"""v465 -- the cross-sector one-parameter seed test: a CMB polarisation angle and a quark mixing angle share the SINGLE seed phi0 (the theta13-independent slice). This is the EXTERNALISED, minimal-observable form of the v306 leave-one-out seed cross-validation, motivated by the observation that the sharpest falsifiable signature of a SHARED-ORIGIN theory is a correlation no non-shared-origin model predicts: in the Standard Model the cosmic-birefringence rotation angle beta (a CMB EB/TB polarisation observable) and the Cabibbo angle lambda_C = |V_us| (a quark-flavour observable) have NOTHING to do with each other. TFPT says they are two readouts of the one axiom seed phi0 = 1/(6 pi) + 48 c3^4: lambda_C = sqrt(phi0 (1 - phi0)), beta = (phi0 / (4 pi)) * (180/pi) deg, sin^2 th12 = 1/3 - phi0/2 (the third, neutrino, channel). [E] 1. plumbing: the axiom phi0 reproduces the three frozen-registry values (lambda_C, sin^2 theta12, beta) to machine precision. [N] 2. CROSS-SECTOR (headline): back-solving phi0 INDEPENDENTLY from beta (CMB) and from lambda_C (quark) yields the SAME seed within < 1 sigma_combined -- a CMB-polarisation angle and a quark-mixing angle point to one number. [N] 3. one-parameter joint fit: the error-weighted mean of the three back-solved phi0 reproduces the AXIOM seed to < 1% (one real seed, not a per-observable fit); the joint chi^2 at the axiom value (ZERO free parameters, 3 dof) has chi^2/dof < 1. [N] 4. combined pull: the axiom seed fits the three-observable cross-sector set within < 1 sigma combined (sqrt(chi^2)); reported with the p-value. [N] 5. theta13 EXCLUSION is declared, not cherry-picked: adding sin^2 theta13 raises the joint chi^2 by ~4 (the documented ~2 sigma pressure point), so this slice is DELIBERATELY the theta13-independent cross-sector statement; the theta13 tension is separately pre-registered (v328/v338). [N] 6. NEGATIVE CONTROL / POWER: shuffling which measured value feeds which inverse map explodes the cluster chi^2 by >100x -- the test fails loudly on a wrong data<->formula assignment (its success is not vacuous). HONEST SCOPE: this is a SUBSET of v306 (three of its five seed observables), restated as the minimal externally-runnable cross-sector falsification for outsiders. It is a CONSISTENCY / out-of-sample statistic, NOT a new prediction and it upgrades NO prediction's status: phi0 is FIXED by the two axioms (the "fit" is adversarial framing). The very low joint chi^2 reflects that these are the three MUTUALLY CONSISTENT channels; the one tensioned channel (sin^2 theta13, ~2 sigma) is excluded ON PURPOSE and separately fenced (v328/v338), which check 5 makes transparent. Measured centrals/sigmas are the repo-documented ones (NuFIT 6.0, ACT DR6, Planck PR4, PDG 2024; same sources as v62/v306 and experiments/cmb-birefringence-seed/data/measurements.json). Python-only (floats), like v100/v306; flagged in the wolfram README. """ import math from tfpt_constants import check, summary, reset, phi0 as phi0_mp PHI0 = float(phi0_mp) # axiom seed = 1/(6 pi) + 48 c3^4 PI = math.pi # frozen-registry values (predictions_frozen.json) for the plumbing check FROZEN = {"lambda_C": 0.2243762368847217731120972, "sin2_th12": 0.3067473572449105696786871, "beta_deg": 0.2424350309009295284924315} def _combine(meas): """Inverse-variance combine a list of (value, sigma) into (mean, sigma).""" w = [1.0 / s ** 2 for _v, s in meas] sw = sum(w) return sum(wi * v for wi, (v, _s) in zip(w, meas)) / sw, math.sqrt(1.0 / sw) # The cross-sector, theta13-independent observables. beta combines the two # INDEPENDENT birefringence measurements (Planck PR3 2020 is superseded by PR4 # and shares its absolute-angle systematic -> NOT independent, excluded). _BETA_M, _BETA_S = _combine([(0.215, 0.074), # ACT DR6 (Diego-Palazuelos & Komatsu 2025) (0.30, 0.11)]) # Planck PR4 (Eskilt 2022) OBS = [ dict(id="lambda_C", sector="quark", m=0.22431, sig=0.00085, src="PDG 2024 |V_us|", fwd=lambda p: math.sqrt(p * (1.0 - p)), inv=lambda y: (1.0 - math.sqrt(max(0.0, 1.0 - 4.0 * y * y))) / 2.0, dinv=lambda y: 2.0 * y / math.sqrt(1.0 - 4.0 * y * y)), dict(id="sin2_th12", sector="neutrino", m=0.307, sig=0.012, src="NuFIT 6.0", fwd=lambda p: 1.0 / 3.0 - p / 2.0, inv=lambda y: 2.0 * (1.0 / 3.0 - y), dinv=lambda y: 2.0), dict(id="beta_deg", sector="CMB", m=_BETA_M, sig=_BETA_S, src="ACT DR6 + Planck PR4 (independent, combined)", fwd=lambda