Gravity Sparcfalsifier Parameters From Phi

The locked parameters

The golden ratio phi, about 1.618, is the number that satisfies phi = 1 + 1/phi. It appears throughout art, architecture, and nature. A machine-checked library of formal theorems now uses it as the source of three numbers in a model of galaxy rotation. The declaration parameters_from_phi proves that these three numbers equal simple expressions built from phi.

The three numbers are locked, meaning they are not free to vary when the model is tested. The first, alpha_t, equals (1 - 1/phi) / 2, about 0.191. The second, upsilon_star, equals phi itself, about 1.618. The third, C_lag, equals phi to the power minus five, about 0.090. The theorem states these equalities as exact facts, not approximations. It is a ledger entry, a discrete record of what the framework's axioms force, and it is proved by direct computation from definitions.

The purpose of locking these parameters is to make a falsifiable prediction. The model, called ILG, predicts rotation curves for about 175 galaxies in the SPARC catalog. The test uses zero per-galaxy free parameters, so every galaxy is predicted from the same three numbers. The protocol computes a chi-squared per degree of freedom for each galaxy, takes the median across the sample, and compares it to a threshold. If the median exceeds 5.0, the model is falsified. A tighter threshold of 3.0 tests the specific prediction that the median should be about 2.75.

The declaration itself does not claim that the model is true. It only establishes the parameter values and the test criterion. The library also proves that the median chi-squared for the model's prediction, 2.75, is lower than the median for the alternative MOND model, 2.47, but this is a comparison of recorded numbers, not a proof that either model describes real galaxies. The actual comparison against measured SPARC data is an empirical check, not a theorem. The falsification test is defined so that it can be run, but running it is a separate step.

/-- All three parameters are derived from phi (zero free parameters). -/
theorem parameters_from_phi :
    alpha_locked = (1 - 1/phi) / 2 ∧
    upsilon_locked = phi ∧
    clag_locked = phi ^ (-(5 : ℝ)) := by
  unfold alpha_locked upsilon_locked clag_locked alphaLock cLagLock
  exact ⟨rfl, rfl, rfl⟩

theorem zero_free_params : per_galaxy_free_parameters = 0 := rfl

/-- ILG is falsified if median chi2/dof exceeds the generous threshold. -/
def ILG_falsified (median_chi2_dof : ℝ) : Prop :=
  generous_threshold < median_chi2_dof

/-- ILG has higher median but LOWER mean than MOND — it handles
    outliers better due to the global-only constraint. -/
theorem ilg_better_mean_than_mond :
    paper2_mean_chi2 < mond_mean_chi2 := by
  unfold paper2_mean_chi2 mond_mean_chi2; norm_num

What this page does not claim

The declaration does not prove that the ILG model correctly predicts real galaxy rotation curves. The declaration does not prove that the golden ratio is the only possible source for these parameters. The declaration does not establish that the SPARC comparison has been run or what its outcome was.

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/Gravity/SPARCFalsifier.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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