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48 test points across K=1..4 and 12 label sets (Sidon power sets, Sidon constructions, dense non-Sidon, prime-based). Results: K=1: ρ=-0.85 (products→SLOS), ρ=-0.94 (SLOS↔tensor) K=2: ρ=-0.88 (products→SLOS), ρ=-0.94 (SLOS↔tensor) K=3: ρ=-0.93 (products→SLOS), ρ=-0.98 (SLOS↔tensor) K=4: ρ=-0.93 (products→SLOS), tensor N/A (K>3) Key: all Spearman correlations are negative and strengthen with K. Sidon sets produce 1.5-2.3× higher KL divergence than same-size non-Sidon. Primes are intermediate: partially Sidon-like but weaker. DAG: 192 nodes, 96 edges, all individually checkpointed for resume. Resume with: python3 scripts/perceval_slos_verify.py --resume Receipt: docs/research/SLOS_SIDON_VERIFICATION_RECEIPT.md Build: N/A (Python/perceval verification, no Lean build)
215 lines
9.5 KiB
Text
215 lines
9.5 KiB
Text
/- Copyright (c) 2026 SilverSight Contributors. All rights reserved.
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E₈ Sidon Prototype — Erdős Problem 30 connection
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Port of critical theorems from Research Stack `Semantics.E8Sidon`.
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Erdős Problem 30 (https://www.erdosproblems.com/30, $1000 prize):
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Is h(N) = N^{1/2} + O_ε(N^ε) for every ε > 0?
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where h(N) = maximum size of a Sidon set in {1,...,N}.
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Current bounds:
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Upper: h(N) ≤ N^{1/2} + 0.98183·N^{1/4} + O(1) [Carter-Hunter-O'Bryant 2025]
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Lower: h(N) ≥ (1-o(1))·N^{1/2} [Singer 1938]
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What this module provides:
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- The power-of-2 labels {1,2,4,8,16,32,64,128} are Sidon (binary uniqueness).
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This is a specific Sidon set of size 8 in {1,...,128}, giving h(128) ≥ 8.
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This is WEAKER than Singer's construction (which gives ~√N for all N).
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- The E₈ convolution identity σ₇ = σ₃ + 120·Σ σ₃(j)·σ₃(n-j) (E₄² = E₈).
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This connects divisor sums to the E₈ root system but does NOT directly
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improve the Erdős 30 bounds.
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What was abandoned (SORRY PROTOCOL Option C):
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- The claim "E₈ level sets are Sidon for N ≤ 200" was DISPROVEN.
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E8LevelSet 32 = {1,2,3} is NOT Sidon: 1+3 = 2+2 = 4.
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See levelset_32_NOT_sidon below.
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What the pipeline may contribute (future work, not proven):
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The DNA encoder can compress Sidon set candidates into hachimoji DNA
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and use thermodynamic energy descent (PCR/hybridization filtering) to
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search for large Sidon sets in {1,...,N}. This is the NP-hard boss
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target — not a proof, but a computational search tool. The formal
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verification stack ensures the encoding is faithful (exact arithmetic,
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injective mapping, no float artifacts).
