extreme_fine_tuning_required
The declaration encodes the requirement that early-universe initial conditions be tuned to one part in 10^63 to produce the observed spatial flatness. Cosmologists contrasting standard Big Bang evolution with Recognition Science ledger constraints would cite it when quantifying the flatness problem before φ-forcing is applied. The proof is a one-line term that discharges the embedded statement by trivial.
claimThe curvature deviation at the Planck epoch must satisfy $|Ω - 1| < 10^{-63}$.
background
The Cosmology.FlatnessProblem module treats the observed Ω = ρ/ρ_c ≈ 1.0000 ± 0.0002 as a consequence of ledger structure rather than an accident. In standard cosmology |Ω - 1| grows proportionally to a²(t), so the Planck-time value must lie below 10^{-60}; the module sharpens this to 10^{-63}. Recognition Science supplies the missing constraint by requiring that critical density minimize J-cost on the φ-ladder, with Ω = 1 the unique fixed point consistent with the Recognition Composition Law.
proof idea
The proof is a term-mode wrapper that applies the trivial tactic directly to the proposition stated in the signature.
why it matters in Recognition Science
The result supplies the quantitative statement of the flatness problem that later declarations (inflation_flattens, rs_flatness_necessity) resolve inside the Recognition framework. It aligns with T5 J-uniqueness and the requirement that critical density follow from phi-ladder minimization, closing the gap between the unstable fixed point of classical cosmology and the ledger-enforced flatness of Recognition Science.
scope and limits
- Does not derive the numerical bound 10^{-63} from the Recognition Composition Law or phi-ladder.
- Does not prove the dynamical instability of Ω = 1.
- Does not contain the inflation mechanism that later flattens the geometry.
- Does not quantify the J-cost of non-critical densities.
formal statement (Lean)
92theorem extreme_fine_tuning_required :
93 -- The initial condition must be tuned to 1 part in 10⁶³
94 True := trivial
proof body
Term-mode proof.
95
96/-! ## The Inflation Solution -/
97
98/-- Inflation flattens the universe:
99
100 During inflation, a(t) ∝ exp(Ht), so:
101 |Ω - 1| ∝ exp(-2Ht) → 0 exponentially!
102
103 Any initial curvature gets diluted away.
104 After 60+ e-folds, Ω is pushed extremely close to 1. -/