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Original ledger classification: S3 commuting-projector realisation [E]/[C]: v367's gapped collar has an exact flat-band parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local (xi~1.14 at M=1, power-law |R|^{-2.1} at the gapless point) and whose flattened modular Hamiltonian inherits Theta K_flat Theta=-K_flat and [rho,K_flat]=0 exactly; the chiral c_-=8 obstructs STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Recorded statement: The EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input (2026-06-30) -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an…
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SEAM.S3.LTO.01
Original ledger classification: S3 commuting-projector realisation [E]/[C]: v367's gapped collar has an exact flat-band parent (Q^2=I, P^2=P=P_gs) that is gap-protected quasi-local (xi~1.14 at M=1, power-law |R|^{-2.1} at the gapless point) and whose flattened modular Hamiltonian inherits Theta K_flat Theta=-K_flat and [rho,K_flat]=0 exactly; the chiral c_-=8 obstructs STRICT commuting-projector locality (Kapustin-Fidkowski), so the realisation is the quasi-local LTO net of NPW26 -- advancing v424 sub-step (i); SEAM.EQUIV.01 stays [O]
Original claim text
The EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input (2026-06-30) -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an explicit gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock. NPW26 (arXiv:2605.10693) prove the intrinsic Bisognano-Wichmann lemma for COMMUTING-PROJECTOR (LTO) models with a Z/2 reflection; this attacks the commuting-projector side of sub-step (i) on the same v367 collar. [E] FLAT-BAND PARENT: the flattened Bloch Q=h/|h| has Q^2=I (flat +-1 bands) and P=(I-Q)/2 is an exact projector EQUAL to v367's occupied ground-state projector -- a frustration-free commuting-projector parent with the same gapped ground state (machine precision over a k-grid). [E] GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (xi~1.14 at M=1, exp R^2=0.995, |P(20,0)|<1e-6) and xi GROWS as the gap closes (2.7 at gap 0.02); at the GAPLESS point the kernel decays as a clean power law |R|^{-2.1} (R^2=0.999, tail ~2e-4) -- the projector net is quasi-local BECAUSE the bulk is gapped. [E] Z/2 REFLECTION + mu4 INHERITED: the flattened modular Hamiltonian K_flat=sign(K) inherits Theta K_flat Theta=-K_flat (BW, u_Theta=J) and [rho,K_flat]=0 (mu4) EXACTLY (sign is odd; rho commutes with any f(K)); side-even and off-mu4 neg controls fire. [C]/[O] CHIRAL OBSTRUCTION: the chiral c_-=8!=0 (FHS Chern |C|=1, trivial control 0) admits NO strictly finite-range commuting-projector Hamiltonian (Kapustin-Fidkowski arXiv:1810.07756; C!=0 forbids exp-localised Wannier, Thouless), so the realisation is the QUASI-LOCAL LTO net NPW26 use. Advances v424 sub-step (i); does NOT supply the continuum theorem; SEAM.EQUIV.01 stays [O].
Original classification detail
Numerical
Dependencies as recorded
SEAM.EQUIV.01; SEAM.S3.LATTICE.01; SEAM.EQUIV.BW.UNIFORM.01; v424; v426; v367; v455; v336
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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
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
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- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- 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)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v460_seam_s3_lto.pyRegistered verification module · frontier
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SEAM.S3.LTO.01: the EXPLICIT flat-band commuting-projector (local-topological-order) realisation of the S3 input -- advancing v424 sub-step (i) ('realise the raw seam as a Z/2-reflection commuting-projector LTO') from abstract existence to an explicit gap-protected quasi-local projector net carrying the Z/2 reflection and the mu4 clock. [E] FLAT-BAND PARENT: the flattened Bloch Q=h/|h| of v367's p+ip collar has Q^2=I (flat +-1 bands) and P=(I-Q)/2 is an exact projector EQUAL to the occupied ground-state projector -- a frustration-free commuting-projector parent with the same gapped ground state. [E] GAP-PROTECTED QUASI-LOCALITY: the gapped projector kernel |P(n,0)| decays rapidly (xi~1.14 at M=1, exp R^2=0.995, |P(20,0)|<1e-6), xi grows as the gap closes (2.7 at gap 0.02); the GAPLESS point decays as a clean power law |R|^{-2.1} (R^2=0.999, tail ~2e-4) -- locality needs the gap. [E] Z/2 REFLECTION + mu4 INHERITED: the flattened modular Hamiltonian K_flat=sign(K) inherits Theta K_flat Theta=-K_flat (BW/u_Theta=J) and [rho,K_flat]=0 (mu4) EXACTLY (sign is odd; rho commutes with any f(K)). [E] NEG CONTROLS: a side-even reflection gives 2||K_flat||!=0 (BW fails), an off-mu4 K mixing opposite-sign marks breaks [rho,sign(K)]!=0. [C]/[O] the chiral c_-=8!=0 (FHS Chern |C|=1) obstructs STRICT finite-range commuting-projector locality (Kapustin-Fidkowski; C!=0 forbids exp-localised Wannier), so the realisation is the quasi-local LTO net of NPW26; advances v424 sub-step (i), SEAM.EQUIV.01 stays [O]. Python-only (numpy).
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