A registered result in its source context
Original ledger classification: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]
Recorded statement: Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of…
Statement, assumptions and sources
Paper context: Research contracts
The exact identifier, original classification and full claim are available below.
The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.
Read the exact registered claim and its dependencies
SEAM.S3.LATTICE.01
Original ledger classification: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]
Original claim text
Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of v356/v366 ('the collar realised as a genuine lattice chiral free-fermion invertible phase') into a RUNNABLE model with a numerically computed topological invariant, not a re-assertion of det K / c_-. [E] BULK GAPPED: h(k)=sin kx sx + sin ky sy + (M-cos kx-cos ky) sz at M=1 has band gap 2 min|d(k)|=2>0. [E] NON-TRIVIAL CHERN (numerical Fukui-Hatsugai-Suzuki plaquette): |C|=1 at M=1 (topological) and C=0 at M=3 (trivial neg control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2; N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam (the edge gravitational anomaly). [E] EDGE K-MATRIX: the 16-Majorana edge is SO(16)_1 (det Cartan D8=4) condensed to (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the index-4 mu4 simple-current condensation SO(16)_1 -> (E8)_1 (v154/v351) is the seam glue on top. [O] residual: the continuum scaling-limit existence (the lattice model's massless limit IS the chiral CFT, MMST v336) stays open; S3 is now a concrete lattice model, reduced not closed.
Original classification detail
[E] the gapped Chern model + the c_-=8 counting + the det K discriminators (numerical Chern + lattice arithmetic); [C] the mu4 condensation to (E8)_1; [O] the continuum scaling limit. Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01.
Dependencies as recorded
SEAM.EQUIV.01; v356; v366; v351; v369; v301; v302; v154
Open the original ledger source, row 81Location recorded in the ledger
These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.
- Research contracts
Sections that explicitly cite this claim
A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.
- 2026-06-23 (Track 1 — the S3 input made CONCRETE: an explicit gapped chiral-Majorana ( [formula in source] ) lattice model with numerical Chern [formula in source] realises the [formula in source] edge ( v367), and a strip shows the chiral edge exists by anomaly inflow ( v368))Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)tfpt research contracts
- 2026-06-30 ( v449– v453 + Lean + Wolfram — the edge CFT named end-to-end: uniform-in- [formula in source] convergence, three independent reads of [formula in source] , the Ising operator tower, the [formula in source] modular data, and [formula in source] -symmetry derived from the four marks)Research changelog; TFPT — Introduction and reading guide
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)tfpt 1 architecture e8
- The closure ledger: what is done, and what genuinely remainsTFPT — Introduction and reading guide
- 2026-07-03 ( v472 — the determinant line over the [formula in source] -twist moduli of the collar model carries curvature [formula in source] the inflow level: the v470 bridge lemma exhibited at the finite/model level; ALPHA.QUILLEN.EXACT.01 stays . Plus: bird's-eye rounds — electron-sector/Cabibbo whitespot probes, Calder\'on UV-universality, parameter-free flat-budget closure (sandbox); the flat-budget closure enters the evidence scorecard)Research changelog; TFPT — Introduction and reading guide
- Honest statusinput 4 origin theory
- Front matter / supportResearch contracts
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
All 54 linked sections
- The unique attractor: a gapped transport selects ONE fixed point
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)
- 2026-06-23 (Status propagation: the v365– v380 [formula in source] FORM.SEAM.MMST.01 status carried into the remaining papers, the README and the Zenodo description)
- 2026-06-30 ( v460 — the explicit flat-band commuting-projector (LTO) realisation of the S3 input: advancing the SEAM.EQUIV.01 sub-step (i) from abstract existence to an explicit gap-protected quasi-local projector net carrying the [formula in source] reflection and the [formula in source] clock)
- 2026-07-20 (II) ( v490 — the [formula in source] spin-structure parity census: [formula in source] witnessed on the lattice, the 16-fold-way bridge [formula in source] on measured numbers, and the honest TEE non-discriminator fence; no gate closes, no marker moves)
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)
- Research Contract 1 — [formula in source]
