import Erdos1135.Tao.Syracuse.AffineReverse import Mathlib.Algebra.Field.GeomSum import Mathlib.Analysis.Complex.ExponentialBounds import Mathlib.Analysis.SpecialFunctions.Log.Basic import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Data.ZMod.Basic import Mathlib.Tactic /-! # Section 6 Corollary 6.3 Star Surface This module records the repaired Corollary 6.3 statement surface and the deterministic final adapter from small congruent cleared offsets to tuple equality. The analytic 3-adic separation and smallness estimates remain open. -/ namespace Erdos1135 namespace Tao /-- Source prefix/interval weight `a_[i+1,j]`, with zero-based Lean endpoints. -/ def taoSection6IntervalWeight (as : List ℕ+) (i j : ℕ) : ℕ := (((as.drop i).take (j - i)).map fun a : ℕ+ => (a : ℕ)).sum /-- The repaired lower reference point `Q = n log_2 3 - C_A^2 log n`. -/ noncomputable def taoCor63StarQ (CA : ℝ) (n : ℕ) : ℝ := (n : ℝ) * (Real.log 3 / Real.log 2) - CA ^ 2 * Real.log (n : ℝ) /-- Repaired range `(6.8 star)`: `Q < l <= Q + 2 C_A log n`. -/ noncomputable def taoCor63StarLRange (CA : ℝ) (n l : ℕ) : Prop := taoCor63StarQ CA n < (l : ℝ) ∧ (l : ℝ) ≤ taoCor63StarQ CA n + 2 * CA * Real.log (n : ℝ) /-- Inclusive repaired form of the local concentration hypothesis `(6.12)`, using zero-based Lean endpoints: source `0 <= i < j <= k+1` is encoded by `i < j` and `j <= as.length`. -/ noncomputable def taoCor63StarInclusiveTypical (CA : ℝ) (n : ℕ) (as : List ℕ+) : Prop := ∀ i j : ℕ, i < j → j ≤ as.length → |((taoSection6IntervalWeight as i j : ℕ) : ℝ) - 2 * (((j - i : ℕ) : ℝ))| ≤ CA * (Real.sqrt ((((j - i : ℕ) : ℝ)) * Real.log (n : ℝ)) + Real.log (n : ℝ)) /-- Cleared offset sum appearing in the proof after multiplying Tao's `F` offset by `2^l`: `sum_j 3^(j-1) 2^(l-a_[1,j])`. -/ def taoCor63ClearedOffsetSumFrom (l idx pref : ℕ) : List ℕ+ → ℕ | [] => 0 | a :: as => 3 ^ idx * 2 ^ (l - (pref + (a : ℕ))) + taoCor63ClearedOffsetSumFrom l (idx + 1) (pref + (a : ℕ)) as /-- Top-level cleared offset sum, starting at source index `j = 1`. -/ def taoCor63ClearedOffsetSum (l : ℕ) (as : List ℕ+) : ℕ := taoCor63ClearedOffsetSumFrom l 0 0 as /-- Repaired Corollary 6.3 statement in cleared-residue form. This records the operator-directed replacement of printed `(6.8)` and strict-range `(6.12)` by `(6.8 star)` and inclusive `(6.12)`. It is deliberately a `Prop` target, not a theorem. -/ noncomputable def TaoCor63StarClearedSeparationStatement : Prop := ∀ CA : ℝ, 17 ≤ CA → ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ k l : ℕ, taoCor63StarLRange CA n l → ∀ as bs : List ℕ+, as.length = k + 1 → bs.length = k + 1 → taoTupleWeight as = l → taoTupleWeight bs = l → taoCor63StarInclusiveTypical CA n as → taoCor63StarInclusiveTypical CA n bs → Nat.ModEq (3 ^ n) (taoCor63ClearedOffsetSum l as) (taoCor63ClearedOffsetSum l bs) → as = bs /-- The error term appearing in repaired `(6.12)` for a prefix of length `j`. -/ noncomputable def taoCor63PrefixError (CA : ℝ) (n j : ℕ) : ℝ := CA * (Real.sqrt ((j : ℝ) * Real.log (n : ℝ)) + Real.log (n : ℝ)) /-- Inclusive typicality specialized to the source prefix interval `a_[1,j]`. -/ theorem taoCor63StarInclusiveTypical_prefix_abs {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) {j : ℕ} (hj_pos : 0 < j) (hj_len : j ≤ as.length) : |((taoSection6IntervalWeight as 0 j : ℕ) : ℝ) - 2 * (j : ℝ)| ≤ taoCor63PrefixError CA n j := by simpa [taoCor63PrefixError] using htyp 0 j hj_pos hj_len /-- Lower prefix-weight bound obtained from repaired inclusive typicality. -/ theorem taoCor63StarInclusiveTypical_prefix_lower {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) {j : ℕ} (hj_pos : 0 < j) (hj_len : j ≤ as.length) : 2 * (j : ℝ) - taoCor63PrefixError CA n j ≤ ((taoSection6IntervalWeight as 0 j : ℕ) : ℝ) := by have h := taoCor63StarInclusiveTypical_prefix_abs (CA := CA) (n := n) (as := as) htyp hj_pos hj_len have hleft := (abs_le.mp h).1 linarith /-- Upper prefix-weight bound obtained from repaired inclusive typicality. -/ theorem taoCor63StarInclusiveTypical_prefix_upper {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) {j : ℕ} (hj_pos : 0 < j) (hj_len : j ≤ as.length) : ((taoSection6IntervalWeight as 0 j : ℕ) : ℝ) ≤ 2 * (j : ℝ) + taoCor63PrefixError CA n j := by have h := taoCor63StarInclusiveTypical_prefix_abs (CA := CA) (n := n) (as := as) htyp hj_pos hj_len have hright := (abs_le.mp h).2 linarith theorem taoSection6IntervalWeight_zero_length (as : List ℕ+) : taoSection6IntervalWeight as 0 as.length = taoTupleWeight as := by simp [taoSection6IntervalWeight, taoTupleWeight, List.map_eq_flatMap] /-- Prefix typicality at the full source length, rewritten as tuple weight. -/ theorem taoCor63StarInclusiveTypical_total_abs {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen_pos : 0 < as.length) : |((taoTupleWeight as : ℕ) : ℝ) - 2 * (as.length : ℝ)| ≤ taoCor63PrefixError CA n as.length := by have h := taoCor63StarInclusiveTypical_prefix_abs (CA := CA) (n := n) (as := as) htyp hlen_pos le_rfl simpa [taoSection6IntervalWeight_zero_length as] using h /-- Lower total-weight consequence of repaired inclusive typicality. -/ theorem taoCor63StarInclusiveTypical_total_lower {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen_pos : 0 < as.length) : 2 * (as.length : ℝ) - taoCor63PrefixError CA n as.length ≤ ((taoTupleWeight as : ℕ) : ℝ) := by have h := taoCor63StarInclusiveTypical_prefix_lower (CA := CA) (n := n) (as := as) htyp hlen_pos le_rfl simpa [taoSection6IntervalWeight_zero_length as] using h /-- Upper total-weight consequence of repaired inclusive typicality. -/ theorem taoCor63StarInclusiveTypical_total_upper {CA : ℝ} {n : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen_pos : 0 < as.length) : ((taoTupleWeight as : ℕ) : ℝ) ≤ 2 * (as.length : ℝ) + taoCor63PrefixError CA n as.length := by have h := taoCor63StarInclusiveTypical_prefix_upper (CA := CA) (n := n) (as := as) htyp hlen_pos le_rfl simpa [taoSection6IntervalWeight_zero_length as] using h /-- Total-weight absolute estimate after fixing the source length and weight. -/ theorem taoCor63StarInclusiveTypical_weight_abs_of_length {CA : ℝ} {n k l : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen : as.length = k + 1) (hweight : taoTupleWeight as = l) : |(l : ℝ) - 2 * ((k + 1 : ℕ) : ℝ)| ≤ taoCor63PrefixError CA n (k + 1) := by have hlen_pos : 0 < as.length := by omega have h := taoCor63StarInclusiveTypical_total_abs (CA := CA) (n := n) (as := as) htyp hlen_pos simpa [hlen, hweight] using h /-- Lower total-weight estimate after fixing the source length and weight. -/ theorem taoCor63StarInclusiveTypical_weight_lower_of_length {CA : ℝ} {n k l : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen : as.length = k + 1) (hweight : taoTupleWeight as = l) : 2 * ((k + 1 : ℕ) : ℝ) - taoCor63PrefixError CA n (k + 1) ≤ (l : ℝ) := by have hlen_pos : 0 < as.length := by omega have h := taoCor63StarInclusiveTypical_total_lower (CA := CA) (n := n) (as := as) htyp hlen_pos simpa [hlen, hweight] using h /-- Upper total-weight estimate after fixing the source length and weight. -/ theorem taoCor63StarInclusiveTypical_weight_upper_of_length {CA : ℝ} {n k l : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) (hlen : as.length = k + 1) (hweight : taoTupleWeight as = l) : (l : ℝ) ≤ 2 * ((k + 1 : ℕ) : ℝ) + taoCor63PrefixError CA n (k + 1) := by have hlen_pos : 0 < as.length := by omega have h := taoCor63StarInclusiveTypical_total_upper (CA := CA) (n := n) (as := as) htyp hlen_pos simpa [hlen, hweight] using h /-- The zero-based prefix interval weight is the tuple weight of `as.take j`. -/ theorem taoSection6IntervalWeight_zero_eq_tupleWeight_take (as : List ℕ+) (j : ℕ) : taoSection6IntervalWeight as 0 j = taoTupleWeight (as.take j) := by simp [taoSection6IntervalWeight, taoTupleWeight, Function.comp_def, ← List.map_eq_flatMap] /-- Tuple weight is monotone under taking a prefix. -/ theorem taoTupleWeight_take_le (as : List ℕ+) (j : ℕ) : taoTupleWeight (as.take j) ≤ taoTupleWeight as := by induction as generalizing j with | nil => simp [taoTupleWeight] | cons a as ih => cases j with | zero => simp [taoTupleWeight] | succ j => exact Nat.add_le_add_left (ih j) (a : ℕ) /-- Prefix interval weights are bounded by total tuple weight. -/ theorem taoSection6IntervalWeight_zero_le_tupleWeight (as : List ℕ+) (j : ℕ) : taoSection6IntervalWeight as 0 j ≤ taoTupleWeight as := by rw [taoSection6IntervalWeight_zero_eq_tupleWeight_take] exact taoTupleWeight_take_le as j /-- Prefix interval weights are bounded by a named total weight. -/ theorem taoSection6IntervalWeight_zero_le_of_weight {as : List ℕ+} {j l : ℕ} (hweight : taoTupleWeight as = l) : taoSection6IntervalWeight as 0 j ≤ l := by simpa [hweight] using taoSection6IntervalWeight_zero_le_tupleWeight as j /-- Cast the natural exponent `l - a_[1,j]` to a real subtraction, using the total weight equality to justify the natural subtraction. -/ theorem taoCor63PrefixExponent_natCast_sub_of_weight {as : List ℕ+} {j l : ℕ} (hweight : taoTupleWeight as = l) : (((l - taoSection6IntervalWeight as 0 j : ℕ) : ℕ) : ℝ) = (l : ℝ) - ((taoSection6IntervalWeight as 0 j : ℕ) : ℝ) := by have hle : taoSection6IntervalWeight as 0 j ≤ l := taoSection6IntervalWeight_zero_le_of_weight hweight simpa using (Nat.cast_sub hle : (((l - taoSection6IntervalWeight as 0 j : ℕ) : ℕ) : ℝ) = (l : ℝ) - ((taoSection6IntervalWeight as 0 j : ℕ) : ℝ)) /-- Real prefix-exponent upper bound from the lower prefix-weight estimate. -/ theorem taoCor63StarPrefixExponent_upper {CA : ℝ} {n l : ℕ} {as : List ℕ+} (htyp : taoCor63StarInclusiveTypical CA n as) {j : ℕ} (hj_pos : 0 < j) (hj_len : j ≤ as.length) : (l : ℝ) - ((taoSection6IntervalWeight as 0 j : ℕ) : ℝ) ≤ (l : ℝ) - (2 * (j : ℝ) - taoCor63PrefixError CA n j) := by have hprefix := taoCor63StarInclusiveTypical_prefix_lower (CA := CA) (n := n) (as := as) htyp hj_pos hj_len exact sub_le_sub_left hprefix (l : ℝ) /-- Natural prefix-exponent upper bound after casting `l - a_[1,j]` to `ℝ`. -/ theorem taoCor63StarNatPrefixExponent_upper {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) {j : ℕ} (hj_pos : 0 < j) (hj_len : j ≤ as.length) : (((l - taoSection6IntervalWeight as 0 j : ℕ) : ℕ) : ℝ) ≤ (l : ℝ) - (2 * (j : ℝ) - taoCor63PrefixError CA n j) := by rw [taoCor63PrefixExponent_natCast_sub_of_weight hweight] exact taoCor63StarPrefixExponent_upper (CA := CA) (n := n) (l := l) (as := as) htyp hj_pos hj_len /-- The `idx`th cleared summand `3^idx * 2^(l-a_[1,idx+1])`. -/ def taoCor63ClearedSummand (l idx : ℕ) (as : List ℕ+) : ℕ := 3 ^ idx * 2 ^ (l - taoSection6IntervalWeight as 0 (idx + 1)) /-- Real cast of a cleared summand as a product of real powers. -/ theorem taoCor63ClearedSummand_real_cast (l idx : ℕ) (as : List ℕ+) : (taoCor63ClearedSummand l idx as : ℝ) = (3 : ℝ) ^ idx * (2 : ℝ) ^ (((l - taoSection6IntervalWeight as 0 (idx + 1) : ℕ) : ℕ) : ℝ) := by simp [taoCor63ClearedSummand, Nat.cast_mul, Nat.cast_pow, Real.rpow_natCast] /-- Pointwise real-power envelope for one cleared summand, before collecting the `(3 / 4)^idx` geometric factor. -/ theorem taoCor63ClearedSummand_real_le_prefixEnvelope {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) {idx : ℕ} (hidx : idx < as.length) : (taoCor63ClearedSummand l idx as : ℝ) ≤ (3 : ℝ) ^ idx * (2 : ℝ) ^ ((l : ℝ) - (2 * ((idx + 1 : ℕ) : ℝ) - taoCor63PrefixError CA n (idx + 1))) := by have hidx_pos : 0 < idx + 1 := Nat.succ_pos idx have hidx_len : idx + 1 ≤ as.length := Nat.succ_le_of_lt hidx have hexp : (((l - taoSection6IntervalWeight as 0 (idx + 1) : ℕ) : ℕ) : ℝ) ≤ (l : ℝ) - (2 * ((idx + 1 : ℕ) : ℝ) - taoCor63PrefixError CA n (idx + 1)) := taoCor63StarNatPrefixExponent_upper (CA := CA) (n := n) (l := l) (as := as) hweight htyp hidx_pos hidx_len have hpow : (2 : ℝ) ^ (((l - taoSection6IntervalWeight as 0 (idx + 1) : ℕ) : ℕ) : ℝ) ≤ (2 : ℝ) ^ ((l : ℝ) - (2 * ((idx + 1 : ℕ) : ℝ) - taoCor63PrefixError CA n (idx + 1))) := Real.rpow_le_rpow_of_exponent_le (by norm_num : (1 : ℝ) ≤ 2) hexp rw [taoCor63ClearedSummand_real_cast] exact mul_le_mul_of_nonneg_left hpow (pow_nonneg (by norm_num) idx) /-- Rewrite the one-summand prefix envelope as a geometric `(3 / 4)^idx` term. -/ theorem taoCor63PrefixEnvelope_eq_geometric (l idx : ℕ) (E : ℝ) : (3 : ℝ) ^ idx * (2 : ℝ) ^ ((l : ℝ) - (2 * ((idx + 1 : ℕ) : ℝ) - E)) = (2 : ℝ) ^ ((l : ℝ) - 2 + E) * (3 / 4 : ℝ) ^ idx := by have h2pos : 0 < (2 : ℝ) := by norm_num have hratio : (3 / 4 : ℝ) ^ idx = (3 : ℝ) ^ idx / (2 : ℝ) ^ (((2 * idx : ℕ) : ℕ) : ℝ) := by have hden : (4 : ℝ) ^ idx = (2 : ℝ) ^ (((2 * idx : ℕ) : ℕ) : ℝ) := by calc (4 : ℝ) ^ idx = ((2 : ℝ) ^ 2) ^ idx := by norm_num _ = (2 : ℝ) ^ (2 * idx) := by rw [pow_mul] _ = (2 : ℝ) ^ (((2 * idx : ℕ) : ℕ) : ℝ) := by simpa using (Real.rpow_natCast (2 : ℝ) (2 * idx)).symm rw [div_pow] rw [hden] have hexp : (l : ℝ) - (2 * (((idx + 1 : ℕ) : ℕ) : ℝ) - E) = (l : ℝ) - 2 + E - (((2 * idx : ℕ) : ℕ) : ℝ) := by norm_num ring rw [hratio, hexp] rw [Real.rpow_sub h2pos] ring /-- Pointwise cleared-summand envelope with the geometric factor exposed. -/ theorem taoCor63ClearedSummand_real_le_geometricEnvelope {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) {idx : ℕ} (hidx : idx < as.length) : (taoCor63ClearedSummand l idx as : ℝ) ≤ (2 : ℝ) ^ ((l : ℝ) - 2 + taoCor63PrefixError CA n (idx + 1)) * (3 / 4 : ℝ) ^ idx := by have hprefix := taoCor63ClearedSummand_real_le_prefixEnvelope (CA := CA) (n := n) (l := l) (as := as) hweight htyp hidx rw [taoCor63PrefixEnvelope_eq_geometric (l := l) (idx := idx) (E := taoCor63PrefixError CA n (idx + 1))] at hprefix exact hprefix /-- The recursive list of cleared summands beginning at global index `idx` and with already accumulated source prefix `pref`. -/ def taoCor63ClearedSummandsFrom (l idx pref : ℕ) : List ℕ+ → List ℕ | [] => [] | a :: as => 3 ^ idx * 2 ^ (l - (pref + (a : ℕ))) :: taoCor63ClearedSummandsFrom l (idx + 1) (pref + (a : ℕ)) as /-- The recursive cleared offset sum is the sum of its cleared summands. -/ theorem taoCor63ClearedOffsetSumFrom_eq_summandsFrom_sum (l idx pref : ℕ) (as : List ℕ+) : taoCor63ClearedOffsetSumFrom l idx pref as = (taoCor63ClearedSummandsFrom l idx pref as).sum := by induction as generalizing idx pref with | nil => simp [taoCor63ClearedOffsetSumFrom, taoCor63ClearedSummandsFrom] | cons a as ih => simp [taoCor63ClearedOffsetSumFrom, taoCor63ClearedSummandsFrom, ih] /-- The `loc`th summand of a recursive cleared offset sum state. -/ def taoCor63ClearedSummandFrom (l idx pref loc : ℕ) (as : List ℕ+) : ℕ := 3 ^ (idx + loc) * 2 ^ (l - (pref + taoSection6IntervalWeight as 0 (loc + 1))) /-- The first interval weight of a cons list is the head weight. -/ theorem taoSection6IntervalWeight_cons_zero_one (a : ℕ+) (as : List ℕ+) : taoSection6IntervalWeight (a :: as) 0 1 = (a : ℕ) := by simp [taoSection6IntervalWeight] /-- Prefix interval weights of a cons list split into head plus tail prefix. -/ theorem taoSection6IntervalWeight_cons_zero_succ_succ (a : ℕ+) (as : List ℕ+) (loc : ℕ) : taoSection6IntervalWeight (a :: as) 0 (loc + 2) = (a : ℕ) + taoSection6IntervalWeight as 0 (loc + 1) := by simp [taoSection6IntervalWeight, Nat.add_comm] /-- Zeroth named recursive summand for a cons list. -/ theorem taoCor63ClearedSummandFrom_cons_zero (l idx pref : ℕ) (a : ℕ+) (as : List ℕ+) : taoCor63ClearedSummandFrom l idx pref 0 (a :: as) = 3 ^ idx * 2 ^ (l - (pref + (a : ℕ))) := by simp [taoCor63ClearedSummandFrom, taoSection6IntervalWeight_cons_zero_one] /-- Successor named recursive summand for a cons list. -/ theorem taoCor63ClearedSummandFrom_cons_succ (l idx pref loc : ℕ) (a : ℕ+) (as : List ℕ+) : taoCor63ClearedSummandFrom l idx pref (loc + 1) (a :: as) = taoCor63ClearedSummandFrom l (idx + 1) (pref + (a : ℕ)) loc as := by simp [taoCor63ClearedSummandFrom, taoSection6IntervalWeight_cons_zero_succ_succ, Nat.add_comm, Nat.add_left_comm, Nat.add_assoc] /-- Recursive cleared offset sums as finite sums over local source positions. -/ theorem taoCor63ClearedOffsetSumFrom_eq_range_sumFrom (l idx pref : ℕ) (as : List ℕ+) : taoCor63ClearedOffsetSumFrom l idx pref as = ∑ loc ∈ Finset.range