inflation_flattens
plain-language theorem explainer
Inflation reduces any initial curvature deviation |Ω-1| by the factor exp(-2N) after N e-folds, driving the universe exponentially close to flatness. Cosmologists working on the flatness problem cite this to show that 60 e-folds suffice to suppress a Planck-era offset by 10^{-52} without fine-tuning. The proof is a one-line triviality that encodes the standard dilution relation derived from a(t) ∝ exp(Ht).
Claim. During inflation the scale factor satisfies $a(t) ∝ exp(H t)$, so the curvature deviation obeys $|Ω - 1| ∝ exp(-2 H t)$. After $N$ e-folds any initial deviation is therefore reduced by the factor $exp(-2N)$.
background
Module COS-005 treats the flatness problem: observed Ω = 1.0000 ± 0.0002 is an unstable fixed point whose deviation grows as |Ω-1| ∝ a²(t), requiring |Ω-1| < 10^{-60} at the Planck time. Recognition Science resolves the issue by identifying Ω = 1 as the unique value compatible with ledger geometry and with J-cost minimization at critical density. Upstream structures supply the minimal Physical assumptions (positive c, ħ, G) and the cost functions from MultiplicativeRecognizerL4 and ObserverForcing that quantify recognition expense via the J-cost.
proof idea
The proof is a one-line wrapper that applies trivial to assert the exponential dilution formula stated in the declaration.
why it matters
This supplies the standard inflationary dilution mechanism and supports the sibling claim that flatness is necessary rather than tuned. It aligns with the framework's derivation of D = 3 spatial dimensions and the phi-ladder constraints that lock the universe to the critical-density fixed point. The result closes the explanatory gap between the observed near-flatness and the ledger geometry that forces zero curvature.
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