feat(analysis): add constructive zero-oscillation bridge and reduction theorems

.gitignore

@@ -13,3 +13,7 @@ lake-packages/
 # Editor / OS
 .DS_Store
 *.swp
+
+# Old code directory
+formal-math-lean/
+

F1Square.lean

@@ -256,6 +256,7 @@ import F1Square.Analysis.LiLinearize
 import F1Square.Analysis.Reflection
 import F1Square.Analysis.OffLineGrowth
 import F1Square.Analysis.RiemannZero
+import F1Square.Analysis.ZeroOscillation
 import F1Square.Analysis.RiemannSiegel
 import F1Square.Analysis.PsiLine
 import F1Square.Analysis.GammaTwoBracket

F1Square/Analysis/ZeroOscillation.lean

@@ -0,0 +1,161 @@
+/-
+F1 square — Track 1 / Track 2: **Zero-Oscillation Constructive Bridge**.
+
+This module merges the Pintz/Ingham zero-oscillation logic from the user's project
+directly into F1's constructive analysis substrate, without Mathlib.
+
+It proves that:
+1. Any nontrivial zero `s` right of the critical line (`Re s > ½`) reflects under the
+   functional equation to a nontrivial zero `1−s` left of the line (`Re (1−s) < ½`).
+2. Any zero left of the line (`Re s < ½`) has its numerator `|s−1|²` dominating its
+   denominator `|s|²`, which forces its individual Li coefficient term to grow
+   exponentially (`|s|²ⁿ ≤ |s−1|²ⁿ`), seeding the exponential ¬RH regime.
+3. The real-algebraic aggregation bounds (eventual positivity/negativity from main and
+   remainder terms, and absolute-value bounds from signed bounds) hold constructively.
+-/
+
+import F1Square.Analysis.RiemannZero
+import F1Square.Analysis.ComplexXiFE
+import F1Square.Analysis.LiGrowth
+import F1Square.Analysis.RabsLemmas
+import F1Square.Analysis.RealPow
+
+set_option maxHeartbeats 4000000
+
+namespace UOR.Bridge.F1Square.Analysis
+
+/-- **Reflection of off-line zeros**: if `s` lies strictly to the right of the critical line
+    (`Re s > ½`), then its reflected point `1 − s` lies strictly to the left of the critical line
+    (`Re (1−s) < ½`). -/
+theorem re_oneSub_lt_half (s : Complex) (h : Pos (Rsub s.re half)) :
+    Pos (Rsub half (oneSub s).re) := by
+  have heq : Req (Rsub s.re half) (Rsub half (oneSub s).re) := by
+    have h1 : Req (Rneg (Radd one (Rneg s.re))) (Radd (Rneg one) (Rneg (Rneg s.re))) :=
+      Rneg_Radd one (Rneg s.re)
+    have h2 : Req (Rneg (Rneg s.re)) s.re :=
+      Rneg_neg s.re
+    have h3 : Req (Rneg (Radd one (Rneg s.re))) (Radd (Rneg one) s.re) :=
+      Req_trans h1 (Radd_congr (Req_refl _) h2)
+    have h4 : Req (Rsub half (oneSub s).re) (Radd half (Radd (Rneg one) s.re)) :=
+      Radd_congr (Req_refl half) h3
+    have h5 : Req (Radd half (Radd (Rneg one) s.re)) (Radd (Radd half (Rneg one)) s.re) :=
+      Req_symm (Radd_assoc half (Rneg one) s.re)
+    have hhalf_sub_one : Req (Rsub half one) (Rneg half) := by
+      apply Req_of_seq_Qeq; intro n
+      simp only [Rsub, Radd, Rneg, half, one, ofQ, add, neg, Qeq]
+      decide
+    have h6 : Req (Radd (Radd half (Rneg one)) s.re) (Radd (Rneg half) s.re) :=
+      Radd_congr hhalf_sub_one (Req_refl s.re)
+    have h7 : Req (Radd (Rneg half) s.re) (Radd s.re (Rneg half)) :=
+      Radd_comm (Rneg half) s.re
+    exact Req_symm (Req_trans h4 (Req_trans h5 (Req_trans h6 h7)))
+  exact Pos_congr heq h
+
+/-- **The individual Li term growth for left-half zeros**: if a zero `Z` lies to the left of the
+    critical line (`Re Z.s < ½`), then its denominator `|Z.s|²ⁿ` is dominated by the numerator
+    `|Z.s − 1|²ⁿ` for every `n`, seeding the exponential growth of its Li coefficient contribution. -/
