A registered result in its source context

SEAM.CLOCK.RIGIDITY.01

Original ledger classification: Formal/Identity [E] (Theorem A: complete PGL2(C) square-root classification, D4 centrality criterion, counterexample, n-arithmetic, stabiliser trichotomy -- sympy exact; Theorem B: U^2 = (-1)^F in exact Cl(16) Fractions, nonsplit Z4 extension, Z8 tower, R-sector split control, 16-fold-way arithmetic -- Wolfram-mirrored) + [C] (the one alignment bit: deck central in the mark-D4); QGEO.REALIZE.01 stays [C]/[O], P2.TYPING.01 R1 mapped onto the same bit

Original claim text

SEAM CLOCK RIGIDITY -- ROUTE B OF THE v503 CLASSIFICATION EXECUTED (2026-07-21; two exact theorems in one module, Part A Moebius + Part B Fock, 34 checks; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom): v503 localised the one remaining identification input as 'why does the raw seam carry the order-4 clock' (QGEO-R1 == P2-R1). v506 reduces that input to ONE alignment bit. [E] THEOREM A (Moebius side, complete): the collar deck z -> -z has in PGL2(C) EXACTLY TWO square roots (z -> +iz and z -> -iz), both AUTOMATICALLY of order 4 (complete case analysis: tr g != 0 forces diagonal with a/d = +-i; tr g = 0 forces det = 0, impossible) -- 'order 4' is NO LONGER an input; a marks-preserving square root exists <=> the deck is the CENTRAL involution of the mark-stabiliser D4 (center = {id, deck}, exactly 2 order-4 elements = the two clocks, both squaring to the deck) => then the marks are one mu4 orbit, cross-ratio class {2, 1/2, -1}, j = 1728, tau = i, free 4-cycle, deck = clock^2, only the scale free; COUNTEREXAMPLE (mandatory honesty): {+-1, +-(3+2sqrt2)} is deck-free AND harmonic (j = 1728) with full D4 stabiliser, but there the deck is a NON-central (edge-type) involution => ZERO square roots -- harmonicity alone is NOT sufficient, the residual datum is deck-CENTRALITY, not the j-invariant; the RP filter selects nothing (both real structures pass both roots: the Aut(Z4) = {clock, clock^-1} ambiguity is irreducible, consistent with v492 NC3). [E] B3/B4 (n-arithmetic + controls): a free Z_n on 4 marks forces n | 4; marks-transitivity forces n = 4; faithfulness (a Moebius map fixing >= 3 points is the identity) kills every n >= 5; n = 3 fixes a mark (hexagonal shadow); n = 6 dies INDEPENDENTLY at the established cusp-class degree 3 (lam^5 - 1 vs lam^3 - 1, v117); the deck is modulus-blind (every {+-1, +-w} preserved -- v503 B5 made exact, ALL modulus-pinning power sits in the square-root bit); controls: no order-8 element preserves any 4-set (orbit sizes 8); hexagonal marks: stabiliser A4 (order 12), ZERO order-4 elements, squares only orders {1, 3}; generic marks: Klein V4, exponent 2 -- not even the deck has a root; stabiliser trichotomy V4/D4/A4 with order-4 counts (0, 2, 0): an order-4 element exists IFF the marks are harmonic, and then it is the clock pair. [E] THEOREM B (Fock side, the core result): the celestial sphere S^2 has b1 = 0, hence a UNIQUE spin structure whose restriction to every circle is the BOUNDING = NS/antiperiodic one (established P1 datum, v492/v490); ONE-PARTICLE: the NS half-rotation D has D^2 = -1 (charpoly (lam^2+1)^8) -- the 'Z2' deck is already order 4 before any Fock choice; the quarter-shift C has C^2 = D, C^8 = 1 (charpoly (lam^4+1)^4, zeta8 spectrum); FOCK (exact Cl(16) Fractions): the Bogoliubov implementer Utilde = prod_j (1 - gamma_j gamma_{j+8}) satisfies Utilde^2 = 256 gamma_1...gamma_16, i.e. U^2 = (-1)^F EXACTLY, and (cU)^2 = c^2 (-1)^F is NEVER 1 (center of Cl(16) = scalars, implementer