"""v503 -- QGEO.EMERGE.LIGHT.01: the deferred emergence test of v215 (lever 7) executed in its first honest LIGHT version, plus the RESIDUAL-CONVERGENCE theorem QGEO-R1 == P2-R1 (one common rest, not two). Analysis / kill-test class (v347/v264 style): no gate closes, NO marker moves anywhere -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom. v215/QGEO.KILL.01 lever 7 said: the decisive (deferred) kill of the bedrock is to compute the raw P1 seam-collar DtN WITHOUT inputting the marks and check whether four SQUARE marks EMERGE. v216 already derives the mark COUNT (4 = 2 chi) and EQUALITY from topology; the genuinely open part is the MODULUS: does the free field on the (2,2,2,2) pillowcase family dynamically select the SQUARE tau = i, or is tau = i selected only by order-4 RIGIDITY (Aut contains Z4 iff tau = i, v214/v168), with the clock as the one carrier input (v493 S1.10)? This module answers the light version of that question. [N] A. NO DYNAMICAL SELECTION (zeta-log-det side; Dedekind-eta formula, mpmath 40 digits): logF(tau) = log(tau2 |eta|^4) is exactly S-invariant (dev < 1e-35), so t = 1 is AUTOMATICALLY critical on the rectangular family tau = it; there g'(1) = 0, g''(1) = -1.298 < 0 (strict maximum of det'), but the tau1-direction has h''(0) = +0.298 > 0: tau = i is a SADDLE of the full moduli space; the critical points of logF are exactly the CM points (E2*(i) = E2*(rho) = 0 to 40 digits, generic |E2*| = 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 is Palais symmetric criticality (the independent heat trace is equally t -> 1/t invariant and equally critical), not dynamics. [N] B. SHARP RIGIDITY (lattice DtN side; v280 machinery, N = 8): the mod-4 off-character indicator of the raw free-field DtN (v215 K3 as a MODULUS FUNCTION) vanishes to 8e-15 EXACTLY at tau = i and is strictly monotonically increasing in both directions (min 8.4e-2 for t != 1): an isolated zero, not a flat valley -- the v215-K3 lever turned into a function of the modulus; 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 (only j = 1728 carries a Z4 automorphism, v214/v168). [E] C. THE Z2 CHAIN IS MODULUS-BLIND: the involution z -> -z has exactly 4 fixed points and [Z2, Delta_t] = 0 on EVERY torus (marks/mark- locality/deck^2 exist on EVERY pillowcase, v216/v264/v280 confirmed without mark input); the pillowcase det' is EXACTLY half the torus det' (+-pair structure, ratio 1 to 1e-12), so all tau conclusions transfer verbatim; ONLY the order-4 clock distinguishes tau = i. Aut(E_rho) = Z6 has element orders {1,2,3,6} -- NO order-4 element: the dynamical winner CANNOT carry the mu4 clock (dynamics and clock are logically independent selectors); GIVEN the clock, P(iZ) = P(Z) forces a3 = a2 = a1 = 0 and j = 1728 identically (v493 S1 mini): the modulus is NOT a second input. [C/O] D. 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: generation space = H^1(P^1 minus mu4) with the cusp connection). 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]. QGEO-R1 == P2-R1: ONE common residual -- 'why does the raw seam carry the order-4 clock' -- not two scattered ones. 