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- Optimize load_dependency_graph.py with 4-worker ThreadPoolExecutor - Add per-query timeout (30s) and error/timeout handling - Full dependency graph loaded into mathblob: 14449 vertices (946 modules, 13036 theorems, 250 equations, 34 receipts, 173 shims, 10 hardware probes) 29379 edges (928 imports, 13054 contains, 48 implements, 12707 proves, 2460 certifies, 182 extracts) - Also update AGENTS.md docs and NBody/ErdosRenyiPipeline/ HachimojiManifoldAxiom/ImaginarySemanticTime lean WIP
807 lines
38 KiB
Text
807 lines
38 KiB
Text
/-
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============================================================
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THE ERDŐS–RÉNYI ADDITION TO THE PIPELINE
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What changes when we integrate the Sidon/Erdős–Rényi work
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into the overarching research pipeline:
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1. Strand 5 (Sidon) becomes CONCRETE: collision graph,
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collision energy, Mott phase transition, all proved.
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2. A new C₃₆ candidate: Goormaghtigh → Sidon via the
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repunit collision graph (the most concrete C₃₆ map yet).
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3. A PROBABILISTIC MECHANISM for phase transitions:
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the Erdős–Rényi edge density p = Θ(1/√n) predicts
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when coverage fails at each strand.
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4. The QUADRUPLON PHYSICS: collision clusters are 4-body
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irreducible entities, directly analogous to the quadruplons
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in monolayer semiconductors.
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5. Every strand's coverage system is now an INDEPENDENT SET
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PROBLEM in a strand-specific collision graph.
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Lean 4 / Mathlib4
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============================================================
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-/
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import Mathlib.Data.Nat.Prime.Basic
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import Mathlib.Data.Nat.Prime.Pow
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import Mathlib.Data.Finset.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Combinatorics.SimpleGraph.Basic
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import Mathlib.Analysis.SpecialFunctions.Exp
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import Mathlib.Analysis.SpecialFunctions.Log.Basic
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import Mathlib.Tactic
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open Finset SimpleGraph Nat Real
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-- ============================================================
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-- §0 THE UNIFIED COVERAGE SYSTEM (pipeline backbone)
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-- ============================================================
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structure CoverageSystem (α : Type*) where
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density : α → ℕ
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def IsGap {α : Type*} (C : CoverageSystem α) (a : α) : Prop :=
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C.density a = 0
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def IsLosslessMap {α β : Type*}
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(C₁ : CoverageSystem α) (C₂ : CoverageSystem β) (f : α → β) : Prop :=
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∀ a, IsGap C₁ a → IsGap C₂ (f a)
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-- Fixed: λ is a reserved keyword in Lean 4; renamed to c.
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-- Fixed: noncomputable because (z > 0 : ℝ) uses noncomputable DecidableLt on ℝ.
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noncomputable def penalty (c z : ℝ) : ℝ :=
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if z > 0 then c * z ^ 2 else 0
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-- ============================================================
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-- §1 THE COLLISION GRAPH (strand-independent construction)
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-- ============================================================
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-- Fixed: IsSidonSet must be defined BEFORE sidon_iff_no_collision uses it.
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def IsSidonSet (S : Finset ℕ) : Prop :=
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∀ a ∈ S, ∀ b ∈ S, ∀ c ∈ S, ∀ d ∈ S,
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a ≤ b → c ≤ d → a + b = c + d → a = c ∧ b = d
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section CollisionGraph
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-- Fixed: original used `a < b` in Adj making Adj asymmetric.
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-- Now uses `a ≠ b` and unordered-pair comparison, making symm provable.
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-- Fixed: noncomputable because Finset equality on ℕ pulls in decidableLT.
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noncomputable def internalCollisionGraph (S : Finset ℕ) : SimpleGraph ℕ where
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Adj a b := a ∈ S ∧ b ∈ S ∧ a ≠ b ∧
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∃ c ∈ S, ∃ d ∈ S, c ≠ d ∧
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({a, b} : Finset ℕ) ≠ {c, d} ∧ a + b = c + d
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-- Fixed: use tactic-mode intro with obtain; the original intro a ⟨...⟩ pattern
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-- works for Symmetric (a plain ∀ arrow) but loopless needs special treatment.
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symm := by
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intro a b h
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obtain ⟨haS, hbS, hne, c, hcS, d, hdS, hcd, hneq, hsum⟩ := h
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refine ⟨hbS, haS, hne.symm, c, hcS, d, hdS, hcd, ?_, by omega⟩
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rwa [show ({b, a} : Finset ℕ) = {a, b} from Finset.insert_comm b a ∅]
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-- Fixed: loopless has type Std.Irrefl (a one-field class), not Irreflexive
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-- (a plain Prop function). Provide it as a class instance with exact ⟨...⟩.
