Follow a result back to its evidence.

SEAM.CLOCK.RIGIDITY.01: clock rigidity, Route B of the v503 classification -- two exact theorems reducing the order-4 clock input to ONE alignment bit (Part A Moebius + Part B Fock in one module; QGEO.REALIZE.01 stays [C]/[O], NO marker moves). Exact sympy + exact Cl(16) Fractions, no floats. [E] THEOREM A (Moebius): the collar deck z -> -z has in PGL2(C) EXACTLY TWO square roots (z -> +iz, z -> -iz), both AUTOMATICALLY of order 4 (complete case analysis: tr != 0 => diagonal => a/d = +-i; tr = 0 impossible) -- 'order 4' is NO input anymore; 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 clocks) => then marks = 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): {+-1, +-(3+2sqrt2)} is deck-free AND harmonic (j = 1728) with full D4 stabiliser, but the deck sits in it non-centrally (edge involution) => 0 roots -- harmonicity alone is NOT sufficient; the RP filter selects nothing (both real structures pass both roots: the Aut(Z4) ambiguity is irreducible, consistent with v492). [E] B3/B4: a free Z_n orbit forces n | 4; mark transitivity forces n = 4; faithfulness kills n >= 5; n = 3 fixes a mark; n = 6 dies independently at cusp degree 3 (v117); the deck is modulus-blind (v503 B5 exact); controls: n = 8 has only orbits of size 8 (impossible on 4 marks); hexagonal A4 (order 12, ZERO order-4 elements, squares only orders {1,3}); generic Klein V4 (exponent 2 -- not even the deck has a root); stabiliser trichotomy V4/D4/A4 with order-4 counts (0, 2, 0). [E] THEOREM B (Fock, the core result): the unique S^2 spin structure gives NS on every circle (established P1 datum); one-particle: the NS half-rotation has D^2 = -1 (charpoly (lam^2+1)^8), the quarter-shift C has C^2 = D, C^8 = 1 (charpoly (lam^4+1)^4); Fock: Utilde = prod_j(1 - gamma_j gamma_{j+8}), Utilde^2 = 256 gamma_1...gamma_16 => U^2 = (-1)^F EXACTLY, and (cU)^2 = c^2 (-1)^F is NEVER 1 (center of Cl(16) = scalars) -- the Z4 lift is FORCED (nonsplit extension 1 -> Z2^{(-1)^F} -> Z4 -> Z2^{deck} -> 1); Z8 tower: V^2 prop Utilde, V^4 prop (-1)^F, projective order 8 = 2|mu4| -- fermionically explained; R control: U_R^2 = +1 (split -- the forcing rests on NS = the established geometry); 16-fold way: theta_v = 1 at nu = 16; robustness: the forcing already holds at 8 sites. [C] THE ONE REST: '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. QGEO-R1 == P2-R1 (v503) now reads: given the bit, order/uniqueness/orbit/tau = i are theorems. Exact (sympy + Cl(16) Fractions).

Linked claims

SEAM.CLOCK.RIGIDITY.01

Paper sections

Source section 1Source section 2Source section 3Source section 4Source section 5Source section 6Source section 7Source section 8Source section 9Source section 10Source section 11Source section 12Source section 13Source section 14Source section 15Source section 16Source section 17Source section 18Source section 19Source section 20Source section 21Source section 22Source section 23Source section 24Source section 25Source section 26Source section 27Source section 28Source section 29Source section 30Source section 31Source section 32