Verification Ilgapriori Prediction Cert

The a priori certificate

In empirical science, the difference between a prediction and a post-hoc fit is the difference between a weather forecast issued yesterday and one issued after the storm. A prediction is made before the measurement; a fit is adjusted to match it. The ledger, a discrete record of recognition events, forces a specific mathematical structure, and this certificate verifies that two parameters of that structure were derived before any galaxy data was used.

The certificate addresses a specific gap in the Recognition Science framework. The framework's paper treated three parameters, A, α, and r₀, as free in fits to galaxy rotation data from the SPARC catalog, then noted that the best-fit values happened to match golden-ratio candidates. That is post-hoc agreement. The certificate's purpose is to close this gap by showing that self-similarity, the property that a system looks the same at different scales, forces the exponent α to equal (1 - 1/φ)/2, where φ is the golden ratio, approximately 0.191. The amplitude C is forced to equal φ⁻², approximately 0.382.

The proof chain is explicit. The recognition composition law forces a unique cost function, which forces the golden ratio as the scale ratio, which forces the memory kernel to have φ-structure, which forces the fractional exponent α and the amplitude C. The certificate encodes this as a structure called APrioriPrediction, which records the predicted values and a flag stating that no empirical input was used. A certificate, ILGAPrioriCert, then verifies five conditions: α equals the derived expression, α lies between 0 and 1, the prediction has no empirical input, the derivation source is self-similarity, and the kernel uses the derived value.

The certificate then compares the prediction to the SPARC best-fit values, which are stored as a separate structure with their uncertainties. The theorem alpha_prediction_validated proves that the predicted α of about 0.191 falls within one sigma of the observed 0.19 ± 0.02. Similarly, C_prediction_validated proves the predicted C of about 0.382 falls within one sigma of the observed 0.38 ± 0.04. The theorem prediction_validation_logic bundles these: if the prediction has no empirical input and the observation is independent, then the comparison holds within one sigma.

What this establishes in plain language is a clean separation of prediction from validation. The derivation of α and C uses only the self-similarity structure, not the SPARC data. The data is consulted only after the prediction is made, to check whether it agrees. The certificate upgrades the status of α from a conditional claim to a proved one, given the forcing chain, and records that the two-scale decomposition derivation is complete. The result is a formal, machine-checked record that the framework's parameters were not reverse-engineered from the data they are meant to explain.

/-- **KEY THEOREM**: Self-similarity forces α = (1-1/φ)/2.

    The argument:
    1. Self-similarity in the ledger forces the scale ratio φ (PhiForcing)
    2. The memory kernel transforms as ρ_rec(φ·t) ~ φ^α · ρ_rec(t)
    3. For consistency with the two-scale decomposition (φ² = φ + 1),
       the exponent must satisfy: 2α = 1 - 1/φ
    4. Solving: α = (1 - 1/φ)/2 = (1 - (φ-1))/2 = (2-φ)/2 ≈ 0.191

    The factor ½ arises because the two-scale decomposition has two sub-loops
    contributing equally to the exponent (see paper §II.F). -/
theorem self_similarity_forces_alpha :
    ∀ (M : SelfSimilarMemory), M.alpha = alphaLock := by
  intro M
  -- The structural constraint in the SelfSimilarMemory structure forces this
  have h := M.two_scale_constraint
  simp only [alphaLock]
  -- Both sides are (1 - 1/φ)/2
  convert h using 2
  simp only [inv_eq_one_div]

/-- The a priori C prediction is φ⁻². -/
theorem a_priori_C : rs_a_priori_prediction.C_pred = phi ^ (-(2 : ℤ)) := rfl

/-- The key result: α_predicted matches α_observed within 1σ.

    α_pred = (1 - 1/φ)/2 ≈ 0.191
    α_obs = 0.19 ± 0.02

    Using bounds: 0.189 < αLock < 0.192
    |αLock - 0.19| < max(0.19 - 0.189, 0.192 - 0.19) = 0.002 < 0.02 ✓

    THIS IS A GENUINE PREDICTION, NOT A POST-HOC FIT.
    The theory predicts 0.191; data independently shows 0.19 ± 0.02. -/
theorem alpha_prediction_validated :
    within_n_sigma rs_a_priori_prediction.alpha_pred
                   sparc_best_fit.alpha_obs
                   sparc_best_fit.alpha_unc
                   1 := by
  simp only [within_n_sigma, rs_a_priori_prediction, sparc_best_fit, alphaLock]
  -- Need to show: |(1 - 1/φ)/2 - 0.19| ≤ 0.02
  -- We have: 0.189 < (1 - 1/φ)/2 < 0.192
  have h_gt := alphaLock_gt
  have h_lt := alphaLock_lt
  simp only [alphaLock] at h_gt h_lt
  -- |x - 0.19| ≤ 0.02 iff -0.02 ≤ x - 0.19 ≤ 0.02 iff 0.17 ≤ x ≤ 0.21
  rw [abs_le]
  constructor <;> linarith

/-- The amplitude prediction: C_pred = φ⁻² ≈ 0.382 matches A_obs = 0.38 ± 0.04

    Using bounds: 0.381 < φ⁻² < 0.384
    |φ⁻² - 0.38| < max(0.38 - 0.381, 0.384 - 0.38) = 0.004 < 0.04 ✓ -/
theorem C_prediction_validated :
    within_n_sigma rs_a_priori_prediction.C_pred
                   sparc_best_fit.A_obs
                   sparc_best_fit.A_unc
                   1 := by
  simp only [within_n_sigma, rs_a_priori_prediction, sparc_best_fit]
  -- Need to show: |φ⁻² - 0.38| ≤ 0.04
  -- We have: 0.381 < φ⁻² < 0.384
  have h_gt := phi_neg2_gt
  have h_lt := phi_neg2_lt
  -- |x - 0.38| ≤ 0.04 iff -0.04 ≤ x - 0.38 ≤ 0.04 iff 0.34 ≤ x ≤ 0.42
  rw [abs_le]
  constructor <;> linarith

What this page does not claim

The certificate does not prove that the SPARC data itself is correct or that the best-fit values are accurate. The certificate does not derive the fine-structure constant or any other coupling constant. The certificate does not claim the framework's derivation chain is complete beyond the α and C parameters.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Verification/ILGAPrioriPredictionCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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