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Original ledger classification: Deeper reduction [E] (finite cores) + [O] (the two sharper irreducible premises)

Recorded statement: DEEPER REDUCTION OF THE TWO OBLIGATIONS (an honest attempt at QGEO.MARKS.01 + QGEO.KERNEL.01; NEITHER is closed -- they are genuine constructive-geometry/AQFT statements, not finite computations -- but each finite/algebraic CORE is now…

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QGEO.REDUCE.01

Original ledger classification: Deeper reduction [E] (finite cores) + [O] (the two sharper irreducible premises)

Original claim text

DEEPER REDUCTION OF THE TWO OBLIGATIONS (an honest attempt at QGEO.MARKS.01 + QGEO.KERNEL.01; NEITHER is closed -- they are genuine constructive-geometry/AQFT statements, not finite computations -- but each finite/algebraic CORE is now closed [E] and each obligation is reduced to ONE sharper irreducible premise [O]): (1) MARKS FINITE CORE [E] -- given a genus-0 double with an order-4 clock rho (conjugate to z->iz, fixed points {0,infinity}) and sheet reflection tau, the parabolic marks are a generic rho-orbit {1,i,-1,-i}=mu4 (|mu4|=4=e1(a), NOT the fixed points), DISTINCT iff b1=(#marks)-1=3=N_fam (any collision drops b1 below N_fam, excluded by the family count), and tau:z->1/z gives tau.rho.tau=rho^{-1} and permutes the marks => <rho,tau>=D4 of order 8, faithful; so MARKS reduces from 'produce the whole marked D4 boundary' to the SINGLE topological/clock premise 'the seam double is genus-0 and the seam clock is an order-4 conformal automorphism whose marks form one orbit'. (2) KERNEL FINITE CORE [E] (dissolves the spectrum-vs-operator worry on the finite block) -- on the 3-dim H^1 character space the mu4 characters (weights (1,2,3) -> i,-1,-i) are DISTINCT, so by Schur a mu4-equivariant operator is forced DIAGONAL (commutant of distinct-eigenvalue U = diagonal algebra) and is COMPLETELY DETERMINED by its eigenvalue per character; hence two such operators with the same forced transfer spectrum {1,(2/3)^6,(1/3)^6} (v162) and the unique residue-normalised basis Phi (v177) are EQUAL as operators, not merely isospectral -- the 'C_Sigma=U^{-1}C_mu4 U vs same spectral list' distinction VANISHES on the multiplicity-free H^1 block; so KERNEL reduces from the full operator identity to the SINGLE full-operator-reduction premise 'the full seam Calderon DtN on L^2 reduces to this 3-dim gapped mu4-moving transfer block plus the fixed rank-8 Calderon polarisation K_Sigma' (principal symbol universally |k|, Lee-Uhlmann/v156; c=8 fixed point isolated, no relevant/marginal drift, v158). NET (honest): neither obligation closed; both sharpened to milder premises, finite cores [E]; the structural residual of the whole theory is exactly {the topological clock premise, the full-operator-reduction premise}

Original classification detail

Deeper reduction [E]/[O]: MARKS + KERNEL finite cores closed; each reduced to one sharper premise (topological clock / full-operator reduction); neither obligation closed

Dependencies as recorded

QGEO.MARKS.01; QGEO.KERNEL.01; QGEO.TREE.01; v156_seam_net_construction.py; v158_fixed_point_stable.py; v162_seam_transport_identification.py; v177_seam_marking_kernel.py

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  • v178_marks_kernel_attempt.pyRegistered verification module · registry
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    an honest ATTEMPT at the two open obligations QGEO.MARKS.01 + QGEO.KERNEL.01 — NOT a closure (they are genuine constructive-geometry/AQFT statements, not finite computations), but a deeper reduction: each finite/algebraic CORE is closed [E] and each obligation reduced to ONE sharper irreducible [O] premise. MARKS finite core [E]: a genus-0 double with an order-4 clock ρ (~z↦iz, fixed points {0,∞}) and sheet reflection τ forces the marks to be a generic ρ-orbit {1,i,−1,−i}=μ₄ (NOT the fixed points), distinct iff b₁=4−1=3=N_fam (collision drops b₁, excluded), τρτ=ρ⁻¹ ⇒ faithful D4 order 8 — so MARKS reduces to the single topological/clock premise. KERNEL finite core [E]: on the 3-dim multiplicity-free H¹ the μ₄ characters (1,2,3) are distinct, so by Schur a μ₄-equivariant operator is forced DIAGONAL and completely determined by its eigenvalue per character — two such with the same forced transfer spectrum {1,(2/3)⁶,(1/3)⁶} (v162) and the unique residue-normalised basis (v177) are EQUAL as operators (the 'spectrum vs operator' worry vanishes on the finite block); so KERNEL reduces to the single full-operator-reduction premise (principal symbol |k| Lee-Uhlmann v156, no drift v158). NET: neither obligation closed; both sharpened to milder premises.

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