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 ('deferred emergence test, not executable') is lifted to 'light version EXECUTED, result: rigidity, not dynamical selection' -- marks emerge (B1), the square does NOT emerge dynamically, it is pinned by the clock, and the clock is the input. NO marker moves: QGEO.REALIZE.01 stays [C]/[O] (the identification premise is untouched -- this module only locates its irreducible rest). Status: [N] the eta/E2*/j evaluations (mpmath 40 digits) and the lattice DtN curves (numpy/scipy, v280-style); [E] the exact structure results (pair structure, permutation identities, sympy forcing, Coxeter charpoly); [C] clock = carrier input; [O] the dynamical realisation of the clock. Python; Wolfram-mirrored (eta/E2*/j identities at the CM points, pillowcase halving, Coxeter charpoly; the lattice DtN curves stay Python-only, flagged), counted per GATE.WOLFRAM.02. Discovery provenance: experiments/tfpt-discovery/qgeo_r1_deferred_emergence_probe.py (2026-07-21, 17/17).""" import mpmath as mp import numpy as np import sympy as sp from scipy.linalg import expm, sqrtm from tfpt_constants import check, summary, reset, N_fam mp.mp.dps = 40 MU4 = N_fam + 1 # 4 = |mu4| = h(A3), the seam clock order # --------------------------------------------------------------------------- # exact modular objects (mpmath, 40 digits) # --------------------------------------------------------------------------- def eta(tau): """Dedekind eta(tau) = e^{pi i tau/12} prod (1 - q^n), q = e^{2 pi i tau}.""" q = mp.exp(2j * mp.pi * tau) return mp.exp(1j * mp.pi * tau / 12) * mp.qp(q) def logF(tau): """log(tau2 |eta(tau)|^4): the scale-invariant log det'(T^2_tau, area 1) up to an additive constant (OPS); pillowcase log det' = 1/2 of this + const.""" return mp.log(mp.im(tau)) + 4 * mp.log(abs(eta(tau))) def E2(tau): """Eisenstein E2(tau) = 1 - 24 sum n q^n/(1-q^n).""" q = mp.exp(2j * mp.pi * tau) s = mp.mpc(0) for n in range(1, 400): term = n * q ** n / (1 - q ** n) s += term if abs(term) < mp.mpf(10) ** (-45): break return 1 - 24 * s def E2star(tau): """The weight-2 almost-holomorphic completion; critical points of logF are exactly E2*(tau) = 0.""" return E2(tau) - 3 / (mp.pi * mp.im(tau)) def jinv(tau): """j-invariant, j(i) = 1728 (mpmath kleinj is normalised J = j/1728).""" return 1728 * mp.kleinj(tau) # --------------------------------------------------------------------------- # v280-style lattice machinery on the anisotropic torus [0,1] x [0,t] # --------------------------------------------------------------------------- N = 8 def _idx(x, y): return (x % N) * N + (y % N) def torus_laplacian(t): """Anisotropic flat-torus Laplacian for tau = i t: spacings a_x = 1/N, a_y = t/N; overall N^2 dropped (scale-irrelevant for commutators/blocks).""" L = np.zeros((N * N, N * N)) wy = 1.0 / t ** 2 for x in range(N): for y in range(N): i = _idx(x, y) L[i, i] = -2.0 - 2.0 * wy L[i, _idx(x + 1, y)] += 1.0 L[i, _idx(x - 1, y)] += 1.0 L[i, _idx(x, y + 1)] += wy L[i, _idx(x, y - 1)] += wy return L def perm_op(f): P = np.zeros((N * N, N * N)) for x in range(N): for y in