"""v306 -- leave-one-out seed cross-validation: phi0 is ONE seed, tested many ways. An adversarial reviewer's correct observation is that a large slice of the TFPT scorecard (theta12, theta13, beta, Omega_b, lambda_C, ...) shares the SINGLE seed phi0 = 1/(6 pi) + 48 c3^4 (v23/tfpt_constants). That is a liability ONLY if the shared seed is hidden; it becomes a FALSIFIABLE asset once stated as an out-of-sample test: phi0 is FIXED by the two axioms (no fit). Pretend, adversarially, that it is a single free parameter. Back-solve phi0 INDEPENDENTLY from each seed observable's measured central value. If the formulas were numerology the back-solved phi0^(i) would scatter; if they are one real seed they cluster at the axiom value. Then LEAVE ONE OUT: fit phi0 from all-but-one observable and PREDICT the held-out one, reporting the out-of-sample residual in sigma. [E] 1. the theory seed equals the texture-inversion of sin^2 theta12 (plumbing). [N] 2. the error-weighted mean of the five back-solved phi0^(i) reproduces the AXIOM seed phi0 to < 1% (the seed is real, not fitted per observable). [N] 3. the precise subset (lambda_C, Omega_b) clusters with small relative spread around phi0 (the single seed is sharply over-determined). [N] 4. LEAVE-ONE-OUT: every held-out observable is predicted within < 3 sigma out of sample; the most-tensioned hold-out is sin^2 theta13 (~2 sigma), the honest current pressure point (cf. v62). [N] 5. NEGATIVE CONTROL / POWER: shuffling which measured value feeds which inversion explodes the cross-validation chi^2 by >100x -- the test fails loudly when the seed assignment is wrong (so its success is not vacuous). HONEST SCOPE: measured centrals/sigmas are the repo-documented ones (NuFIT 6.0, ACT DR6, Planck 2018, PDG 2024 -- the same sources as v62/v100); nothing is imported fresh. This is a consistency / out-of-sample statistic, not a new prediction. Python-only (floats), like v100/v62; 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 # ---- the five "seed grammar" observables (each a pure function of phi0) ------- # id, measured central, experimental sigma, source, and the inverse map y -> phi0. # lambda_C(phi0)=sqrt(phi0(1-phi0)) so its inversion takes the small root. def _inv_lambda(y): return (1.0 - math.sqrt(max(0.0, 1.0 - 4.0 * y * y))) / 2.0 OBS = [ dict(id="sin2_theta12", 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=2.0), dict(id="sin2_theta13", m=0.02195, sig=0.00058, src="NuFIT 6.0", fwd=lambda p: p * math.exp(-5.0 / 6.0), inv=lambda y: y * math.exp(5.0 / 6.0), dinv=math.exp(5.0 / 6.0)), dict(id="beta_deg", m=0.215, sig=0.074, src="ACT DR6", fwd=lambda p: (p / (4.0 * PI)) * (180.0 / PI), inv=lambda y: y * (PI / 180.0) * 4.0 * PI, dinv=(PI / 180.0) * 4.0 * PI), dict(id="Omega_b", m=0.0493, sig=0.0006, src="Planck 2018", fwd=lambda p: (1.0 - 1.0 / (4.0 * PI)) * p, inv=lambda y: y / (1.0 - 1.0 / (4.0 * PI)), dinv=1.0 / (1.0 - 1.0 / (4.0 * PI))), dict(id="lambda_C", m=0.2245, sig=0.0005, src="PDG 2024", fwd=lambda p: math.sqrt(p * (1.0 - p)), inv=_inv_lambda, dinv=lambda y: 2.0 * y / math.sqrt(1.0 - 4.0 * y * y)), ] def back_solve(obs): """phi0 estimate and its propagated sigma from one observable.""" phi = obs["inv"](obs["m"]) d = obs["dinv"](obs["m"]) if callable(obs["dinv"]) else obs["dinv"] return phi, abs(d) * obs["sig"] def weighted_mean(items): """Inverse-variance weighted mean and its sigma of (value, sigma) pairs.""" w = [1.0 / s ** 2 for _v, s in items] sw = sum(w) mean = sum(wi * v for wi, (v, _s) in zip(w, items)) / sw