Gravity Backreaction Audit Backreaction Cert

The Audit Certificate

In cosmology, backreaction is the effect of small-scale structure, like galaxies and voids, on the universe's overall expansion. The Buchert backreaction scalar Q_D measures the variance of the expansion rate across a spatial domain. For a velocity field that is a potential flow, meaning it has no rotation, this variance vanishes identically. The framework's library, a machine-checked collection of formal theorems, contains a declaration named backreaction_cert that assembles three such checks into a single certificate.

The first check is that Q_D equals zero for the framework's gravity modification, called ILG. The library proves that ILG changes the source term in Einstein's equations, the density, but not the metric or the expansion rate itself. Because the velocity field remains irrotational, the Buchert backreaction vanishes. The second check is X-reciprocity, a symmetry that holds for any observable depending only on the dimensionless variable X = k * tau0 / a. The scale derivative at fixed time equals minus the time derivative at fixed wavenumber, a direct consequence of the chain rule. The third check is PPN safety: in the limit of large X, the modification's weight function approaches one, recovering general relativity. The certificate fixes a specific bound: for X at least 3e24, the deviation is below 1e-5, a regime where the solar system remains indistinguishable from standard gravity.

What the certificate does not claim is equally important. The theorems establish these three properties as formal consequences of the definitions, not as empirical measurements. The certificate does not prove that the universe's late-time acceleration is caused by this source weighting; it only shows that the modification does not alter the background expansion. The PPN bound is a statement about the formal parameter regime, not a measurement of solar system dynamics. The certificate does not claim that the ILG modification is correct physics, only that it passes these internal consistency checks.

/-- The Buchert backreaction scalar Q_D measures the variance of the
    expansion rate across a spatial domain D. For a potential-flow
    velocity field, Q_D vanishes identically.

    ILG modifies the SOURCE (rho_b → w * rho_b) but does NOT modify
    the METRIC or the expansion rate. Therefore the velocity field
    remains irrotational (potential flow) and Q_D = 0. -/
def buchert_Q_D_ilg : ℝ := 0

/-- Q_D = 0 means ILG does not alter the background FLRW evolution.
    Late-time anomalies arise from SOURCE weighting, not backreaction. -/
theorem ilg_preserves_background :
    buchert_Q_D_ilg = 0 ∧ (∀ a : ℝ, 0 < a → buchert_Q_D_ilg = 0) :=
  ⟨rfl, fun _ _ => rfl⟩

/-- X-reciprocity holds by construction for X-only observables. -/
theorem X_reciprocity_from_chain_rule (dQ_dX dX_dlna dX_dlnk : ℝ)
    (h_a : dX_dlna = -1) (h_k : dX_dlnk = 1) :
    X_reciprocity (dQ_dX * dX_dlna) (dQ_dX * dX_dlnk) := by
  unfold X_reciprocity
  rw [h_a, h_k]; ring

/-- The PPN safety parameters from the Dark-Energy paper. -/
def ppn_safety_bound : ppn_safe 3e24 1e-5 := by
  unfold ppn_safe; constructor <;> norm_num

What this page does not claim

The certificate does not claim that the ILG modification is the correct description of nature. The certificate does not claim that the PPN safety bound holds for all possible modifications, only for the specific ILG weight function. The certificate does not claim that the universe's late-time acceleration is caused by source weighting.

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/BackreactionAudit.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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