range(N): xx, yy = f(x, y) P[_idx(xx, yy), _idx(x, y)] = 1 return P R_OP = perm_op(lambda x, y: (-y, x)) # order-4 clock z -> i z Z2_OP = perm_op(lambda x, y: (-x, -y)) # elliptic involution z -> -z PROJ4 = [sum((1j) ** (-k * j) * np.linalg.matrix_power(R_OP, j) for j in range(4)) / 4.0 for k in range(4)] def rel(x, y): return np.linalg.norm(x) / np.linalg.norm(y) def offblock4(M): """Off-character weight of M w.r.t. the mu4 clock decomposition (v215 K3).""" off = sum(np.linalg.norm(PROJ4[k] @ M @ PROJ4[l]) for k in range(4) for l in range(4) if k != l) return off / np.linalg.norm(M) def dtn_indicators(t): """(comm, offH, offC): [R, Delta_t] norm, mod-4 off-block of H = sqrt(-Delta) and of the quasi-free covariance C = (1 + e^H)^{-1}.""" L = torus_laplacian(t) H = sqrtm(-L + 1e-6 * np.eye(N * N)).real C = np.linalg.inv(np.eye(N * N) + expm(H)) return (rel(R_OP @ L - L @ R_OP, L), offblock4(H), offblock4(C)) # --------------------------------------------------------------------------- # continuum spectral functions on tau = i t at fixed area 1 (for C1) # --------------------------------------------------------------------------- def heat_trace(t, s=mp.mpf("0.05"), M=40): """theta(t) = sum_{(m,n) != 0} exp(-s lambda_{mn}), lambda_{mn}(it, area 1) = 4 pi^2 (m^2 t + n^2 / t); invariant under t -> 1/t by (m,n) -> (n,m).""" tot = mp.mpf(0) for m in range(-M, M + 1): for n in range(-M, M + 1): if m == 0 and n == 0: continue tot += mp.exp(-s * 4 * mp.pi ** 2 * (m * m * t + n * n / t)) return tot def run(): reset() print("v503 QGEO.EMERGE.LIGHT.01: does the SQUARE modulus emerge " "dynamically from the free field, or only by order-4 rigidity? " "(v215 lever-7 light + residual convergence QGEO-R1 == P2-R1)") # ================================================== A. exact eta/log-det side # A1. modular S-invariance of the target functional (exact identity check) t0 = mp.mpf("1.7") dev = abs(logF(1j * t0) - logF(1j / t0)) gen_tau = mp.mpc("0.31", "1.13") dev_gen = abs(logF(gen_tau) - logF(-1 / gen_tau)) check("A1 ETA/S-INVARIANCE [exact to 1e-35]: logF(tau) = log(tau2|eta|^4) obeys " "logF(it) = logF(i/t) (dev %.1e at t=1.7) and full S logF(-1/tau) = " "logF(tau) (dev %.1e at generic tau) -- the rectangular family is " "t -> 1/t symmetric, so t = 1 (tau = i) is AUTOMATICALLY critical for " "any such functional" % (dev, dev_gen), dev < mp.mpf(10) ** -35 and dev_gen < mp.mpf(10) ** -35) # A2. the log-det family on tau = i t: extremum location + curvature + table g = lambda tt: logF(1j * tt) g1 = mp.diff(g, mp.mpf(1), 1) g2 = mp.diff(g, mp.mpf(1), 2) print(" t logF(it) = log det'(T^2) + const (pillowcase = 1/2 of this)") ts = [mp.mpf(x) / 100 for x in (50, 60, 70, 85, 100, 120, 141, 170, 200)] vals = [] for tt in ts: v = g(tt) vals.append(v) print(" %-6s %s" % (mp.nstr(tt, 4), mp.nstr(v, 12))) vmax = max(vals) check("A2 RECTANGULAR FAMILY EXTREMUM [num, 40 digits]: on tau = it the " "functional has g'(1) = %s (= 0) and g''(1) = %s < 0: tau = i is a " "strict local MAXIMUM of det' (= max of the pillowcase log det') within " "the clock-symmetric rectangular family; table max at t = 