Gravity Backreaction Audit Buchert Backreaction Zero

The backreaction result

In cosmology, backreaction (the effect of small-scale structure on the large-scale expansion of the universe) is often quantified by a scalar Q_D that measures how much the expansion rate varies across a spatial domain. When Q_D is zero, the average expansion follows the smooth background model exactly, with no correction from clumping. The Recognition Science framework formalizes a claim about its own modified-gravity model, called ILG, in a machine-checked library of formal theorems: the declaration buchert_backreaction_zero proves that for this model, Q_D equals zero.

The reason is structural. ILG changes the source term in Einstein's equations, multiplying the matter density by a weight function, but it does not alter the metric or the expansion rate itself. The velocity field therefore remains irrotational, a potential flow, and for any potential flow the Buchert backreaction scalar vanishes identically. The theorem is proved by definitional equality: the library defines buchert_Q_D_ilg to be 0, so the statement that it equals 0 is true by reflexivity, the simplest kind of proof. A companion theorem, ilg_preserves_background, extends this to state that Q_D remains zero for every positive scale factor a.

What this establishes is a consistency check, not a discovery. It says that within the ILG model, late-time cosmic acceleration anomalies must come from how sources are weighted, not from backreaction altering the background expansion. The framework's library also records that in the large-X limit the weight function approaches 1, recovering general relativity, with a safety bound at X = 3 × 10^24 for an epsilon of 10^-5. None of this is a measurement of the real universe; it is a formal statement about a model's internal coherence.

The declaration does not claim that backreaction is zero in the actual cosmos, nor that ILG is correct physics. It does not prove that any observed dark-energy signal is explained, and it does not compare against observational data. The theorem is a definitional identity inside a specific formalism, valuable as a check that the model does not secretly alter the background it claims to preserve, but silent on whether that model matches nature.

theorem buchert_backreaction_zero : buchert_Q_D_ilg = 0 := rfl

/-- 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⟩

/-- 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 theorem does not claim backreaction is zero in the actual universe. It does not claim ILG is a correct or observationally validated theory of dark energy. It does not claim any measurement or comparison with cosmological data.

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)

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Derived articles

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