"""v590 -- QGEO.INVOL.01: the sheet-involution existence question of GATE.QGEO.01, narrowed to a two-demand selection with UNCONDITIONAL existence. THE RIGIDITY [E]: the integral involutions commuting with the sheet flow V form EXACTLY the spectral sign group (Z/2)^3 -- all eight sign choices epsilon on V's eigenspaces {0, 1, 2} give INTEGER matrices that STABILIZE the integral order Z (nontrivial: for a generic integral operator the spectral projectors are not integral), and the same holds verbatim on the alt twin. THE NARROWING CHAIN [E]: (i) the v574 rank theorem (a two-dim-positive involution cannot carry the binary trace code) kills the trace +1 half: 6 nontrivial involutions -> 3 of Sigma-signature (1,-1,-1); (ii) the seam-reversal demand S e3 = -e3 narrows 3 -> 2; (iii) the zero-mode-fixing demand (+1 on ker V, i.e. the involution leaves the inert direction alone and flips exactly the dynamical sheet pair {1,2}) singles out S0 = [[1,-1,0],[0,-1,0],[0,0,-1]] (trace -1, S0^2 = I) UNIQUELY. Existence is therefore UNCONDITIONAL (S0 is explicit, integral, order-stabilizing, of exact Sigma-signature); the residual freedom of GATE.QGEO.01's involution input is the explicit, finite selection above -- two geometric demands, each machine-checked, no continuum input. HONEST SCOPE: this is the V-COMPATIBLE (automorphism) involution class -- the corpus RP reflection theta(X) = Sigma X^T Sigma is an anti-automorphism and does NOT commute with V (checked); the two structures play different roles; the gate demand addressed here is existence/selection of the sheet involution, not the RP reflection. The alt twin admits the same classification (the involution does not separate the twins). FIREWALL: exact sympy arithmetic throughout; the two demands are declared, not derived from continuum geometry (that remains the open [C] of the gate); no marker moves beyond the narrowing note. Verdict enums (frozen): EXISTENCE-UNCONDITIONAL, AMBIGUOUS, FAILS. PROVENANCE: discovery probe involution_existence_probe.py (2026-08-01, 7/7, EXISTENCE-UNCONDITIONAL); exact sympy throughout. Python-only, counted per GATE.WOLFRAM.02. """ import itertools import time import sympy as sp from sympy.matrices.normalforms import hermite_normal_form T0 = time.time() FAILS = [] N_CHK = 0 I3 = sp.eye(3) IDX7 = ((0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)) E3 = sp.Matrix([0, 0, 1]) S0_EXPECT = sp.Matrix([[1, -1, 0], [0, -1, 0], [0, 0, -1]]) def check(name, ok, detail=""): global N_CHK N_CHK += 1 if not ok: FAILS.append(name.split()[0]) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (": " + detail) if detail else "")) def Qc(c): return sp.Matrix([[3, 1, 0], [3, 2, 0], [3, c, 1]]) def order_data(c): Q = Qc(c) U, V = Q * sp.diag(1, 0, 0), Q * sp.diag(0, 1, 1) ws, fr = [I3], [I3] for _ in range(5): fr = [w * G for w in fr for G in (U, V)] ws += fr Mrows = sp.Matrix([[w[i, j] for (i, j) in IDX7] for w in ws]) H = hermite_normal_form(Mrows.T) basis = [] for k in range(H.cols): Mx = sp.zeros(3, 3) for r, (i, j) in enumerate(IDX7): Mx[i, j] = H[r, k] basis.append(Mx) Bm = sp.Matrix([[b[i, j] for (i, j) in IDX7] for b in basis]).T return U, V, Bm def in_order(X, Bm): if X[0, 2] != 0 or X[1, 2] != 0: return False v = sp.Matrix([X[i, j] for (i, j) in IDX7]) coef = Bm.solve(v) return all(x.is_integer for x in coef) def spectral_involutions(V): evs = sorted(V.eigenvals().keys(), key=lambda e: abs(e)) Ps = [] for e in evs: P = sp.eye(3) for e2 in evs: if e2 != e: P = P * (V - e2 * sp.eye(3)) / (e - e2) Ps.append(sp.simplify(P)) out = [] for eps in itertools.product((1, -1), repeat=3): S = sp.simplify(eps[0] * Ps[0] + eps[1] * Ps[1] + eps[2] * Ps[2]) out.append((eps, S)) return evs, Ps, out def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("QGEO.INVOL -- existence and selection of the sheet " "involution") print("=" * 78) results = {} for c in (2, 0): U, V, Bm = order_data(c) evs, Ps, cands = spectral_involutions(V) all_integral = all(all(x.is_integer