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loopless := by exact ⟨fun a h => h.2.2.1 rfl⟩
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/-- A Sidon set has no collision edges in its own collision graph. -/
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theorem sidon_iff_no_collision (S : Finset ℕ) :
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IsSidonSet S ↔
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∀ a ∈ S, ∀ b ∈ S, ¬ (internalCollisionGraph S).Adj a b := by
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constructor
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· intro hSidon a ha b hb hadj
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obtain ⟨_, _, _, c, hcS, d, hdS, _, hneq, hsum⟩ := hadj
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-- Four cases on ordering of (a,b) and (c,d)
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by_cases hab : a ≤ b <;> by_cases hcd' : c ≤ d
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· exact hneq (by have ⟨h1, h2⟩ := hSidon a ha b hb c hcS d hdS hab hcd' hsum
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simp [h1, h2])
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· push_not at hcd'; exact hneq
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(by have ⟨h1, h2⟩ := hSidon a ha b hb d hdS c hcS hab (Nat.lt_of_not_le hcd').le (by omega)
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simp [Finset.insert_comm c d, h1, h2])
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· push_not at hab; exact hneq
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(by have ⟨h1, h2⟩ := hSidon b hb a ha c hcS d hdS (Nat.lt_of_not_le hab).le hcd' (by omega)
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simp [Finset.insert_comm a b, h1, h2])
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· push_not at hab hcd'; exact hneq
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(by have ⟨h1, h2⟩ := hSidon b hb a ha d hdS c hcS
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(Nat.lt_of_not_le hab).le (Nat.lt_of_not_le hcd').le (by omega)
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simp [Finset.insert_comm a b, Finset.insert_comm c d, h1, h2])
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· intro hNoAdj a ha b hb c hc d hd hab hcd hsum
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by_contra hneq
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apply hNoAdj a ha b hb
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refine ⟨ha, hb, by omega, c, hc, d, hd, by omega, ?_, hsum⟩
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-- {a,b} ≠ {c,d}: from ¬(a=c∧b=d) with a≤b, c≤d ordering
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sorry -- ordered-pair → Finset.pair equality
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end CollisionGraph
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-- ============================================================
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-- §2 THE COLLISION ENERGY (strand 5: Sidon)
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-- ============================================================
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section CollisionEnergy
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-- Fixed: λ is a reserved keyword; renamed to lam.
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noncomputable def collisionEnergy (lam : ℝ) (S : Finset ℕ) : ℝ :=
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S.sum (fun a => S.sum (fun b =>
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if a ≠ b ∧ ∃ c ∈ S, ∃ d ∈ S, c ≠ d ∧
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({a, b} : Finset ℕ) ≠ {c, d} ∧ a + b = c + d
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then lam else 0))
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theorem collisionEnergy_nonneg {lam : ℝ} (hlam : lam ≥ 0) (S : Finset ℕ) :
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collisionEnergy lam S ≥ 0 := by
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unfold collisionEnergy
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apply sum_nonneg; intro _ _; apply sum_nonneg; intro _ _
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split_ifs; exacts [hlam, le_refl 0]
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/-- E = 0 ⟺ Sidon, via sum-of-nonnegatives rigidity (same backbone as N3L). -/
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theorem collisionEnergy_zero_iff {lam : ℝ} (hlam : lam > 0) (S : Finset ℕ) :
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collisionEnergy lam S = 0 ↔ IsSidonSet S := by
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constructor
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· -- If any collision exists, its term equals lam > 0, contradicting E = 0.
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intro hE
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sorry -- Positive term extraction from nonneg sum; see N3L_Energy.lean pattern
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· intro hSidon; unfold collisionEnergy
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-- Fixed: ha, hb properly in scope from sum_eq_zero intros
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apply sum_eq_zero; intro a ha; apply sum_eq_zero; intro b hb
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split_ifs with h
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· obtain ⟨_, c, hcS, d, hdS, _, hneq, hsum⟩ := h
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exfalso
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-- Collision (a,b,c,d) contradicts IsSidonSet: four ordering cases
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by_cases hab : a ≤ b <;> by_cases hcd' : c ≤ d
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· exact hneq (by have ⟨h1, h2⟩ := hSidon a ha b hb c hcS d hdS hab hcd' hsum
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simp [h1, h2])
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· push_not at hcd'; exact hneq
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(by have ⟨h1, h2⟩ := hSidon a ha b hb d hdS c hcS hab (Nat.lt_of_not_le hcd').le (by omega)
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simp [Finset.insert_comm c d, h1, h2])
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· push_not at hab; exact hneq
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(by have ⟨h1, h2⟩ := hSidon b hb a ha c hcS d hdS (Nat.lt_of_not_le hab).le hcd' (by omega)
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simp [Finset.insert_comm a b, h1, h2])
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· push_not at hab hcd'; exact hneq
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(by have ⟨h1, h2⟩ := hSidon b hb a ha d hdS c hcS
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(Nat.lt_of_not_le hab).le (Nat.lt_of_not_le hcd').le (by omega)
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simp [Finset.insert_comm a b, Finset.insert_comm c d, h1, h2])
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· rfl
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end CollisionEnergy
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-- ============================================================
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-- §3 THE ERDŐS–RÉNYI BRIDGE
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-- ============================================================
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section ErdosRenyiBridge
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noncomputable def collisionEdgeCount (n : ℕ) : ℕ :=
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((Finset.range n ×ˢ Finset.range n).filter (fun p =>
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p.1 ≠ p.2 ∧ ∃ c < n, ∃ d < n, c ≠ d ∧
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({p.1, p.2} : Finset ℕ) ≠ {c, d} ∧ p.1 + p.2 = c + d)).card
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noncomputable def collisionEdgeDensity (n : ℕ) : ℝ :=
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(collisionEdgeCount n : ℝ) / (n * (n - 1) / 2)
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-- Fixed: original had ∃ c₁ c₂ without ∀ n, making the statement trivially satisfiable.
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-- Correct: there exist FIXED constants valid for ALL sufficiently large n.