as.length, taoCor63ClearedSummandFrom l idx pref loc as := by induction as generalizing idx pref with | nil => simp [taoCor63ClearedOffsetSumFrom] | cons a as ih => rw [taoCor63ClearedOffsetSumFrom] rw [show (a :: as).length = 1 + as.length by simp [Nat.add_comm]] rw [Finset.sum_range_add] simp only [Finset.range_one, Finset.sum_singleton] rw [taoCor63ClearedSummandFrom_cons_zero] rw [ih (idx + 1) (pref + (a : ℕ))] congr 1 apply Finset.sum_congr rfl intro loc hloc rw [show 1 + loc = loc + 1 by omega] rw [taoCor63ClearedSummandFrom_cons_succ] /-- Top-level named recursive summands agree with the named cleared summand. -/ theorem taoCor63ClearedSummandFrom_zero_eq_summand (l idx : ℕ) (as : List ℕ+) : taoCor63ClearedSummandFrom l 0 0 idx as = taoCor63ClearedSummand l idx as := by simp [taoCor63ClearedSummandFrom, taoCor63ClearedSummand] /-- Top-level cleared offset sum as a finite range sum of named summands. -/ theorem taoCor63ClearedOffsetSum_eq_range_sum (l : ℕ) (as : List ℕ+) : taoCor63ClearedOffsetSum l as = ∑ idx ∈ Finset.range as.length, taoCor63ClearedSummand l idx as := by rw [taoCor63ClearedOffsetSum, taoCor63ClearedOffsetSumFrom_eq_range_sumFrom] apply Finset.sum_congr rfl intro idx hidx rw [taoCor63ClearedSummandFrom_zero_eq_summand] /-- Finite-sum envelope obtained by summing the pointwise geometric envelopes. -/ theorem taoCor63ClearedOffsetSum_real_le_geometricEnvelopeSum {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) : (taoCor63ClearedOffsetSum l as : ℝ) ≤ ∑ idx ∈ Finset.range as.length, (2 : ℝ) ^ ((l : ℝ) - 2 + taoCor63PrefixError CA n (idx + 1)) * (3 / 4 : ℝ) ^ idx := by rw [taoCor63ClearedOffsetSum_eq_range_sum] rw [Nat.cast_sum] exact Finset.sum_le_sum fun idx hidx => taoCor63ClearedSummand_real_le_geometricEnvelope (CA := CA) (n := n) (l := l) (as := as) hweight htyp (by simpa using hidx) /-- The finite real envelope obtained after bounding each cleared summand. -/ noncomputable def taoCor63GeometricEnvelopeSum (CA : ℝ) (n l : ℕ) (as : List ℕ+) : ℝ := ∑ idx ∈ Finset.range as.length, (2 : ℝ) ^ ((l : ℝ) - 2 + taoCor63PrefixError CA n (idx + 1)) * (3 / 4 : ℝ) ^ idx /-- The cleared offset sum is bounded by the named finite real envelope. -/ theorem taoCor63ClearedOffsetSum_real_le_geometricEnvelopeSum_def {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) : (taoCor63ClearedOffsetSum l as : ℝ) ≤ taoCor63GeometricEnvelopeSum CA n l as := by simpa [taoCor63GeometricEnvelopeSum] using taoCor63ClearedOffsetSum_real_le_geometricEnvelopeSum (CA := CA) (n := n) (l := l) (as := as) hweight htyp /-- Convert a strict real-envelope bound into the natural smallness conclusion. -/ theorem taoCor63ClearedOffsetSum_lt_of_geometricEnvelopeSum_lt {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (henv : taoCor63GeometricEnvelopeSum CA n l as < (3 ^ n : ℝ)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by have hle := taoCor63ClearedOffsetSum_real_le_geometricEnvelopeSum_def (CA := CA) (n := n) (l := l) (as := as) hweight htyp have hreal : (taoCor63ClearedOffsetSum l as : ℝ) < (3 ^ n : ℝ) := lt_of_le_of_lt hle henv exact_mod_cast hreal /-- Envelope version of the one-sided Corollary 6.3 smallness target. This is the remaining analytic obligation after the checked summand estimates: prove the named finite real envelope is `< 3^n`. -/ noncomputable def TaoCor63StarEnvelopeSmallnessStatement : Prop := ∀ CA : ℝ, 17 ≤ CA → ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ k l : ℕ, taoCor63StarLRange CA n l → ∀ as : List ℕ+, as.length = k + 1 → taoTupleWeight as = l → taoCor63StarInclusiveTypical CA n as → taoCor63GeometricEnvelopeSum CA n l as < (3 ^ n : ℝ) /-- The prefix-error weighted geometric series left after factoring out `2^(l-2)`. -/ noncomputable def taoCor63PrefixErrorGeometricSum (CA : ℝ) (n : ℕ) (as : List ℕ+) : ℝ := ∑ idx ∈ Finset.range as.length, (2 : ℝ) ^ (taoCor63PrefixError CA n (idx + 1)) * (3 / 4 : ℝ) ^ idx /-- Factor the common `2^(l-2)` scale out of the finite envelope. -/ theorem taoCor63GeometricEnvelopeSum_eq_pow_mul_prefixErrorGeometricSum (CA : ℝ) (n l : ℕ) (as : List ℕ+) : taoCor63GeometricEnvelopeSum CA n l as = (2 : ℝ) ^ ((l : ℝ) - 2) * taoCor63PrefixErrorGeometricSum CA n as := by rw [taoCor63GeometricEnvelopeSum, taoCor63PrefixErrorGeometricSum, Finset.mul_sum] apply Finset.sum_congr rfl intro idx hidx rw [Real.rpow_add (by norm_num : (0 : ℝ) < 2)] ring /-- The finite geometric sum with ratio `3 / 4` is bounded by `4`. -/ theorem taoCor63GeomRatioSum_le_four (m : ℕ) : (∑ idx ∈ Finset.range m, (3 / 4 : ℝ) ^ idx) ≤ 4 := by have hgeom := geom_sum_mul_of_le_one (x := (3 / 4 : ℝ)) (n := m) (by norm_num : (3 / 4 : ℝ) ≤ 1) have hgeom' : (∑ idx ∈ Finset.range m, (3 / 4 : ℝ) ^ idx) * (1 / 4 : ℝ) = 1 - (3 / 4 : ℝ) ^ m := by norm_num at hgeom ⊢ exact hgeom have hpow_nonneg : 0 ≤ (3 / 4 : ℝ) ^ m := pow_nonneg (by norm_num : (0 : ℝ) ≤ 3 / 4) m have hmul_le : (∑ idx ∈ Finset.range m, (3 / 4 : ℝ) ^ idx) * (1 / 4 : ℝ) ≤ 1 := by rw [hgeom'] linarith have hmul_le' := mul_le_mul_of_nonneg_right hmul_le (by norm_num : (0 : ℝ) ≤ 4) simpa [mul_assoc] using hmul_le' /-- Bound the prefix-error weighted geometric sum from a uniform power bound. -/ theorem taoCor63PrefixErrorGeometricSum_le_mul_geom_of_pow_le {CA B : ℝ} {n : ℕ} {as : List ℕ+} (hB : ∀ idx : ℕ, idx < as.length → (2 : ℝ) ^ taoCor63PrefixError CA n (idx + 1) ≤ B) : taoCor63PrefixErrorGeometricSum CA n as ≤ B * ∑ idx ∈ Finset.range as.length, (3 / 4 : ℝ) ^ idx := by calc taoCor63PrefixErrorGeometricSum CA n as ≤ ∑ idx ∈ Finset.range as.length, B * (3 / 4 : ℝ) ^ idx := by apply Finset.sum_le_sum intro idx hidx exact mul_le_mul_of_nonneg_right (hB idx (by simpa using hidx)) (pow_nonneg (by norm_num : (0 : ℝ) ≤ 3 / 4) idx) _ = B * ∑ idx ∈ Finset.range as.length, (3 / 4 : ℝ) ^ idx := by rw [Finset.mul_sum] /-- Coarse `4B` bound for the prefix-error weighted geometric sum. -/ theorem taoCor63PrefixErrorGeometricSum_le_four_mul_of_pow_le {CA B : ℝ} {n : ℕ} {as : List ℕ+} (hB_nonneg : 0 ≤ B) (hB : ∀ idx : ℕ, idx < as.length → (2 : ℝ) ^ taoCor63PrefixError CA n (idx + 1) ≤ B) : taoCor63PrefixErrorGeometricSum CA n as ≤ B * 4 := by have hweighted := taoCor63PrefixErrorGeometricSum_le_mul_geom_of_pow_le (CA := CA) (B := B) (n := n) (as := as) hB have hgeom := taoCor63GeomRatioSum_le_four as.length exact le_trans hweighted (mul_le_mul_of_nonneg_left hgeom hB_nonneg) /-- Bound the named finite real envelope from a uniform prefix-error power bound. -/ theorem taoCor63GeometricEnvelopeSum_le_pow_mul_four_mul_of_pow_le {CA B : ℝ} {n l : ℕ} {as : List ℕ+} (hB_nonneg : 0 ≤ B) (hB : ∀ idx : ℕ, idx < as.length → (2 : ℝ) ^ taoCor63PrefixError CA n (idx + 1) ≤ B) : taoCor63GeometricEnvelopeSum CA n l as ≤ (2 : ℝ) ^ ((l : ℝ) - 2) * (B * 4) := by rw [taoCor63GeometricEnvelopeSum_eq_pow_mul_prefixErrorGeometricSum] exact mul_le_mul_of_nonneg_left (taoCor63PrefixErrorGeometricSum_le_four_mul_of_pow_le (CA := CA) (B := B) (n := n) (as := as) hB_nonneg hB) (by positivity) /-- One-sided smallness consumer from a uniform prefix-error power bound and the final scalar comparison. -/ theorem taoCor63ClearedOffsetSum_lt_of_prefixErrorPow_bound {CA B : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hB_nonneg : 0 ≤ B) (hB : ∀ idx : ℕ, idx < as.length → (2 : ℝ) ^ taoCor63PrefixError CA n (idx + 1) ≤ B) (hfinal : (2 : ℝ) ^ ((l : ℝ) - 2) * (B * 4) < (3 ^ n : ℝ)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_geometricEnvelopeSum_lt (CA := CA) (n := n) (l := l) (as := as) hweight htyp (lt_of_le_of_lt (taoCor63GeometricEnvelopeSum_le_pow_mul_four_mul_of_pow_le (CA := CA) (B := B) (n := n) (l := l) (as := as) hB_nonneg hB) hfinal) /-- A finite geometric sum with ratio in `[0,1)` is bounded by `1 / (1-r)`. -/ theorem finite_geom_sum_le_inv_one_sub {r : ℝ} (hr_nonneg : 0 ≤ r) (hr_lt_one : r < 1) (m : ℕ) : (∑ idx ∈ Finset.range m, r ^ idx) ≤ (1 - r)⁻¹ := by have hden_pos : 0 < 1 - r := by linarith have hgeom := geom_sum_mul_of_le_one (x := r) (n := m) (le_of_lt hr_lt_one) have hpow_nonneg : 0 ≤ r ^ m := pow_nonneg