Gravity Sparcfalsifier

The SPARC falsifier

The SPARC falsifier is a formal criterion for testing a prediction about galaxy rotation. SPARC is a real catalog of about 175 galaxies with measured rotation curves, the speeds of stars and gas at various distances from the center. The framework's gravity model, called ILG, predicts those curves using only three global constants, all fixed by the golden ratio phi. The falsifier asks a simple question: if you compute the predicted curves with zero per-galaxy adjustments, does the median error exceed a chosen threshold? If it does, the model is falsified.

Galaxy rotation curves are a classic test for gravity theories. Stars far from a galaxy's center orbit faster than visible mass alone would explain, and the usual response is to add dark matter. The ILG model instead claims the extra speed comes from the framework's own structure. The falsifier makes that claim testable. It defines two thresholds: a generous one at 5.0, meant to allow for systematic errors, and a tight one at 3.0, which tests the specific prediction. The model's own predicted median chi-squared per degree of freedom is 2.75, so the tight threshold sits just above it. The generous threshold is the official pass/fail line.

The module also records a comparison against MOND, a different modified-gravity approach. The mean chi-squared for ILG on the SPARC sample is 4.23, while MOND's is 4.65. A theorem in the library proves ILG's mean is better. This is not a claim that ILG is correct; it is a statement about which model fits the current data better under the same zero-free-parameter protocol.

In Recognition Science, the falsifier is part of the framework's commitment to disprovability. The library proves that the pass/fail decision is decidable: for any median chi-squared value, either the model is falsified or it passes. It also proves the protocol uses zero per-galaxy free parameters and that all three constants come from phi. The structure bundles these facts into a certificate, a single object that carries the whole protocol's guarantees.

The plain-language takeaway is this: the framework has written down, in machine-checked form, the exact experiment that would kill its gravity model. The threshold is generous, the parameters are locked, and the decision is binary. If the real SPARC data ever produces a median error above 5.0, the model is done. That is what a falsifier is for.

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

/-- The generous falsification threshold for ILG chi-squared per dof.
    If median chi2/dof > 5.0 across the SPARC sample, ILG is falsified. -/
def generous_threshold : ℝ := 5.0

/-- The tight threshold matching the RS prediction (median ~ 2.75).
    If median chi2/dof > 3.0, the specific RS prediction is refuted
    (though ILG as a framework might survive with different parameters). -/
def tight_threshold : ℝ := 3.0

/-- RS-predicted median chi2/dof across SPARC. -/
def predicted_median : ℝ := 2.75

/-- 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

/-- 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)

What this page does not claim

The ILG model has been tested against the actual SPARC data in this module. The framework's gravity model is correct or even plausible. The SPARC falsifier proves anything about dark matter.

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)

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.

Derived articles

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