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Original ledger classification: Exact [E] torsion/separation identities + [C] typed frame caveat + [X] honest arithmetic negative (exact sympy; Smith transforms over Z)
Recorded statement: The mode-separation round (one module, 7 checks, ~2 s; discovery probe mode_separation_probe.py 7/7, verdict MODES-SEPARATED). ALL THREE MODES ARE PURE TORSION AT INFINITY [E]: the three eigenlines of the sheet operator V (eigenvalues 0,…
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Paper context: TFPT — Introduction and reading guide
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QGEO.MODESEP.01
Original ledger classification: Exact [E] torsion/separation identities + [C] typed frame caveat + [X] honest arithmetic negative (exact sympy; Smith transforms over Z)
Original claim text
The mode-separation round (one module, 7 checks, ~2 s; discovery probe mode_separation_probe.py 7/7, verdict MODES-SEPARATED). ALL THREE MODES ARE PURE TORSION AT INFINITY [E]: the three eigenlines of the sheet operator V (eigenvalues 0, 1, 2), lifted to the unreduced integral module along the unique intertwiner (1-parameter, rank 3), all have ZERO free part in coker(1 - T_inf) = Z^2 + (Z/3)^3 -- like the seam (v605), the zero mode and the top mode are pure 3-torsion classes at infinity. THE SEPARATION [E]: the three classes are PAIRWISE DISTINCT elements of (Z/3)^3 (fixed Smith frame: (2,0,2), (2,2,0), (1,2,0) for V = 0, 1, 2; distinctness is frame-invariant) and none is trivial -- the infinity-cusp torsion is a SELECTION CARRIER distinguishing the zero mode from the top mode: exactly the separation the real structure could NOT provide (only the seam is R-fixed; the 0/2 eigenlines are neither R-fixed nor R-swapped, v605-followup). HONEST SCOPE [C]: the specific coordinate triples are Smith-frame-dependent; the canonical rule singling out the zero mode's class (the '+1' of the v590 demand) from the structure alone is the named remaining step. THE ARITHMETIC ROUTE FOR c IS CLOSED [X, honest negative]: the canonical cubic invariant char(7 Gamma(c) c) = y^3 + 3y^2 + 6y - 1 (integer monic, irreducible) has discriminant -783 = -3^3 x 29, NOT a square -- Galois group S_3, not abelian, so no cyclotomic-subfield identity for c exists (Kronecker-Weber); c's geometric identity stays the named open. GATE.QGEO does not move; no marker changes. [2026-08-01 XXV: the canonical rule LANDS -- v607: the deck rotation has exactly ONE projective eigenline on the torsion (char (x+1)(x^2+1) mod 3) and it is the top mode; the zero mode is the only unlabeled mode; see QGEO.SELRULE.01.] [2026-08-01 XXVII: the c-identity question is resolved geometrically -- v609 (factorization through the unique mirror intertwiner); the arithmetic negative of this module stands as the reason the cyclotomic route had to fail; see QGEO.CFACTOR.01.]
Original classification detail
MODES-SEPARATED: all three V-eigenlines pure 3-torsion at infinity, classes pairwise distinct ((2,0,2)/(2,2,0)/(1,2,0)); canonical '+1' rule open; cubic field of c disc -783, S_3, not abelian (arithmetic route closed); 7/7 checks
Dependencies as recorded
QGEO.TRANSLATOR.01; QGEO.SEAMMARKS.01; QGEO.EQORDER.01; v590; v603; v604; v605
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- Architecture, E8 and the electromagnetic fixed point
- Standard Model and flavor structure
- E8 audit and bootstrap
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- =150 2026-08-01 (XXVI) (the translator factorization lands ( v609 promoted): the geometric identity of [formula in source] is CLOSED — the mirror intertwiner [formula in source] ( [formula in source] ) is unique, unimodular ( [formula in source] ), [formula in source] -integral and [formula in source] -involutive ( [formula in source] exactly), and the v605 element factorizes exactly as [formula in source] with [formula in source] : [formula in source] — all three v605 residuals are now resolved (seam reversal v603, zero-mode rule v607, [formula in source] -identity v609))Research changelog; TFPT — Introduction and reading guide
- Front matter / supportE8 audit and bootstrap
- =150 2026-08-01 (XXIV) (the selection rule lands ( v607 promoted): the infinity torsion is a genuine [formula in source] space, the deck rotation acts with char [formula in source] mod [formula in source] — EXACTLY ONE projective eigenline, and it IS the top mode; the three modes carry mutually exclusive canonical labels (seam [formula in source] [formula in source] -real, top [formula in source] deck-eigen, zero [formula in source] the only unlabeled) and [formula in source] flips exactly the labeled pair: BOTH v590 demands expressed through cover structure (typed reading; the gate residual is the continuum derivation))Research changelog; TFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v606_mode_separation.pyRegistered verification module · core
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QGEO.MODESEP.01: die Moden-Separations-Runde — DIE ∞-TORSION TRENNT DIE DREI V-EIGENLINIEN: alle drei Moden (0, 1, 2), auf das unreduzierte integrale Modul geliftet, sind REINE 3-Torsions-Klassen bei ∞ (null freier Anteil), und die drei Klassen sind PAARWEISE VERSCHIEDEN ((2,0,2)/(2,2,0)/(1,2,0) im fixierten Smith-Rahmen; Verschiedenheit rahmen-invariant, keine trivial) — der Selektions-Träger, den die reelle Struktur NICHT liefern konnte (nur die Seam ist R-fix); ehrlich: die kanonische Regel, welche Klasse das „+1“ der v590-Forderung bekommt, bleibt offen; EHRLICHES NEGATIV: der kubische Körper von c (char(7Γ(c)c) = y³+3y²+6y−1) hat Diskriminante −783 = −3³·29, kein Quadrat → Galois S₃, nicht abelsch → keine Kreisteilungs-Identität für c (Kronecker-Weber); die arithmetische Route ist geschlossen (7 checks, ~2 s)
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