return mean, math.sqrt(1.0 / sw) def run(): reset() print("v306 leave-one-out seed cross-validation (phi0 is one seed, tested many ways)") # ---- 1. plumbing: theory seed = texture inversion of sin^2 theta12 ---- th12 = next(o for o in OBS if o["id"] == "sin2_theta12") check("theory seed phi0 = 1/(6pi)+48 c3^4 = %.7f reproduces sin^2 theta12 = " "1/3 - phi0/2 = %.7f (plumbing)" % (PHI0, th12["fwd"](PHI0)), abs(th12["fwd"](PHI0) - (1.0 / 3.0 - PHI0 / 2.0)) < 1e-12) # ---- 2/3. back-solve phi0 from each observable; cluster around the axiom ---- ests = {o["id"]: back_solve(o) for o in OBS} print(" --- per-observable back-solved phi0 (adversarial: phi0 as free dial) ---") for o in OBS: phi, s = ests[o["id"]] ns = (phi - PHI0) / s print(f" {o['id']:<14} phi0 = {phi:.6f} +- {s:.6f} " f"({ns:+.2f} sigma vs axiom {PHI0:.6f}) [{o['src']}]") items = list(ests.values()) mean, msig = weighted_mean(items) rel = abs(mean - PHI0) / PHI0 print(f" error-weighted mean phi0 = {mean:.6f} +- {msig:.6f} " f"(axiom {PHI0:.6f}; {rel*100:.3f}% off)") check("CLUSTER [N]: the error-weighted mean of the 5 back-solved phi0 " "reproduces the AXIOM seed to < 1%% (%.3f%%) -- one real seed, not a " "per-observable fit" % (rel * 100), rel < 0.01) precise = [ests["lambda_C"], ests["Omega_b"]] pmean, _ = weighted_mean(precise) pspread = max(abs(v - pmean) / pmean for v, _s in precise) check("OVERDETERMINATION [N]: the precise subset {lambda_C, Omega_b} clusters " "within 1.5% relative spread around phi0 (sharp over-determination)", pspread < 0.015) # ---- 4. leave-one-out: predict each held-out observable out of sample ---- print(" --- leave-one-out out-of-sample predictions ---") worst_id, worst_ns = None, 0.0 loo_ok = True for o in OBS: rest = [ests[k] for k in ests if k != o["id"]] phi_loo, sphi = weighted_mean(rest) pred = o["fwd"](phi_loo) ns = (pred - o["m"]) / o["sig"] loo_ok = loo_ok and abs(ns) < 3.0 if abs(ns) > abs(worst_ns): worst_ns, worst_id = ns, o["id"] print(f" hold out {o['id']:<14} predict {pred:.6f} vs measured " f"{o['m']:.6f} ({ns:+.2f} sigma)") check("LEAVE-ONE-OUT [N]: every held-out observable is predicted within 3 " "sigma out of sample (one seed survives cross-validation)", loo_ok) check("PRESSURE POINT [N]: the most-tensioned hold-out is sin^2 theta13 " "(~2 sigma), the honest current pressure point (cf. v62); |ns|=%.2f" % abs(worst_ns), worst_id == "sin2_theta13" and 1.5 < abs(worst_ns) < 2.6) # ---- 5. negative control: shuffle data<->formula, chi^2 must explode ---- def chi2(assign): # assign: map obs-index -> measured-value-index; back-solve & test cluster vals = [] for i, o in enumerate(OBS): mj = OBS[assign[i]]["m"] try: phi = o["inv"](mj) except (ValueError, ZeroDivisionError): return 1e18 d = o["dinv"](mj) if callable(o["dinv"]) else o["dinv"] vals.append((phi, abs(d) * o["sig"])) mu, _ = weighted_mean(vals) return sum((v - mu) ** 2 / s ** 2 for v, s in vals) ident = list(range(len(OBS))) chi2_true = chi2(ident) cyc = ident[1:] + ident[:1] # cyclic shuffle (derangement) chi2_shuf = chi2(cyc) print(f" shuffle control: chi^2(correct assignment) = {chi2_true:.2f}, " f"chi^2(shuffled) = {chi2_shuf:.1f}") check("NEG CONTROL / POWER [N]: shuffling data<->formula explodes the " "cross-validation chi^2 by >100x -- the test fails loudly on a wrong " "seed assignment (success in 2-4 is not vacuous)", chi2_shuf > 100.0 * max(chi2_true, 1e-6)) return summary("v306 leave-one-out seed cross-validation") if __name__ == "__main__": raise SystemExit(1 if run() else 0)