A registered result in its source context

SEAM.TAU.FLAG.01

Original ledger classification: Formal/Identity [E] (complete V4/D4 symmetry census, flag/arc orbit arithmetic, odd-split and K3-indicator closed forms, twist mode identity, H4 Schur rigidity, negative controls -- sympy exact + mpmath 30 digits, Wolfram-mirrored) + [C] (the symmetry-lift bit itself: flag transitivity stays a postulate; the gain is the FORM, not a derivation)

Original claim text

THE FLAG-TRANSITIVITY EQUIVALENCE WEB -- THE tau = i ATTACK EXECUTED ON THE FREE RP SEAM CIRCLE (2026-07-22; one module, 29 checks; NO marker moves -- the bit stays [C], QGEO.REALIZE.01 and P2.TYPING.01/AX.P2.01 text-precised only, SEAM.BIT.FREEDOM.01 cross-referenced): after v510 the last identification input is the square-modulus datum tau = i; which ESTABLISHED seam datum excludes the free-but-non-harmonic configurations? Test family: the deck-invariant 4-mark configurations on the free circle are, up to rotation, M(delta) = {+-1, +-e^(i delta)}, delta in (0, pi/2]; the v510 counterwitness {+-1, +-(3+4i)/5} is the member delta_cw = arctan(4/3). [E] FALSIFICATION OF BARE TRANSITIVITY: on the free RP circle the symmetry group is G(delta) = {z, -z, b/z, -b/z} ~= V4 for EVERY delta and the two inversions are pair-EXCHANGING -- V4 acts SIMPLY TRANSITIVELY on the four marks for every delta (complete census: any Moebius map commuting with the deck z -> -z is diagonal or antidiagonal); the expectation 'transitivity <= v216 local equality selects tau = i' is REFUTED; hardening: ALL undirected per-mark data (arc-pair multiset {delta, pi-delta}, chordal distance multiset, cone germ) are identical at the four marks for EVERY delta -- no local jet sees the bit. [E] THE SHARP SELECTOR: FLAG transitivity (marks AND their two sides indistinguishable) <=> delta = pi/2 <=> tau = i <=> the clock -- stabiliser jump 1 -> 2, flag/arc orbit collapse 2 -> 1, order-4 mark rotation (b^2 = -1 solveset {pi/2} exactly), mark-fixing mirror z -> 1/z <=> cos delta = 0. [E] 12-FOLD EQUIVALENCE WEB (every link exact): clock rotation, D4 (vs V4), mark mirror, flag transitivity, three spectral degeneracies (odd-doublet split of the seam covariance = (2/pi) ln cot(delta/2), zero only at pi/2 -- counterwitness exactly (2/pi) ln 2; arc-weighted seam-Laplacian degeneracy 4/pi; the v215-K3/v503 mod-4 indicator in CLOSED FORM (2 sqrt 2/pi)|ln tan(delta/2)| -- the v503 lattice curve analytically), rho-twist existence (mode identity (m^2 - n^2)(t - 1/t) = 0 <=> t = 1; Atiyah-Bott rationality is delta-BLIND -- Tr(sigma e^(t Delta)) = 1 for ALL delta, j_cw = 148176/25 rational non-CM: rationality does NOT select), cusp-4-cycle implementability (H4 rigidity: V4 with element orders {1,2,2,2} cannot induce the cusp 4-cycle; joint centraliser scalar, U^2 nonscalar => no projective involution -- the implementer has projective order exactly 4, the Z8-shadow consistent v510), harmonic, j = 1728. [E] NEGATIVE CONTROLS: hexagonal (marks not concyclic, fails the PRECONDITIONS of the battery -- consistent v503/v510); silver (harmonic but topologically dead -- position and modulus halves independent, consistent v510); 3 marks (impossible on the free circle: 0 invariant 3-subsets vs 28 invariant 4-subsets on Z16; on a plain circle bare transitivity ALREADY forces equal spacing at n = 3 -- '4 counts': the deck-pairing is exactly what makes bare transitivity free of charge at n = 4). [C] THE REST (verbatim, no overclaim): the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4: side-indistinguishability of the four equally-decorated marks); v216 local equality does NOT deliver it (no local jet can see the side bit); its negation is now concretely measurable (odd-split (2/pi) ln cot(delta/2) != 0). Scoreboard: the counterwitness passes all 7 established unsided tests and fails all 9 clock-equivalent ones; the free RP/Theta battery (v519/v521) is since documented as the EIGHTH side-blind test (7 -> 8), face #13 (the OS-reflection group closes to D4 exactly at the clock point) extends the equivalence web 12 -> 13 (v521), and the mark-decorated twist-state class (everything Wick-computable) is since documented as the NINTH side-blind test (8 -> 9, v525) -- the free-plus-twist class is exhausted. No gate closes, no marker moves.

Original classification detail

Flag transitivity [E]/[C]: bare mark-transitivity is FALSIFIED as a selector (pair-exchanging V4 simply transitive for every delta, no local jet sees the bit); the sharp selector is FLAG transitivity (marks AND sides) <=> delta = pi/2 <=> tau = i <=> the clock, welded into a 12-fold exact equivalence web (D4 enhancement, mark mirror, odd-split (2/pi) ln cot(delta/2), arc-Laplacian 4/pi, K3 indicator closed form, rho-twist existence, cusp-4-cycle implementability, harmonic, j = 1728) -- the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4), concretely measurable in its negation; the bit stays the one [C] carrier input; the free RP/Theta battery (v521) is the eighth and the twist-state class (v525) the ninth side-blind test (the free-plus-twist class is exhausted); face #13 extends the web to 13; no marker moves

Dependencies as recorded

SEAM.BIT.FREEDOM.01; SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; AX.P2.01; v117; v215; v216; v483; v503; v506; v507; v510