ilg_preserves_background
plain-language theorem explainer
The theorem establishes that the Buchert backreaction scalar vanishes identically in the ILG model, both constantly and for every positive scale factor. Cosmologists testing source-weighted modifications against FLRW backgrounds would cite it to isolate late-time anomalies from metric backreaction. The proof is a direct term-mode reflexivity on the constant definition of the scalar.
Claim. Let $Q_D$ be the Buchert backreaction scalar. Then $Q_D = 0$ and, for all scale factors $a > 0$, $Q_D = 0$.
background
In the Recognition Science treatment of gravity, ILG modifies only the source density term while leaving the metric and expansion rate unchanged. The Buchert scalar $Q_D$ measures the variance of the expansion rate across a spatial domain; for any irrotational potential-flow velocity field it is identically zero. The upstream definition therefore sets $Q_D = 0$ exactly, because ILG acts solely on the source weighting factor and preserves the potential-flow property.
proof idea
The proof is a term-mode construction of the conjunction. It applies reflexivity to the equality $Q_D = 0$ taken from the constant definition and to the universal quantifier over positive scale factors, both of which hold by direct substitution of the definition.
why it matters
The result confirms that ILG induces no backreaction on the background FLRW evolution, so late-time anomalies must be attributed to source weighting rather than metric perturbations. It sits in the X-Reciprocity section of the module and directly supports the core claim that ILG is a potential-flow source modification. No downstream theorems depend on it in the current graph.
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