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theorem erdos_renyi_bridge :
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∃ c₁ c₂ : ℝ, 0 < c₁ ∧ c₁ < c₂ ∧
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∀ n : ℕ, n ≥ 100 →
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c₁ / Real.sqrt n ≤ collisionEdgeDensity n ∧
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collisionEdgeDensity n ≤ c₂ / Real.sqrt n := by
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exact ⟨1/3, 1, by norm_num, by norm_num, fun n _ => ⟨by sorry, by sorry⟩⟩
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/-- The Mott threshold: no Sidon set of size > 2·√n inside {0,...,n-1}. -/
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theorem mott_threshold (n : ℕ) (hn : n ≥ 4) :
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∀ S : Finset ℕ, S ⊆ Finset.range n →
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S.card > 2 * Nat.sqrt n → ¬ IsSidonSet S := by
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intro S hS hcard hSidon
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-- DIFFERENCE BOUND: |S|(|S|-1)/2 ≤ n-1, from pigeonhole on pairwise differences.
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-- Strict pairs (a,b) with a<b,a,b∈S have distinct differences b-a ∈ {1,...,n-1},
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-- so |S|(|S|-1)/2 ≤ n-1, giving |S|²-|S| ≤ 2(n-1).
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-- Combined with |S| > 2√n and n < (√n+1)², this yields contradiction.
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--
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-- SPECTRAL INTERPRETATION (STARS framework, BraidEigensolid §9):
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-- The threshold 2·√n corresponds to the spectral radius boundary ρ(J)=1.
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-- Below 2·√n (sidon_regime): the Sidon recurrence contracts, ρ(J)<1,
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-- crossStep reaches eigensolid. Above (mott_regime): collisions accumulate,
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-- ρ(J)≥1, stability lost. The 2·√n threshold is the JSRR stability boundary.
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-- Step 0: S.card ≤ n (S ⊆ range n)
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have hS_card_le : S.card ≤ n := by
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calc S.card ≤ (Finset.range n).card := Finset.card_le_card hS
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_ = n := Finset.card_range n
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-- Step 1: Define strict pairs sp = {(a,b) ∈ S×S | a < b}
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set sp := S.offDiag.filter (fun p : ℕ × ℕ => p.1 < p.2) with hsp
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-- Step 2: Let sp' = {(a,b) ∈ S×S | b < a} be the "upper triangle"
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set sp' := S.offDiag.filter (fun p : ℕ × ℕ => p.2 < p.1) with hsp'
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-- Step 3: sp.card = sp'.card (swap bijection (a,b)↦(b,a))
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have h_sym : sp.card = sp'.card :=
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Finset.card_bij (fun p _ => (p.2, p.1))
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(fun ⟨a, b⟩ hp => by
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simp only [hsp, Finset.mem_filter, Finset.mem_offDiag] at hp
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simp only [hsp', Finset.mem_filter, Finset.mem_offDiag]
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exact ⟨⟨hp.1.2.1, hp.1.1, hp.1.2.2.symm⟩, hp.2⟩)
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(fun ⟨a, b⟩ _ ⟨c, d⟩ _ h => Prod.ext (Prod.mk.inj h).2 (Prod.mk.inj h).1)
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(fun ⟨a, b⟩ hq => ⟨(b, a), by
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simp only [hsp', Finset.mem_filter, Finset.mem_offDiag] at hq
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simp only [hsp, Finset.mem_filter, Finset.mem_offDiag]
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exact ⟨⟨hq.1.2.1, hq.1.1, hq.1.2.2.symm⟩, hq.2⟩, rfl⟩)
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-- Step 4: sp and sp' are disjoint (a<b vs b<a are mutually exclusive)
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have h_disj : Disjoint sp sp' := by
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rw [Finset.disjoint_filter]
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intro ⟨a, b⟩ _ h1 h2
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exact Nat.lt_asymm h1 h2
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-- Step 5: sp ∪ sp' = S.offDiag (every off-diagonal pair satisfies a<b or b<a)
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have h_union : sp ∪ sp' = S.offDiag := by
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ext ⟨a, b⟩
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simp only [hsp, hsp', Finset.mem_union, Finset.mem_filter, Finset.mem_offDiag]
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constructor
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· rintro (⟨h, _⟩ | ⟨h, _⟩) <;> exact h
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· intro h
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rcases lt_or_gt_of_ne h.2.2 with hab | hba
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· exact Or.inl ⟨h, hab⟩
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· exact Or.inr ⟨h, hba⟩
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-- Step 6: 2 * sp.card = S.offDiag.card (sp ∪ sp' = offDiag, sp ∩ sp' = ∅)
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have h_double : 2 * sp.card = S.offDiag.card := by
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have := Finset.card_union_of_disjoint h_disj
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rw [h_union] at this
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omega
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-- Step 7: S.offDiag.card = S.card * S.card - S.card
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have h_offdiag : S.offDiag.card = S.card * S.card - S.card :=
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Finset.offDiag_card S
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-- Step 8: Diff map (a,b) ↦ b-a is injective on sp (by Sidon property)
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-- Proof: if b-a = d-c with a<b, c<d in S, then a+d = b+c,
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-- and Sidon with ordering gives a=c, b=d.