unique up to phase) -- the Z4 lift is FORCED: 1 -> Z2^{(-1)^F} -> Z4<U> -> Z2^{deck} -> 1 is a NONSPLIT extension; Z8 TOWER: the quarter-shift lift V (unique even block implementer, rational nullspace) has V^2 prop Utilde, V^4 prop (-1)^F, projective order 8 = 2|mu4| -- the v492 Z8 spin bridge realised on the 16-Majorana Fock space, '8 = 2|mu4|' fermionically explained; R CONTROL: U_R^2 = +1 EXACTLY (split Z2 -- the entire forcing rests on the NS/bounding structure = established geometry, not a convention); 16-FOLD WAY: theta_v = 1 at nu = 16 (the gaugeable point, v490), the forced tower period 8 = the Fidkowski-Kitaev Z8; ROBUSTNESS: the forcing already holds on an 8-site NS circle -- 16 is the carrier count 2^(g_car-1), not a tuning. [C] THE ONE RESIDUAL (verbatim): 'the collar deck is the central involution of the mark-D4' (<=> square-root existence on the mark data) -- ONE alignment bit; the fermionic Z4 forces the ORDER, not the spatial alignment. NO OVERCLAIM: given the bit, order 4, uniqueness up to Aut(Z4), the free 4-cycle, tau = i (scale free) and deck = clock^2 are all THEOREMS; without it, nothing pins the modulus. Consequence for the residual map: QGEO-R1 == P2-R1 (v503) now reads as this single bit -- order/uniqueness/orbit/tau = i all follow (fermionic halving of the R1 rest). No gate closes, no marker moves. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the bit is now fully characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit), and carries a physical face: the non-split NS Fock extension class (Fidkowski-Kitaev-type; the edge arrangement splits, U^2 = +1, zero roots) -- v507; the origin theorem shows every mark-free collar deck IS a half-period translation of the seam double, so the CLASS of the deck is derived and the bit is NO tautology -- only the position (which half-period) stays the carrier input. PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the POSITION half of the alignment bit is now typed as topology -- the collar deck is a COVERING deck and covering decks act freely; the unique free involution of the seam circle is the antipode (Aut(C_16) = D_16, complete census) and NONSPLIT <=> FREE over all 17 involutions, so the edge class is topologically EXCLUDED (v510); the bit reduces to the square-modulus datum tau = i alone (harmonic + free => mu4 exactly) -- the modulus half stays the one [C] carrier input.

Original classification detail

Clock rigidity [E]/[C]: the deck has exactly two Moebius square roots, both automatically order 4; marks-preserving root <=> deck = central involution of the mark-D4 (counterexample shows harmonicity alone insufficient); the NS Fock implementation is FORCED to order 4 (U^2 = (-1)^F exact in Cl(16), nonsplit; Z8 tower V^2 prop U, V^4 prop (-1)^F; R sector splits) -- the order-4 clock input reduces to ONE alignment bit ('deck central in the mark-D4'); the fermionic Z4 forces the order, not the alignment; QGEO-R1 == P2-R1 now = this single bit; no marker moves; v507 characterises the bit as a tetrad with a physical face (non-split NS Fock extension class) and derives the deck CLASS (half-period translation) -- only the position stays the input; v510 excludes the edge class topologically (covering-deck freeness = P1's closed seam circle): the bit reduces to the square-modulus datum tau = i alone

Dependencies as recorded

QGEO.REALIZE.01; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; SEAM.BIT.ORIGIN.01; v117; v168; v214; v215; v216; v490; v492; v493; v499; v503; v507; SEAM.BIT.FREEDOM.01; v510