A registered result in its source context

Original ledger classification: Identity/Lattice [E] (Gauss-Bonnet count + Euclidean-orbifold uniqueness) + premise [O] (square modulus)

Recorded statement: THE FOUR MARKS EMERGE FROM GAUSS-BONNET (2026-06-15; executes the K1 lever of QGEO.KILL.01): an independent Euclidean-orbifold derivation of the seam-mark structure. (1) [E] MARK COUNT = 2 chi: Z2 branch points (cone angle pi, deficit pi)…

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Paper context: TFPT Gesamtsynthese 2026-09-28

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QGEO.MARKS.03

Original ledger classification: Identity/Lattice [E] (Gauss-Bonnet count + Euclidean-orbifold uniqueness) + premise [O] (square modulus)

Original claim text

THE FOUR MARKS EMERGE FROM GAUSS-BONNET (2026-06-15; executes the K1 lever of QGEO.KILL.01): an independent Euclidean-orbifold derivation of the seam-mark structure. (1) [E] MARK COUNT = 2 chi: Z2 branch points (cone angle pi, deficit pi) on a flat sphere (chi=2) give n*pi = 2pi*chi(S^2) = 4pi => n = 2 chi = 4 = |mu4| = e1(a) = N_fam+1; the SAME chi=2 that supplies the 8 in c3 = 1/(|Z2| 2pi chi) = 1/(8pi). (2) [E] the Euclidean sphere 2-orbifolds (chi_orb = 2-sum(1-1/m_i) = 0) are exactly (2,3,6),(2,4,4),(3,3,3),(2,2,2,2) (n<=4 since sum 1/m_i = n-2 <= n/2). (3) [E] ALL-ORDER-2 (= the |Z2| sheet branch) selects (2,2,2,2) uniquely. (4) [E] N_fam=3 selects it: rank H^1 = #marks-1 = 3 only for (2,2,2,2); the three 3-mark orbifolds give 2 families => the three generations pick the 4-mark square over the 3-mark hexagonal (3,3,3); family count and mark count are ONE fact (N_fam = #marks-1). (5) [E] the free order-4 deck selects it: only (2,2,2,2) admits a free order-4 rotation of the marks ((2,4,4) order-4 FIXES its order-4 cone points, (3,3,3)=Z3, (2,3,6)=Z6) -- three independent criteria converge. (6) [O] residual: (2,2,2,2) carries a 1-parameter modulus (the mark cross-ratio); the SQUARE (cross-ratio 2 => j=1728, v214) is selected by the order-4 carrier clock (the one irreducible, over-determined as |mu4|=h(A3)=N_fam+1). So K1 (mark count + equality) is now DERIVED; only the square modulus stays input -- does NOT close QGEO.REALIZE.01, but removes the mark COUNT from the inputs. Exact (sympy); Wolfram-mirrored

Original classification detail

Identity/Lattice [E] + premise [O]: the mark COUNT (4 = 2 chi) and EQUALITY (all order-2) are DERIVED from Gauss-Bonnet + Z2 branch + flatness (three independent selectors: all-order-2, N_fam=3, free order-4); only the square modulus (cross-ratio 2, the order-4 clock) stays the carrier input; executes K1 of QGEO.KILL.01, does NOT close QGEO.REALIZE.01

Dependencies as recorded

QGEO.MARKS.02; QGEO.PILLOW.01; QGEO.KILL.01; AX.P1.01; v195_marks_lefschetz_character.py; v214_seam_pillowcase.py

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2026-06-22 (the geometric bridge made explicit: the keystone is ONE arrow – raw-seam [formula in source] is a six-link chain, five theorems [formula in source] one open orbifold realisation, the group forced by the [formula in source] atoms: v346)Research changelog; TFPT — Introduction and reading guide
Externalization Contract — SEAM.EQUIV.01 (the seam scaling-limit theorem, packaged for external specialists)tfpt research contracts
How to read this documenttfpt 1 architecture e8
The whole integer skeleton from one pair [formula in source]input 4 origin theory
The EM closure: [formula in source] as an explicit fixed point (made explicit here)tfpt 1 architecture e8
Honest statusinput 4 origin theory
The unique attractor: a gapped transport selects ONE fixed pointorigin theory
2026-06-18 (the two proof routes decomposed: Route (i) and Route (ii) into lemma chains, v284/ v285)Research changelog; TFPT — Introduction and reading guide

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Registered verification modules

v216_marks_gauss_bonnet.pyRegistered verification module · registry
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the FOUR seam marks EMERGE from Gauss-Bonnet (QGEO.MARKS.03), executing the K1 lever of the v215 kill-test. [E] mark count = 2χ: Z2 branch points (deficit π) on a flat sphere (χ=2) give n·π = 2π·χ = 4π ⟹ n = 2χ = 4 = |μ4| = e1(a) = N_fam+1 (the same χ=2 as the 8 in c3=1/(8π)). [E] the Euclidean sphere 2-orbifolds (χ_orb=0) are exactly (2,3,6),(2,4,4),(3,3,3),(2,2,2,2). [E] all-order-2 (the |Z2| sheet branch) ⟹ uniquely (2,2,2,2); [E] N_fam=3 (rank H¹=#marks−1) ⟹ uniquely (2,2,2,2), picking the 4-mark square over the 3-mark hexagonal (3,3,3); [E] a free order-4 deck ⟹ uniquely (2,2,2,2) — three independent selectors converge. [O] residual: the square modulus (cross-ratio 2 ⟹ j=1728, v214) needs the one order-4 carrier input. The mark COUNT + EQUALITY are now derived; does NOT close QGEO.REALIZE.01.

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