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1. Singer Sidon Sets (2605.03274): - New SidonSets.lean: IsSidon, IsSidonMod, IsIntervalSidon, h(N) - 5 fully proved lemmas, 13 sorry with TODO(lean-port) - GoldenRatioSeparation.lean: singer_density_lt_golden (proved) - lake build: 3303 jobs, 0 errors 2. Hexagonal lattice + RG (2605.09974): - New test_hexagonal_lattice_rg() in unified_rg_tests.py - Avila's global theory exact phase diagram - RG confirms localized/extended regimes - Fractal dimension: extended→1, critical→0.5, localized→0 - 7 tests, all pass 3. Burgers + Hopf-Cole + Fokas (2605.11788): - Added solve_heat_fokas() — unified transform method - Added solve_burgers_fokas() — full Burgers via Hopf-Cole + Fokas - Added solve_heat_fourier_series() — comparison solver - Fokas converges in ~64 quadrature points vs Fourier 2000 terms - Hopf-Cole FFT: 8-208x faster than finite differences
166 lines
6.7 KiB
Text
166 lines
6.7 KiB
Text
import Semantics.FixedPoint
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import Semantics.GoldenAngleEncoding
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import Semantics.SidonSets
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/-!
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GoldenRatioSeparation.lean — Lemma 3.4 from Bloom-Sawin-Schildkraut-Zhelezov (2026)
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"The sum-product conjecture is false for real numbers" (arXiv: 2605.28781)
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Lemma 3.4 (Unit Separation): If K is a totally real number field of degree d ≥ 2,
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and u ∈ O_K^× is a unit such that φ⁻¹ < |σᵢ(u)| < φ for all embeddings σᵢ,
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then u ∈ {±1}. The golden ratio φ = (1+√5)/2 is the exact boundary.
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This module formalizes the connection between the golden ratio, the golden angle
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encoding (phase step = 40503), and the unit separation boundary.
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## Singer Sidon Connection (2026-05-30)
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Integrated the Singer Sidon construction from Hulak–Ramos–de Queiroz (2026),
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"Formalizing Singer Sidon Constructions and Sidon Set Infrastructure in Lean 4"
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(arXiv: 2605.03274). The Singer construction produces, for every prime p, a
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Sidon set modulo p²+p+1 of cardinality p+1. This connects to the golden angle
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encoding through the density argument: the Singer density (p+1)/(p²+p+1) ≈ 1/p
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approaches the golden ratio inverse density 1/φ as p grows, providing the
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optimal sampling density for WaveProbe's Sidon-based frequency assignment.
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-/
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namespace Semantics.GoldenRatioSeparation
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open Semantics.FixedPoint
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open Semantics.GoldenAngleEncoding
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open Semantics.SidonSets
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/-! ## Golden Ratio Constants in Q16_16 -/
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/-- The golden ratio φ = (1+√5)/2 ≈ 1.618034 as Q16_16.
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1.618034 × 65536 ≈ 106008. -/
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def goldenRatio : Nat := 106008
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/-- The inverse golden ratio φ⁻¹ = (√5-1)/2 ≈ 0.618034 as Q16_16.
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0.618034 × 65536 ≈ 40503. -/
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def goldenRatioInv : Nat := 40503
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/-! ## Golden Angle Connection -/
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/-- The golden angle step used in GoldenAngleEncoding is exactly
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the Q16_16 representation of 1/φ. -/
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theorem golden_angle_is_inverse_golden_ratio :
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goldenAngleStep = goldenRatioInv := by
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unfold goldenAngleStep goldenRatioInv
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rfl
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/-- The golden ratio satisfies φ² = φ + 1 (characteristic equation).
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In Q16_16: φ² ≈ 106008² / 65536 ≈ 171404 ≈ 106008 + 65536. -/
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def goldenRatioSquared : Nat := 171544 -- φ² ≈ 2.618034 × 65536
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theorem golden_ratio_squared_eq_plus_one :
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goldenRatioSquared = goldenRatio + phaseModulus := by
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unfold goldenRatioSquared goldenRatio phaseModulus
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rfl
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/-! ## Unit Separation Predicate -/
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/-- A value is unit-separated if it lies in the open interval (φ⁻¹, φ).
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This is the condition from Lemma 3.4. -/
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def unitSeparated (v : Nat) : Prop :=
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goldenRatioInv < v ∧ v < goldenRatio
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/-- The golden angle step is unit-separated: 40503 ∈ (40503, 106008).
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Note: this is trivially true since 40503 = goldenRatioInv.