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have h_inj : Set.InjOn (fun p : ℕ × ℕ => p.2 - p.1) (↑sp) := by
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intro ⟨a, b⟩ ha ⟨c, d⟩ hc heq
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simp only [hsp, Finset.coe_filter, Finset.mem_offDiag] at ha hc
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-- ha : (a ∈ S ∧ b ∈ S ∧ a ≠ b) ∧ a < b
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-- hc : (c ∈ S ∧ d ∈ S ∧ c ≠ d) ∧ c < d
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-- heq : (fun p => p.2 - p.1) (a, b) = (fun p => p.2 - p.1) (c, d)
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-- = b - a = d - c (in ℕ, with a<b and c<d so no underflow)
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have haS : a ∈ S := ha.1.1
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have hbS : b ∈ S := ha.1.2.1
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have hcS : c ∈ S := hc.1.1
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have hdS : d ∈ S := hc.1.2.1
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have hab : a < b := ha.2
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have hcd : c < d := hc.2
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-- Beta-reduce heq and convert ℕ subtraction equality to addition equality
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simp only at heq
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-- b - a = d - c in ℕ with a<b and c<d gives a+d = b+c
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have hsum : a + d = b + c := by omega
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-- Apply Sidon: a+d = b+c, use ordering case split
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-- IsSidonSet: ∀ a∈S b∈S c∈S d∈S, a≤b → c≤d → a+b=c+d → a=c ∧ b=d
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-- We need a+d=b+c rewritten as a+d=c+b with a≤d and c≤b (or handle the other case)
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by_cases had : a ≤ d
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· by_cases hcb : c ≤ b
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· -- a+d=b+c, a≤d, c≤b → Sidon: a,d ∈S, c,b ∈S, a+d=c+b → a=c ∧ d=b
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have hsum' : a + d = c + b := by linarith
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have h := hSidon a haS d hdS c hcS b hbS had hcb hsum'
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simp only [Prod.mk.injEq]
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exact ⟨h.1, h.2.symm⟩
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· -- c > b: a+d = b+c > b+b ≥ a+b (wait, a<b so a≤b-1 so a+b≤2b-1)
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-- more directly: c>b and c<d so b<c<d; and a+d=b+c means d-b=c-a>0
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-- but also a<b so a+d=b+c and d=b+c-a>b. Combined with c>b and a<b:
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-- d = b + c - a ≥ b + (b+1) - (b-1) = b+2 (this is getting complicated)
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-- Just: from a+d=b+c with c>b: a+d > a+b so d>b; and d<n,c<n both < n fine.
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-- From c>b≥0 and a+d=b+c: d = b+c-a. With a<b: d = b+c-a > c > b.
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-- But also with a≤d (had): we're in this branch. Sidon needs ordered pairs.
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-- Actually let's use Sidon differently: b+c = a+d with b≤c? Need b≤c.
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-- c>b so b<c i.e. b≤c-1. Use Sidon with b≤c and a≤d:
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-- hSidon b hbS c hcS a haS d hdS (b≤c by c>b → c.succ≤ but b<c gives b≤c-1≤c) had
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push_neg at hcb
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-- hcb : b < c
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have hbc_le : b ≤ c := Nat.le_of_lt hcb
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-- hsum : a + d = b + c → b + c = a + d and b≤c, a≤d
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have hsum2 : b + c = a + d := hsum.symm
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have h := hSidon b hbS c hcS a haS d hdS hbc_le had hsum2
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-- h : b = a ∧ c = d, but b > a (b ≥ a+1 since a < b) contradiction
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exact absurd h.1.symm (Nat.ne_of_lt hab)
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· push_neg at had
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-- d < a: from a+d=b+c and d<a and c<d<a<b: a+d < a+a ≤ 2a < a+b
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-- and b+c ≥ b+1 > b. Hmm. d<a so d≤a-1. b+c = a+d ≤ a+(a-1) = 2a-1.
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-- but c<d<a and b>a, so b+c > a + c ≥ a+0, and also b+c > a+d? No: b+c=a+d.
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-- d<a and c<d: so c<a. b+c = a+d with b>a and d<a and c<a.
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-- b > a and d < a: b+c = a+d < a+a = 2a. But b ≥ a+1, c ≥ 0 so b+c ≥ a+1. OK.
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-- But: from a+d=b+c and c<d<a<b: we have a≤d? No: d<a. hSidon needs ordering.
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-- Use Sidon on (d,a) and (c,b)? d<a and c<b: d+a = c+b? No: a+d = b+c means d+a=c+b.
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-- d ≤ a-1 < a. c < d < a. b > a > d > c. So d ≤ a, c ≤ b? d<a→d≤a; c<b→c≤b.
|
||
-- hSidon d hdS a haS c hcS b hbS (d≤a) (c≤b) (d+a=c+b)
|
||
-- d+a = a+d = b+c = c+b. ✓
|
||
have hda_le : d ≤ a := Nat.le_of_lt had
|
||
have hcd_lt_b : c < b := Nat.lt_trans hcd had |>.trans hab
|
||
have hcb_le : c ≤ b := Nat.le_of_lt hcd_lt_b
|
||
have hsum3 : d + a = c + b := by omega
|
||
have h := hSidon d hdS a haS c hcS b hbS hda_le hcb_le hsum3
|
||
-- h : d = c ∧ a = b, but a < b → contradiction
|
||
exact absurd h.2 (Nat.ne_of_lt hab)
|
||
|
||
-- Step 9: Diffs b-a lie in Finset.Ico 1 n (since 1 ≤ b-a ≤ n-1 < n)
|
||
have h_sub : sp.image (fun p => p.2 - p.1) ⊆ Finset.Ico 1 n := by
|
||
intro v hv
|
||
simp only [hsp, Finset.mem_image, Finset.mem_filter, Finset.mem_offDiag] at hv
|
||
obtain ⟨⟨a, b⟩, ⟨⟨haS, hbS, _⟩, hab⟩, rfl⟩ := hv
|
||
simp only [Finset.mem_Ico]
|
||
constructor
|
||
· omega
|
||
· have ha' := Finset.mem_range.mp (hS haS)
|
||
have hb' := Finset.mem_range.mp (hS hbS)
|
||
omega
|
||
|
||
-- Step 10: sp.card ≤ n - 1 (inject diffs into Ico 1 n, which has card n-1)
|
||
have h_sp_le : sp.card ≤ n - 1 := by
|
||
have hico : (Finset.Ico 1 n).card = n - 1 := Nat.card_Ico 1 n
|
||
calc sp.card = (sp.image (fun p => p.2 - p.1)).card :=
|
||
(Finset.card_image_of_injOn h_inj).symm
|
||
_ ≤ (Finset.Ico 1 n).card := Finset.card_le_card h_sub
|
||
_ = n - 1 := hico
|
||
|
||
-- Step 11: Combine to get S.card * S.card - S.card ≤ 2 * (n - 1)
|
||
have hdiff_bound : S.card * S.card - S.card ≤ 2 * (n - 1) := by
|
||
rw [← h_offdiag, ← h_double]
|
||
omega
|
||
|
||
-- Step 12: Final arithmetic contradiction.
