phi_neg2_eq

In Recognition Science the scale ratio φ is forced by self-similarity of the discrete ledger (T6 in the forcing chain). Its algebraic hallmark is the quadratic identity φ² = φ + 1, proved in Constants from the closed form (1+√5)/2.

This module (ILG A Priori Prediction Certificate) closes the paper's post-hoc gap: α and the amplitude C must be derived before SPARC comparison. The derivation chain runs RCL → unique J-cost → self-similarity → φ → memory kernel → α = (1-1/φ)/2 and C = φ^{-2}.

Negative integer powers of φ therefore appear as explicit constants. The identity here rewrites φ^{-2} as a simple rational expression in φ, which is the form needed for decimal bounds.

Invoke the upstream lemma phi_sq_eq giving φ² = φ + 1. Positivity of φ (Constants.phi_pos) yields positivity of φ². Rewrite the left-hand side by zpow_neg and zpow_ofNat so φ^{-2} becomes (φ²)^{-1}. Substitute the quadratic identity, then convert the field inverse to the division form 1/(φ+1) via one_div.

The a priori ILG amplitude is C = φ^{-2} ≈ 0.382. Downstream lemmas phi_neg2_gt and phi_neg2_lt immediately rewrite through this identity to prove 0.381 < φ^{-2} < 0.384, sandwiching the decimal window used in the certificate.

That sandwich supports the module claim that C is forced by self-similarity before any fit, completing the chain from RCL and T5–T6 to the SPARC-facing prediction structure APrioriPrediction. Without the exact rewrite, the decimal bounds would rest on floating-point evaluation rather than the algebraic relation φ² = φ + 1.