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allaun
77488ac0ae
Lean proof fixes: - N3L_Energy.lean: fully close gaussian_line_integral_unit_dir (nlinarith+hab for unit-circle quadratic, sqrt_mul+neg_div for integral_gaussian_1d match, exp_sum_of_sq order fix, add_assoc for h_gauss_shift, sq_sqrt for field_simp, sq_abs for perpDistance hd) - Add Adapters/AlphaProofNexus: 12 Erdos/graph adapter stubs (AlphaProof nexus) - Add Adapters/ErgodicAdditive.lean, SidonMatroid.lean - Add AntiDiophantine.lean, EffectiveBoundDQ.lean, PVGS_DQ_Bridge.lean - Add FormalConjectures/Util/ProblemImports.lean - Add RRC/EntropyCandidates/Candidates.lean - Add OTOM external project (lakefile.toml, lake-manifest.json, lean-toolchain) Infrastructure: - Add 4-Infrastructure/shim/: 17 Python probes (RRC manifold, Sidon kernel, Wannier, arxiv harvest, math_symbols DB, coverage density, geometric entropy) - Add 4-Infrastructure/NoDupeLabs/: Node server + package files - Add 6-Documentation/docs/specs/DP_RRC_RECEIPT_ENCODING_SPEC.md - Add fix_offloat.py Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
154 lines
5.6 KiB
Text
154 lines
5.6 KiB
Text
import Mathlib.Data.Set.Basic
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import Mathlib.Data.Finset.Basic
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import Mathlib.Data.Finset.Sort
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import Mathlib.Data.Int.Basic
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import Mathlib.Tactic
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import Semantics.SidonSets
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import Semantics.FixedPoint
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open Semantics
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open FixedPoint
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open Semantics.SidonSets
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/-!
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# Anti-Diophantine Constructions and the Sidon Intersection
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A **Diophantine** equation `P(x) = 0` has finitely many integer solutions (Baker).
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An **Anti-Diophantine** construction has infinitely many or dense solutions.
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Sidon sets `aᵢ + aⱼ = aₖ + aₗ ⇒ {i,j} = {k,l}` live at the intersection:
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- **Diophantine**: the sum equation has only trivial solutions (finiteness)
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- **Anti-Diophantine**: maximal Sidon size `~√N` (positive density)
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- **Slack σ** = M − max(label): large σ → Anti-Diophantine, small σ → Diophantine
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-/
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/-! ## 1. Dual Predicates -/
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/-- A family `F` is **Diophantine**: each instance has finitely many solutions. -/
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def IsDiophantineFamily (F : ℕ → Set ℕ) : Prop :=
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∀ p, Set.Finite (F p)
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/-- A family is **Anti-Diophantine**: each instance has infinitely many solutions. -/
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def IsAntiDiophantineFamily (F : ℕ → Set ℕ) : Prop :=
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∀ p, Set.Infinite (F p)
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/-- Agreement zone: both Diophantine finiteness AND Anti-Diophantine density hold. -/
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structure Agreement (F : ℕ → Set ℕ) where
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diophantine : IsDiophantineFamily F
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antiDiophantine : IsAntiDiophantineFamily F
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/-- Disagreement zone: both fail. -/
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structure Disagreement (F : ℕ → Set ℕ) where
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notDiophantine : ¬ IsDiophantineFamily F
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notAntiDiophantine : ¬ IsAntiDiophantineFamily F
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/-! ## 2. Sidon Slack -/
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/--
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The Sidon slack σ = M − max(label) measures address headroom.
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Large slack → Anti-Diophantine regime (many embeddings).
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Small slack → Diophantine regime (tight constraints).
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-/
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def sidonSlack (labels : Finset ℕ) (M : ℕ) : ℕ :=
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M - (if h : labels.Nonempty then labels.max' h else 0)
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theorem sidonSlack_eq (labels : Finset ℕ) (h : labels.Nonempty) (M : ℕ) :
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sidonSlack labels M = M - labels.max' h := by
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unfold sidonSlack
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simp [h]
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/-- Slack ≥ 128 → Anti-Diophantine regime. -/
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def isAntiDiophantineSlack (σ : ℕ) : Prop :=
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σ ≥ 128
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/-- Slack < 8 → Diophantine regime. -/
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def isDiophantineSlack (σ : ℕ) : Prop :=
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σ < 8
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/-! ## 3. Canonical 8-strand Labels -/
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/--
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Canonical 8-strand Sidon labels (powers of 2): {1,2,4,8,16,32,64,128}.
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These achieve σ = M − 128 for address budget M.
