Follow a result back to its evidence.

QGEO.EMERGE.LIGHT.01: the v215 deferred emergence test (lever 7) EXECUTED in its first honest LIGHT version, plus the RESIDUAL-CONVERGENCE result QGEO-R1 == P2-R1 (analysis/kill-test class, v347/v264 style; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom). [N] NO DYNAMICAL SELECTION (zeta-log-det via the Dedekind-eta formula, mpmath 40 digits): logF = log(tau2 |eta|^4) exactly S-invariant (< 1e-35); on the rectangular family tau = it a strict local MAXIMUM (g'(1) = 0, g''(1) = -1.298) but a MINIMUM in the tau1-direction (h''(0) = +0.298): tau = i is a SADDLE of the full moduli space; critical points = the CM points (E2*(i) = E2*(rho) = 0 to 40 digits, generic 0.163 != 0); the GLOBAL det' winner at fixed area is the HEXAGONAL torus (logF(rho) = -1.03352 > logF(i) = -1.05469, diff 0.021169; Osgood-Phillips-Sarnak 1988) -- free-field dynamics alone does NOT select tau = i; criticality at the square = Palais symmetric criticality (the independent heat trace is equally critical). [N] SHARP RIGIDITY (v280 lattice DtN, N = 8): the mod-4 off-character indicator (the v215-K3 lever as a MODULUS FUNCTION) vanishes to 8e-15 EXACTLY at tau = i and rises strictly monotonically in both directions (min 8.4e-2 for t != 1) -- an isolated zero, not a flat valley; R^2 = Z2 exact as permutation matrices (the v492 Z8 bridge on the lattice); j(it) >= 1728 with equality ONLY at t = 1, j(rho) = 0. [E] THE Z2 CHAIN IS MODULUS-BLIND: the involution has exactly 4 fixed points and [Z2, Delta_t] = 0 on EVERY torus (v216 confirmed without mark input); pillowcase det' = 1/2 torus det' EXACTLY (+- pair structure, ratio 1 to 1e-12) so all tau conclusions transfer verbatim; [Z2, H_t] = 0 for all t -- marks / mark-locality / deck^2 exist on EVERY pillowcase, ONLY the order-4 clock distinguishes tau = i; Aut(E_rho) = Z6 (element orders 1,2,3,6) contains NO order-4 element -- dynamics and clock are logically INDEPENDENT selectors; GIVEN the clock, P(iZ) = P(Z) forces a3 = a2 = a1 = 0 and j = 1728 identically (v493 mini): the modulus is NOT a second input. [C/O] RESIDUAL CONVERGENCE (the core finding): the order-4 element on H^1 is a COXETER element of W(A3) (char poly lam^3+lam^2+lam+1, order h(A3) = 4 = |mu4| = N_fam + 1) -- exactly the P2 cusp-class/carrier datum of the P2 hardening (v499 R1) => QGEO-R1 == P2-R1: ONE common residual, 'why does the raw seam carry the order-4 clock', not two scattered ones; chain: (i) mark count 4 derived (v216 [E]), (ii) square derived GIVEN the clock (v493 [E]), (iii) clock = carrier input [C], (iv) rest = the dynamical realisation of the clock [O]; route classification (v347 analogue): B-rigidity in the sharpened form 'RP seam data => order-4 clock' is the live route -- free dynamics alone does NOT deliver the clock (the hexagonal finding). LEVER UPDATE (documented here, v215 itself unchanged): v215 lever 7 lifted from 'deferred, not executable' to 'light version EXECUTED, result: rigidity, not dynamical selection' -- marks emerge, the square does NOT emerge dynamically, it is pinned by the clock, and the clock is the input. Python (mpmath 40 digits + numpy/scipy lattice + sympy exact); the eta/E2*/j CM identities, the pillowcase halving and the Coxeter charpoly are Wolfram-mirrored, the lattice DtN curves stay Python-only (flagged).

Linked claims

QGEO.EMERGE.LIGHT.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 21