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Original ledger classification: Geometric realisation premise [O] (the single residual; component reductions [I] established)

Recorded statement: THE SEAM CONFORMAL-REALISATION PREMISE -- the SINGLE structural residual of the whole theory (the two obligations QGEO.MARKS.01 + QGEO.KERNEL.01 collapse onto it; honest unification, the premise itself stays OPEN). The two residual…

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QGEO.CONF.01

Original ledger classification: Geometric realisation premise [O] (the single residual; component reductions [I] established)

Original claim text

THE SEAM CONFORMAL-REALISATION PREMISE -- the SINGLE structural residual of the whole theory (the two obligations QGEO.MARKS.01 + QGEO.KERNEL.01 collapse onto it; honest unification, the premise itself stays OPEN). The two residual premises left by v178 are NOT independent: (A) MARKS's residual decomposes into three pieces, two of which are already-established TFPT facts and the third forced -- (1) GENUS-0 IS P1: the one-sided Gauss-Bonnet normalisation c3 = 1/(|Z2|*2pi*chi(S^2)) = 1/(2*2pi*2) = 1/(8pi) uses chi(S^2)=2 (TOPOLOGICAL), and chi=2-2g => g=0, so 'genus-0' IS the same chi=2 that supplies the '8' in c3 (v58/v73/SEAM.KHALF.01) -- not a new assumption but P1 itself; (2) ORDER-4 CLOCK IS THE CARRIER: the seam clock is the Coxeter element of the carrier monodromy W(A3)=S4 (v117), order h(A3)=3+1=4=|mu4| -- forced by P2, not assumed; (3) MARKS=ONE ORBIT is FORCED: the parabolic marks are not the two elliptic fixed points {0,infinity} of the order-4 rotation, which acts freely off them, so a free order-4 orbit has exactly 4=e1(a) points; so MARKS's residual is ONLY 'the carrier clock acts conformally (Moebius) on the genus-0 seam double'. (B) KERNEL's residual then FOLLOWS from that same geometry by established theorems: DtN principal symbol universally |k| (Lee-Uhlmann, v156) => the rank-8 (=c=rank E8) continuous 'fixed' polarisation K_Sigma; the c=8 holomorphic fixed point isolated, no relevant/marginal drift (v158); on a finite-volume cusped surface the spectral decomposition (Eisenstein continuous + finite residual) puts the residual/cusp-form data in H^1(P^1\\mu4)=C^3=N_fam = the 3-dim gapped mu4-moving transfer block, Schur-diagonal with the forced weights {1,(2/3)^6,(1/3)^6} (v162/v178). (C) UNIFICATION: both reduce to QGEO.CONF.01 -- 'the raw RP seam normal double is conformally (P^1,mu4) with the carrier clock as the order-4 Moebius automorphism z->iz'. Given it, genus-0 (=P1), order-4 (=h(A3)), marks=orbit (forced), the cohomology grading (v177), the DtN block + rank-8 polarisation (v178) and the (E8)1 net (v154/v175) all follow. [O] IRREDUCIBLE: QGEO.CONF.01 is a constructive-geometry statement (the conformal class of the raw seam double IS the cusped sphere with the carrier clock as a Moebius map), NOT a finite computation; it is the single structural residual of the theory, left honestly OPEN

Original classification detail

Geometric [O]: the SINGLE residual is the seam conformal realisation (P^1,mu4) with the carrier clock as the order-4 Moebius map; MARKS+KERNEL both reduce to it

Dependencies as recorded

QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.REDUCE.01; AX.P1.01; SEAM.KHALF.01; v58_seam_horizon_chain.py; v73_k_c3_half.py; v117_monodromy_weyl_a3.py; v156_seam_net_construction.py; v158_fixed_point_stable.py

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  • v179_conformal_realisation.pyRegistered verification module · registry
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    the two residual premises (MARKS-residual + KERNEL-residual) collapse to ONE geometric premise QGEO.CONF.01 (honest unification, the conformal realisation stays OPEN). MARKS's residual is NOT three independent assumptions: (1) genus-0 IS P1 — c3=1/(|Z2|·2π·χ(S²))=1/(8π) with χ(S²)=2 ⇒ g=0, the same χ=2 that gives the '8' (v58/v73); (2) the order-4 clock IS the carrier — Coxeter element of W(A3)=S4, order h(A3)=4=|μ4| (v117); (3) marks=one orbit is FORCED — parabolic marks ≠ the 2 elliptic fixed points {0,∞}, a free order-4 orbit has exactly 4=e1(a) points. So MARKS reduces to 'the carrier clock acts conformally on the genus-0 double'. KERNEL then FOLLOWS from that geometry: DtN symbol |k| (Lee-Uhlmann, v156) ⇒ rank-8 c=8 fixed polarisation; no drift (v158); the cusped-surface residual data is H¹=ℂ³=N_fam = the 3-dim gapped transfer block (Schur, v162/v178). Both ⇒ ONE premise QGEO.CONF.01: 'the raw RP seam normal double is conformally (ℙ¹,μ4) with the carrier clock as the order-4 Möbius map'. The two doors of v177/v178 are one geometric realisation — the single structural residual of the whole theory, left honestly OPEN.

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