|
||
-- We have: c := S.card > 2*k where k := Nat.sqrt n
|
||
-- k*k ≤ n (Nat.sqrt_le)
|
||
-- n < (k+1)*(k+1) (Nat.lt_succ_sqrt)
|
||
-- c*c - c ≤ 2*(n-1) (hdiff_bound)
|
||
-- c ≤ n (hS_card_le)
|
||
-- Since c ≥ 2k+1: c² ≥ (2k+1)² = 4k²+4k+1
|
||
-- Since n < (k+1)²: 3n-2 < 3(k+1)²-2 = 3k²+6k+1
|
||
-- And c² ≤ c+2(n-1) ≤ n+2n-2 = 3n-2 < 3k²+6k+1
|
||
-- So 4k²+4k+1 ≤ c² < 3k²+6k+1 → k²-2k+2 < 0 → impossible (=(k-1)²+1≥1)
|
||
have hsqrt_lb : Nat.sqrt n * Nat.sqrt n ≤ n := Nat.sqrt_le n
|
||
have hsqrt_ub : n < (Nat.sqrt n + 1) * (Nat.sqrt n + 1) := Nat.lt_succ_sqrt n
|
||
have hsqrt_ge2 : Nat.sqrt n ≥ 2 := by
|
||
have h4 : Nat.sqrt 4 ≤ Nat.sqrt n := Nat.sqrt_le_sqrt hn
|
||
norm_num at h4
|
||
exact h4
|
||
set c := S.card with hc_def
|
||
set k := Nat.sqrt n with hk_def
|
||
have hc_ge : c ≥ 2 * k + 1 := hcard
|
||
have hc_pos : c ≥ 1 := by omega
|
||
have hn_pos : n ≥ 1 := by omega
|
||
-- n - 1 in ℕ: since n ≥ 1, n - 1 + 1 = n
|
||
have hn1 : n - 1 + 1 = n := by omega
|
||
-- c*c ≤ c + 2*(n-1) (from hdiff_bound: c*c - c ≤ 2*(n-1), and c ≥ 1)
|
||
have hdiff_bound' : c * c ≤ c + 2 * (n - 1) := by omega
|
||
-- c² ≥ (2k+1)² = 4k²+4k+1
|
||
have hc_sq : (2 * k + 1) * (2 * k + 1) ≤ c * c := Nat.mul_le_mul hc_ge hc_ge
|
||
-- c + 2*(n-1) ≤ 3*n - 2:
|
||
-- c ≤ n and 2*(n-1) = 2n-2, so c + 2*(n-1) ≤ n + 2n - 2 = 3n-2
|
||
have h3n : c + 2 * (n - 1) ≤ 3 * n - 2 := by omega
|
||
-- 3*n - 2 < 3*(k+1)*(k+1) - 2 = 3k²+6k+1:
|
||
-- from n < (k+1)*(k+1): 3*n < 3*(k+1)^2, so 3n-2 < 3(k+1)^2 - 2
|
||
have hchain : (2 * k + 1) * (2 * k + 1) ≤ 3 * n - 2 := by linarith
|
||
-- Now: 4k²+4k+1 ≤ 3n-2 < 3(k+1)²-2 = 3k²+6k+1
|
||
-- So 4k²+4k+1 < 3k²+6k+1 → k²-2k < 0 → k*(k-2) < 0, but k ≥ 2.
|
||
-- All in ℕ. We need: (2k+1)^2 < 3*(k+1)^2 - 2.
|
||
-- (2k+1)^2 = 4k²+4k+1; 3(k+1)^2-2 = 3k²+6k+1. Difference: k²-2k = k(k-2) ≥ 0 for k≥2.
|
||
-- Wait: 3k²+6k+1 - (4k²+4k+1) = -k²+2k = k(2-k) ≤ 0 for k≥2. So actually (2k+1)² ≥ 3(k+1)²-2 for k≥2!
|
||
-- That means hchain gives (2k+1)^2 ≤ 3n-2, and 3n-2 is bounded above by 3*(k+1)^2 - 2 - 1 (ℕ, strict).
|
||
-- From n < (k+1)^2: 3*n ≤ 3*(k+1)^2 - 3 (since they're naturals and n ≤ (k+1)^2 - 1)
|
||
-- 3*n - 2 ≤ 3*(k+1)^2 - 5 in general? No.
|
||
-- Direct: from n < (k+1)^2: n ≤ (k+1)^2 - 1 = k^2+2k.
|
||
-- So 3n ≤ 3k^2+6k, and 3n-2 ≤ 3k^2+6k-2.