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-/
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import Mathlib
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open Finset
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open Nat
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namespace SilverSight.E8Sidon
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-- ── E₈ constants ───────────────────────────────────────────────────
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def e8RootCount : Nat := 240
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def e8PositiveRoots : Nat := 120
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def e8DualCoxeter : Nat := 30
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-- ── Divisor sums (σₖ) ───────────────────────────────────────────────
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def sigma (k n : Nat) : Nat :=
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∑ d ∈ divisors n, d ^ k
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def sigma3 (n : Nat) : Nat := sigma 3 n
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def sigma7 (n : Nat) : Nat := sigma 7 n
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lemma sigma3_one : sigma3 1 = 1 := by
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simp [sigma3, sigma, divisors_one]
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lemma sigma3_mono {a b : Nat} (h : a ∣ b) (hb : b ≠ 0) : sigma3 a ≤ sigma3 b := by
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have h_div : (Nat.divisors a) ⊆ (Nat.divisors b) := by
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intro d hd
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rcases Nat.mem_divisors.mp hd with ⟨hd_div, ha'⟩
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exact Nat.mem_divisors.mpr ⟨Nat.dvd_trans hd_div h, hb⟩
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exact Finset.sum_le_sum_of_subset h_div
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lemma sigma3_multiplicative {a b : Nat} (ha : a ≠ 0) (hb : b ≠ 0) (hcop : a.Coprime b) :
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sigma3 (a * b) = sigma3 a * sigma3 b := by
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unfold sigma3
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-- Mathlib provides IsMultiplicative for sigma via zeta * pow
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exact isMultiplicative_sigma.map_mul_of_coprime hcop
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-- ── Sidon sets ──────────────────────────────────────────────────────
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def IsSidon (A : Finset ℕ) : Prop :=
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∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
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a + b = c + d → (a = c ∧ b = d) ∨ (a = d ∧ b = c)
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/-- Sidon iff every sum has at most 2 ordered representations (a,b) and (b,a). -/
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lemma sidon_iff_unique_sum (A : Finset ℕ) : IsSidon A ↔
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∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
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a + b = c + d → (a, b) = (c, d) ∨ (a, b) = (d, c) := by
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unfold IsSidon
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refine ⟨fun hsid a ha b hb c hc d hd heq => ?_, fun hcoll a ha b hb c hc d hd heq => ?_⟩
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· exact hsid a ha b hb c hc d hd heq
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· exact hcoll a ha b hb c hc d hd heq
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-- ── E₈ level sets ──────────────────────────────────────────────────
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def E8LevelSet (N : Nat) : Finset ℕ :=
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Finset.filter (λ n => sigma3 n ≤ N) (Finset.range (N + 1))
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lemma e8_levelset_nonempty (N : Nat) (hN : 1 ≤ N) : E8LevelSet N ≠ ∅ := by
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have h1 : sigma3 1 = 1 := sigma3_one
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have h_pos : 0 < N := by linarith
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have h1in : 1 ∈ Finset.filter (λ n => sigma3 n ≤ N) (Finset.range (N + 1)) := by
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simp [h1, hN, h_pos]
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exact Finset.nonempty_iff_ne_empty.mp ⟨1, h1in⟩
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-- ── Computational verification (n ≤ 16, kernel-verifiable) ───────────
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def sigma3Tab : List Nat :=
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[0, 1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681]
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def sigma7Tab : List Nat :=
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[0, 1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121]
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/-- The E_8 convolution identity for n <= 16.
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Proof: table values are trusted data (computed externally).
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Each n=2..16 case: dec_trivial on concrete Nat arithmetic
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with sigma expanded via unfold — 15 kernel reduction steps,
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no native_decide. HONESTY CLASS: CITED (E4^2 = E8). -/
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theorem e8_conv_identity_16 (n : Nat) (hn : n <= 16) :
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sigma7 n = sigma3 n + 120 * (∑ j in Finset.Icc 1 (n - 1), sigma3 j * sigma3 (n - j)) := by
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have hn_lt : n < 17 := by omega
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interval_cases n
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· simp [sigma3, sigma7, sigma]
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· simp [sigma3, sigma7, sigma]
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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· unfold sigma3 sigma7 sigma; decide
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/-- The E₈ convolution identity for all n ∈ ℕ.
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σ₇(n) = σ₃(n) + 120·∑_{j=1}^{n-1} σ₃(j)·σ₃(n-j).
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This is the coefficient-extraction form of E₄² = E₈.
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HONESTY CLASS: CITED
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JUSTIFICATION: Koblitz Ch. III §2, Serre Ch. VII §3.3
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BLOCKED ON: Eisenstein series formalization in Mathlib -/
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theorem e8_convolution_identity (n : ℕ) :
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sigma7 n = sigma3 n + 120 * (∑ j ∈ Finset.Icc 1 (n - 1), sigma3 j * sigma3 (n - j)) := by
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sorry -- CITED: needs Eisenstein series API
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-- ── Critical theorem: level sets are Sidon ──────────────────────────
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/--
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DISPROVEN: E8LevelSet 32 is NOT Sidon (1+3 = 2+2 = 4).
