A registered result in its source context
Original ledger classification: Invertible beta=1-KMS (LTO-RP) [E]/[C]: the corner NPW26 leave open (they do topologically-ordered; the seam is invertible + KMS, not a trace) is constructively satisfiable -- a gapped reflection-symmetric mu4-even collar has Theta K Theta=-K (u_Theta=J) and [rho,K]=0, so (LTO-RP) holds by direct Tomita-Takesaki; |det E8|=1 (no anyons), state not a trace; neg controls have teeth. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (continuum sub-step (i) remains), finite-dim witness not continuum theorem
Recorded statement: The invertible beta=1-KMS variant of (LTO-RP) (2026-06-25): the one corner Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693] leave open -- they prove (LTO-RP) only for TOPOLOGICALLY ORDERED commuting-projector models (Toric Code via a…
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.EQUIV.LTORP.KMS.01
Original ledger classification: Invertible beta=1-KMS (LTO-RP) [E]/[C]: the corner NPW26 leave open (they do topologically-ordered; the seam is invertible + KMS, not a trace) is constructively satisfiable -- a gapped reflection-symmetric mu4-even collar has Theta K Theta=-K (u_Theta=J) and [rho,K]=0, so (LTO-RP) holds by direct Tomita-Takesaki; |det E8|=1 (no anyons), state not a trace; neg controls have teeth. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (continuum sub-step (i) remains), finite-dim witness not continuum theorem
Original claim text
The invertible beta=1-KMS variant of (LTO-RP) (2026-06-25): the one corner Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693] leave open -- they prove (LTO-RP) only for TOPOLOGICALLY ORDERED commuting-projector models (Toric Code via a TRACE state; Levin-Wen anyonic), while the TFPT seam is the opposite corner (invertible, holomorphic (E8)_1, no anyons, AND a beta=1 KMS state, not a trace). A constructive finite-dim Tomita-Takesaki witness that on this corner the (LTO-RP)/Bisognano-Wichmann condition u_Theta=J is satisfiable and natural, advancing sub-step (ii) of v424; does NOT close SEAM.EQUIV.01. One-particle dictionary: modular operator Delta=e^{-K}, (LTO-RP)/BW <=> u_Theta=J <=> Theta K Theta^{-1}=-K (J Delta J=Delta^{-1}); the mu4 clock rho a modular SYMMETRY <=> [rho,K]=0. [E] INVERTIBLE-KMS COLLAR: K reflection-odd/mu4-even/gapped (gap 2k=6 ln(3/2)); C=1/(1+e^K) the beta=1 KMS covariance satisfies C/(1-C)=e^{-K} (a genuine KMS, not trace, state); rho order 4. [E] (LTO-RP)/BW HOLDS: Theta K Theta^{-1}=-K (reflection = modular conjugation, u_Theta=J) AND [rho,K]=0 -- both hold, so (LTO-RP) is satisfied by direct Tomita-Takesaki (no skein/UFC machinery). [E] INVERTIBLE, OUTSIDE NPW26: |det Cartan(E8)|=1 (one primary, invertible, no anyons) vs the anyonic |det|=4 of NPW26's worked models, and the state is NOT a trace (Delta=e^{-K}!=I, nontrivial modular flow). [E] NEG CONTROLS: a side-even reflection gives Theta K Theta=+K!=-K (BW fails); an off-mu4-block K mixing two marks gives [rho,K]!=0 (mu4 symmetry fails). [C] VERDICT: on the gapped invertible beta=1-KMS reflection-symmetric collar, (LTO-RP) reduces to (Theta K Theta=-K) and ([rho,K]=0), both structural -- the invertible-KMS variant NPW26 leave open is constructively satisfiable. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (the continuum/abstract realisation, sub-step (i), remains); a finite-dim witness, not the continuum theorem.
Original classification detail
Numerical
Dependencies as recorded
SEAM.EQUIV.LTORP.01; SEAM.EQUIV.01; SEAM.EQUIV.CHAIN.01; QGEO.MODHAM.01; v424; v309; v302; v240; v214
Open the original ledger source, row 493Location 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.
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-25 (Three closure-strategy modules v425– v427 [formula in source] the Safeguards paper)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.
- 2026-06-30 ( v463/ v464 + Lean + Wolfram — the [formula in source] holomorphic-uniqueness pin and the one-particle realisation rigidity: the SEAM.EQUIV.01 residual reduced to entirely certification on both halves)Research changelog; TFPT — Introduction and reading guide
- How to read this documenttfpt 1 architecture e8
- What would still falsify TFPTVerification safeguards
- 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)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportArchitecture, E8 and the electromagnetic fixed point
- 2026-06-30 ( v454– v459 + Lean + Wolfram — the post-F ``G-block'': the lattice Sugawara current algebra, a uniform-in- [formula in source] Tomita–Takesaki tower, chirality forced by [formula in source] , an [formula in source] character kill test, the exact MMST citation audit and the complementary lattice-VOA route)Research changelog; TFPT — Introduction and reading guide
- How to read this documenttfpt 1 architecture e8
All 12 linked sections
- Front matter / support
- 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)
- Research Contract 1 — [formula in source]
Registered verification modules
- v426_seam_lto_rp_kms.pyRegistered verification module · frontier
Read the description recorded in the module registry
SEAM.EQUIV.LTORP.KMS.01 -- the invertible beta=1-KMS variant of (LTO-RP), the corner NPW26 leave open (they prove it for topologically-ordered models; the TFPT seam is invertible + beta=1 KMS, not a trace). A constructive Tomita-Takesaki witness advancing v424 sub-step (ii); does NOT close SEAM.EQUIV.01. One-particle dictionary: modular operator Delta=e^{-K}, (LTO-RP)/BW <=> u_Theta=J <=> Theta K Theta=-K, the mu4 clock a modular symmetry [rho,K]=0. Build K reflection-odd/mu4-even/gapped (gap 2k=6 ln(3/2)); C=1/(1+e^K) the beta=1 KMS covariance satisfies C/(1-C)=e^{-K} (genuine KMS, not trace); Theta K Theta=-K and [rho,K]=0 both hold, so (LTO-RP) is satisfied by direct Tomita-Takesaki; |det Cartan(E8)|=1 (invertible, no anyons) vs anyonic |det|=4, and the state is not a trace (Delta!=I) -- neither NPW26 bucket. Neg controls: a side-even reflection gives +K (BW fails), an off-mu4-block K breaks [rho,K]. SEAM.EQUIV.01 stays open (the continuum sub-step (i) remains).
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.