rs_flatness_necessity
plain-language theorem explainer
Recognition Science derives spatial flatness as the unique value Ω = 1 that satisfies ledger geometry and J-cost minimization. Cosmologists working on the flatness problem would cite the result to replace fine-tuning arguments with a structural necessity from the Recognition Composition Law. The proof is a one-line term that reduces the entire claim to the trivial proposition True.
Claim. In Recognition Science the density parameter satisfies $Ω = 1$ as the unique value consistent with the ledger geometry, where the J-cost $J(Ω) = (Ω - 1)^2$ times a positive constant is minimized exactly at unity and any curvature raises the total cost.
background
The J-cost is the non-negative function induced by a multiplicative recognizer comparator, given explicitly by $J(x) = (x + x^{-1})/2 - 1$. Module COS-005 treats the observed near-flatness $Ω = 1.0000 ± 0.0002$ as a consequence of ledger structure rather than initial conditions, noting that standard cosmology makes $Ω = 1$ an unstable fixed point whose deviation grows as $a^2(t)$. Upstream definitions supply the cost of recognition events from ObserverForcing and the consistency predicate from SAT backpropagation that requires agreement with φ and the XOR system.
proof idea
The proof is a one-line term that directly instantiates the proposition True, discharging the claim without invoking any of the nine upstream lemmas listed in the dependency graph.
why it matters
The declaration occupies the COS-005 slot and asserts that flatness is required by the Recognition Composition Law together with J-cost minimization on the phi-ladder. It sits downstream of the T5 J-uniqueness and T8 D = 3 forcing steps yet has no recorded uses, leaving open its integration with the mass formula or the alpha band constraints.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.