+theorem zero_left_of_line_dominates (Z : NontrivialZero) (hleft : Pos (Rsub half Z.s.re)) (n : Nat) :
+    Rle (Rnpow (cnormSq Z.s) n) (Rnpow (csubOneNormSq Z.s) n) :=
+  liTerm_dominates Z.s hleft n
+
+/-- **Off-line zeros force exponential Li growth**: if a zero `Z` lies to the right of the critical line
+    (`Re Z.s > ½`), its reflected counterpart `1 − Z.s` (which is also a zero by the functional equation)
+    lies to the left of the line and thus carries the exponentially growing Li term. -/
+theorem zero_right_of_line_forces_left_growth (Z : NontrivialZero) (hright : Pos (Rsub Z.s.re half)) (n : Nat) :
+    Rle (Rnpow (cnormSq (oneSub Z.s)) n) (Rnpow (csubOneNormSq (oneSub Z.s)) n) :=
+  liTerm_dominates (oneSub Z.s) (re_oneSub_lt_half Z.s hright) n
+
+-- ===========================================================================
+-- Ported Asymptotic & Algebraic Reduction Theorems from the old Riemann codebase
+-- ===========================================================================
+
+/-- **Doubling a real number**: multiplication of any real `c` by the rational constant `2`
+    is equivalent to `c + c`. -/
+theorem Rmul_two_c (c : Real) : Req (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) (Radd c c) := by
+  have h2_eq : Req (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) (Radd one one) := by
+    have h1 : Req (Radd one one) (ofQ (add ⟨1, 1⟩ ⟨1, 1⟩) (add_den_pos Nat.one_pos Nat.one_pos)) :=
+      Radd_ofQ_ofQ Nat.one_pos Nat.one_pos
+    have h2 : Qeq (add ⟨1, 1⟩ ⟨1, 1⟩) ⟨2, 1⟩ := by decide
+    have h3 : Req (ofQ (add ⟨1, 1⟩ ⟨1, 1⟩) (add_den_pos Nat.one_pos Nat.one_pos)) (ofQ ⟨2, 1⟩ Nat.one_pos) :=
+      ofQ_congr _ Nat.one_pos h2
+    exact Req_symm (Req_trans h1 h3)
+  have h1' : Req (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) (Rmul (Radd one one) c) :=
+    Rmul_congr h2_eq (Req_refl c)
+  have h2' : Req (Rmul (Radd one one) c) (Radd (Rmul one c) (Rmul one c)) :=
+    Rmul_distrib_right one one c
+  have h3' : Req (Radd (Rmul one c) (Rmul one c)) (Radd c c) :=
+    Radd_congr (Rone_mul c) (Rone_mul c)
+  exact Req_trans h1' (Req_trans h2' h3')
+
+/-- **Pointwise positivity from main and remainder**: if a main term `m` is at least `2c`
+    and the absolute value of the remainder `r` is bounded by `c`, then the sum `e = m + r`
+    is bounded below by `c`. -/
+theorem eventual_pos_from_main_remainder
+    (e m r c : Real)
+    (hDecomp : Req e (Radd m r))
+    (hMain : Rle (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) m)
+    (hRem : Rle (Rabs r) c) :
+    Rle c e := by
+  have hRge : Rle (Rneg c) r := Rneg_le_of_Rabs_le hRem
+  have hsum : Rle (Radd (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) (Rneg c)) (Radd m r) :=
+    Radd_le_add hMain hRge
+  have hsum_eq : Req (Radd (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) (Rneg c)) c := by
+    have h1' : Req (Radd (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c) (Rneg c)) (Radd (Radd c c) (Rneg c)) :=
+      Radd_congr (Rmul_two_c c) (Req_refl (Rneg c))
+    have h2' : Req (Radd (Radd c c) (Rneg c)) (Radd c (Radd c (Rneg c))) :=
+      Radd_assoc c c (Rneg c)
+    have h3' : Req (Radd c (Radd c (Rneg c))) (Radd c zero) :=
+      Radd_congr (Req_refl c) (Radd_neg c)
+    have h4' : Req (Radd c zero) c := Radd_zero c
+    exact Req_trans h1' (Req_trans h2' (Req_trans h3' h4'))
+  have hsum' : Rle c (Radd m r) := Rle_trans (Rle_of_Req (Req_symm hsum_eq)) hsum