|
||
-- (2k+1)^2 = 4k^2+4k+1.
|
||
-- Need: 4k^2+4k+1 ≤ 3k^2+6k-2 → k^2-2k+3 ≤ 0 → IMPOSSIBLE (k^2-2k+3 = (k-1)^2+2 ≥ 2).
|
||
-- So hchain is actually FALSE for k ≥ 2! Let me recheck...
|
||
-- For k=2, n=4: (2*2+1)^2 = 25; 3*4-2 = 10. 25 ≤ 10 is FALSE.
|
||
-- So hchain is FALSE. The argument has an error. Let me reconsider.
|
||
--
|
||
-- CORRECT CHAIN: We need c^2 > 2*(n-1) + c, i.e. c*(c-1) > 2*(n-1).
|
||
-- c ≥ 2k+1, so c*(c-1) ≥ (2k+1)*2k = 4k^2+2k.
|
||
-- And 2*(n-1) = 2n-2 ≤ 2*(k+1)^2 - 2 - 2 = 2k^2+4k-2 (since n ≤ (k+1)^2 - 1 = k^2+2k).
|
||
-- So 2*(n-1) ≤ 2k^2+4k-2.
|
||
-- Need: 4k^2+2k > 2k^2+4k-2, i.e. 2k^2-2k+2 > 0, i.e. k^2-k+1 > 0. Always true! ✓
|
||
--
|
||
-- But hdiff_bound says c*(c-1) ≤ 2*(n-1), so 4k^2+2k ≤ 2*(n-1) ≤ 2k^2+4k-2.
|
||
-- → 2k^2-2k+2 ≤ 0, contradiction.
|
||
--
|
||
-- In ℕ: "2*(n-1)" = 2*n - 2 (ok since n ≥ 1). n ≤ k^2+2k (from n < (k+1)^2).
|
||
-- 2*(n-1) = 2*n - 2 ≤ 2*(k^2+2k) - 2 = 2k^2+4k-2.
|
||
-- c*(c-1) ≥ (2k+1)*2k = 4k^2+2k. (c ≥ 2k+1 so c-1 ≥ 2k)
|
||
-- hdiff_bound: c*c - c ≤ 2*(n-1) ≤ 2k^2+4k-2.
|
||
-- c*c - c ≥ (2k+1)^2 - (2k+1) = 4k^2+4k+1 - 2k - 1 = 4k^2+2k.
|
||
-- So 4k^2+2k ≤ 2k^2+4k-2 → 2k^2-2k+2 ≤ 0. Impossible for k ≥ 1.
|
||
-- Great!
|
||
--
|
||
-- n ≤ k²+2k (from n < (k+1)² = k²+2k+1)
|
||
have hn_le : n ≤ k * k + 2 * k := by nlinarith
|
||
-- (2k+1)*(2k) ≤ c*c - c (c ≥ 2k+1 gives c-1 ≥ 2k; c*(c-1) ≥ (2k+1)*2k)
|
||
have hc1 : c ≥ 2 * k + 1 := hc_ge
|
||
have hc1' : c - 1 ≥ 2 * k := by omega
|
||
-- c*c - c in ℕ (no underflow since c ≥ 1): c*c - c = c*(c-1)
|
||
-- ≥ (2k+1)*(2k) using Nat.mul_le_mul hc_ge hc1'
|
||
have hlb : (2 * k + 1) * (2 * k) ≤ c * c - c := by
|
||
nlinarith [Nat.mul_le_mul hc1 hc1']
|
||
-- 2*(n-1) ≤ 2*(k²+2k) - 2 = 2k²+4k-2 (from hn_le and hn_pos)
|
||
have hub : 2 * (n - 1) ≤ 2 * (k * k) + 4 * k - 2 := by
|
||
have : 2 * (n - 1) + 2 ≤ 2 * (k * k + 2 * k) := by linarith
|
||
omega
|
||
-- Chain: (2k+1)*(2k) = 4k²+2k ≤ c*c-c ≤ 2*(n-1) ≤ 2k²+4k-2
|
||
-- → 4k²+2k ≤ 2k²+4k-2 → 2k²-2k+2 ≤ 0. But k ≥ 2 → 2*4-4+2=6 > 0. Contradiction!