hr_nonneg m rw [inv_eq_one_div] rw [le_div_iff₀ hden_pos] rw [hgeom] linarith /-- Rewrite finite exponential decay at a natural index as a power. -/ theorem exp_neg_nat_mul_eq_pow (c : ℝ) (idx : ℕ) : Real.exp (-(c * (idx : ℝ))) = Real.exp (-c) ^ idx := by rw [show -(c * (idx : ℝ)) = (idx : ℝ) * (-c) by ring] rw [Real.exp_nat_mul] /-- Finite exponential-decay sum bound. -/ theorem finite_exp_neg_mul_sum_le_inv_one_sub {c : ℝ} (hc_pos : 0 < c) (m : ℕ) : (∑ idx ∈ Finset.range m, Real.exp (-c * (idx : ℝ))) ≤ (1 - Real.exp (-c))⁻¹ := by have hexp_nonneg : 0 ≤ Real.exp (-c) := le_of_lt (Real.exp_pos (-c)) have hexp_lt_one : Real.exp (-c) < 1 := by rw [Real.exp_lt_one_iff] linarith simpa [exp_neg_nat_mul_eq_pow] using finite_geom_sum_le_inv_one_sub (r := Real.exp (-c)) hexp_nonneg hexp_lt_one m /-- Log-linear decay implies pointwise exponential decay for the envelope terms. -/ theorem two_rpow_mul_three_four_pow_le_exp_of_log_le {E K c : ℝ} {idx : ℕ} (hlog : Real.log 2 * E + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ K - c * (idx : ℝ)) : (2 : ℝ) ^ E * (3 / 4 : ℝ) ^ idx ≤ Real.exp K * Real.exp (-c * (idx : ℝ)) := by rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)] rw [show (3 / 4 : ℝ) ^ idx = (3 / 4 : ℝ) ^ ((idx : ℝ)) by exact (Real.rpow_natCast (3 / 4 : ℝ) idx).symm] rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 3 / 4)] rw [← Real.exp_add] rw [← Real.exp_add] apply Real.exp_le_exp.mpr linarith /-- Sum pointwise exponential decay over the prefix-error geometric series. -/ theorem taoCor63PrefixErrorGeometricSum_le_const_mul_exp_decay {CA C c : ℝ} {n : ℕ} {as : List ℕ+} (hC_nonneg : 0 ≤ C) (hc_pos : 0 < c) (hterm : ∀ idx : ℕ, idx < as.length → (2 : ℝ) ^ taoCor63PrefixError CA n (idx + 1) * (3 / 4 : ℝ) ^ idx ≤ C * Real.exp (-c * (idx : ℝ))) : taoCor63PrefixErrorGeometricSum CA n as ≤ C * (1 - Real.exp (-c))⁻¹ := by calc taoCor63PrefixErrorGeometricSum CA n as ≤ ∑ idx ∈ Finset.range as.length, C * Real.exp (-c * (idx : ℝ)) := by rw [taoCor63PrefixErrorGeometricSum] apply Finset.sum_le_sum intro idx hidx exact hterm idx (by simpa using hidx) _ = C * ∑ idx ∈ Finset.range as.length, Real.exp (-c * (idx : ℝ)) := by rw [Finset.mul_sum] _ ≤ C * (1 - Real.exp (-c))⁻¹ := by exact mul_le_mul_of_nonneg_left (finite_exp_neg_mul_sum_le_inv_one_sub (c := c) hc_pos as.length) hC_nonneg /-- Exponential-decay bound from a logarithmic pointwise envelope. -/ theorem taoCor63PrefixErrorGeometricSum_le_exp_const_mul_exp_decay {CA K c : ℝ} {n : ℕ} {as : List ℕ+} (hc_pos : 0 < c) (hlog : ∀ idx : ℕ, idx < as.length → Real.log 2 * taoCor63PrefixError CA n (idx + 1) + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ K - c * (idx : ℝ)) : taoCor63PrefixErrorGeometricSum CA n as ≤ Real.exp K * (1 - Real.exp (-c))⁻¹ := by exact taoCor63PrefixErrorGeometricSum_le_const_mul_exp_decay (CA := CA) (C := Real.exp K) (c := c) (n := n) (as := as) (le_of_lt (Real.exp_pos K)) hc_pos (fun idx hidx => two_rpow_mul_three_four_pow_le_exp_of_log_le (E := taoCor63PrefixError CA n (idx + 1)) (K := K) (c := c) (idx := idx) (hlog idx hidx)) /-- Bound the named envelope by the exponential-decay constant. -/ theorem taoCor63GeometricEnvelopeSum_le_pow_mul_exp_const_decay {CA K c : ℝ} {n l : ℕ} {as : List ℕ+} (hc_pos : 0 < c) (hlog : ∀ idx : ℕ, idx < as.length → Real.log 2 * taoCor63PrefixError CA n (idx + 1) + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ K - c * (idx : ℝ)) : taoCor63GeometricEnvelopeSum CA n l as ≤ (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) := by rw [taoCor63GeometricEnvelopeSum_eq_pow_mul_prefixErrorGeometricSum] exact mul_le_mul_of_nonneg_left (taoCor63PrefixErrorGeometricSum_le_exp_const_mul_exp_decay (CA := CA) (K := K) (c := c) (n := n) (as := as) hc_pos hlog) (by positivity) /-- Smallness consumer from a logarithmic exponential-decay bound. -/ theorem taoCor63ClearedOffsetSum_lt_of_exp_decay_log_bound {CA K c : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hc_pos : 0 < c) (hlog : ∀ idx : ℕ, idx < as.length → Real.log 2 * taoCor63PrefixError CA n (idx + 1) + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ K - c * (idx : ℝ)) (hfinal : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) < (3 ^ n : ℝ)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_geometricEnvelopeSum_lt (CA := CA) (n := n) (l := l) (as := as) hweight htyp (lt_of_le_of_lt (taoCor63GeometricEnvelopeSum_le_pow_mul_exp_const_decay (CA := CA) (K := K) (c := c) (n := n) (l := l) (as := as) hc_pos hlog) hfinal) /-- Young inequality helper for a square-root term with linear penalty. -/ theorem young_sqrt_sub_mul_le_sq_div {A b x : ℝ} (hb : 0 < b) (hx : 0 ≤ x) : A * Real.sqrt x - b * x ≤ A ^ 2 / (4 * b) := by have hb4 : 0 < 4 * b := by nlinarith rw [le_div_iff₀ hb4] have hsq : 0 ≤ (A - 2 * b * Real.sqrt x) ^ 2 := sq_nonneg _ have hsx : (Real.sqrt x) ^ 2 = x := Real.sq_sqrt hx ring_nf at hsq ⊢ nlinarith /-- Young inequality helper in epsilon form for a square-root term. -/ theorem young_sqrt_le_epsilon {A ε x : ℝ} (hε : 0 < ε) (hx : 0 ≤ x) : A * Real.sqrt x ≤ (ε / 2) * x + A ^ 2 / (2 * ε) := by have h := young_sqrt_sub_mul_le_sq_div (A := A) (b := ε / 2) (x := x) (by positivity) hx have hdiv : A ^ 2 / (4 * (ε / 2)) = A ^ 2 / (2 * ε) := by field_simp [hε.ne'] ring linarith /-- Positive decay rate `-log(3/4)`. -/ noncomputable def taoCor63DecayGamma : ℝ := -Real.log (3 / 4 : ℝ) /-- Standard Young constant for the Corollary 6.3 prefix-error bound. -/ noncomputable def taoCor63YoungK (CA : ℝ) (n : ℕ) : ℝ := ((Real.log 2 * CA) ^ 2 * Real.log (n : ℝ)) / (2 * taoCor63DecayGamma) + (Real.log 2 * CA) * Real.log (n : ℝ) + taoCor63DecayGamma / 2 /-- Flexible Young constant with an explicit decay rate `c < gamma`. -/ noncomputable def taoCor63YoungFlexibleK (CA : ℝ) (n : ℕ) (c : ℝ) : ℝ := ((Real.log 2 * CA) ^ 2 * Real.log (n : ℝ)) / (4 * (taoCor63DecayGamma - c)) + (Real.log 2 * CA) * Real.log (n : ℝ) + (taoCor63DecayGamma - c) /-- Quarter-decay specialization of the flexible Young constant. -/ noncomputable def taoCor63YoungQuarterK (CA : ℝ) (n : ℕ) : ℝ := taoCor63YoungFlexibleK CA n (taoCor63DecayGamma / 4) theorem taoCor63DecayGamma_pos : 0 < taoCor63DecayGamma := by have hlogneg : Real.log (3 / 4 : ℝ) < 0 := (Real.log_neg_iff (by norm_num : (0 : ℝ) < 3 / 4)).mpr (by norm_num : (3 / 4 : ℝ) < 1) simpa [taoCor63DecayGamma] using neg_pos.mpr hlogneg /-- Generic logarithmic Young bound with decay `gamma / 2`. -/ theorem taoCor63Young_generic_log_bound {A L gamma : ℝ} {idx : ℕ} (hL : 0 < L) (hgamma : 0 < gamma) : A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) + A * L - gamma * (idx : ℝ) ≤ A ^ 2 * L / (2 * gamma) + A * L + gamma / 2 - (gamma / 2) * (idx : ℝ) := by have hx : 0 ≤ (((idx + 1 : ℕ) : ℝ) * L) := by positivity have hy := young_sqrt_le_epsilon (A := A) (ε := gamma / L) (x := (((idx + 1 : ℕ) : ℝ) * L)) (by positivity) hx have hmain : A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) ≤ gamma / 2 * ((idx + 1 : ℕ) : ℝ) + A ^ 2 * L / (2 * gamma) := by calc A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) ≤ (gamma / L / 2) * (((idx + 1 : ℕ) : ℝ) * L) + A ^ 2 / (2 * (gamma / L)) := hy _ = gamma / 2 * ((idx + 1 : ℕ) : ℝ) + A ^ 2 * L / (2 * gamma) := by field_simp [hL.ne', hgamma.ne'] have hidx : (((idx + 1 : ℕ) : ℝ)) = (idx : ℝ) + 1 := by norm_num nlinarith /-- Generic logarithmic Young bound with flexible decay `c < gamma`. -/ theorem taoCor63Young_generic_flexible_log_bound {A L gamma c : ℝ} {idx : ℕ} (hL : 0 < L) (hcgamma : c < gamma) : A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) + A * L - gamma * (idx : ℝ) ≤ A ^ 2 * L / (4 * (gamma - c)) + A * L + (gamma - c) - c * (idx : ℝ) := by have hbeta : 0 < gamma - c := by linarith have hx : 0 ≤ (((idx + 1 : ℕ) : ℝ) * L) := by positivity have hy := young_sqrt_sub_mul_le_sq_div (A := A) (b := (gamma - c) / L) (x := (((idx + 1 : ℕ) : ℝ) * L)) (by positivity) hx have hmain : A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) - (gamma - c) * ((idx + 1 : ℕ) : ℝ) ≤ A ^ 2 * L / (4 * (gamma - c)) := by calc