'a path that turns at every hole of its k-line forces curl-free' proved by Theorem LC; the corner case and (V) both have depth-1 probing refutations in the cage-law calculus (L = 3, 4, 5), but not uniform ones. Not proved: the corner case, (E′), (V) 2026-09-27-opus-uwall.md | |
| 2026-09-27 | (RP) at distance 2 proved modulo three sign laws — Lemma O (every hole of an odd state oriented, ⟸ Theorem U), (R⁺) and the gap ≥ 1 line-sign law, the last two laws of every frozen state — through a new global identity: at an oriented hole the moment field obeys the cage law Pᵏ, so signed line counts are separable, and with the line-sign law the + and − counts of a two-signed sublattice depend on one transverse coordinate (stripes). Walls between stripes are k-free planes at which (E′) holds automatically, so Theorem D applies without (E′) as a hypothesis. The same argument gives every two-way axis a uniform chain, hence H ⇐ K_u + Lemma O + (R⁺) + (S≥1). The type-only part of (RP) (pattern ⇒ k-free plane) holds without oddness (solver) and is LP-positive; 'three types' enters only in Theorem D(iii) 2026-09-27-opus-rphand.md | thm-1-first-form |
| 2026-09-27 | (S≥1) is a corollary of (R⁺) and Lemma O (proved), so H ⇐ K_u + Lemma O + (R⁺). Neither law is tube-local: without Lemma O both are global in all three directions (the class of (V)); with Lemma O both have a light-cone reach, R*(s) = (s−1)/2, so no tube automaton exists. With Lemma O, (R⁺) is a counting statement on the causal diamond between the two rods (exact LP certificates for every separation up to 19, valid for every L), and a new two-cage law (conservation of τ·in and τ·out across every hexagon, plus a trail rule; proved by exhaustive check) gives (R⁺) by hand for |s| ≤ 5 2026-09-27-opus-lines.md | |
| 2026-09-27 | (V) at g = 2 (agent gtwo): the linear programme is positive on every slab and on every cell whose rod axis has even length, and exactly 0 on every cell with odd rod-axis length tested, so at g = 2 the integrality is a parity-type step along the rod axis that a closed odd x-circle supplies linearly; the thinnest integral family found that is uniform in the geometry of the pattern is one x-chain, the chain next to the rod's chain in the slab of the rod and the lower bridgeheads, and the same family closes at g = 3 and 4; the case split lives on a contiguous run of that chain beside the cage, lengthens with the torus, and is a caterpillar only on the thinnest tori; the degree-two Sherali-Adams lift refutes (V) on the slab over (2,2) and (2,4), gave no verdict on (4,6) and is feasible for every thin lift tried at g = 3. Nothing about (V) is proved 2026-09-27-opus-gtwo.md | thm-11 |
| 2026-09-27 | (V) is proved by hand on every cell whose transverse extents are coprime, for every length of the rod axis: in chain coordinates two pairs of pattern vertices give X(-1) - X(1) = 1 + (X(-2) - X(2))/2, one vertex P of the rod line found by the Chinese remainder theorem gives X(-2) - X(2) = t_y(P) + t_z(P) >= 0, and the grid sums close it. The proof is linear, uses the whole x-circle, and is one explicit identity. The slab certificates, whose reach grows with the torus, are a different and still unexplained family 2026-09-27-opus-coprime.md | thm-10 |
| 2026-09-27 | (R⁺_O) is proved for every separation: the Farkas certificates are one closed-form family (Lemma O +1 on the diamond, (T2) −½ on the hexagons that touch it, one-hot ½·min(2, μ_b), bond rule ¼·min(4, n_E)), and the identity they prove is a corner law, exactly one of the six corner holes of the diamond takes its in-rod from outside. The same certificate proves (S) at every gap. Its validity is a check of nine local star types plus one lemma, 'no vertex sees exactly two bridge holes of the diamond', and that lemma is where the transverse offset ≤ 1 enters. Orientation is needed at every hole of the diamond, straight holes included, so the cone proof does not make (R⁺) independent of Theorem U 2026-09-27-opus-cone.md | thm-9-diagonal-law |