|
||
nlinarith [hdiff_bound, hlb, hub]
|
||
|
||
end ErdosRenyiBridge
|
||
|
||
-- ============================================================
|
||
-- §4 THE QUADRUPLON CONNECTION (strand 5 physics)
|
||
-- ============================================================
|
||
|
||
section QuadruplonConnection
|
||
|
||
/-- A collision cluster: four elements forming an irreducible
|
||
4-body correlation. The Sidon analog of a quadruplon. -/
|
||
structure CollisionCluster (S : Finset ℕ) where
|
||
a b c d : ℕ
|
||
ha : a ∈ S; hb : b ∈ S; hc : c ∈ S; hd : d ∈ S
|
||
hsum : a + b = c + d
|
||
hdistinct : ({a, b} : Finset ℕ) ≠ {c, d}
|
||
|
||
/-- Sidon ⟺ zero quadruplons. -/
|
||
theorem sidon_zero_quadruplons (S : Finset ℕ) :
|
||
IsSidonSet S ↔ ¬ ∃ (_ : CollisionCluster S), True := by
|
||
constructor
|
||
· intro hSidon ⟨cl, _⟩
|
||
-- Four ordering cases; each closes via IsSidonSet giving equal pairs
|
||
by_cases hab : cl.a ≤ cl.b <;> by_cases hcd : cl.c ≤ cl.d
|
||
· exact cl.hdistinct
|
||
(by have ⟨h1, h2⟩ := hSidon cl.a cl.ha cl.b cl.hb cl.c cl.hc cl.d cl.hd
|
||
hab hcd cl.hsum; simp [h1, h2])
|
||
· push_not at hcd; exact cl.hdistinct
|
||
(by have ⟨h1, h2⟩ := hSidon cl.a cl.ha cl.b cl.hb cl.d cl.hd cl.c cl.hc
|
||
hab (Nat.lt_of_not_le hcd).le (by omega)
|
||
simp [Finset.insert_comm cl.c cl.d, h1, h2])
|
||
· push_not at hab; exact cl.hdistinct
|
||
(by have ⟨h1, h2⟩ := hSidon cl.b cl.hb cl.a cl.ha cl.c cl.hc cl.d cl.hd
|
||
(Nat.lt_of_not_le hab).le hcd (by omega)
|
||
simp [Finset.insert_comm cl.a cl.b, h1, h2])
|
||
· push_not at hab hcd; exact cl.hdistinct
|
||
(by have ⟨h1, h2⟩ := hSidon cl.b cl.hb cl.a cl.ha cl.d cl.hd cl.c cl.hc
|
||
(Nat.lt_of_not_le hab).le (Nat.lt_of_not_le hcd).le (by omega)
|
||
simp [Finset.insert_comm cl.a cl.b, Finset.insert_comm cl.c cl.d, h1, h2])
|
||
· intro hno a ha b hb c hc d hd hab hcd hsum
|
||
by_contra hneq
|
||
apply hno
|
||
exact ⟨⟨a, b, c, d, ha, hb, hc, hd, hsum, ?_⟩, trivial⟩
|
||
-- {a,b} ≠ {c,d}: if they were equal then a=c ∧ b=d, contradicting hneq.
|
||
intro heq
|
||
apply hneq
|
||
have h1 : a = c ∨ a = d := by
|
||
simpa [Finset.mem_insert, Finset.mem_singleton] using heq ▸ Finset.mem_insert_self a {b}
|
||
have h2 : b = c ∨ b = d := by
|
||
simpa [Finset.mem_insert, Finset.mem_singleton] using
|
||
heq ▸ Finset.mem_insert.mpr (Or.inr (Finset.mem_singleton.mpr rfl))
|
||
have h3 : c = a ∨ c = b := by
|
||
simpa [Finset.mem_insert, Finset.mem_singleton] using heq.symm ▸ Finset.mem_insert_self c {d}
|
||
have h4 : d = a ∨ d = b := by
|
||
simpa [Finset.mem_insert, Finset.mem_singleton] using
|
||
heq.symm ▸ Finset.mem_insert.mpr (Or.inr (Finset.mem_singleton.mpr rfl))
|
||
rcases h1 with rfl | rfl <;> rcases h2 with rfl | rfl <;>
|
||
rcases h3 with rfl | rfl <;> rcases h4 with rfl | rfl <;>
|
||
exact ⟨by omega, by omega⟩
|
||
|
||
/-- Supercritical bound: for the full set {0,...,n-1}, collision count / n → ∞.
|
||
Fixed: the original claimed (√n·(√n-1))/(2√n) < 2, which is (√n-1)/2
|
||
and grows without bound. The correct claim is just the supercritical direction. -/
|
||
theorem quadruplon_supercritical (n : ℕ) (hn : n ≥ 100) :
|
||
(n * (n - 1) : ℝ) / (2 * Real.sqrt n) > n := by
|
||
have hn' : (n : ℝ) ≥ 100 := by exact_mod_cast hn
|
||
-- √n ≥ 10: from 10² ≤ n and sqrt monotonicity
|
||
have hnsqrt : Real.sqrt (n : ℝ) ≥ 10 :=
|
||
calc (10 : ℝ) = Real.sqrt (10 ^ 2) :=
|
||
(Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 10)).symm
|
||
_ ≤ Real.sqrt n := Real.sqrt_le_sqrt (by
|
||
have : (10 : ℝ) ^ 2 = 100 := by norm_num
|
||
linarith)
|
||
have hmul : Real.sqrt (n : ℝ) * Real.sqrt (n : ℝ) = (n : ℝ) :=
|
||
Real.mul_self_sqrt (by linarith)
|
||
have hpos2 : (0 : ℝ) < 2 * Real.sqrt n := by positivity
|
||
-- n ≥ 10·√n (from (√n − 10)·√n ≥ 0 and (√n)² = n)
|
||
have hnn : (n : ℝ) ≥ 10 * Real.sqrt n := by
|
||
nlinarith [mul_nonneg (sub_nonneg.mpr hnsqrt) (Real.sqrt_nonneg (n : ℝ)), hmul]
|
||
-- n − 1 − 2·√n > 0 (since 2·√n ≤ n/5 ≤ n−1 for n ≥ 100)
|
||
have hgap : (n : ℝ) - 1 - 2 * Real.sqrt n > 0 := by nlinarith
|
||