1" % (mp.nstr(g1, 3), mp.nstr(g2, 8)), abs(g1) < mp.mpf(10) ** -25 and g2 < 0 and vals.index(vmax) == ts.index(mp.mpf(1))) # A3. but tau = i is a SADDLE of the full moduli space h = lambda x: logF(x + 1j) h1 = mp.diff(h, mp.mpf(0), 1) h2 = mp.diff(h, mp.mpf(0), 2) check("A3 SADDLE [num]: along the tau1-direction h(x) = logF(x + i) has " "h'(0) = %s (= 0, tau -> -taubar symmetry) and h''(0) = %s > 0: tau = i " "is a MINIMUM in the tau1-direction, hence a SADDLE of the full moduli " "space -- the 'extremum at the square' is family-relative, not global" % (mp.nstr(h1, 3), mp.nstr(h2, 8)), abs(h1) < mp.mpf(10) ** -25 and h2 > 0) # A4. criticality is exactly the CM points: E2*(i) = E2*(rho) = 0, generic != 0 rho = mp.exp(1j * mp.pi / 3) e_i, e_rho = abs(E2star(1j)), abs(E2star(rho)) e_gen = abs(E2star(gen_tau)) check("A4 CRITICAL POINTS = CM POINTS [num, 40 digits]: E2*(i) = %.1e and " "E2*(rho) = %.1e (both 0: the two exceptional-automorphism points are " "the ONLY critical points, both symmetry-forced), generic " "|E2*(0.31+1.13i)| = %.3f != 0 -- no critical point at any generic tau" % (e_i, e_rho, e_gen), e_i < mp.mpf(10) ** -35 and e_rho < mp.mpf(10) ** -35 and e_gen > 0.01) # A5. the FULL moduli space dynamically selects HEXAGONAL, not the square F_i, F_rho = logF(1j), logF(rho) check("A5 GLOBAL SELECTOR = HEXAGONAL [num]: logF(rho) = %s > logF(i) = %s " "(diff %s > 0): the free-field det' at fixed area is maximised at the " "hexagonal torus (OPS 1988), NOT the square -- free-field dynamics " "alone does NOT select tau = i on the full (2,2,2,2) moduli space" % (mp.nstr(F_rho, 10), mp.nstr(F_i, 10), mp.nstr(F_rho - F_i, 6)), F_rho > F_i) # A6. pillowcase log det' = 1/2 torus log det' (orbifold spectrum = one copy # of each +/- pair): exact pair structure on the discrete torus L1 = torus_laplacian(1.0) ev_torus = np.sort(np.linalg.eigvalsh(-L1)) w_z2, v_z2 = np.linalg.eigh(Z2_OP) Veven = v_z2[:, w_z2 > 0.5] ev_even = np.sort(np.linalg.eigvalsh(Veven.T @ (-L1) @ Veven)) # self-paired Fourier modes (m, n) with 2m = 2n = 0 mod N: 4 of them, # exactly the 4 cone points in frequency space -- all land in the even sector self_modes = [(0, 0), (N // 2, 0), (0, N // 2), (N // 2, N // 2)] lam = lambda m, n: (2 - 2 * np.cos(2 * np.pi * m / N)) + (2 - 2 * np.cos(2 * np.pi * n / N)) prod_self = np.prod([lam(m, n) for (m, n) in self_modes if (m, n) != (0, 0)]) det_torus = np.prod(ev_torus[1:]) det_even = np.prod(ev_even[1:]) ratio = det_even ** 2 / (det_torus * prod_self) dim_ok = Veven.shape[1] == (N * N + 4) // 2 check("A6 PILLOWCASE = HALF TORUS [num, exact pair structure]: dim(even) = %d " "= (N^2+4)/2 (4 self-paired cone modes); det'_even^2 / (det'_torus * " "prod_self) = %.12f = 1 exactly -- the orbifold spectrum is one copy of " "each +/- pair, so zeta_pillow(s) = zeta_torus(s)/2 in the continuum " "and log det'_pillow = (1/2) log det'_torus: ALL tau-conclusions " "(A2-A5) transfer verbatim to the pillowcase" % (Veven.shape[1], ratio), dim_ok and abs(ratio - 1) < 1e-9) # A7. rigidity witness on the family: j(it) touches 1728 exactly once (t = 1) js = [(tt, mp.re(jinv(1j * tt))) for tt in ts] j_dev = [(tt, jj - 1728) for tt, jj in js] only_t1 = all((abs(d) < mp.mpf(10) ** -25) == (tt == 1) for tt, d in j_dev) all_ge = all(d > -mp.mpf(10) ** -25 for _, d in j_dev) j_rho = abs(jinv(rho)) check("A7 RIGIDITY WITNESS [num, 40 digits]: on tau = it the j-invariant is " "real with j(it) >= 1728 and j = 1728 ONLY at t = 1 (checked on the " "9-point family; min at the square); j(rho) = %.1e = 0 (hexagonal) -- " "Aut(E) > Z2 happens only at j in {0, 1728}, and the order-4 clock " "(Z4) exists ONLY at j = 1728, i.e. tau = i (v214/v168)" % j_rho, only_t1 and all_ge and j_rho < mp.mpf(10) ** -30) # ================================================== B. lattice DtN mod-4 side # B1. topology: 4 involution fixed points and [Z2, Delta_t] = 0 for EVERY t fixed = [(x, y) for x in range(N) for y in range(N) if ((-x) % N, (-y) % N) == (x, y)] comm_z2 = [] for tt in (0.5, 0.7, 1.0, 1.3, 2.0): Lt = torus_laplacian(tt) comm_z2.append(rel(Z2_OP @ Lt - Lt @ Z2_OP, Lt)) check("B1 TOPOLOGY IS tau-INDEPENDENT [num]: the involution z -> -z has " "exactly %d fixed points (the 2-torsion = the 4 marks) on EVERY torus, " "and [Z2, Delta_t] = 0 for all t (max %.1e): mark COUNT and equality " "need NO modulus input -- v216 confirmed without inputting marks" % (len(fixed), max(comm_z2)), len(fixed) == 4 and max(comm_z2) < 1e-12) # B2. clock^2 = deck on the lattice (the v492/v493 Z8 bridge, structural) check("B2 CLOCK^2 = DECK [exact permutation identity]: R^2 = Z2 as N^2 x N^2 " "permutation matrices (%s) and R^4 = 1 -- the elliptic involution IS " "the square of the order-4 clock (v492 WP1 bridge on the lattice)" % np.allclose(np.linalg.matrix_power(R_OP, 2), Z2_OP), np.allclose(np.linalg.matrix_power(R_OP, 2), Z2_OP) and np.allclose(np.linalg.matrix_power(R_OP, 4), np.eye(N * N))) # B3. the DtN mod-4 indicator as a function of the modulus (v215 K3 curve) print(" t ||[R,Delta_t]||/||Delta_t|| off4(H_t) off4(C_t)") curve = {} for tt in (0.5, 0.7, 0.85, 1.0, 1.2, 1.5, 2.0): c, oh, oc = dtn_indicators(tt) curve[tt] = (c, oh, oc) print(" %-6.2f %-28.3e %-14.3e %-14.3e" % (tt, c, oh, oc)) at1 = curve[1.0] away = {tt: v for tt, v in curve.items() if tt != 1.0} check("B3 MOD-4 BLOCK-DIAGONALITY IFF SQUARE [num, v215 K3 as a modulus " "function]: at t = 1 the commutator and BOTH off-character weights " "vanish (%.1e, %.1e, %.1e); for every t != 1 all three are > 1e-3 " "(min %.1e) -- the raw free-field DtN is mu4 block-diagonal EXACTLY at " "the square modulus, at no other point of the family" % (at1[0], at1[1], at1[2], min(min(v) for v in away.values())), max(at1) < 1e-6 and all(min(v) > 1e-3 for v in away.values())) # B4. the indicator grows monotonically in |log t| (sharp selection, no # spurious flat directions), and respects t -> 1/t up to the anisotropy scale seq = [curve[tt][1] for tt in (1.0, 1.2, 1.5, 2.0)] seq_dn = [curve[tt][1] for tt in (1.0, 0.85, 0.7, 0.5)] check("B4 SHARP WELL [num]: off4(H_t) increases strictly away from t = 1 in " "both directions (up: %s; down: %s) -- t = 1 is an isolated zero, not " "a flat valley: the square is PINNED by the clock-symmetry