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Original ledger classification: Exact [E] F_3/rigidity/labeling identities + [C] typed selection reading (exact sympy / F_3 arithmetic; Smith transforms over Z)

Recorded statement: The selection rule round (one module, 10 checks, ~2 s; discovery probe selection_rule_probe.py 10/10, verdict SELECTION-RULE-LANDED). THE F_3 STRUCTURE [E]: the torsion of coker(1 - T_inf) is a genuine F_3 vector space --…

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Paper context: TFPT — Introduction and reading guide

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QGEO.SELRULE.01

Original ledger classification: Exact [E] F_3/rigidity/labeling identities + [C] typed selection reading (exact sympy / F_3 arithmetic; Smith transforms over Z)

Original claim text

The selection rule round (one module, 10 checks, ~2 s; discovery probe selection_rule_probe.py 10/10, verdict SELECTION-RULE-LANDED). THE F_3 STRUCTURE [E]: the torsion of coker(1 - T_inf) is a genuine F_3 vector space -- omega-multiplication acts as the IDENTITY ((1-omega) acts as zero) -- and the deck rotation r (commuting with T_inf = r^4) acts F_3-linearly on it. THE RIGIDITY [E]: the induced action is the explicit matrix [[0,1,0],[1,0,2],[2,2,2]] mod 3; its only fixed class is 0; char mod 3 = (x+1)(x^2+1) with x^2+1 IRREDUCIBLE over F_3: the action has EXACTLY ONE projective eigenline (eigenvalue -1) -- structural uniqueness, census-confirmed. THE LABELING [E]: the unique deck eigenline IS the top-mode class (V = 2); the zero mode and the seam are NOT projectively fixed (neither under r nor r^2). With v603/v605 the three modes carry MUTUALLY EXCLUSIVE canonical labels: seam (V = 1) = the R-REAL line; top (V = 2) = the unique DECK-EIGEN torsion class; zero (V = 0) = the only UNLABELED mode. THE RULE [C, typed reading]: the two-demand involution S_0 acts as -1 exactly on the two labeled modes and +1 on the unlabeled one -- the v590 zero-mode demand ('+1 on ker V') is the statement 'fix the unlabeled mode', the seam-reversal demand is 'reverse the R-real line' (v603): BOTH demands are now expressed through canonical cover structure. Honest scope: a typed reading of the selection, not a continuum derivation; eigen-ness, uniqueness and distinctness are Smith-frame-invariant, the coordinate triples are not. GATE.QGEO does not move; no marker changes.

Original classification detail

SELECTION-RULE-LANDED: torsion = F_3 space (omega trivial); r-action explicit, char (x+1)(x^2+1) mod 3, unique projective eigenline = top mode; seam/zero not fixed; labels mutually exclusive (R-real / deck-eigen / unlabeled); S_0 flips the labeled pair; 10/10 checks

Dependencies as recorded

QGEO.MODESEP.01; QGEO.TRANSLATOR.01; QGEO.INVOL.01; v590; v603; v605; v606

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  • v607_selection_rule.pyRegistered verification module · core
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    QGEO.SELRULE.01: die Selektions-Regel-Runde — DIE KANONISCHE „+1“-REGEL IST GEFUNDEN: die Torsion des ∞-Kokerns ist ein echter F₃-Raum (ω wirkt als Identität), die Deck-Rotation wirkt F₃-linear mit der expliziten Matrix [[0,1,0],[1,0,2],[2,2,2]]; char mod 3 = (x+1)(x²+1) mit x²+1 IRREDUZIBEL über F₃ — GENAU EINE projektive Eigenlinie (λ = −1), strukturell — und sie IST die Top-Mode-Klasse (V = 2); Nullmode und Seam sind nicht projektiv fixiert (auch nicht unter r²): mit v603/v605 tragen die drei Moden paarweise ausschließende kanonische Labels (Seam = R-REELL, Top = DECK-EIGEN, Nullmode = UNMARKIERT) — S₀ = −1 exakt auf den markierten, +1 auf der unmarkierten: BEIDE v590-Forderungen durch Cover-Struktur ausgedrückt (getypte Lesart, keine Kontinuums-Herleitung) (10 checks, ~2 s)

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