A registered result in its source context
Original ledger classification: beta-interpolation [E]/[C]: the (LTO-RP)/BW data (Theta K Theta=-K, [rho,K]=0) are beta-independent; along the whole path C_beta=1/(1+e^{beta K}) from the NPW26 trace (beta=0) to the seam KMS (beta=1) every covariance is reflection-related, mu4-invariant and valid, norm-continuous, no obstruction -- a PATH argument for v424 (ii), stronger than v426. SEAM.EQUIV.01 stays [O]
Recorded statement: The beta-interpolation connecting the NPW26-PROVEN trace case (beta=0) to the seam's beta=1 KMS case along an obstruction-free homotopy (2026-06-30) -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument.…
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Paper context: Research contracts
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SEAM.EQUIV.LTORP.BETA.01
Original ledger classification: beta-interpolation [E]/[C]: the (LTO-RP)/BW data (Theta K Theta=-K, [rho,K]=0) are beta-independent; along the whole path C_beta=1/(1+e^{beta K}) from the NPW26 trace (beta=0) to the seam KMS (beta=1) every covariance is reflection-related, mu4-invariant and valid, norm-continuous, no obstruction -- a PATH argument for v424 (ii), stronger than v426. SEAM.EQUIV.01 stays [O]
Original claim text
The beta-interpolation connecting the NPW26-PROVEN trace case (beta=0) to the seam's beta=1 KMS case along an obstruction-free homotopy (2026-06-30) -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument. NPW26 prove (LTO-RP) for the TRACE state (beta=0, C=1/2); the TFPT seam is the beta=1 KMS state; v426 showed (LTO-RP) holds AT beta=1; this module shows it holds ALONG the whole interpolation C_beta=1/(1+e^{beta K}), beta in [0,inf). [E] beta-INDEPENDENT (LTO-RP) DATA: Theta K Theta=-K and [rho,K]=0 are conditions on K alone, independent of beta. [E] THE KMS PATH: beta=0 -> C=1/2 I (the NPW26 TRACE state), beta=1 -> the seam KMS state, beta->inf -> the ground-state projection; at every beta Theta C_beta Theta=1-C (reflection), [rho,C_beta]=0 (mu4), 0<=C_beta<=1 (valid covariance/RP). [E] NO OBSTRUCTION: beta->C_beta is norm-continuous and stays in the (LTO-RP)-valid set -- a continuous homotopy from the proven trace endpoint to the seam KMS point, no topological wall. [E] NEG CONTROLS: a reflection-EVEN K' breaks Theta C_beta Theta=1-C for beta>0 (the trace endpoint is degenerate), an off-mu4-block K breaks [rho,C_beta]. [C]/[O] VERDICT: (LTO-RP)/BW connects the NPW26 trace case to the seam beta=1 KMS case along an obstruction-free homotopy -- a PATH argument for v424 sub-step (ii), stronger than the single point of v426; SEAM.EQUIV.01 stays [O], the residual being closedness of (LTO-RP) along the path in the continuum (uniform in the limit).
Original classification detail
Numerical
Dependencies as recorded
SEAM.EQUIV.LTORP.01; SEAM.EQUIV.LTORP.KMS.01; SEAM.EQUIV.01; v424; v426
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- Research contracts
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 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 changelog; TFPT — Introduction and reading guide
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- How to read this documenttfpt 1 architecture e8
- Firewall Contract — GRAV.NONCIRCULAR.01 (the Bisognano–Wichmann direction discipline)tfpt research contracts
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportArchitecture, E8 and the electromagnetic fixed point
- How to read this documenttfpt 1 architecture e8
- Front matter / supportResearch contracts
- Firewall Contract — GRAV.NONCIRCULAR.01 (the Bisognano–Wichmann direction discipline)Research contracts
- Firewall Contract — GRAV.NONCIRCULAR.01 (the Bisognano–Wichmann direction discipline)tfpt research contracts
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Registered verification modules
- v440_seam_lto_rp_beta.pyRegistered verification module · frontier
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SEAM.EQUIV.LTORP.BETA.01: the β-interpolation connecting the NPW26-PROVEN trace case (β=0) to the seam's β=1 KMS case along an obstruction-free homotopy -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument. [E] β-INDEPENDENT (LTO-RP) DATA: ΘKΘ=-K and [ρ,K]=0 are conditions on K alone. [E] THE KMS PATH: C_β=1/(1+e^{βK}) has β=0→C=½I (NPW26 TRACE), β=1→seam KMS, β→∞→ground state; at every β ΘC_βΘ=1-C, [ρ,C_β]=0, 0≤C_β≤1. [E] NO OBSTRUCTION: β→C_β norm-continuous, stays in the (LTO-RP)-valid set -- a continuous homotopy, no wall. [E] NEG CONTROLS: reflection-EVEN K' breaks ΘC_βΘ=1-C for β>0; off-mu4 K breaks [ρ,C_β]. [C]/[O] a PATH argument for v424 (ii), stronger than v426; SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy).
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