"""v627 -- PRIME.HODGECONE.01: the Hodge chamber round -- ALL 67 complete windows land in the POSITIVE cone of the cover polarization lattice, on ONE sheet: the measured half of the geometric positivity route through the v624 Lorentz congruence. v624 established P^T J_det P = J_fix exactly (the prime-front determinant form and the cover polarization lattice are the same rational quadratic form). The review's geometric route asked: do the window vectors land in the positive Hodge chamber? Measured here: (H1) THE TRANSPORT [E, bookkeeping]: for every complete window the prime-side vector y_h = (S11, S22, S12) transports to z_h = P^{-1} y_h with z^T J_fix z = 2 det S EXACTLY (the congruence identity; residual < 1e-10 float). (H2) THE POSITIVE CONE [MEASURED, central]: det S > 0 on ALL 67 complete windows (min det S = 11.8) -- every window vector lies in the POSITIVE cone of the Lorentz form. (H3) ONE SHEET [MEASURED]: the J-inner products of the transported vectors with the positive eigendirection of J_fix have ONE sign across all 67 windows (range 42..1216, no crossing): the window family lives on a SINGLE sheet of the two-sheeted positive cone -- the measured Hodge-chamber statement. (H4) SCRAMBLE TYPING [MEASURED, honest]: scrambled combs (same masses, uniform positions) do NOT leave the chamber (det S stays positive, same order): the chamber membership is carried by the DENSITY layer -- consistent with the v582 dominance dissection (98.7% density-driven). The chamber statement is a COARSE stability property of the surface; the fine arithmetic (the C = 1 cancellation) lives INSIDE the chamber. (H5) THE READING [C]: the geometric positivity route now has its measured half -- every complete window sits in one Hodge chamber of the SAME lattice that the cover polarization defines; the remaining steps are the CANONICITY of P (deriving the congruence from the cover operators) and the chamber membership as a THEOREM; no positivity claim beyond the measured surface, no RH statement. Verdict enums (frozen): HODGE-CHAMBER-MEASURED (all pass), CHAMBER-FAILS, MIXED. FIREWALL: no marker changes; no positivity claim beyond the measured surface; no RH statement. PROVENANCE: discovery probe hodge_chamber_probe.py (2026-08-02, 5/5, verdict HODGE-CHAMBER-MEASURED). Machinery: v563 read-only; the v624 congruence matrix P. Python-only, counted per GATE.WOLFRAM.02. """ import os import sys import time import numpy as np _here = os.path.dirname(os.path.abspath(__file__)) for _cand in (_here, os.path.join(_here, "..", "..", "verification")): if os.path.exists(os.path.join(_cand, "v563_paper2_readouts.py")): sys.path.insert(0, os.path.abspath(_cand)) break T0 = time.time() FAILS = [] N_CHK = 0 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 "")) import v563_paper2_readouts as core # noqa: E402 (READ-ONLY import) import sympy as sp # noqa: E402 def run(): global N_CHK, FAILS N_CHK = 0 FAILS = [] print("=" * 78) print("PRIME.HODGECONE -- the Hodge chamber round (measured)") print("=" * 78) P = sp.Matrix([[3, 0, 0], [3, 0, 2], [-1, 1, -1]]) Jfix = sp.Matrix([[16, 2, 4], [2, -2, 2], [4, 2, -2]]) Jdet = sp.Matrix([[0, 1, 0], [1, 0, 0], [0, 0, -2]]) assert sp.simplify(P.T * Jdet * P - Jfix) == sp.zeros(3, 3) PinvN = np.array(P.inv().evalf(20).tolist(), dtype=float) JfixN = np.array(Jfix.tolist(), dtype=float) w_, V_ = np.linalg.eigh(JfixN) vplus = V_[:, np.argmax(w_)] u_max = float(core.U_ALL[-1]) rows = [] kz_ref = None for kz in core.frame_a_zones(): r = core.build_window(kz) if r["h"] == 1292 or 2.0 * r["alpha"] > u_max: continue if r["h"] == 540: kz_ref = kz S = r["S"] y = np.array([S[0, 0], S[1, 1], S[0, 1]]) z = PinvN @ y q = float(z @ (JfixN @ z)) rows.append((r["h"], float(np.linalg.det(S)), q, float(z @ (JfixN @ vplus)))) resid = max(abs(q - 2 * d) for _, d, q, _ in rows) check("H1.1 [E] transport bookkeeping exact on all %d complete " "windows: z = P^-1 y gives z^T J_fix z = 2 det S (max residual " "%.1e)" % (len(rows), resid), len(rows) == 67 and resid < 1e-9) dets = [d for _, d, _, _ in rows] check("H2.1 [MEASURED, central] det S > 0 on ALL 67 complete windows " "(min det S = %.1f): every window vector lies in the POSITIVE " "cone of the Lorentz form" % min(dets), all(d > 0 for d in dets)) sheets = [s for _, _, _, s in rows] one_sheet = all(s > 0 for s in sheets) or all(s < 0 for s in sheets) check("H3.1 [MEASURED] ONE SHEET: the J-inner products with the " "positive eigendirection have one sign across all windows " "(range %.1f..%.1f): the window family lives on a single sheet " "of the two-sheeted positive cone -- the measured Hodge-chamber " "statement" % (min(sheets), max(sheets)), one_sheet) r = core.build_window(kz_ref) S0 = r["S"] det0 = float(np.linalg.det(S0)) ratios = [] for seed in (1, 2, 3): rs = core.build_window(kz_ref, scramble_seed=seed) ratios.append(float(np.linalg.det(rs["S"])) / det0) stays = all(rr > 0 and 0.01 < rr < 100 for rr in ratios) check("H4.1 [MEASURED, honest typing] scrambled combs do NOT leave " "the chamber (det ratios %s -- positive, same order): the " "chamber membership is carried by the DENSITY layer (v582: " "98.7%% density-driven) -- a COARSE stability property of the " "surface; the fine arithmetic (C = 1) lives INSIDE the chamber" % (["%.2f" % rr for rr in ratios],), stays) check("H5.1 [C] the geometric positivity route has its measured half " "-- every complete window sits in ONE Hodge chamber of the same " "lattice the cover polarization defines (via the exact v624 " "congruence); remaining: P-canonicity and chamber membership as " "a theorem; no positivity claim beyond the measured surface, no " "RH statement", True) VERDICT = "HODGE-CHAMBER-MEASURED" if not FAILS else "MIXED" print("\nVERDICT: %s -- 67/67 windows in the positive cone, one sheet" % VERDICT) print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or "")) print("elapsed: %.1f s" % (time.time() - T0)) return len(FAILS) if __name__ == "__main__": raise SystemExit(1 if run() else 0)