p: (p / (4.0 * PI)) * (180.0 / PI), inv=lambda y: y * (PI / 180.0) * 4.0 * PI, dinv=lambda y: (PI / 180.0) * 4.0 * PI), ] # the deliberately EXCLUDED channel (the documented ~2 sigma pressure point) S13 = dict(id="sin2_th13", m=0.02195, sig=0.00058, fwd=lambda p: p * math.exp(-5.0 / 6.0)) def _back(o): """Back-solved phi0 and its propagated sigma from one observable.""" return o["inv"](o["m"]), abs(o["dinv"](o["m"])) * o["sig"] def run(): reset() print("v465 cross-sector one-parameter seed test (CMB beta + quark lambda_C, " "theta13-independent slice of v306)") # ---- 1. plumbing: axiom phi0 reproduces the three frozen values ---- ok_plumb = all(abs(o["fwd"](PHI0) - FROZEN[o["id"]]) <= 1e-9 * FROZEN[o["id"]] for o in OBS) check("plumbing [E]: axiom phi0 = %.7f reproduces the frozen lambda_C, " "sin^2 theta12, beta to 1e-9" % PHI0, ok_plumb) est = {o["id"]: _back(o) for o in OBS} print(" --- per-observable back-solved phi0 (phi0 as adversarial free dial) ---") for o in OBS: phi, s = est[o["id"]] print(f" {o['id']:<10} [{o['sector']:<8}] phi0 = {phi:.6f} +- {s:.6f} " f"({(phi - PHI0) / s:+.2f} sigma vs axiom) [{o['src']}]") # ---- 2. CROSS-SECTOR headline: beta (CMB) vs lambda_C (quark) ---- pb, sb = est["beta_deg"] pl, sl = est["lambda_C"] dsig = abs(pb - pl) / math.sqrt(sb ** 2 + sl ** 2) print(f" CROSS-SECTOR: beta(CMB) -> phi0 = {pb:.6f} +- {sb:.6f} vs " f"lambda_C(quark) -> phi0 = {pl:.6f} +- {sl:.6f} (diff {dsig:.2f} sigma)") check("CROSS-SECTOR [N]: a CMB-polarisation angle (beta) and a quark-mixing " "angle (lambda_C) back-solve to the SAME seed within 1 sigma_combined " "(diff %.2f sigma) -- a correlation only a shared-origin theory predicts" % dsig, dsig < 1.0) # ---- 3. one-parameter joint fit: weighted mean = axiom; chi^2/dof < 1 ---- mean, _msig = _combine(list(est.values())) rel = abs(mean - PHI0) / PHI0 chi2 = sum(((o["fwd"](PHI0) - o["m"]) / o["sig"]) ** 2 for o in OBS) dof = len(OBS) print(f" error-weighted mean phi0 = {mean:.6f} (axiom {PHI0:.6f}; " f"{rel * 100:.3f}% off); joint chi^2 = {chi2:.4f} (dof {dof}, " f"chi^2/dof = {chi2 / dof:.4f})") check("ONE-PARAMETER FIT [N]: the error-weighted mean of the 3 back-solved " "phi0 reproduces the AXIOM seed to < 1%% (%.3f%%) and the joint chi^2 " "at the axiom (0 free params) has chi^2/dof < 1" % (rel * 100), rel < 0.01 and chi2 / dof < 1.0) # ---- 4. combined pull + p-value ---- combined_pull = math.sqrt(chi2) for o in OBS: print(f" {o['id']:<10} pull = {(o['fwd'](PHI0) - o['m']) / o['sig']:+.3f} sigma") print(f" combined pull sqrt(chi^2) = {combined_pull:.3f} sigma") check("COMBINED PULL [N]: the axiom seed (zero free parameters) fits the " "three-observable cross-sector set within 1 sigma combined " "(sqrt(chi^2) = %.3f)" % combined_pull, combined_pull < 1.0) # ---- 5. theta13 exclusion is declared, not cherry-picked ---- chi2_with = chi2 + ((S13["fwd"](PHI0) - S13["m"]) / S13["sig"]) ** 2 d_chi2 = chi2_with - chi2 print(f" theta13 exclusion: adding sin^2 theta13 raises chi^2 " f"{chi2:.3f} -> {chi2_with:.3f} (delta {d_chi2:.2f} = the ~2 sigma " f"pressure point, pre-registered v328/v338)") check("THETA13 EXCLUSION [N]: including sin^2 theta13 adds delta chi^2 ~ 4 " "(sqrt ~ 2 sigma) -- the excluded channel carries the documented " "pressure, so this slice is the DECLARED theta13-independent statement, " "not cherry-picking (%.2f, sqrt %.2f)" % (d_chi2, math.sqrt(d_chi2)), 3.0 < d_chi2 < 5.0 and 1.7 < math.sqrt(d_chi2) < 2.3) # ---- 6. negative control: shuffle data<->formula, cluster chi^2 explodes ---- def cluster_chi2(assign): vals = [] for i, o in enumerate(OBS): mj = OBS[assign[i]]["m"] try: phi = o["inv"](mj) d = abs(o["dinv"](mj)) * o["sig"] except (ValueError, ZeroDivisionError): return 1e18 vals.append((phi, d)) mu, _ = _combine(vals) return sum((v - mu) ** 2 / s ** 2 for v, s in vals) c_true = cluster_chi2([0, 1, 2]) ratios = [cluster_chi2(p) / max(c_true, 1e-9) for p in ([1, 2, 0], [2, 0, 1])] print(f" shuffle control: true chi^2 = {c_true:.4f}; derangement ratios " f"= {[f'{r:.0f}x' for r in ratios]}") check("NEG CONTROL / POWER [N]: shuffling data<->formula explodes the cluster " "chi^2 by >100x -- the cross-sector agreement fails on a wrong " "assignment (min ratio %.0fx)" % min(ratios), min(ratios) > 100.0) return summary("v465 cross-sector one-parameter seed test") if __name__ == "__main__": raise SystemExit(1 if run() else 0)