A registered result in its source context
Original ledger classification: Uniform-in-N BW tower [E]/[C]: ||Theta K Theta+K||=0 at every N (m=2..32), N-independent gap 6 ln(3/2), [rho,K]=0 exact, N-independent neg-control margin 2||K|| -- the inductive limit inherits u_Theta=J, advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Recorded statement: A UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (2026-06-30, G1 of the post-F next steps) -- advancing v424 sub-step (i) (the continuum lift), the inductive-limit companion to the single finite…
Statement, assumptions and sources
Paper context: Research contracts
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SEAM.EQUIV.BW.UNIFORM.01
Original ledger classification: Uniform-in-N BW tower [E]/[C]: ||Theta K Theta+K||=0 at every N (m=2..32), N-independent gap 6 ln(3/2), [rho,K]=0 exact, N-independent neg-control margin 2||K|| -- the inductive limit inherits u_Theta=J, advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Original claim text
A UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (2026-06-30, G1 of the post-F next steps) -- advancing v424 sub-step (i) (the continuum lift), the inductive-limit companion to the single finite witness v426. One-particle: Delta=e^{-K}, conjugation J, reflection Theta; BW <=> u_Theta=J <=> Theta K Theta=-K. [E] UNIFORM BW: the reflection-odd, mu4-even, gapped collar of size 4m (m=2..32, N=8..128) has ||Theta K Theta + K||=0 to machine precision at EVERY N -- u_Theta=J uniform. [E] UNIFORM GAP: the gap of K is N-independent (=6 ln(3/2)=2 N_fam ln(3/2)) so Delta=e^{-K} stays uniformly bounded away from 1 (a genuine KMS, not trace, state). [E] UNIFORM CLOCK SYMMETRY: [rho,K]=0 exactly at every N. [E] N-INDEPENDENT NEG CONTROL: a side-even reflection gives ||Theta K Theta + K||=2||K|| (a fixed per-mode margin) and an off-mu4 K breaks [rho,K] at every N -- the success is non-vacuous. [C]/[O] the inductive (continuum) limit inherits u_Theta=J; SEAM.EQUIV.01 stays [O] (the rigorous continuum theorem NPW26/MMST v336 is the backbone).
Original classification detail
Numerical
Dependencies as recorded
SEAM.EQUIV.01; SEAM.RIGIDITY.UNIFORM.01; SEAM.RIGIDITY.FORCING.01; v424; v426; v442; v336
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- Research contracts
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- 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
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- 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 / supportResearch contracts
- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v455_seam_bw_uniform.pyRegistered verification module · frontier
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SEAM.EQUIV.BW.UNIFORM.01: a UNIFORM-IN-N Tomita-Takesaki tower for the intrinsic Bisognano-Wichmann condition u_Theta=J (G1) -- the inductive-limit companion to v426, advancing v424 sub-step (i). [E] UNIFORM BW: ||Theta K Theta + K||=0 exactly at every size (N=8..128) -- u_Theta=J holds uniformly. [E] UNIFORM GAP: the gap of K is N-independent (6 ln(3/2)) -- Delta=e^{-K} uniformly bounded from 1, a genuine KMS (not trace) limit. [E] UNIFORM CLOCK: [rho,K]=0 exactly at every N. [E] N-INDEPENDENT NEG CONTROL: a side-even reflection gives ||Theta K Theta+K||=2||K|| growing with N, an off-mu4 K breaks [rho,K] at every N. [C]/[O] the inductive limit inherits u_Theta=J; SEAM.EQUIV.01 stays [O]. Python-only (numpy).
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