- Honest status
- =150 2026-08-20 (CXI) (the externalization contracts — the two open non-RH kind-(A) external-math targets SEAM.EQUIV.01 and ALPHA.QUILLEN.EXACT.01 packaged as short, self-contained EXTERNALIZATION CONTRACTS for outside specialist mathematicians (conformal nets/VOA/AQFT for the seam; determinant lines/ [formula in source] -invariants/index theory for alpha), added as two new sections at the end of tfpt_research_contracts (before ``Certifiability and recommended order''), plus a one-sentence pointer in the head keybox; DOCUMENTATION ONLY — no module touched, no number changed, no status marker moved, no gate closed or narrowed, NO RH claim: (1) the SEAM.EQUIV.01 contract states the minimal theorem (the scaling limit of the gapped [formula in source] -equivariant CAR collar exists in local-net topology; the index-4 simple-current extension functor commutes with the limit; the OS reconstruction is the holomorphic [formula in source] lattice net) broken into the four review-named lemmata L1 (uniform energy/nuclearity bounds — honestly typed: NO uniform nuclearity estimate exists in the corpus), L2 (convergence of local algebras, the cited Osborne–Stottmeister scaling-limit theorem with the [formula in source] -equivariant structure retained), L3 (crossed product commutes with the limit; the finite index-4 Q-system is explicit, v125/ v154/ v469) and L4 (uniqueness of the holomorphic [formula in source] reconstruction; [formula in source] vs the same- [formula in source] rival [formula in source] ), with the complete typed hypothesis list (12 PROVEN-FINITE rows citing v302/ v113/ v155/ v160/ v356/ v366/ v367/ v368/ v461/ v376/ v377/ v378/ v379/ v125/ v154/ v216/ v286, 5 CITED-CLASSICAL rows, 1 OPEN target), the runnable finite models, the HONEST Lean axiom surface ( SeamScalingLimit.lean kernel-proves only the arithmetic hypotheses by decide; the analytic content is named cited axioms over opaque Props; no sorry; explicitly flagged: L1–L4 have ZERO Lean content beyond the composition audit), five must-fail controls (MF-S1: [formula in source] alone must NOT suffice — the red-team point; MF-S2: index-2 glue must yield [formula in source] ; MF-S3: the trivial [formula in source] collar must give no chiral limit; MF-S4: non-order-4 clock breaks the marks; MF-S5: the v286 import firewall) and four acceptance tests; (2) the ALPHA.QUILLEN.EXACT.01 contract states the two-step theorem T1 (gap-stable continuum limit [formula in source] under norm-resolvent convergence with the Fermi gap uniformly open over the twist torus — the Dai–Freed section has no zero; finite side computed, v472) and T2 ( [formula in source] [formula in source] Bismut–Freed curvature [formula in source] inflow response with the exact corpus coefficients [formula in source] , [formula in source] , [formula in source] , [formula in source] , exponent [formula in source] ), the two named open faces (the analytic diagonal- [formula in source] /multiplicity face v484/ v485 and the topological EM1 lemma v48/ v435), the typed hypothesis list (10 PROVEN-FINITE rows citing v3/ v48/ v341/ v382/ v434/ v470/ v435/ v342/ v367/ v472/ v484/ v485/ v391/ v433, 5 CITED-CLASSICAL rows, 1 OPEN target), the runnable finite models, the HONEST Lean statement (there is NO Lean formalisation of this target — stated plainly), four must-fail controls (MF-A1: no experimental [formula in source] anywhere in the construction; MF-A2: deck/carrier/Chern-level ablations must destroy the CODATA match; MF-A3: [formula in source] , [formula in source] ; MF-A4: the [formula in source] [formula in source] -power/metric-independence partition) and the review's three acceptance tests verbatim (no experimental [formula in source] input; unique physical fixed point [formula in source] ; ablations destroy the hit) plus one structural test; (3) the website paper mirror of tfpt_research_contracts ( papers.ts) gains one section entry covering both contracts; tfpt_research_contracts.pdf and changelog.pdf recompiled and mirrored via bash build.sh website; maps and changelog.ts regenerated via bash build.sh gen; bash build.sh audit [formula in source] AUDIT OK; manifests regenerated LAST. Both targets stay [O]; NO RH claim anywhere)
- What this means for the theory-of-everything question
- Outcome
- The hard reduction: the live residual as typed interfaces
- Honest status
- Front matter / support
- 2026-06-30 ( v461/ v462 + Lean axiom merge — the strict-locality obstruction made explicit, the 128-spinor extension exhibited at character level, and the SEAM.EQUIV.01 residual reduced to ONE realisation axiom [formula in source] ONE combined cited theorem)
- Front matter / support
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)