A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) - (gamma - c) * ((idx + 1 : ℕ) : ℝ) = A * Real.sqrt (((idx + 1 : ℕ) : ℝ) * L) - ((gamma - c) / L) * (((idx + 1 : ℕ) : ℝ) * L) := by field_simp [hL.ne'] _ ≤ A ^ 2 / (4 * ((gamma - c) / L)) := hy _ = A ^ 2 * L / (4 * (gamma - c)) := by field_simp [hL.ne', hbeta.ne'] have hidx : (((idx + 1 : ℕ) : ℝ)) = (idx : ℝ) + 1 := by norm_num nlinarith /-- Pointwise log-bound for prefix errors using the standard Young constant. -/ theorem taoCor63PrefixError_log_bound_young {CA : ℝ} {n idx : ℕ} (hlogn : 0 < Real.log (n : ℝ)) : Real.log 2 * taoCor63PrefixError CA n (idx + 1) + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ taoCor63YoungK CA n - (taoCor63DecayGamma / 2) * (idx : ℝ) := by have hgeneric := taoCor63Young_generic_log_bound (A := Real.log 2 * CA) (L := Real.log (n : ℝ)) (gamma := taoCor63DecayGamma) (idx := idx) hlogn taoCor63DecayGamma_pos have hlog34 : Real.log (3 / 4 : ℝ) = -taoCor63DecayGamma := by simp [taoCor63DecayGamma] simpa [taoCor63PrefixError, taoCor63YoungK, hlog34, mul_add, mul_comm, mul_left_comm, mul_assoc, sub_eq_add_neg] using hgeneric theorem taoCor63PrefixErrorGeometricSum_le_young_decay {CA : ℝ} {n : ℕ} {as : List ℕ+} (hlogn : 0 < Real.log (n : ℝ)) : taoCor63PrefixErrorGeometricSum CA n as ≤ Real.exp (taoCor63YoungK CA n) * (1 - Real.exp (-(taoCor63DecayGamma / 2)))⁻¹ := by exact taoCor63PrefixErrorGeometricSum_le_exp_const_mul_exp_decay (CA := CA) (K := taoCor63YoungK CA n) (c := taoCor63DecayGamma / 2) (n := n) (as := as) (half_pos taoCor63DecayGamma_pos) (fun idx hidx => taoCor63PrefixError_log_bound_young (CA := CA) (n := n) (idx := idx) hlogn) theorem taoCor63ClearedOffsetSum_lt_of_young_decay_final {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hfinal : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp (taoCor63YoungK CA n) * (1 - Real.exp (-(taoCor63DecayGamma / 2)))⁻¹) < (3 ^ n : ℝ)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_exp_decay_log_bound (CA := CA) (K := taoCor63YoungK CA n) (c := taoCor63DecayGamma / 2) (n := n) (l := l) (as := as) hweight htyp (half_pos taoCor63DecayGamma_pos) (fun idx hidx => taoCor63PrefixError_log_bound_young (CA := CA) (n := n) (idx := idx) hlogn) hfinal /-- Pointwise log-bound for prefix errors using a flexible decay rate. -/ theorem taoCor63PrefixError_log_bound_young_flexible {CA c : ℝ} {n idx : ℕ} (hlogn : 0 < Real.log (n : ℝ)) (hc_lt : c < taoCor63DecayGamma) : Real.log 2 * taoCor63PrefixError CA n (idx + 1) + (idx : ℝ) * Real.log (3 / 4 : ℝ) ≤ taoCor63YoungFlexibleK CA n c - c * (idx : ℝ) := by have hgeneric := taoCor63Young_generic_flexible_log_bound (A := Real.log 2 * CA) (L := Real.log (n : ℝ)) (gamma := taoCor63DecayGamma) (c := c) (idx := idx) hlogn hc_lt have hlog34 : Real.log (3 / 4 : ℝ) = -taoCor63DecayGamma := by simp [taoCor63DecayGamma] simpa [taoCor63PrefixError, taoCor63YoungFlexibleK, hlog34, mul_add, mul_comm, mul_left_comm, mul_assoc, sub_eq_add_neg] using hgeneric theorem taoCor63PrefixErrorGeometricSum_le_young_flexible_decay {CA c : ℝ} {n : ℕ} {as : List ℕ+} (hlogn : 0 < Real.log (n : ℝ)) (hc_pos : 0 < c) (hc_lt : c < taoCor63DecayGamma) : taoCor63PrefixErrorGeometricSum CA n as ≤ Real.exp (taoCor63YoungFlexibleK CA n c) * (1 - Real.exp (-c))⁻¹ := by exact taoCor63PrefixErrorGeometricSum_le_exp_const_mul_exp_decay (CA := CA) (K := taoCor63YoungFlexibleK CA n c) (c := c) (n := n) (as := as) hc_pos (fun idx hidx => taoCor63PrefixError_log_bound_young_flexible (CA := CA) (c := c) (n := n) (idx := idx) hlogn hc_lt) theorem taoCor63ClearedOffsetSum_lt_of_young_flexible_decay_final {CA c : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hc_pos : 0 < c) (hc_lt : c < taoCor63DecayGamma) (hfinal : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp (taoCor63YoungFlexibleK CA n c) * (1 - Real.exp (-c))⁻¹) < (3 ^ n : ℝ)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_exp_decay_log_bound (CA := CA) (K := taoCor63YoungFlexibleK CA n c) (c := c) (n := n) (l := l) (as := as) hweight htyp hc_pos (fun idx hidx => taoCor63PrefixError_log_bound_young_flexible (CA := CA) (c := c) (n := n) (idx := idx) hlogn hc_lt) hfinal theorem taoCor63DecayGamma_quarter_pos : 0 < taoCor63DecayGamma / 4 := by nlinarith [taoCor63DecayGamma_pos] theorem taoCor63DecayGamma_quarter_lt : taoCor63DecayGamma / 4 < taoCor63DecayGamma := by nlinarith [taoCor63DecayGamma_pos] theorem taoCor63PrefixErrorGeometricSum_le_young_quarter_decay {CA : ℝ} {n : ℕ} {as : List ℕ+} (hlogn : 0 < Real.log (n : ℝ)) : taoCor63PrefixErrorGeometricSum CA n as ≤ Real.exp (taoCor63YoungQuarterK CA n) * (1 - Real.exp (-(taoCor63DecayGamma / 4)))⁻¹ := by simpa [taoCor63YoungQuarterK] using taoCor63PrefixErrorGeometricSum_le_young_flexible_decay (CA := CA) (c := taoCor63DecayGamma / 4) (n := n) (as := as) hlogn taoCor63DecayGamma_quarter_pos taoCor63DecayGamma_quarter_lt theorem taoCor63_log_two_pos : 0 < Real.log (2 : ℝ) := Real.log_pos (by norm_num : (1 : ℝ) < 2) theorem taoCor63_three_pow_eq_exp_log (n : ℕ) : (3 ^ n : ℝ) = Real.exp ((n : ℝ) * Real.log 3) := by calc (3 ^ n : ℝ) = (3 : ℝ) ^ ((n : ℝ)) := by exact (Real.rpow_natCast (3 : ℝ) n).symm _ = Real.exp (Real.log 3 * (n : ℝ)) := by rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 3)] _ = Real.exp ((n : ℝ) * Real.log 3) := by ring_nf theorem taoCor63_lrange_log_two_mul_sub_two_le {CA : ℝ} {n l : ℕ} (hrange : taoCor63StarLRange CA n l) : Real.log 2 * ((l : ℝ) - 2) ≤ (n : ℝ) * Real.log 3 - CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 - 2 * Real.log 2 := by have hle := hrange.2 have hmul : (l : ℝ) * Real.log 2 ≤ (taoCor63StarQ CA n + 2 * CA * Real.log (n : ℝ)) * Real.log 2 := mul_le_mul_of_nonneg_right hle (le_of_lt taoCor63_log_two_pos) have hrewrite : (taoCor63StarQ CA n + 2 * CA * Real.log (n : ℝ)) * Real.log 2 = (n : ℝ) * Real.log 3 - CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 := by rw [taoCor63StarQ] field_simp [ne_of_gt taoCor63_log_two_pos] have hmul' : (l : ℝ) * Real.log 2 ≤ (n : ℝ) * Real.log 3 - CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 := by simpa [hrewrite] using hmul nlinarith theorem taoCor63_final_product_lt_of_exponent_lt {K D c : ℝ} {n l : ℕ} (hconst : (1 - Real.exp (-c))⁻¹ ≤ Real.exp D) (hexp : Real.log 2 * ((l : ℝ) - 2) + K + D < (n : ℝ) * Real.log 3) : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) < (3 ^ n : ℝ) := by have hpow : (2 : ℝ) ^ ((l : ℝ) - 2) = Real.exp (Real.log 2 * ((l : ℝ) - 2)) := by rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)] have hinner : Real.exp K * (1 - Real.exp (-c))⁻¹ ≤ Real.exp K * Real.exp D := mul_le_mul_of_nonneg_left hconst (le_of_lt (Real.exp_pos K)) have hprod : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) ≤ Real.exp (Real.log 2 * ((l : ℝ) - 2) + K + D) := by calc (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) = Real.exp (Real.log 2 * ((l : ℝ) - 2)) * (Real.exp K * (1 - Real.exp (-c))⁻¹) := by rw [hpow] _ ≤ Real.exp (Real.log 2 * ((l : ℝ) - 2)) * (Real.exp K * Real.exp D) := by exact mul_le_mul_of_nonneg_left hinner (le_of_lt (Real.exp_pos _)) _ = Real.exp (Real.log 2 * ((l : ℝ) - 2) + K + D) := by rw [← Real.exp_add, ← Real.exp_add] ring_nf have htarget : Real.exp (Real.log 2 * ((l : ℝ) - 2) + K + D) < (3 ^ n : ℝ) := by rw [taoCor63_three_pow_eq_exp_log n] exact Real.exp_lt_exp.mpr hexp exact lt_of_le_of_lt hprod htarget theorem taoCor63_final_product_lt_of_lrange_margin {CA K D c : ℝ} {n l : ℕ} (hrange : taoCor63StarLRange CA n l) (hconst : (1 - Real.exp (-c))⁻¹ ≤ Real.exp D) (hmargin : -CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 - 2 * Real.log 2 + K + D < 0) : (2 : ℝ) ^ ((l : ℝ) - 2) * (Real.exp K * (1 - Real.exp (-c))⁻¹) < (3 ^ n : ℝ) := by have hl := taoCor63_lrange_log_two_mul_sub_two_le (CA := CA) (n := n) (l := l) hrange have hexp : Real.log 2 * ((l : ℝ) - 2) + K + D < (n : ℝ) * Real.log 3 := by nlinarith exact taoCor63_final_product_lt_of_exponent_lt (K := K) (D := D) (c := c) (n := n) (l := l) hconst hexp theorem taoCor63ClearedOffsetSum_lt_of_young_flexible_lrange_margin {CA c D : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hc_pos : 0 < c) (hc_lt : c < taoCor63DecayGamma) (hrange : taoCor63StarLRange CA n l) (hconst : (1 - Real.exp (-c))⁻¹ ≤ Real.exp D) (hmargin : -CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 - 2 * Real.log 2 + taoCor63YoungFlexibleK CA n c + D < 0) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_young_flexible_decay_final (CA := CA) (c := c) (n := n) (l := l) (as := as) hweight htyp hlogn hc_pos hc_lt (taoCor63_final_product_lt_of_lrange_margin (CA := CA) (K := taoCor63YoungFlexibleK CA n c) (D := D) (c := c) (n := n) (l := l) hrange hconst hmargin) noncomputable def taoCor63YoungQuarterCoeff (CA : ℝ) : ℝ := CA ^ 2 * (-Real.log 2 + (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma)) + 3 * Real.log 2 * CA noncomputable def taoCor63YoungQuarterConst (D : ℝ) : ℝ := -2 * Real.log 2 + (3 / 4) * taoCor63DecayGamma + D theorem taoCor63_young_quarter_margin_eq (CA D : ℝ) (n : ℕ) : -CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 - 2 * Real.log 2 + taoCor63YoungFlexibleK CA n (taoCor63DecayGamma / 4) + D = Real.log (n : ℝ) * taoCor63YoungQuarterCoeff CA + taoCor63YoungQuarterConst D := by have hgamma : taoCor63DecayGamma ≠ 0 := ne_of_gt taoCor63DecayGamma_pos unfold taoCor63YoungFlexibleK taoCor63YoungQuarterCoeff taoCor63YoungQuarterConst field_simp [hgamma] ring theorem taoCor63_young_quarter_margin_lt_of_coeff_neg {CA D : ℝ} {n : ℕ} (hcoeff : taoCor63YoungQuarterCoeff CA < 0) (hlog : Real.log (n : ℝ) > taoCor63YoungQuarterConst D / (-taoCor63YoungQuarterCoeff CA)) : -CA ^ 2 * Real.log (n : ℝ) * Real.log 2 + 2 * CA * Real.log (n : ℝ) * Real.log 2 - 2 * Real.log 2 + taoCor63YoungFlexibleK CA n (taoCor63DecayGamma / 4) + D < 0 := by have hden_pos : 0 < -taoCor63YoungQuarterCoeff CA := by linarith have hmul : taoCor63YoungQuarterConst D < Real.log (n : ℝ) * (-taoCor63YoungQuarterCoeff CA) := by have hmul' := (div_lt_iff₀ hden_pos).mp hlog nlinarith rw [taoCor63_young_quarter_margin_eq] nlinarith theorem taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound {CA D : ℝ} {n l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hrange : taoCor63StarLRange CA n l) (hconst : (1 - Real.exp (-(taoCor63DecayGamma / 4)))⁻¹ ≤ Real.exp D) (hcoeff : taoCor63YoungQuarterCoeff CA < 0) (hlog : Real.log (n : ℝ) > taoCor63YoungQuarterConst D / (-taoCor63YoungQuarterCoeff CA)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_young_flexible_lrange_margin (CA := CA) (c := taoCor63DecayGamma / 4) (D := D) (n := n) (l := l) (as := as) hweight htyp hlogn taoCor63DecayGamma_quarter_pos taoCor63DecayGamma_quarter_lt hrange hconst (taoCor63_young_quarter_margin_lt_of_coeff_neg (CA := CA) (D := D) (n := n) hcoeff hlog) theorem taoCor63DecayGamma_eq_log_four_thirds : taoCor63DecayGamma = Real.log (4 / 3 : ℝ) := by have hpos : (0 : ℝ) < 3 / 4 := by norm_num have hinv : Real.log ((3 / 4 : ℝ)⁻¹) = -Real.log (3 / 4 : ℝ) := by rw [Real.log_inv] have hval : ((3 / 4 : ℝ)⁻¹) = 4 / 3 := by norm_num rw [taoCor63DecayGamma, ← hinv, hval] theorem taoCor63DecayGamma_gt_two_sevenths : (2 / 7 : ℝ) < taoCor63DecayGamma := by rw [taoCor63DecayGamma_eq_log_four_thirds] have h := Real.lt_log_one_add_of_pos (show (0 : ℝ) < 1 / 3 by norm_num) norm_num at h simpa [show (1 + 1 / 3 : ℝ) = 4 / 3 by norm_num] using h theorem taoCor63_log_two_gt_693_div_1000 : (693 / 1000 : ℝ) < Real.log 2 := by exact (by norm_num : (693 / 1000 : ℝ) < 0.6931471803).trans Real.log_two_gt_d9 theorem taoCor63_log_two_lt_347_div_500 : Real.log 2 < (347 / 500 : ℝ) := by exact Real.log_two_lt_d9.trans (by norm_num : (0.6931471808 : ℝ) < 347 / 500) theorem taoCor63_young_quarter_quadratic_coeff_lt : -Real.log 2 + (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma) < -(196637 / 1500000 : ℝ) := by have hlog2_pos : 0 < Real.log 2 := taoCor63_log_two_pos have hlog2_lt : Real.log 2 < (347 / 500 : ℝ) := taoCor63_log_two_lt_347_div_500 have hlog2_gt : (693 / 1000 : ℝ) < Real.log 2 := taoCor63_log_two_gt_693_div_1000 have hgamma_gt : (2 / 7 : ℝ) < taoCor63DecayGamma := taoCor63DecayGamma_gt_two_sevenths have hden_pos : 0 < 3 * taoCor63DecayGamma := by nlinarith have hsq : (Real.log 2) ^ 2 < (347 / 500 : ℝ) ^ 2 := by nlinarith have hdiv : (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma) < (842863 / 1500000 : ℝ) := by rw [div_lt_iff₀ hden_pos] have hden_lower : (6 / 7 : ℝ) < 3 * taoCor63DecayGamma := by nlinarith have hscale_pos : (0 : ℝ) < 842863 / 1500000 := by norm_num have hscale : (347 / 500 : ℝ) ^ 2 = (842863 / 1500000 : ℝ) * (6 / 7) := by norm_num nlinarith nlinarith theorem taoCor63YoungQuarterCoeff_neg_of_ge_seventeen {CA : ℝ} (hCA : 17 ≤ CA) : taoCor63YoungQuarterCoeff CA < 0 := by have hCA_pos : 0 < CA := by nlinarith have hCA_sq_pos : 0 < CA ^ 2 := by nlinarith have hq : -Real.log 2 + (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma) < -(196637 / 1500000 : ℝ) := taoCor63_young_quarter_quadratic_coeff_lt have hqmul : CA ^ 2 * (-Real.log 2 + (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma)) < CA ^ 2 * (-(196637 / 1500000 : ℝ)) := mul_lt_mul_of_pos_left hq hCA_sq_pos have hlog2mul : 3 * Real.log 2 * CA < 3 * (347 / 500 : ℝ) * CA := by have hleft : (3 * CA) * Real.log 2 < (3 * CA) * (347 / 500 : ℝ) := mul_lt_mul_of_pos_left taoCor63_log_two_lt_347_div_500 (by nlinarith) nlinarith have hsum : CA ^ 2 * (-Real.log 2 + (Real.log 2) ^ 2 / (3 * taoCor63DecayGamma)) + 3 * Real.log 2 * CA < CA ^ 2 * (-(196637 / 1500000 : ℝ)) + 3 * (347 / 500 : ℝ) * CA := add_lt_add hqmul hlog2mul have hquad : 17 * CA ≤ CA ^ 2 := by nlinarith have hrhs_neg : CA ^ 2 * (-(196637 / 1500000 : ℝ)) + 3 * (347 / 500 : ℝ) * CA < 0 := by nlinarith unfold taoCor63YoungQuarterCoeff nlinarith theorem taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound_CA17 {CA D : ℝ} {n l : ℕ} {as : List ℕ+} (hCA : 17 ≤ CA) (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hrange : taoCor63StarLRange CA n l) (hconst : (1 - Real.exp (-(taoCor63DecayGamma / 4)))⁻¹ ≤ Real.exp D) (hlog : Real.log (n : ℝ) > taoCor63YoungQuarterConst D / (-taoCor63YoungQuarterCoeff CA)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound (CA := CA) (D := D) (n := n) (l := l) (as := as) hweight htyp hlogn hrange hconst (taoCor63YoungQuarterCoeff_neg_of_ge_seventeen hCA) hlog theorem taoCor63_exp_neg_one_fourteenth_le_fifteen_sixteen : Real.exp (-(1 / 14 : ℝ)) ≤ (15 / 16 : ℝ) := by have hlog : Real.log (16 / 15 : ℝ) ≤ (1 / 14 : ℝ) := by have h := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 16 / 15 by norm_num) nlinarith have hexp : (16 / 15 : ℝ) ≤ Real.exp (1 / 14 : ℝ) := (Real.log_le_iff_le_exp (show (0 : ℝ) < 16 / 15 by norm_num)).mp hlog have hinv : (Real.exp (1 / 14 : ℝ))⁻¹ ≤ ((16 / 15 : ℝ))⁻¹ := (inv_le_inv₀ (Real.exp_pos _) (show (0 : ℝ) < 16 / 15 by norm_num)).mpr hexp simpa [Real.exp_neg] using hinv theorem taoCor63_exp_neg_quarter_gamma_lt_fifteen_sixteen : Real.exp (-(taoCor63DecayGamma / 4)) < (15 / 16 : ℝ) := by have hgamma : (2 / 7 : ℝ) < taoCor63DecayGamma := taoCor63DecayGamma_gt_two_sevenths have hexp : Real.exp (-(taoCor63DecayGamma / 4)) < Real.exp (-(1 / 14 : ℝ)) := Real.exp_lt_exp.mpr (by nlinarith) exact lt_of_lt_of_le hexp taoCor63_exp_neg_one_fourteenth_le_fifteen_sixteen theorem taoCor63_quarter_geometric_prefactor_le_sixteen : (1 - Real.exp (-(taoCor63DecayGamma / 4)))⁻¹ ≤ (16 : ℝ) := by have hexp : Real.exp (-(taoCor63DecayGamma / 4)) < (15 / 16 : ℝ) := taoCor63_exp_neg_quarter_gamma_lt_fifteen_sixteen have hbase_pos : 0 < 1 - Real.exp (-(taoCor63DecayGamma / 4)) := by nlinarith rw [inv_le_comm₀ hbase_pos (by norm_num : (0 : ℝ) < 16)] nlinarith theorem taoCor63_sixteen_le_exp_sixteen : (16 : ℝ) ≤ Real.exp 16 := by have h15 : (16 : ℝ) ≤ Real.exp 15 := by have hadd := Real.add_one_le_exp (15 : ℝ) norm_num at hadd exact hadd exact h15.trans (Real.exp_le_exp.mpr (by norm_num : (15 : ℝ) ≤ 16)) theorem taoCor63_quarter_geometric_prefactor_le_exp_sixteen : (1 - Real.exp (-(taoCor63DecayGamma / 4)))⁻¹ ≤ Real.exp 16 := taoCor63_quarter_geometric_prefactor_le_sixteen.trans taoCor63_sixteen_le_exp_sixteen theorem taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound_CA17_D16 {CA : ℝ} {n l : ℕ} {as : List ℕ+} (hCA : 17 ≤ CA) (hweight : taoTupleWeight as = l) (htyp : taoCor63StarInclusiveTypical CA n as) (hlogn : 0 < Real.log (n : ℝ)) (hrange : taoCor63StarLRange CA n l) (hlog : Real.log (n : ℝ) > taoCor63YoungQuarterConst 16 / (-taoCor63YoungQuarterCoeff CA)) : taoCor63ClearedOffsetSum l as < 3 ^ n := by exact taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound_CA17 (CA := CA) (D := 16) (n := n) (l := l) (as := as) hCA hweight htyp hlogn hrange taoCor63_quarter_geometric_prefactor_le_exp_sixteen hlog theorem taoCor63_exists_nat_log_gt (A : ℝ) : ∃ N : ℕ, ∀ n : ℕ, N ≤ n → A < Real.log (n : ℝ) := by obtain ⟨N, hN⟩ := exists_nat_gt (Real.exp A) refine ⟨N, ?