Gravity Sparcfalsifier Falsification Decidable

The decidable falsifier

In astronomy, a model of galaxy rotation curves is judged by how well its predicted velocities match observed ones. The SPARC sample is a widely used collection of 175 disk galaxies with measured rotation curves. A model's quality is scored by chi-squared per degree of freedom, a standard statistical measure where smaller means a better fit. The Recognition Science framework has a specific prediction for these curves, and its machine-checked library of formal theorems contains a declaration, falsification_decidable, that pins down the logic of testing that prediction.

The declaration proves a simple but important fact: for any possible median chi-squared value, the model is either falsified or it passes. The formal statement reads: for every real number x, either ILG_falsified x or ILG_passes x. Here ILG stands for the framework's galaxy model, and the dividing line is set at a generous threshold of 5.0. If the median chi-squared per degree of freedom across the sample exceeds 5.0, the model is falsified; if it is at or below 5.0, the model passes. This is a theorem, not a hope: the library proves it by the elementary fact that any real number is either greater than 5.0 or not. The practical consequence is that the model cannot hide behind ambiguity. A definite answer always exists once the data are in.

The theorem sits inside a larger certificate that bundles the model's testable commitments. The certificate, SPARCFalsifierCert, records that the model uses zero free parameters per galaxy, that all its parameters derive from the golden ratio phi, and that its mean chi-squared of 4.23 beats the alternative MOND model's mean of 4.65. These are separate declarations, each with its own proof. The decidable theorem itself does not say the model is true, does not say the data have been collected, and does not say what the median chi-squared actually is. It only guarantees that the test has a definite outcome.

What the declaration does not claim is as important as what it proves. It does not assert that the SPARC data have been run through the test, or that the model passes. The predicted median chi-squared of 2.75 and the thresholds are definitions, not measurements. The theorem also does not claim that a passing result confirms the framework's physics; a passing test only means the model survives one check. The library's own honesty controls, such as rotating galaxies by 180 degrees or swapping gas and star contributions, are defined but not yet evaluated. The declaration is a logical guarantee about the test's structure, not a report on the universe.

/-- The falsification criterion is decidable (for any real number). -/
theorem falsification_decidable (x : ℝ) :
    ILG_falsified x ∨ ILG_passes x := by
  unfold ILG_falsified ILG_passes generous_threshold
  exact le_or_gt x 5.0 |>.elim (Or.inr) (Or.inl)

theorem zero_free_params : per_galaxy_free_parameters = 0 := rfl

/-- 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⟩

/-- 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 SPARC data have been run through the test and the model's fate is known. A passing result would confirm the framework's physics rather than merely surviving one check. The declaration says anything about the actual median chi-squared value that will be observed.

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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