- The unique attractor: a gapped transport selects ONE fixed point
- Ausführbare Ergebnisse der Sitzung
- Front matter / support
- Research Contract — SEAM.SIMPLECURRENT.GENERATOR.01 (one generator instead of 128 currents)
- 2026-07-21 (XI) ( v501 — WP5d- [formula in source] of CELEST.SEAM.01 ``the two-interval index from the lattice'' promoted as CELEST.WP5D.01: the KLM Haag-duality discriminator measured entropically on the seam edge — the [formula in source] -layer carrier saturates the [formula in source] offset at machine precision, [formula in source] the v490 census, and the condensation arithmetic [formula in source] closes with [formula in source] ; the preregistered KILL (`` [formula in source] -offset [formula in source] after condensation'') does NOT fire; contract now ``WP1–WP5d executed; WP5e the single remaining milestone''; Wolfram extension [formula in source] (verified engine run); SEAM.EQUIV.01 stays , no marker moves)
- 2026-06-23 (S3 closure stack — the target of the seam scaling limit pinned at EVERY computable level: central charge [formula in source] from the lattice ( v376), the [formula in source] character ( v377), the genus-1 torus count [formula in source] ( v378), and reflection positivity ( v379); only the abstract continuum existence (cited MMST) stays external)
- Front matter / support
- Honest status
- Front matter / support
- 2026-06-30 (six convergent attacks on the one SEAM.EQUIV.01 continuum residual — intrinsic Bisognano–Wichmann, a Lieb–Robinson continuum motor, the [formula in source] -homotopy, an applicability audit, uniform rigidity, and falsifiable kill tests; none closes the keystone, together they reduce it to a single analytic fact and make the claim falsifiable)
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)
- Research Contract — SEAM.SIMPLECURRENT.GENERATOR.01 (one generator instead of 128 currents)
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)
- The [formula in source] compiler: [formula in source] as a closure, not an input
- 2026-06-30 (the chiral-edge scaling motor v444 — the edge half of the one SEAM.EQUIV.01 continuum residual: the lattice edge already carries the conformal data of a chiral [formula in source] Majorana CFT and converges under refinement, grounding the EDGE the way v439 grounded the BULK; the keystone stays )
- 2026-07-02 ( v469/ v470 — the closure-route round on the two named targets: the SEAM.EQUIV.01 extension leg re-founded on 1995–2001 peer-reviewed crossed-product theorems [formula in source] the realisation axiom reduced to invariant level; the [formula in source] level [formula in source] the computed bulk Chern invariant [formula in source] the seam [formula in source] -normalisation [formula in source] the embedding index [formula in source] ; both targets stay )
- The [formula in source] compiler: [formula in source] as a closure, not an input
- Research Contract 1 — [formula in source]
- The compact status formula: three honest layers
- Research Contract — SEAM.SIMPLECURRENT.GENERATOR.01 (one generator instead of 128 currents)
- Front matter / support
- The [formula in source] compiler: [formula in source] as a closure, not an input
- Ein wirklicher topologischer Zusammenhang: Warum der Faktor 16 erscheint
- Full quantum gravity — induced from the boundary kernel
- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)
- 2026-07-20 ( v489 — the chiral central charge [formula in source] measured DIRECTLY from the bulk ground state via the Kim–Shehab–Kim modular commutator: the first ground-state witness for premise (i) of SEAM.EQUIV.01, independent of the Chern route ( v367) and the KLM count ( v378); no gate closes, no marker moves)
- The unique attractor: a gapped transport selects ONE fixed point
- In plain words
- Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)
Registered verification modules
- v367_seam_s3_lattice.pyRegistered verification module · core
Read the description recorded in the module registry
Track 1: the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1; turns the abstract S3 residual (v356/v366) into a RUNNABLE model with a numerically computed topological invariant. [E] bulk gapped (band gap 2>0 at M=1). [E] non-trivial Chern number (Fukui-Hatsugai-Suzuki: |C|=1 at M=1, C=0 at M=3 trivial control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2, N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam. [E] edge K-matrix SO(16)_1 (det Cartan D8=4) -> (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the mu4 condensation SO(16)_1 -> (E8)_1 (v154/v351). [O] residual: the continuum scaling-limit existence (MMST v336). Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01.
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.