indicator" % (["%.2e" % x for x in seq], ["%.2e" % x for x in seq_dn]), all(seq[i] < seq[i + 1] for i in range(3)) and all(seq_dn[i] < seq_dn[i + 1] for i in range(3))) # B5. the Z2 chain (v264/v280) holds for EVERY t: only the CLOCK pins t = 1 off_z2 = [] for tt in (0.5, 1.0, 2.0): Lt = torus_laplacian(tt) Ht = sqrtm(-Lt + 1e-6 * np.eye(N * N)).real off_z2.append(rel(Z2_OP @ Ht - Ht @ Z2_OP, Ht)) check("B5 INVOLUTION CHAIN IS MODULUS-BLIND [num]: [Z2, H_t] = 0 for all t " "(max %.1e) -- the whole v264/v280 Z2-chain (4 marks, mark-locality, " "omega o (deck)^2-invariance) is available on EVERY pillowcase; ONLY " "the order-4 clock distinguishes tau = i. The emergence question is " "therefore exactly 'why the clock', nothing else" % max(off_z2), max(off_z2) < 1e-8) # ================================================== C. extremum vs rigidity # C1. Palais symmetric criticality: EVERY t -> 1/t invariant functional is # critical at t = 1 -- criticality there carries no dynamical information th1 = mp.diff(lambda tt: heat_trace(tt), mp.mpf(1), 1) th2 = mp.diff(lambda tt: heat_trace(tt), mp.mpf(1), 2) inv_dev = abs(heat_trace(mp.mpf("1.31")) - heat_trace(1 / mp.mpf("1.31"))) check("C1 SYMMETRY-GENERIC CRITICALITY [num]: the (independent) heat trace " "theta(t) = sum exp(-s lambda_mn(it)) is t -> 1/t invariant (dev %.1e) " "and hence ALSO critical at t = 1 (theta'(1) = %s, theta''(1) = %s " "!= 0) -- criticality of the log-det at the square is a Palais " "symmetric-criticality consequence of the R-symmetry of the FAMILY, " "not evidence of dynamical selection" % (inv_dev, mp.nstr(th1, 3), mp.nstr(th2, 4)), inv_dev < mp.mpf(10) ** -35 and abs(th1) < mp.mpf(10) ** -20 and abs(th2) > mp.mpf(10) ** -6) # C2. the hexagonal winner has NO order-4 element: Z6 contains no Z4 z6_orders = sorted({sp.Integer(6) / sp.gcd(k, 6) for k in range(1, 7)}) check("C2 HEXAGONAL HAS NO CLOCK [exact]: Aut(E_rho) = Z6, element orders " "%s -- no order-4 element (4 does not divide 6); so the dynamically " "selected hexagonal point CANNOT carry the mu4 clock, and conversely " "the clock excludes the dynamical winner: dynamics and clock-symmetry " "point at DIFFERENT tau -- they are logically independent selectors" % z6_orders, 4 not in z6_orders and sp.Integer(4) not in z6_orders and z6_orders == [1, 2, 3, 6]) # C3. sympy mini-replication of the v493 link: GIVEN the clock, the square is # frozen automatically (the modulus is NOT an extra input beyond the clock) Z, a0s, a1s, a2s, a3s = sp.symbols('Z a0 a1 a2 a3') P = Z ** 4 + a3s * Z ** 3 + a2s * Z ** 2 + a1s * Z + a0s forced = sp.solve([sp.Poly(sp.expand(P.subs(Z, sp.I * Z) - P), Z).coeff_monomial(Z ** k) for k in range(4)], [a3s, a2s, a1s], dict=True) Iq = 12 * a0s # binary-quartic invariants of Z^4 + a0 Jq = sp.Integer(0) jfrozen = sp.simplify(6912 * Iq ** 3 / (4 * Iq ** 3 - Jq ** 2)) check("C3 CLOCK => SQUARE, AUTOMATICALLY [exact, v493 S1 mini]: P(iZ) = P(Z) " "forces a3 = a2 = a1 = 0 (%s), and the surviving family Z^4 + a0 has " "I = 12 a0, J = 0 identically => j = %s = 1728 for ALL a0: GIVEN the " "order-4 clock, the (2,2,2,2) modulus is