A registered result in its source context
Original ledger classification: Half-sided modular inclusion [E]/[C]: a Longo standard pair (P>=0, [K,P]=P, J:JKJ=-K, JPJ=P_-) is a +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY (no covariance presupposed); face (ii) becomes a structure consequence. Finite witnesses: ladder algebra exact, sl(2,R) positivity, BW reflection; neg control indefinite boost. SEAM.EQUIV.01 stays [O] (continuum standard pair = continuum existence, same wall as MMST/NPW26)
Recorded statement: The half-sided modular inclusion (Borchers/Wiesbrock) as an INTRINSIC Bisognano-Wichmann route (2026-06-30) for face (ii) of SEAM.EQUIV.01 -- geometric modular flow WITHOUT presupposing covariance. Complementary to v323/v426 (which assume…
Statement, assumptions and sources
Paper context: Research contracts
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SEAM.EQUIV.BW.HSMI.01
Original ledger classification: Half-sided modular inclusion [E]/[C]: a Longo standard pair (P>=0, [K,P]=P, J:JKJ=-K, JPJ=P_-) is a +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY (no covariance presupposed); face (ii) becomes a structure consequence. Finite witnesses: ladder algebra exact, sl(2,R) positivity, BW reflection; neg control indefinite boost. SEAM.EQUIV.01 stays [O] (continuum standard pair = continuum existence, same wall as MMST/NPW26)
Original claim text
The half-sided modular inclusion (Borchers/Wiesbrock) as an INTRINSIC Bisognano-Wichmann route (2026-06-30) for face (ii) of SEAM.EQUIV.01 -- geometric modular flow WITHOUT presupposing covariance. Complementary to v323/v426 (which assume a rotation-covariant covariance) and to the Adamo-Giorgetti-Tanimoto covariance theorem. The standard-pair criterion (Borchers/Wiesbrock/Longo): a pair (K,P) with K self-adjoint (Delta=e^{-K}), P>=0 (positive-energy lightlike translation), [K,P]=P, and a reflection J with JKJ=-K and JPJ=-P is a +half-sided modular inclusion, by Borchers' theorem the modular flow is then GEOMETRIC. [E] STANDARD-PAIR ALGEBRA (ladder, exact): K=diag(n), P=raising shift give [K,P]=P and the Borchers scaling Delta^{it}P Delta^{-it}=e^{-t}P (the ax+b/Moebius relation). [E] BW REFLECTION: J(n->-n) gives J^2=I, JKJ=-K (u_Theta=J) and JPJ=P_- (opposite translation), the standard-pair reflection J U(s)J=U(-s). [E] POSITIVE ENERGY: in the unitary lowest-weight sl(2,R) rep the line translation P_line=L0-(L+ +L-)/2 is POSITIVE (min eig>0), the positive-energy generator the c3 one-sided Gauss-Bonnet origin demands; the boost direction is the indefinite negative control. [C] SYNTHESIS: P>=0 + [K,P]=P + J is a Longo STANDARD PAIR / +half-sided modular inclusion, so by Borchers the seam modular flow is geometric INTRINSICALLY, without presupposing the rotation-covariance of the vacuum; face (ii) becomes a STRUCTURE consequence, not a state condition. [C]/[O] VERDICT: an intrinsic-BW REDUCTION via Borchers/Wiesbrock; SEAM.EQUIV.01 stays [O] -- assembling the finite algebra and the positive-energy rep into ONE continuum standard pair is the infinite-dim continuum existence (the same wall as MMST/NPW26), a finite witness + cited theorem, not a closure.
Original classification detail
Numerical
Dependencies as recorded
SEAM.EQUIV.01; SEAM.EQUIV.LTORP.01; SEAM.EQUIV.LTORP.KMS.01; QGEO.MODHAM.01; v323; v426; v309
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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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- Firewall Contract — GRAV.NONCIRCULAR.01 (the Bisognano–Wichmann direction discipline)tfpt research contracts
- Research Contract 1 — [formula in source]tfpt research contracts
- 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
- Research Contract 1 — [formula in source]tfpt research contracts
- 2026-06-30 (Lean kernel-hardening of two of the seven SEAM.EQUIV.01 routes — the Borchers/Wiesbrock standard-pair relations and the applicability-ledger count, machine-verified with clean axioms; rigor without new mathematics)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v438_seam_hsmi_borchers.pyRegistered verification module · frontier
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SEAM.EQUIV.BW.HSMI.01: the half-sided modular inclusion (Borchers/Wiesbrock) as an INTRINSIC Bisognano-Wichmann route for face (ii) of SEAM.EQUIV.01 -- geometric modular flow WITHOUT presupposing covariance. [E] STANDARD-PAIR ALGEBRA (ladder, exact): K=diag(n), P=raising shift give [K,P]=P and the Borchers scaling Δ^{it}PΔ^{-it}=e^{-t}P. [E] BW REFLECTION: J(n→-n) gives JKJ=-K (u_Θ=J) and JPJ=P_- (opposite translation). [E] POSITIVE ENERGY: in the unitary lowest-weight sl(2,R) rep P_line=L0-(L+ +L-)/2 is POSITIVE (min eig>0); the boost direction is the indefinite neg control. [C] SYNTHESIS: P≥0 + [K,P]=P + J is a Longo STANDARD PAIR / +half-sided modular inclusion, so by Borchers the flow is geometric INTRINSICALLY (complementary to Adamo-Giorgetti-Tanimoto); face (ii) becomes a STRUCTURE consequence. [C]/[O] assembling the finite algebra + positive-energy rep into ONE continuum standard pair is the continuum existence; SEAM.EQUIV.01 stays [O]. Python-only (numpy/scipy).
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