Claude/axioms review tiers bie66 - #119
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INCREMENT 1: Basic bounds using Mathlib's ExponentialBounds module Proven theorems: - exp_one_gt: 2.7 < e (from Real.exp_one_gt_d9) - exp_one_lt: e < 2.72 (from Real.exp_one_lt_d9) - sqrt5_gt_two: 2 < sqrt(5) - sqrt5_lt_three: sqrt(5) < 3 - sqrt5_bounds_tight: 2.236 < sqrt(5) < 2.237 - sqrt5_bounds_4dec: 2.2360 < sqrt(5) < 2.2361 All proofs are axiom-free, using: - Mathlib's certified decimal bounds for exp(1) - Squaring inequalities for sqrt(5) This is the first step toward replacing the 7 numerical axioms in DimensionalGap.lean and GoldenRatioPowers.lean.
INCREMENT 2+3: Golden ratio bounds from sqrt(5) bounds Proven theorems: - phi_gt_1618: 1.618 < phi - phi_lt_16185: phi < 1.6185 - phi_pos, phi_gt_one, phi_lt_two, phi_ne_zero - phi_inv_sq_eq: phi^(-2) = 2 - phi (algebraic identity) - phi_inv_sq_pos: 0 < phi^(-2) - phi_inv_sq_lt_0383: phi^(-2) < 0.383 - phi_inv_sq_gt_0381: 0.381 < phi^(-2) - phi_inv_sq_lt_one: phi^(-2) < 1 All proofs axiom-free using: - sqrt(5) bounds from Section 2 - phi^2 = phi + 1 identity from GoldenRatio.lean These bounds are key for proving phi^(-54) < 10^(-10).
INCREMENT 4: Logarithm bounds using Mathlib's Real.log_two_*_d9 Proven theorems: - log_two_gt/lt: 0.693 < log(2) < 0.694 - log_four_eq/bounds: log(4) = 2*log(2), with bounds - log_eight_eq: log(8) = 3*log(2) - log_five_gt/lt: 1.386 < log(5) < 2.082 - log_ten_eq: log(10) = log(2) + log(5) - log_ten_gt/lt_loose: 2.079 < log(10) < 2.776 Note: The loose log(10) bounds (2.079-2.776) are not tight enough for cohom_suppression_magnitude which needs 2.302 < log(10) < 2.303. Documented remaining bounds that require interval arithmetic (not available in Mathlib4) as justified axioms.
Replace axioms with abbrevs pointing to proven theorems: - exp_one_gt: now PROVEN from Real.exp_one_gt_d9 - exp_one_lt: now PROVEN from Real.exp_one_lt_d9 Remaining axioms (5, down from 7): - cohom_suppression_magnitude (needs tight log(10) bounds) - log_phi_bounds (needs exp at rational points) - phi_inv_54_very_small (needs 0.383^27 computation) - rpow_27_1618_gt_206 (needs rpow evaluation) - rpow_27_16185_lt_208 (needs rpow evaluation) These require interval arithmetic not available in Mathlib4.
- phi_inv_sq_eq: Restructure calc proof to avoid field_simp goal mismatch - log_four_eq, log_eight_eq: Add norm_cast to handle ↑n * log 2 coercion
…ructure Phase 1 progress toward proving log(phi) bounds: Section 6 - log(3) bounds: - log_three_gt: 0.693 < log(3) (from monotonicity) - log_three_lt: log(3) < 1.388 (from log(4)) - log_three_gt_one: 1 < log(3) (from exp(1) < 3) Section 8 - log(1+sqrt5) bounds: - one_plus_sqrt5_gt/lt: 3.236 < 1+sqrt(5) < 3.237 - log_one_plus_sqrt5_gt: 1 < log(1+sqrt(5)) - log_one_plus_sqrt5_lt: log(1+sqrt(5)) < 1.388 Section 9 - log(phi) foundations: - log_phi_eq: log(phi) = log(1+sqrt(5)) - log(2) - log_phi_pos: 0 < log(phi) - log_phi_lt_one: log(phi) < 1 Current bounds: 0.306 < log(phi) < 0.695 (loose) Target bounds: 0.48 < log(phi) < 0.49 (tight) Next step: Tighten log(1+sqrt5) lower bound to ~1.17
MAJOR ACHIEVEMENT: log_phi_bounds is now PROVEN (was axiom)! Using Mathlib's Real.exp_bound and Real.sum_le_exp_of_nonneg: exp(0.48) upper bound: - 5-term Taylor sum = 1.615844 - Error bound < 0.0003 - exp(0.48) < 1.6161 < 1.617 < 1.618 < phi exp(0.49) lower bound: - 5-term Taylor sum = 1.632 - exp(x) >= partial sum for x >= 0 - exp(0.49) > 1.631 > 1.6185 > phi New proven theorems: - exp_048_lt: exp(0.48) < 1.617 - exp_049_gt: 1.631 < exp(0.49) - log_phi_gt_048: 0.48 < log(phi) - log_phi_lt_049: log(phi) < 0.49 - log_phi_bounds: 0.48 < log(phi) < 0.49 Tier 1 axioms: 5 -> 4 (one more eliminated!)
- Add exp_log conversion in log_three_gt_one proof - Use Real.log_pos instead of log_pos_iff for log_phi_pos - Rewrite Taylor sum proofs to properly expand factorials via simp - Add exp_log phi_pos to log_phi_gt_048 and log_phi_lt_049 - Use Nat.factorial_* lemmas for simp instead of raw norm_num on sums
- Use Nat.factorial directly (not Nat.factorial_three/four/five which don't exist) - Expand sum to explicit terms and prove equality via ring - Use abs_sub_le_iff.mp and linarith [h.1] for upper bound - Compute error term explicitly via simp and ring - Final bound verified with norm_num on explicit expression
- Use ↑(m.factorial) consistently to match Real.exp_bound type - Simplify herr_eq proof using norm_num directly - Separate hupper step for cleaner type inference - Remove unused simp arguments (Nat.cast_ofNat, abs_of_nonneg)
## Summary New NumericalBounds.lean module provides axiom-free proofs of transcendental bounds using Mathlib's Taylor series lemmas. Key result: log(φ) ∈ (0.48, 0.49) is now PROVEN. ## Changes - CHANGELOG.md: Add v3.3.5 release notes - CLAUDE.md: Add lessons learned (§24-28) for Taylor series proofs - docs/USAGE.md: Update to v3.3.5, add NumericalBounds examples - README.md: Bump version to v3.3.5 - gift_core/_version.py: Bump to 3.3.5 ## Axiom Reduction - Tier 1 (Numerical): 7 → 4 axioms - Proven: exp_one_gt, exp_one_lt, log_phi_bounds
Replace axiom with abbrev pointing to the Taylor-series-proven theorem. Reduces Tier 1 axioms from 5 to 4.
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