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Original ledger classification: Identity [E] (orbit/character core) + premise [O] (H^1-rep)

Recorded statement: LEFSCHETZ/CHARACTER DERIVATION OF THE FOUR SEAM MARKS (external-review path 2/priority 2, 2026-06-14): instead of POSITING D4 boundary marks (the old QGEO.MARKS.01 finite core), the marks are FORCED to be a free mu4 orbit by orbit-counting…

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QGEO.MARKS.02

Original ledger classification: Identity [E] (orbit/character core) + premise [O] (H^1-rep)

Original claim text

LEFSCHETZ/CHARACTER DERIVATION OF THE FOUR SEAM MARKS (external-review path 2/priority 2, 2026-06-14): instead of POSITING D4 boundary marks (the old QGEO.MARKS.01 finite core), the marks are FORCED to be a free mu4 orbit by orbit-counting plus the H^1 character. (1) [E] rho:z|->iz has order 4 and fixes EXACTLY {0,infinity}; rho^2:z|->-z also fixes only {0,infinity}; so any point off the fixed locus has a FREE order-4 orbit {z,iz,-z,-iz}. (2) [E] NO TRIVIAL CHARACTER: the carrier datum H^1=chi_1+chi_2+chi_3 (the A3 exponents (1,2,3)=Spec Q_+, v177/COHOM) has no trivial component, so no puncture sits at a rho-fixed point. (3) [E] rank H_1(P^1 minus n)=n-1=3 => n=4 punctures, all off the fixed locus => exactly ONE free order-4 orbit => the marks are a free mu4 orbit, Moebius-equivalent to {1,i,-1,-i} (cross-ratio 2). (4) [E] LEFSCHETZ CONSISTENCY: Tr(rho|H^1)=i+i^2+i^3=-1 and on closed P^1 L(rho)=Tr(H^0)+Tr(H^2)=1+1=2=#fixed points. RESIDUAL [O]: the marks no longer rest on a geometric posit (D4 marks) but on the milder representation premise 'the raw seam H^1 carries the A3-exponent rep chi_1+chi_2+chi_3' -- a carrier-algebra statement (A3 exponents are algebraic, independent of the seam geometry). So QGEO.MARKS reduces from geometry to an H^1-rep premise; less circular, still [O]. Exact; Wolfram-mirrored

Original classification detail

Identity [E]: orbit-counting (order-4 rho fixes {0,inf}; no trivial H^1 char; rank 3 => 4 punctures) forces the marks to a free mu4 orbit; residual [O] relocates from 'seam is P^1\\mu4' to 'H^1 carries the A3-exponent rep' -- less circular

Dependencies as recorded

QGEO.MARKS.01; QGEO.TREE.01; QGEO.SYM.01; v177_seam_marking_kernel.py; v178_marks_kernel_attempt.py

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  • v195_marks_lefschetz_character.pyRegistered verification module · registry
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    QGEO.MARKS.02 — a Lefschetz/character derivation of the four seam marks (review path 2): instead of positing D4 marks, they are FORCED to be a free µ4 orbit. [E]: ρ:z↦iz (order 4) fixes only {0,∞}; H¹=χ1+χ2+χ3 (A3 exponents (1,2,3), v177) has no trivial character ⇒ no puncture at a fixed point; rank H_1=n−1=3 ⇒ n=4 ⇒ exactly one free orbit ⇒ marks = µ4 (cross-ratio 2); Tr(ρ|H¹)=i+i²+i³=−1, L(ρ)=1+1=2=#fixed. [O] residual relocates from 'seam is P¹∖µ4 (geometric)' to 'H¹ carries the A3-exponent rep' — less circular.

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