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-/
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def canonicalSidonLabels : Finset ℕ :=
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{1, 2, 4, 8, 16, 32, 64, 128}
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theorem canonicalSidonLabels_nonempty : canonicalSidonLabels.Nonempty := by
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refine ⟨1, ?_⟩
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simp [canonicalSidonLabels]
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theorem canonicalSidonLabels_max : canonicalSidonLabels.max' canonicalSidonLabels_nonempty = 128 := by
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native_decide
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theorem canonical_labels_sidon : Semantics.SidonSets.IsSidon (canonicalSidonLabels.image (λ (n : ℕ) => (n : ℤ))) := by
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native_decide
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theorem canonical_sidon_slack (M : ℕ) (hM : M ≥ 128) :
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sidonSlack canonicalSidonLabels M = M - 128 := by
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rw [sidonSlack_eq canonicalSidonLabels canonicalSidonLabels_nonempty M]
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rw [canonicalSidonLabels_max]
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/-! ## 4. Intersection Bounds -/
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/--
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**Agreement upper bound** (Diophantine side): `h(N) ≤ √(2N) + 1` for `N ≥ 1`.
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This is the sumset double-counting bound from SidonSets.lean.
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-/
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theorem sidon_agreement_upper (N : ℕ) (hN : 1 ≤ N) :
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Semantics.SidonSets.sidonMaximum N ≤ Nat.sqrt (2 * N) + 1 :=
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Semantics.SidonSets.sidonMaximum_le_sqrt_two N hN
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/--
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**Agreement lower bound** (Anti-Diophantine side): `√N / 2 ≤ h(N)` for `N ≥ 5`.
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This uses the Singer/Bose-Chowla construction (SidonSets.lean:2990).
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-/
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theorem sidon_agreement_lower (N : ℕ) (hN : 5 ≤ N) :
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Nat.sqrt N / 2 ≤ Semantics.SidonSets.sidonMaximum N := by
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have h_strong : (Nat.sqrt N + 1) / 2 < Semantics.SidonSets.sidonMaximum N :=
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Semantics.SidonSets.sidonMaximum_gt_sqrt_div_two N hN
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have h_weak : Nat.sqrt N / 2 ≤ (Nat.sqrt N + 1) / 2 := by
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omega
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omega
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/--
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**Full agreement theorem**: for `N ≥ 5`,
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`√N / 2 ≤ h(N) ≤ √(2N) + 1`.
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Thus Sidon sets live in the agreement zone: Diophantine upper bound meets
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Anti-Diophantine lower bound at `Θ(√N)`.
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-/
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theorem sidon_agreement_theorem (N : ℕ) (hN : 5 ≤ N) :
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Nat.sqrt N / 2 ≤ Semantics.SidonSets.sidonMaximum N ∧
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Semantics.SidonSets.sidonMaximum N ≤ Nat.sqrt (2 * N) + 1 := by
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have h_upper : Semantics.SidonSets.sidonMaximum N ≤ Nat.sqrt (2 * N) + 1 := by
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have h1 : 1 ≤ N := by omega
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exact sidon_agreement_upper N h1
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exact ⟨sidon_agreement_lower N hN, h_upper⟩
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/-! ## 5. Slack Regime Transition -/
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theorem slack_regime_transition (M : ℕ) (hM : M ≥ 256) :
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sidonSlack canonicalSidonLabels M ≥ 128 := by
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have hM128 : M ≥ 128 := by omega
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rw [canonical_sidon_slack M hM128]
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omega
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/-! ## 6. Diophantine vs Anti-Diophantine Comparison -/
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/--
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The equation `a + b = c + d` over a Sidon set `A` has only trivial solutions.
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This means the sumset `A + A` grows quadratically in `|A|`.
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**Diophantine constraint**: `|A + A| ≥ |A|·(|A|−1)/2` (all non-trivial sums distinct).
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**Anti-Diophantine density**: `|A| ≥ √N/2` (Singer construction gives large sets).
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The comparison: Diophantine says `|A|` is bounded by `√(2N)`, Anti-Diophantine says
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`|A|` is at least `√N/2`. Together they pin `|A|` to `Θ(√N)`.
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-/
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theorem diophantine_antiDiophantine_comparison (N : ℕ) (hN : 5 ≤ N) :
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let diophantineBound := Nat.sqrt (2 * N) + 1; let antiDiophantineBound := Nat.sqrt N / 2;
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antiDiophantineBound ≤ Semantics.SidonSets.sidonMaximum N ∧
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Semantics.SidonSets.sidonMaximum N ≤ diophantineBound := by
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intro diophantineBound antiDiophantineBound
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exact sidon_agreement_theorem N hN
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