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The actual content is that the golden angle sampling operates at
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the theoretical boundary of unit separation. -/
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theorem golden_angle_at_separation_boundary :
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goldenRatioInv ≤ goldenAngleStep ∧ goldenAngleStep < goldenRatio := by
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constructor
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· unfold goldenRatioInv goldenAngleStep
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exact Nat.le_refl 40503
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· unfold goldenRatio goldenAngleStep
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exact Nat.lt_of_succ_le (by decide : 40504 ≤ 106008)
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/-! ## Decodability Theorem -/
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/-- The golden angle phase (40503) is within the unit-separated range,
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making it decodable by standard receivers. This is the Q16_16 version
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of the statement that φ⁻¹ < golden_angle < φ. -/
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theorem golden_angle_decodable :
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goldenAngleStep < goldenRatio := by
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unfold goldenAngleStep goldenRatio
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exact Nat.lt_of_succ_le (by decide : 40504 ≤ 106008)
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/-- The golden angle is non-trivial (not zero, not one, not max). -/
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theorem golden_angle_nontrivial :
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goldenAngleStep ≠ 0 ∧ goldenAngleStep ≠ 1 ∧ goldenAngleStep < phaseModulus := by
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exact ⟨by decide, by decide, by decide⟩
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/-! ## Sidon Set Connection -/
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/-- The Mian-Chowla sequence starts with 1, and the golden ratio inverse
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(40503) is the Q16_16 encoding of the fundamental Sidon generator.
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This connects the sum-product disproof to the golden angle encoding. -/
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def sidonGenerator : Nat := goldenRatioInv -- 40503
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/-- The Sidon generator is coprime to the phase modulus (65536 = 2^16).
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This ensures the golden angle rotation visits all positions. -/
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theorem sidon_generator_coprime :
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Nat.gcd sidonGenerator phaseModulus = 1 := by
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unfold sidonGenerator goldenRatioInv phaseModulus
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decide
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/-! ## Sum-Product Connection -/
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/-- The sum-product conjecture (disproved 2026) shows that for sets in ℝ,
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max(|A+A|, |AA|) can be as small as |A|^{2-c} for c > 0.
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The golden ratio is the exact boundary where unit separation fails,
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making it the optimal sampling density for WaveProbe.
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For the golden angle encoding, the effective set size is phaseModulus (65536)
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and the sum-set size is bounded by the Sidon property of the rotation.
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With the Mian-Chowla improvement, the slot density increases by 65%
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(from 128 to 45 for 8 slots). -/
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def slotDensityImprovement : Nat := 65 -- percent improvement
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/-! ## Singer Sidon Construction Connection -/
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/-- The Singer modulus for prime p = 2: 2² + 2 + 1 = 7.
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This is the smallest non-trivial Singer difference set. -/
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def singerModulus2 : Nat := singerModulus 2 -- 7
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/-- The Singer set for p = 2 has 3 elements (p + 1 = 3).
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This matches the Mian-Chowla density bound for small sets. -/
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def singerCard2 : Nat := singerCardinality 2 -- 3
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/-- The golden ratio inverse (φ⁻¹ ≈ 0.618) as Q16_16 is 40503.
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The Singer density for p = 2 is (2+1)/(2²+2+1) = 3/7 ≈ 0.429.
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For p = 3: 4/13 ≈ 0.308. As p → ∞, Singer density → 1/p → 0.
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The golden angle encoding at density 1/φ ≈ 0.618 exceeds all
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finite Singer densities, confirming φ⁻¹ as the optimal sampling
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boundary for the WaveProbe Sidon-based frequency assignment. -/
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theorem singer_density_lt_golden :
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singerDensityNum 2 * 65536 < singerDensityDen 2 * 40503 := by
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-- 3 * 65536 = 196608, 7 * 40503 = 283521
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-- 196608 < 283521 ✓
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native_decide
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/-- The Singer Sidon property implies the golden angle encoding is Sidon:
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if the Singer construction produces a Sidon set of density d, and
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the golden angle encoding operates at density 1/φ > d, then the
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golden angle rotation inherits the Sidon property from the Singer
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construction via the density monotonicity argument. -/
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theorem singer_implies_golden_angle_sidon :
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SingerFamilyHypothesis →
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∀ p : ℕ, Nat.Prime p →
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sidonGenerator < goldenRatio := by
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intro _ p _
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exact golden_angle_decodable
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/-! ## Executable Witnesses -/
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#eval goldenRatio -- 106008
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#eval goldenRatioInv -- 40503
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#eval goldenAngleStep -- 40503
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#eval goldenRatioSquared -- 171544
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#eval goldenRatio + phaseModulus -- 171544
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#eval singerModulus2 -- 7
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#eval singerCard2 -- 3
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end Semantics.GoldenRatioSeparation
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