| 2026-09-27 | Independent check of the cone certificate: (R⁺_O) and (S) confirmed with corrections for every separation, no gap; the closed form, Lemmas 1, A, 3, B and 5, and the evaluation at the all-z state all hold, and the periodisation onto any torus is sound 2026-09-27-opus-check2.md | thm-9-diagonal-law |
| 2026-09-27 | Independent check of H ⇐ K_u + Lemma O + (R⁺) + (S≥1): confirmed with corrections (no gap; one mis-cited input, one unexercised case, two standing remarks) 2026-09-27-opus-check.md | thm-1-first-form |
| 2026-09-27 | DRAT certificates for Theorem U′, (R⁺) and (S≥1) on the (L,L,L) torus at L = 4, 5 and 6, with controls, an independent encoding and a corpus check; Theorem H at L = 4, 5, 6 modulo the 2026-09-27 reduction and K_u 2026-09-27-opus-cert.md | u-prime-certificates-2026-09-27 |
| 2026-09-27 | (V) on non-cubic cells: the reach along the rod axis is set by the gcd of the two transverse extents and not by the length of the rod axis, so (V) does not need the x-circle; on cells with coprime transverse extents (V) is a counting statement. A critique of the six ideas for Theorem U in the light of this, and a proposal for the next round 2026-09-27-aniso.md | v-reach-law |
| 2026-09-22 | Lemma (V) by hand: the cage's four hexagons give exactly law C and nothing else, and the neighbouring B-cages give nothing at all. Two exact delimitations of the residual: (V) has no proof by local propagation (one-hexagon closure plus depth-1 case analysis derives only the six cage exclusions, L = 3, 4, 5), and none by counting either — (V) is false in the LP relaxation of the type model, with an exactly verified rational witness, so no sum of P and C laws round the x-circle can prove it however it is weighted. What closes the gap is one sixth of the integrality: 0/1 on a single (axis, sublattice) family of indicators makes the LP optimum 0. Also: the forcing needs the transverse torus as well as the x-circle 2026-09-22-uhand.md | |
| 2026-09-22 | The shape of the proof of (V): its deletion-minimal cores are thin but global — they touch every x-grid and every transverse residue, no tube round the x-circle suffices, there is no bounded-x induction step (the next cage's bridgehead pattern is not determined at all), and the statement survives both the affine and the LP relaxation, so no linear certificate exists. What is new and cheap: (V) needs only four literals — both rods type x, one lower bridgehead type y, one upper bridgehead type z — and Theorem U holds under that weakened hypothesis too 2026-09-22-ucore.md | |
| 2026-09-22 | Oriented loops: in every odd frozen state the moment signs orient the rod loops (one rod in, one out at every hole), and this fails in curl-free states; the unoriented hole is a type-only pattern (a straight cage with bridgeheads yy/zz), and Theorem U — one such cage forces the opposite sublattice x-free, hence the state curl-free — is solver-true at L = 3, 4, 5 in seconds. Also: the loops report's wall picture withdrawn; corner counts; the sign field is always a function of the type field 2026-09-22-orient.md | |
| 2026-09-17 | P1 proved for every cell: a curl-free frozen state has a slab-type type field (linear algebra in the chain-weight picture, the kernel step proved by the stratified Fourier method), and the bond state is then a slab sum because the profiles must close on the torus 2026-09-17-p1-proof.md | |
| 2026-09-17 | Completeness of the chain modes, proved for every cell: a stratified symbolic kernel computation on the unit torus 2026-09-17-completeness.md | parametrisation-c |
| 2026-09-04 | Subagent report: stage 4f — a helicity (pseudoscalar) mode in cubic geometry (2026-09-04) 2026-09-04-stage4f.md | |
| 2026-09-04 | Subagent report: stage 4d — the thin-cell even mode (2026-09-04) 2026-09-04-stage4d.md | |
| 2026-09-04 | Subagent report: stage 4, items 4a / 4b / 4c (2026-09-04) 2026-09-04-stage4abc.md | |
| 2026-09-04 | Subagent report: Isabelle/HOL certification of the stage-4a real-root verdicts (2026-09-04) 2026-09-04-isabelle-sturm.md | |
| 2026-09-04 | Machine-checkable certificates for the finite results, and the frozen census at the first cubic cell 2026-09-04-certificates-sat.md | |
| 2026-08-11 | Subagent report: stage-1 falsifiers (2026-08-11) 2026-08-11-stage1-falsifiers.md | |