pmns_theta13_match
plain-language theorem explainer
The theorem shows that the geometric prediction sin²θ₁₃ = φ^{-8} from the eight-tick structure lies within 0.002 of the measured reactor angle 0.022. Neutrino phenomenologists and Recognition Science modelers cite it when assembling the full PMNS matrix from ledger overlaps. The tactic proof imports tight numeric bounds on φ^8, inverts them via reciprocal properties, and closes the absolute-value inequality with linarith after rewriting to abs_lt.
Claim. $|sin^2 θ_{13}^{pred} - 0.022| < 0.002$, where $sin^2 θ_{13}^{pred}$ is the leading-order geometric prediction φ^{-8} obtained from the eight-tick octave in the cubic ledger.
background
Module Physics.MixingDerivation formalizes mixing angles from edge-dual couplings in the 8-tick windows of the cubic ledger, replacing ad-hoc numerics with topological ratios. The phi-ladder supplies mass and angle predictions via φ^rung scaling; sin²θ₁₃ is identified with the φ^{-8} rung. Upstream lemmas include the reciprocal automorphism (CostAlgebra.reciprocal) that inverts positive ratios and the strict bounds phi_pow8_gt and phi_pow8_lt (Numerics.Interval.PhiBounds) that pin φ^8 inside (46.97, 46.99). The local setting is Phase 7.2, where unitarity of the mixing matrix follows from 8-tick closure.
proof idea
The tactic unfolds sin2_theta13_pred to expose the φ^{-8} expression, then invokes phi_pow8_gt and phi_pow8_lt to bound φ^8. Inversion uses one_div_lt_one_div_of_lt together with positivity and zpow_neg_coe_of_pos to obtain the reciprocal bounds 1/46.99 < φ^{-8} < 1/46.97. The proof rewrites the target to abs_lt, splits into two inequalities, and discharges each with lt_trans followed by linarith.
why it matters
The result populates the theta13_match field inside the MixingCert of mixing_verified, completing the certificate that all CKM and PMNS elements arise from ledger geometry. It realizes the T7 eight-tick octave prediction for the reactor angle and sits inside the alpha band of Recognition Science constants. The parent theorem mixing_verified assembles the full set of derived ratios and angles; the downstream row_pmns_theta13 simply records the same bound for scorecard use.
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