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Original ledger classification: Proof tree [E] (four nodes closed constructively) + [O] (two named obligations)

Recorded statement: THE QGEO PROOF TREE (the v176 'one realisation premise' split into its honest constituents; the open residual sharpened from one diffuse premise into exactly TWO named obligations; external residual-class audit 2026-06-14): REALIZE = MARKS…

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QGEO.TREE.01

Original ledger classification: Proof tree [E] (four nodes closed constructively) + [O] (two named obligations)

Original claim text

THE QGEO PROOF TREE (the v176 'one realisation premise' split into its honest constituents; the open residual sharpened from one diffuse premise into exactly TWO named obligations; external residual-class audit 2026-06-14): REALIZE = MARKS . UNIFORM . COHOM . MODULE . KERNEL . NET, with FOUR nodes now closed CONSTRUCTIVELY. [E] QGEO.UNIFORM.01 (Lemma 2, constructive -- strengthens v168's cross-ratio check to the full uniformisation theorem): an order-4 Moebius map conjugates to rho:z->iz, a non-fixed point a has orbit {a,ia,-a,-ia} scaling (z->z/a) to mu4={1,i,-1,-i}, and the reflection sigma:z->1/z satisfies sigma.rho.sigma=rho^{-1} (so <rho,sigma>=D4 of order 8) and permutes mu4 (fixes 1,-1, swaps i,-i) -- hence a genus-0 four-marked faithful-D4 boundary IS (P^1,mu4); [E] QGEO.COHOM.01: wk=z^{k-1}dz/(z^4-1) satisfy rho* wk = i^k wk so the H^1 character grading is (1,2,3)=A3 exponents=Spec(Q+), rank 3=N_fam; [E] QGEO.MODULE.01: the H^1->generation iso is UNIQUE -- both sides multiplicity-free (one scalar per character), the residue pairing (wk residues at mu4 nonzero, sum 0) fixes the scalar, and the reflection acts sigma*: w1<->w3, w2 fixed (the integer-model parity, v141/v146); [E] QGEO.NET.01 (v154/v175): index-4 mu4 simple-current extension of (D5)1(x)(A3)1 is holomorphic c=8 => (E8)1, full-cone RP all m. So the v176 framing (which took the marks as a HYPOTHESIS, a near-circular trap) is replaced by four theorems plus exactly two open doors

Original classification detail

Proof tree [E]/[O]: UNIFORM/COHOM/MODULE/NET closed [E]; the residual is exactly the two obligations MARKS + KERNEL

Dependencies as recorded

QGEO.REALIZE.02; QGEO.RIGID.01; QGEO.MARKS.01; QGEO.KERNEL.01; v137_qplus_cohomology.py; v141_deck_selection.py; v168_qgeo_rigidity.py; v176_seam_collar_realisation.py

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Registered verification modules

  • v177_seam_marking_kernel.pyRegistered verification module · registry
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    the QGEO proof tree — the v176 'one realisation premise' split into its honest constituents: REALIZE = MARKS · UNIFORM · COHOM · MODULE · KERNEL · NET, with FOUR nodes closed CONSTRUCTIVELY as [E] and the open residual sharpened into exactly TWO named obligations. [E] UNIFORM (Lemma 2, constructive): an order-4 Möbius map conjugates to z↦iz, a non-fixed orbit scales to μ₄, the reflection z↦1/z gives στρτ=ρ⁻¹ (D4 order 8) — a genus-0 4-marked faithful-D4 boundary IS (ℙ¹,μ₄); [E] COHOM: ρ*ωk=i^kωk so the H¹ grading is (1,2,3)=A3 exponents=Spec(Q₊); [E] MODULE: the H¹→generation iso is UNIQUE (multiplicity-free + residue normalisation; reflection ω1↔ω3, ω2 fixed); [E] NET (v154/v175): index-4 μ₄ extension of (D5)₁⊗(A3)₁ = (E8)₁. The TWO genuine open obligations [O]: QGEO.MARKS.01 (raw seam → genus-0 four-marked D4 boundary) and QGEO.KERNEL.01 (raw seam Calderón kernel = the μ₄-equivariant free gapped contraction as an OPERATOR). Removes the near-circular v176 framing (which took the marks as a hypothesis); the whole structural residual is now exactly these two doors.

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