for x in S) for _, S in cands) all_stab = all(in_order(S * U * S, Bm) and in_order(S * V * S, Bm) for _, S in cands) all_invol = all(sp.simplify(S * S - I3) == sp.zeros(3, 3) for _, S in cands) all_commute = all(sp.simplify(S * V - V * S) == sp.zeros(3, 3) for _, S in cands) results[c] = (V, cands, all_integral, all_stab, all_invol, all_commute) check("I1.1 [E, THE RIGIDITY] the integral V-commuting involutions " "are EXACTLY the spectral sign group (Z/2)^3: all eight sign " "choices on V's eigenspaces {0,1,2} are INTEGER matrices, " "genuine involutions, commute with V, and STABILIZE the " "integral order Z -- on BOTH twins (nontrivial: generic " "integral operators do not have integral spectral " "projectors)", all(results[c][2] and results[c][3] and results[c][4] and results[c][5] for c in (2, 0))) V2, cands2 = results[2][0], results[2][1] nontriv = [(eps, S) for eps, S in cands2 if S != I3 and S != -I3] sig = [(eps, S) for eps, S in nontriv if sp.trace(S) == -1] check("I2.1 [E, narrowing step 1 -- the v574 type theorem] the " "trace +1 half (two-dim-positive type, killed by the v574 " "rank theorem: it cannot carry the binary trace code) drops " "%d -> %d candidates, all of exact Sigma-signature " "(1,-1,-1)" % (len(nontriv), len(sig)), len(nontriv) == 6 and len(sig) == 3) seam = [(eps, S) for eps, S in sig if sp.simplify(S * E3 + E3) == sp.zeros(3, 1)] check("I2.2 [E, narrowing step 2 -- seam reversal] the demand " "S e3 = -e3 (the involution reverses the seam line) narrows " "%d -> %d" % (len(sig), len(seam)), len(seam) == 2) ker_fix = [(eps, S) for eps, S in seam if eps[0] == 1] check("I2.3 [E, narrowing step 3 -- the zero mode] the demand " "that the involution FIX ker V (act as +1 on the inert " "direction, flipping exactly the dynamical sheet pair " "{1,2}) singles out ONE candidate: S0 = %s -- explicit, " "integral, order-stabilizing, trace -1: EXISTENCE IS " "UNCONDITIONAL" % (ker_fix[0][1].tolist() if ker_fix else None), len(ker_fix) == 1 and ker_fix[0][1] == S0_EXPECT) SIG_RP = sp.diag(1, -1, -1) rp_commutes = sp.simplify(SIG_RP * V2 - V2 * SIG_RP) == sp.zeros(3, 3) check("I3.1 [E, honest scope] the corpus RP reflection Sigma = " "diag(1,-1,-1) (used as the ANTI-automorphism theta(X) = " "Sigma X^T Sigma) does NOT commute with V (%s): the sheet " "involution S0 and the RP reflection are different " "structures with different roles -- the gate demand " "addressed here is the sheet involution, not the RP slot" % rp_commutes, not rp_commutes) V0, cands0 = results[0][0], results[0][1] s0_alt = [S for eps, S in cands0 if eps == (1, -1, -1)][0] check("I4.1 [E] the alt twin admits the same classification and " "its selected involution is %s: the involution does NOT " "separate the twins -- consistent with v567/v572 (every " "classical certificate coincides on the pair); the " "narrowing is a GATE.QGEO.01 statement, not a bit selector" % s0_alt.tolist(), sp.simplify(s0_alt * s0_alt - I3) == sp.zeros(3, 3) and sp.trace(s0_alt) == -1) check("I5.1 [C, the relocation] GATE.QGEO.01's involution-" "existence input relocates: existence is now DERIVED " "(unconditional, explicit S0), and the residual freedom is " "the finite two-demand selection (seam reversal + zero-mode " "fixing), machine-checked; deriving those two demands from " "the continuum geometry (mu4/D4/H1) is the remaining open " "[C] of the gate -- named, not claimed", True) VERDICT = "EXISTENCE-UNCONDITIONAL" if not FAILS else "AMBIGUOUS" print("\nVERDICT: %s -- rigidity (Z/2)^3 on both twins; narrowing " "6 -> 3 -> 2 -> 1; S0 explicit with Sigma-signature; the " "two demands named" % VERDICT) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) print("FIREWALL: exact sympy; demands declared; continuum " "derivation stays open; no marker moves beyond the " "narrowing note") print("--- QGEO.INVOL.01 sheet involution existence: %d passed, " "%d failed ---" % (N_CHK - len(FAILS), len(FAILS))) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)