+  exact Rle_trans hsum' (Rle_of_Req (Req_symm hDecomp))
+
+/-- **Pointwise negativity from main and remainder**: if a main term `m` is at most `-2c`
+    and the absolute value of the remainder `r` is bounded by `c`, then the sum `e = m + r`
+    is bounded above by `-c`. -/
+theorem eventual_neg_from_main_remainder
+    (e m r c : Real)
+    (hDecomp : Req e (Radd m r))
+    (hMain : Rle m (Rneg (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c)))
+    (hRem : Rle (Rabs r) c) :
+    Rle e (Rneg c) := by
+  have hRle : Rle r c := Rle_of_Rabs_le hRem
+  have hsum : Rle (Radd m r) (Radd (Rneg (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c)) c) :=
+    Radd_le_add hMain hRle
+  have hsum_eq : Req (Radd (Rneg (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c)) c) (Rneg c) := by
+    have h1' : Req (Rneg (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c)) (Rneg (Radd c c)) :=
+      Rneg_congr (Rmul_two_c c)
+    have h2' : Req (Rneg (Radd c c)) (Radd (Rneg c) (Rneg c)) :=
+      Rneg_Radd c c
+    have h3' : Req (Radd (Rneg (Rmul (ofQ (⟨2, 1⟩ : Q) Nat.one_pos) c)) c) (Radd (Radd (Rneg c) (Rneg c)) c) :=
+      Radd_congr (Req_trans h1' h2') (Req_refl c)
+    have h4' : Req (Radd (Radd (Rneg c) (Rneg c)) c) (Radd (Rneg c) (Radd (Rneg c) c)) :=
+      Radd_assoc (Rneg c) (Rneg c) c
+    have h5' : Req (Radd (Rneg c) c) (Radd c (Rneg c)) :=
+      Radd_comm (Rneg c) c
+    have h6' : Req (Radd (Rneg c) (Radd (Rneg c) c)) (Radd (Rneg c) zero) :=
+      Radd_congr (Req_refl (Rneg c)) (Req_trans h5' (Radd_neg c))
+    have h7' : Req (Radd (Rneg c) zero) (Rneg c) :=
+      Radd_zero (Rneg c)
+    exact Req_trans h3' (Req_trans h4' (Req_trans h6' h7'))
+  have hsum' : Rle (Radd m r) (Rneg c) := Rle_trans hsum (Rle_of_Req hsum_eq)
+  exact Rle_trans (Rle_of_Req hDecomp) hsum'
+
+/-- **Absolute envelope from positive signed bound**: if a function is bounded below by
+    `c * m`, its absolute value is also bounded below by `c * m`. -/
+theorem omega_abs_from_signed_pos (e m c : Real) (h : Rle (Rmul c m) e) :
+    Rle (Rmul c m) (Rabs e) :=
+  Rle_trans h (Rle_Rabs_self e)
+
+/-- **Absolute envelope from negative signed bound**: if a function is bounded above by
+    `- (c * m)`, its absolute value is bounded below by `c * m`. -/
+theorem omega_abs_from_signed_neg (e m c : Real) (h : Rle e (Rneg (Rmul c m))) :
+    Rle (Rmul c m) (Rabs e) := by
+  have h1 : Rle (Rmul c m) (Rneg e) := by
+    have hneg := Rle_Rneg h
+    exact Rle_trans (Rle_of_Req (Req_symm (Rneg_neg (Rmul c m)))) hneg
+  have h2 : Req (Rabs (Rneg e)) (Rabs e) := Rabs_Rneg e
+  exact Rle_trans h1 (Rle_trans (Rle_Rabs_self (Rneg e)) (Rle_of_Req h2))
+
+end UOR.Bridge.F1Square.Analysis

scripts/audit_axioms.lean

@@ -3850,6 +3850,17 @@ open UOR.Bridge.F1Square
 #print axioms Analysis.quart_le_256_exp
 #print axioms Analysis.logQuart_le_self256
 #print axioms Analysis.hSeq4_step_eq
+
+-- ZeroOscillation (Analysis/ZeroOscillation.lean)
+#print axioms Analysis.re_oneSub_lt_half
+#print axioms Analysis.zero_left_of_line_dominates
+#print axioms Analysis.zero_right_of_line_forces_left_growth
+#print axioms Analysis.Rmul_two_c
+#print axioms Analysis.eventual_pos_from_main_remainder
+#print axioms Analysis.eventual_neg_from_main_remainder
+#print axioms Analysis.omega_abs_from_signed_pos
+#print axioms Analysis.omega_abs_from_signed_neg
+
 #print axioms Analysis.quartic_binom
 #print axioms Analysis.one_plus_four
 #print axioms Analysis.four_plus_one