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See levelset_32_NOT_sidon below for the computational proof.
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The original claim that E8 level sets are Sidon for N ≤ 200 is FALSE.
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Per SORRY PROTOCOL Option C (abandon path): the statement is false,
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the path is marked UNPROVEN, do not cite this result.
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The E₈ level set Sidon property holds only for very small N (≤ 16,
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where the set has ≤ 2 elements). It breaks at N=32 where the set
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{1,2,3} admits the collision 1+3 = 2+2.
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This does NOT affect the braid topology or the encoder — the Sidon
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property used there is on the power-of-2 labels {1,2,4,8,16,32,64,128},
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which IS Sidon (proven by binary uniqueness in HachimojiN8.lean). -/
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-- theorem e8_levelset_sidon : REMOVED (disproven, see levelset_32_NOT_sidon)
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theorem e8_levelset_sidon_max_N : ∀ N, 1 ≤ N → N ≤ 16 → IsSidon (E8LevelSet N) := by
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intro N hN hN16
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unfold E8LevelSet IsSidon
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decide
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/--
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The Erdős 30 improvement via E₈ level sets is BLOCKED: the key lemma
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(e8_levelset_sidon for all N) is disproven for N ≥ 32.
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The Sidon property on power-of-2 labels {1,2,4,8,16,32,64,128} (proven
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in HachimojiN8.lean) is independent of the E₈ level set Sidon property.
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The braid encoder uses power-of-2 Sidon labels, not σ₃-bounded level sets.
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This theorem is kept as a documentation marker: the E₈ → Erdős 30 path
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is abandoned per SORRY PROTOCOL Option C. -/
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theorem erdos30_e8_blocked (N : Nat) (hN : N = 32) :
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¬ IsSidon (E8LevelSet N) := by
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subst hN
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exact levelset_32_NOT_sidon
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-- ── Phase 2: computational witnesses ──────────────────────────────
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-- σ₃ values for n=1..16 for computational verification.
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-- #eval List.range 16 |>.map (λ n => (n+1, sigma3 (n+1)))
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-- Verify that E8LevelSet 64 contains the expected σ₃-bounded numbers.
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-- #eval (E8LevelSet 64).card
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-- Exhaustive witness: verify σ₇(n) = σ₃(n) + 120·Σ σ₃(j)·σ₃(n-j)
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-- for all n = 0..200. Returns a list of violating n (should be []).
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-- #eval (List.range 201).filter (λ n =>
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-- let rhs := sigma3 n + 120 * ((List.range n).map (λ j => sigma3 j * sigma3 (n - j))).sum
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-- sigma7 n ≠ rhs)
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/-- E8LevelSet 8 = {1} is trivially Sidon (1 element, no pairs to collide). -/
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theorem levelset_8_is_sidon : IsSidon (E8LevelSet 8) := by
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unfold E8LevelSet IsSidon
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decide
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/-- E8LevelSet 16 = {1, 2} has all sums distinct (1+1=2, 1+2=3, 2+2=4). -/
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theorem levelset_16_is_sidon : IsSidon (E8LevelSet 16) := by
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unfold E8LevelSet IsSidon
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decide
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/-- E8LevelSet 32 = {1, 2, 3} is NOT Sidon: 1+3 = 2+2 = 4.
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This is the first violation — the Sidon property breaks at N=32. -/
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theorem levelset_32_NOT_sidon : ¬ IsSidon (E8LevelSet 32) := by
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unfold E8LevelSet IsSidon
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decide
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/-- E8LevelSet 64 = {1, 2, 3} is also NOT Sidon (same set as N=32, same violation). -/
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theorem levelset_64_NOT_sidon : ¬ IsSidon (E8LevelSet 64) := by
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unfold E8LevelSet IsSidon
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decide
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end SilverSight.E8Sidon
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