frozen to the square with " "only the scale free -- the modulus is NOT a second input" % (forced == [{a3s: 0, a2s: 0, a1s: 0}], jfrozen), forced == [{a3s: 0, a2s: 0, a1s: 0}] and jfrozen == 1728) # C4. the clock arithmetic is carrier data: h(A3) = 4 = |mu4| = N_fam + 1 A = sp.Matrix([[0, 0, -1], [1, 0, -1], [0, 1, -1]]) lamx = sp.Symbol('lam') chi = sp.expand(A.charpoly(lamx).as_expr()) check("C4 THE CLOCK IS THE CARRIER INPUT [exact]: the order-4 element acting " "on H^1 is a Coxeter element of W(A3) (char poly %s = lam^3 + lam^2 + " "lam + 1, order h(A3) = 4 = |mu4| = N_fam + 1 = %d) -- exactly the P2 " "cusp-class/carrier datum of the P2 hardening v499 (its R1: " "'generation space = H^1(P^1 minus mu4) with the cusp connection'): " "the QGEO modulus rest and the P2 typing rest are the SAME order-4 input" % (chi, MU4), sp.expand(chi - (lamx ** 3 + lamx ** 2 + lamx + 1)) == 0 and sp.simplify(A ** 4 - sp.eye(3)) == sp.zeros(3, 3) and MU4 == 4) # C5. the honest verdict, tied to the computed facts verdict_facts = (F_rho > F_i # dynamics -> hexagonal and max(at1) < 1e-6 # clock indicator zero at i and all(min(v) > 1e-3 for v in away.values()) and forced == [{a3s: 0, a2s: 0, a1s: 0}]) check("C5 VERDICT [typed]: NO dynamical selection of the square (the free " "field's global det' winner is hexagonal, A5; the extremum at tau = i " "within the rectangular family is Palais-symmetry, C1). The square is " "selected by RIGIDITY alone: Aut contains Z4 iff tau = i (A7), the " "DtN is mod-4 block-diagonal iff tau = i (B3), and GIVEN the clock the " "modulus freezes automatically (C3). Therefore R1's irreducible rest " "is precisely: 'WHY does the raw seam carry the order-4 clock' -- the " "same carrier input as the P2-hardening R1 (C4): ONE common residual, " "not two. The deferred emergence test (v215 lever 7) light version " "returns: marks emerge (B1), square does NOT emerge dynamically -- it " "is pinned by the clock, and the clock is the input. NO marker moves: " "QGEO.REALIZE.01 stays [C]/[O]", verdict_facts) return summary("v503 QGEO.EMERGE.LIGHT.01: v215 lever 7 executed light -- NO " "dynamical selection of the square (global det' winner is the " "HEXAGONAL torus, logF(rho) = -1.03352 > logF(i) = -1.05469; " "tau = i is a saddle, g''(1) = -1.298 rectangular vs h''(0) = " "+0.298 in tau1; critical points = CM points E2*(i) = E2*(rho) " "= 0; criticality = Palais symmetry). The square is pinned by " "RIGIDITY: the DtN mod-4 indicator has an isolated zero (8e-15) " "exactly at tau = i (strictly monotone both ways, min 8.4e-2 " "away); j(it) = 1728 only at t = 1; Aut(hexagonal) = Z6 has NO " "order-4 element. The Z2 chain is modulus-blind (marks emerge " "on EVERY pillowcase; pillowcase det' = half torus det' " "exactly); GIVEN the clock the modulus freezes (j = 1728 " "identically). RESIDUAL CONVERGENCE: the clock = Coxeter of " "W(A3) = the P2 cusp-class carrier datum (v499 R1) -- " "QGEO-R1 == P2-R1, ONE common rest: 'why does the raw seam " "carry the order-4 clock'. No marker moves; QGEO.REALIZE.01 " "stays [C]/[O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)