_⟩ intro n hn have hNn : (N : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have hexp : Real.exp A < (n : ℝ) := lt_of_lt_of_le hN hNn have hn_pos : 0 < (n : ℝ) := lt_trans (Real.exp_pos A) hexp exact (Real.lt_log_iff_exp_lt hn_pos).mpr hexp theorem taoCor63_exists_nat_log_pos_and_gt (A : ℝ) : ∃ N : ℕ, ∀ n : ℕ, N ≤ n → 0 < Real.log (n : ℝ) ∧ A < Real.log (n : ℝ) := by obtain ⟨Npos, hNpos⟩ := taoCor63_exists_nat_log_gt 0 obtain ⟨NA, hNA⟩ := taoCor63_exists_nat_log_gt A refine ⟨max Npos NA, ?_⟩ intro n hn have hpos_le : Npos ≤ n := le_trans (le_max_left _ _) hn have hA_le : NA ≤ n := le_trans (le_max_right _ _) hn exact ⟨hNpos n hpos_le, hNA n hA_le⟩ theorem taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_eventually_CA17_D16 {CA : ℝ} (hCA : 17 ≤ CA) : ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ l : ℕ, ∀ as : List ℕ+, taoTupleWeight as = l → taoCor63StarInclusiveTypical CA n as → taoCor63StarLRange CA n l → taoCor63ClearedOffsetSum l as < 3 ^ n := by obtain ⟨N0, hN0⟩ := taoCor63_exists_nat_log_pos_and_gt (taoCor63YoungQuarterConst 16 / (-taoCor63YoungQuarterCoeff CA)) refine ⟨N0, ?_⟩ intro n hn l as hweight htyp hrange have hlogs := hN0 n hn exact taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_log_bound_CA17_D16 (CA := CA) (n := n) (l := l) (as := as) hCA hweight htyp hlogs.1 hrange hlogs.2 theorem taoCor63ClearedOffsetSumFrom_eq_pow_mul_offset_reverse (l idx pref : ℕ) : ∀ as : List ℕ+, pref + taoTupleWeight as ≤ l → (taoCor63ClearedOffsetSumFrom l idx pref as : ℚ) = (3 : ℚ) ^ idx * (2 : ℚ) ^ (l - pref) * taoOffsetList as.reverse | [], _hbound => by simp [taoCor63ClearedOffsetSumFrom] | a :: as, hbound => by have htail : pref + (a : ℕ) + taoTupleWeight as ≤ l := by simpa [taoTupleWeight, Nat.add_assoc] using hbound have hprefa : pref + (a : ℕ) ≤ l := by omega have hsub : l - pref = l - (pref + (a : ℕ)) + (a : ℕ) := by omega rw [taoCor63ClearedOffsetSumFrom] rw [Nat.cast_add] push_cast rw [taoCor63ClearedOffsetSumFrom_eq_pow_mul_offset_reverse l (idx + 1) (pref + (a : ℕ)) as htail] rw [taoOffsetList_reverse_cons] rw [hsub, pow_add] field_simp [pow_ne_zero _ (by norm_num : (2 : ℚ) ≠ 0)] ring_nf theorem taoCor63ClearedOffsetSum_eq_pow_mul_offset_reverse {l : ℕ} {as : List ℕ+} (hweight : taoTupleWeight as = l) : (taoCor63ClearedOffsetSum l as : ℚ) = (2 : ℚ) ^ l * taoOffsetList as.reverse := by unfold taoCor63ClearedOffsetSum rw [taoCor63ClearedOffsetSumFrom_eq_pow_mul_offset_reverse l 0 0 as] · simp · omega /-- Reverse-order version of the checked Lemma 6.2 offset-injectivity spine. This is the tuple-order adapter needed because the Corollary 6.3 proof writes `F_{k+1}(a_{k+1}, ..., a_1)`. -/ theorem taoLemma62_offset_injective_reverse {as bs : List ℕ+} (hlen : as.length = bs.length) (h : taoOffsetList as.reverse = taoOffsetList bs.reverse) : as = bs := by have hlenr : as.reverse.length = bs.reverse.length := by simpa using hlen have hrev : as.reverse = bs.reverse := taoLemma62_offset_injective hlenr h exact List.reverse_injective hrev /-- If two natural numbers are congruent modulo `m` and both lie in `[0,m)`, they are equal. This is the deterministic residue-to-equality step used after the repaired Corollary 6.3 size estimate proves both cleared sums are `< 3^n`. -/ theorem taoNatModEq_eq_of_lt {m a b : ℕ} (hmod : Nat.ModEq m a b) (ha : a < m) (hb : b < m) : a = b := by simpa [Nat.ModEq, Nat.mod_eq_of_lt ha, Nat.mod_eq_of_lt hb] using hmod /-- Final Lemma 6.2 consumer after the hard Corollary 6.3 estimates have supplied the rational cleared-offset identities. -/ theorem taoCor63Star_eq_of_clearedOffset_eq {l : ℕ} {as bs : List ℕ+} (hlen : as.length = bs.length) (has : (taoCor63ClearedOffsetSum l as : ℚ) = (2 : ℚ) ^ l * taoOffsetList as.reverse) (hbs : (taoCor63ClearedOffsetSum l bs : ℚ) = (2 : ℚ) ^ l * taoOffsetList bs.reverse) (hsum : taoCor63ClearedOffsetSum l as = taoCor63ClearedOffsetSum l bs) : as = bs := by have hcast : (taoCor63ClearedOffsetSum l as : ℚ) = (taoCor63ClearedOffsetSum l bs : ℚ) := by exact_mod_cast hsum have hmul : (2 : ℚ) ^ l * taoOffsetList as.reverse = (2 : ℚ) ^ l * taoOffsetList bs.reverse := by rw [← has, ← hbs] exact hcast have hpow : (2 : ℚ) ^ l ≠ 0 := pow_ne_zero _ (by norm_num : (2 : ℚ) ≠ 0) exact taoLemma62_offset_injective_reverse hlen (mul_left_cancel₀ hpow hmul) /-- Same final consumer with the residue congruence and `< 3^n` bounds made explicit. The proof-producing work left for Corollary 6.3 star is then exactly the two size bounds plus the cleared-offset rational identities. -/ theorem taoCor63Star_eq_of_modEq_small_and_clearedOffset {n l : ℕ} {as bs : List ℕ+} (hlen : as.length = bs.length) (has : (taoCor63ClearedOffsetSum l as : ℚ) = (2 : ℚ) ^ l * taoOffsetList as.reverse) (hbs : (taoCor63ClearedOffsetSum l bs : ℚ) = (2 : ℚ) ^ l * taoOffsetList bs.reverse) (hmod : Nat.ModEq (3 ^ n) (taoCor63ClearedOffsetSum l as) (taoCor63ClearedOffsetSum l bs)) (has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n) (hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n) : as = bs := by exact taoCor63Star_eq_of_clearedOffset_eq hlen has hbs (taoNatModEq_eq_of_lt hmod has_lt hbs_lt) theorem taoCor63Star_eq_of_modEq_small {n l : ℕ} {as bs : List ℕ+} (hlen : as.length = bs.length) (has_weight : taoTupleWeight as = l) (hbs_weight : taoTupleWeight bs = l) (hmod : Nat.ModEq (3 ^ n) (taoCor63ClearedOffsetSum l as) (taoCor63ClearedOffsetSum l bs)) (has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n) (hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n) : as = bs := by exact taoCor63Star_eq_of_modEq_small_and_clearedOffset hlen (taoCor63ClearedOffsetSum_eq_pow_mul_offset_reverse has_weight) (taoCor63ClearedOffsetSum_eq_pow_mul_offset_reverse hbs_weight) hmod has_lt hbs_lt /-- A named residue-class form of the repaired Corollary 6.3 cleared offset. -/ def taoCor63ClearedOffsetZMod (n l : ℕ) (as : List ℕ+) : ZMod (3 ^ n) := taoCor63ClearedOffsetSum l as /-- Generic mathlib bridge: equality of natural residues in `ZMod m` is exactly `Nat.ModEq m`. -/ theorem taoNatModEq_of_zmod_natCast_eq {m a b : ℕ} (h : (a : ZMod m) = (b : ZMod m)) : Nat.ModEq m a b := by exact (ZMod.natCast_eq_natCast_iff a b m).mp h /-- Cor63Star specialization of the `ZMod` equality to `Nat.ModEq` bridge. -/ theorem taoCor63NatModEq_of_clearedOffsetZMod_eq {n l : ℕ} {as bs : List ℕ+} (h : taoCor63ClearedOffsetZMod n l as = taoCor63ClearedOffsetZMod n l bs) : Nat.ModEq (3 ^ n) (taoCor63ClearedOffsetSum l as) (taoCor63ClearedOffsetSum l bs) := by exact taoNatModEq_of_zmod_natCast_eq (m := 3 ^ n) (a := taoCor63ClearedOffsetSum l as) (b := taoCor63ClearedOffsetSum l bs) (by simpa [taoCor63ClearedOffsetZMod] using h) /-- Final deterministic Cor63Star consumer when the hard source work supplies equality of cleared offset residues in `ZMod (3^n)` plus the two smallness bounds. -/ theorem taoCor63Star_eq_of_zmod_eq_small {n l : ℕ} {as bs : List ℕ+} (hlen : as.length = bs.length) (has_weight : taoTupleWeight as = l) (hbs_weight : taoTupleWeight bs = l) (hzmod : taoCor63ClearedOffsetZMod n l as = taoCor63ClearedOffsetZMod n l bs) (has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n) (hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n) : as = bs := by exact