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Original ledger classification: Numerical (finite MTC + cited HR) [E]/[O]

Recorded statement: SCATTERING: HAAG-RUELLE SKELETON + BRAIDING S-MATRIX (2026-06-17): the S-matrix layer of the emergent QFT. The seam net has a mass gap (v64/v238), so Haag-Ruelle scattering theory constructs asymptotic in/out states; and for the abelian…

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DHR.SMATRIX.01

Original ledger classification: Numerical (finite MTC + cited HR) [E]/[O]

Original claim text

SCATTERING: HAAG-RUELLE SKELETON + BRAIDING S-MATRIX (2026-06-17): the S-matrix layer of the emergent QFT. The seam net has a mass gap (v64/v238), so Haag-Ruelle scattering theory constructs asymptotic in/out states; and for the abelian anyon MTC (v241/v242) the 2-particle S-matrix is the monodromy (double braiding), a pure phase. Same MTC: Z4 x Z4, q=(5x^2+3y^2)/8, theta=e^{2 pi i q}, M(a,b)=theta(a+b)/(theta(a)theta(b))=e^{2 pi i B(a,b)}. (1) [E] mass gap Delta=6 log(3/2)>0 (v64/v238) is the verified input; HR then constructs asymptotic states (standard gapped-net theorem). (2) [E] 2-particle S = monodromy: |M|=1 (unitary), diagonal in the conserved anyon labels -> NO particle production, factorised (integrable) scattering. (3) [E] Yang-Baxter/factorisation: M is a bicharacter (M(a+a',b)=M(a,b)M(a',b)), checked on all triples -> multi-particle factorisation. (4) [E] crossing+symmetry: M(a,b)=M(b,a) and M(a,-b)=conj(M(a,b)). (5) [E] holomorphic point: on the condensed (E8)_1 (single deconfined sector, v241/v235/v237) M=1, trivial braiding -- a closed anomaly-free bulk (det 1/SRE); scattering non-trivial only in the un-condensed matter sectors. (6) [O] residual: the 4d S-matrix and physical cross-sections are the open holographic uplift (Phase 3 / the spectral-action route) -- this exhibits the EXACT 2d topological scattering data (braiding = S-matrix) with the mass gap from v238

Original classification detail

[E] mass gap -> HR asymptotic states; 2-particle S = monodromy (unitary, diagonal, factorised); Yang-Baxter bicharacter; crossing; trivial S on condensed E8. [O] the 4d S-matrix / cross-sections = the open holographic uplift

Dependencies as recorded

DHR.MODULAR.01; DYN.SEMIGROUP.01; v64

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  • v243_haag_ruelle_braiding.pyRegistered verification module · registry
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    DHR.SMATRIX.01 — scattering: the Haag–Ruelle skeleton + the braiding S-matrix. The seam net has a mass gap (v64/v238) so Haag–Ruelle constructs asymptotic states; for the abelian anyon MTC (v241/v242) the 2-particle S-matrix is the monodromy M(a,b)=θ(a+b)/(θ(a)θ(b))=e^{2πi B(a,b)}. [E] mass gap Δ=6 log(3/2)>0 → HR in/out states (gapped-net theorem). [E] 2-particle S = monodromy: |M|=1, diagonal in conserved anyon labels → no particle production, factorised (integrable) scattering. [E] Yang–Baxter: M is a bicharacter → multi-particle factorisation. [E] crossing+symmetry: M(a,b)=M(b,a), M(a,−b)=M(a,b)*. [E] holomorphic point: on the condensed (E8)₁ (single sector) M=1, trivial braiding (det 1/SRE). [O] the 4d S-matrix / cross-sections are the open holographic uplift (Phase 3).

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