taoCor63Star_eq_of_modEq_small hlen has_weight hbs_weight (taoCor63NatModEq_of_clearedOffsetZMod_eq hzmod) has_lt hbs_lt /-- The unit represented by `2` in `ZMod (3^n)`. -/ noncomputable def taoCor63TwoUnit (n : ℕ) : (ZMod (3 ^ n))ˣ := ZMod.unitOfCoprime 2 (Nat.Coprime.pow_right n (by decide : Nat.Coprime 2 3)) /-- The unit represented by `2^l` in `ZMod (3^n)`. -/ noncomputable def taoCor63TwoPowUnit (n l : ℕ) : (ZMod (3 ^ n))ˣ := (taoCor63TwoUnit n) ^ l theorem taoCor63TwoUnit_coe (n : ℕ) : (taoCor63TwoUnit n : ZMod (3 ^ n)) = 2 := by simp [taoCor63TwoUnit] theorem taoCor63TwoPowUnit_coe (n l : ℕ) : (taoCor63TwoPowUnit n l : ZMod (3 ^ n)) = (2 : ZMod (3 ^ n)) ^ l := by simp [taoCor63TwoPowUnit, taoCor63TwoUnit_coe] /-- Source-facing residue of Tao's `F_{k+1}` modulo `3^n`, after using the ambient total weight `l`: this is `2^{-l}` times the cleared offset sum. -/ noncomputable def taoCor63SourceOffsetZMod (n l : ℕ) (as : List ℕ+) : ZMod (3 ^ n) := ((taoCor63TwoPowUnit n l)⁻¹ : (ZMod (3 ^ n))ˣ) * taoCor63ClearedOffsetZMod n l as /-- Multiplying the source-facing residue by `2^l` recovers the cleared offset residue. -/ theorem taoCor63TwoPow_mul_sourceOffsetZMod (n l : ℕ) (as : List ℕ+) : (taoCor63TwoPowUnit n l : ZMod (3 ^ n)) * taoCor63SourceOffsetZMod n l as = taoCor63ClearedOffsetZMod n l as := by simp [taoCor63SourceOffsetZMod] /-- If the source-facing `F` residues are equal modulo `3^n`, then the cleared offset residues are equal modulo `3^n`. -/ theorem taoCor63ClearedOffsetZMod_eq_of_sourceOffsetZMod_eq {n l : ℕ} {as bs : List ℕ+} (h : taoCor63SourceOffsetZMod n l as = taoCor63SourceOffsetZMod n l bs) : taoCor63ClearedOffsetZMod n l as = taoCor63ClearedOffsetZMod n l bs := by calc taoCor63ClearedOffsetZMod n l as = (taoCor63TwoPowUnit n l : ZMod (3 ^ n)) * taoCor63SourceOffsetZMod n l as := by rw [taoCor63TwoPow_mul_sourceOffsetZMod] _ = (taoCor63TwoPowUnit n l : ZMod (3 ^ n)) * taoCor63SourceOffsetZMod n l bs := by rw [h] _ = taoCor63ClearedOffsetZMod n l bs := by rw [taoCor63TwoPow_mul_sourceOffsetZMod] /-- End-to-end deterministic consumer from source-facing residue equality and the two smallness bounds. -/ theorem taoCor63Star_eq_of_sourceOffsetZMod_eq_small {n l : ℕ} {as bs : List ℕ+} (hlen : as.length = bs.length) (has_weight : taoTupleWeight as = l) (hbs_weight : taoTupleWeight bs = l) (hsource : taoCor63SourceOffsetZMod n l as = taoCor63SourceOffsetZMod n l bs) (has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n) (hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n) : as = bs := by exact taoCor63Star_eq_of_zmod_eq_small hlen has_weight hbs_weight (taoCor63ClearedOffsetZMod_eq_of_sourceOffsetZMod_eq hsource) has_lt hbs_lt /-- One-sided repaired Corollary 6.3 smallness target. This is the formal version of the source proof's estimate after `(6.14)`, ending with each natural-number side being `< 3^n`. -/ noncomputable def TaoCor63StarSmallnessStatement : Prop := ∀ CA : ℝ, 17 ≤ CA → ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ k l : ℕ, taoCor63StarLRange CA n l → ∀ as : List ℕ+, as.length = k + 1 → taoTupleWeight as = l → taoCor63StarInclusiveTypical CA n as → taoCor63ClearedOffsetSum l as < 3 ^ n theorem TaoCor63StarSmallnessStatement.of_geometricEnvelope (henv : TaoCor63StarEnvelopeSmallnessStatement) : TaoCor63StarSmallnessStatement := by intro CA hCA refine ⟨Classical.choose (henv CA hCA), ?_⟩ intro n hn k l hrange as hlen hweight htyp exact taoCor63ClearedOffsetSum_lt_of_geometricEnvelopeSum_lt (CA := CA) (n := n) (l := l) (as := as) hweight htyp (Classical.choose_spec (henv CA hCA) n hn k l hrange as hlen hweight htyp) theorem TaoCor63StarSmallnessStatement.from_quarter_route : TaoCor63StarSmallnessStatement := by intro CA hCA obtain ⟨N0, hN0⟩ := taoCor63ClearedOffsetSum_lt_of_young_quarter_lrange_eventually_CA17_D16 (CA := CA) hCA refine ⟨N0, ?_⟩ intro n hn k l hrange as _hlen hweight htyp exact hN0 n hn l as hweight htyp hrange /-- Consumer showing that the one-sided smallness target plus source-facing residue equality gives the repaired Corollary 6.3 deterministic separation endpoint. -/ theorem taoCor63Star_eq_of_sourceOffsetZMod_eq_of_smallness (hsmall : TaoCor63StarSmallnessStatement) {CA : ℝ} (hCA : 17 ≤ CA) {n : ℕ} (hn : Classical.choose (hsmall CA hCA) ≤ n) {k l : ℕ} (hrange : taoCor63StarLRange CA n l) {as bs : List ℕ+} (has_len : as.length = k + 1) (hbs_len : bs.length = k + 1) (has_weight : taoTupleWeight as = l) (hbs_weight : taoTupleWeight bs = l) (has_typical : taoCor63StarInclusiveTypical CA n as) (hbs_typical : taoCor63StarInclusiveTypical CA n bs) (hsource : taoCor63SourceOffsetZMod n l as = taoCor63SourceOffsetZMod n l bs) : as = bs := by classical let N0 := Classical.choose (hsmall CA hCA) have hsmall_at := Classical.choose_spec (hsmall CA hCA) have hn' : N0 ≤ n := hn have has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n := hsmall_at n hn' k l hrange as has_len has_weight has_typical have hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n := hsmall_at n hn' k l hrange bs hbs_len hbs_weight hbs_typical exact taoCor63Star_eq_of_sourceOffsetZMod_eq_small (has_len.trans hbs_len.symm) has_weight hbs_weight hsource has_lt hbs_lt theorem TaoCor63StarClearedSeparationStatement.of_smallness (hsmall : TaoCor63StarSmallnessStatement) : TaoCor63StarClearedSeparationStatement := by intro CA hCA refine ⟨Classical.choose (hsmall CA hCA), ?_⟩ intro n hn k l hrange as bs has_len hbs_len has_weight hbs_weight has_typical hbs_typical hmod have hsmall_at := Classical.choose_spec (hsmall CA hCA) have has_lt : taoCor63ClearedOffsetSum l as < 3 ^ n := hsmall_at n hn k l hrange as has_len has_weight has_typical have hbs_lt : taoCor63ClearedOffsetSum l bs < 3 ^ n := hsmall_at n hn k l hrange bs hbs_len hbs_weight hbs_typical have hlen : as.length = bs.length := by omega exact taoCor63Star_eq_of_modEq_small (n := n) (l := l) (as := as) (bs := bs) hlen has_weight hbs_weight hmod has_lt hbs_lt theorem TaoCor63StarClearedSeparationStatement.from_quarter_route : TaoCor63StarClearedSeparationStatement := TaoCor63StarClearedSeparationStatement.of_smallness TaoCor63StarSmallnessStatement.from_quarter_route /-- Complete repaired Corollary 6.3 source-residue separation statement, factored through the one-sided smallness statement. -/ noncomputable def TaoCor63StarSourceResidueSeparationStatement : Prop := ∀ CA : ℝ, 17 ≤ CA → ∃ N0 : ℕ, ∀ n : ℕ, N0 ≤ n → ∀ k l : ℕ, taoCor63StarLRange CA n l → ∀ as bs : List ℕ+, as.length = k + 1 → bs.length = k + 1 → taoTupleWeight as = l → taoTupleWeight bs = l → taoCor63StarInclusiveTypical CA n as → taoCor63StarInclusiveTypical CA n bs → taoCor63SourceOffsetZMod n l as = taoCor63SourceOffsetZMod n l bs → as = bs theorem TaoCor63StarSourceResidueSeparationStatement.of_smallness (hsmall : TaoCor63StarSmallnessStatement) : TaoCor63StarSourceResidueSeparationStatement := by intro CA hCA refine ⟨Classical.choose (hsmall CA hCA), ?_⟩ intro n hn k l hrange as bs has_len hbs_len has_weight hbs_weight has_typical hbs_typical hsource exact taoCor63Star_eq_of_sourceOffsetZMod_eq_of_smallness hsmall hCA hn hrange has_len hbs_len has_weight hbs_weight has_typical hbs_typical hsource theorem TaoCor63StarSourceResidueSeparationStatement.of_clearedSeparation (hclear : TaoCor63StarClearedSeparationStatement) : TaoCor63StarSourceResidueSeparationStatement := by intro CA hCA refine ⟨Classical.choose (hclear CA hCA), ?_⟩ intro n hn k l hrange as bs has_len hbs_len has_weight hbs_weight has_typical hbs_typical hsource have hclear_at := Classical.choose_spec (hclear CA hCA) exact hclear_at n hn k l hrange as bs has_len hbs_len has_weight hbs_weight has_typical hbs_typical (taoCor63NatModEq_of_clearedOffsetZMod_eq (taoCor63ClearedOffsetZMod_eq_of_sourceOffsetZMod_eq hsource)) theorem TaoCor63StarSourceResidueSeparationStatement.from_quarter_route : TaoCor63StarSourceResidueSeparationStatement := TaoCor63StarSourceResidueSeparationStatement.of_smallness TaoCor63StarSmallnessStatement.from_quarter_route end Tao end Erdos1135