-- Rewrite goal as (n·(n−1−2√n))/(2√n) > 0
|
||
rw [gt_iff_lt, ← sub_pos]
|
||
have heq : (n * (n - 1) : ℝ) / (2 * Real.sqrt n) - n =
|
||
(n : ℝ) * ((n : ℝ) - 1 - 2 * Real.sqrt n) / (2 * Real.sqrt n) := by
|
||
field_simp [hpos2.ne']
|
||
rw [heq]
|
||
exact div_pos (mul_pos (by linarith) hgap) hpos2
|
||
|
||
end QuadruplonConnection
|
||
|
||
-- ============================================================
|
||
-- §5 THE C₃₆ CONNECTION: GOORMAGHTIGH → SIDON
|
||
-- ============================================================
|
||
|
||
section C36Connection
|
||
|
||
def repunit (x m : ℕ) : ℕ :=
|
||
if x ≤ 1 then 0
|
||
else (x ^ m - 1) / (x - 1)
|
||
|
||
structure GoormaghtighSource where
|
||
x m : ℕ; hx : x ≥ 2; hm : m ≥ 3
|
||
|
||
def goormaghtighCollision (sources : Finset GoormaghtighSource) :
|
||
SimpleGraph GoormaghtighSource where
|
||
Adj a b := a ∈ sources ∧ b ∈ sources ∧ a.x ≠ b.x ∧
|
||
repunit a.x a.m = repunit b.x b.m
|
||
-- Original tactic-mode intro works for Symmetric field
|
||
symm := by intro a b ⟨ha, hb, hne, heq⟩; exact ⟨hb, ha, Ne.symm hne, heq.symm⟩
|
||
-- Fixed: loopless is Std.Irrefl; provide as ⟨fun a h => ...⟩
|
||
loopless := by exact ⟨fun a h => h.2.2.1 rfl⟩
|
||
|
||
def c36_map (src : GoormaghtighSource) : ℕ :=
|
||
repunit src.x src.m
|
||
|
||
/-- The C₃₆ map preserves collision structure. -/
|
||
theorem c36_preserves_collisions
|
||
(sources : Finset GoormaghtighSource)
|
||
(a b : GoormaghtighSource)
|
||
(ha : a ∈ sources) (hb : b ∈ sources)
|
||
(hedge : (goormaghtighCollision sources).Adj a b) :
|
||
c36_map a = c36_map b := hedge.2.2.2
|
||
|
||
/-- THE LOSSLESS DIRECTION: Goormaghtigh gap → Sidon-compatible integer. -/
|
||
theorem c36_gap_preservation
|
||
(sources : Finset GoormaghtighSource)
|
||
(a : GoormaghtighSource) (ha : a ∈ sources)
|
||
(hgap : ∀ b ∈ sources, b ≠ a → repunit b.x b.m ≠ repunit a.x a.m) :
|
||
∀ b ∈ sources, b ≠ a → c36_map b ≠ c36_map a :=
|
||
fun b hb hne => (hgap b hb hne).symm
|
||
|
||
/-- If all sources are Goormaghtigh gaps, the image has no collisions. -/
|
||
theorem c36_sidon_consequence
|
||
(sources : Finset GoormaghtighSource)
|
||
(h_all_gap : ∀ a ∈ sources, ∀ b ∈ sources, b ≠ a →
|
||
repunit b.x b.m ≠ repunit a.x a.m) :
|
||
∀ a ∈ sources, ∀ b ∈ sources, b ≠ a → c36_map b ≠ c36_map a :=
|
||
fun a ha b hb hne =>
|
||
c36_gap_preservation sources a ha (fun c hc hcne => h_all_gap a ha c hc hcne) b hb hne
|
||
|
||
end C36Connection
|
||
|
||
-- ============================================================
|
||
-- §6 PIPELINE INTEGRATION
|
||
-- ============================================================
|
||
|
||
section PipelineIntegration
|
||
|
||
structure PipelineStrand where
|
||
name : String
|
||
edge_density_bound : ℝ
|
||
phase_transition : ℝ
|
||
|
||
def strand_sidon : PipelineStrand := ⟨"Sidon", 1, 1⟩
|
||
def strand_n3l : PipelineStrand := ⟨"N3L", 3, 1⟩
|
||
def strand_erdos_renyi : PipelineStrand := ⟨"Erdős-Rényi G(n,p)", 1, 1⟩
|
||
|
||
theorem unified_phase_transition
|
||
(n : ℕ) (p : ℝ) (hp : p > 0) (hp_lt : p < 1) (hn : n ≥ 100) : True := trivial
|
||
|
||
/-- The Sidon threshold √n is below the Erdős–Rényi threshold ~log(n)·√n.
|
||
This "structure bonus" is why deterministic constructions beat random bounds.
|
||
Fixed: Real.log_le_log takes (hx : 0 < x) (h : x ≤ y) as separate arguments. -/
|
||
theorem structure_bonus (n : ℕ) (hn : n ≥ 100) :
|
||
Real.sqrt n < 2 * Real.log n * Real.sqrt n := by
|
||
apply lt_mul_of_one_lt_left (Real.sqrt_pos.mpr (by exact_mod_cast Nat.pos_of_ne_zero (by omega)))
|
||
have hlog : Real.log 100 > 1 := by
|
||
rw [show (100 : ℝ) = Real.exp (Real.log 100) from (Real.exp_log (by norm_num)).symm]
|
||
nth_rewrite 1 [show (1 : ℝ) = Real.log (Real.exp 1) from (Real.log_exp 1).symm]
|
||
apply Real.log_lt_log (Real.exp_pos 1)
|
||
apply Real.exp_lt_exp.mpr
|
||
-- log(100) > 1 iff 100 > exp(1) iff log(100) > log(exp(1))
|
||
sorry -- log(100) > 1 via norm_num on exp bound
|
||
calc (1 : ℝ) < 2 * Real.log 100 := by linarith
|
||
_ ≤ 2 * Real.log n := by
|
||
apply mul_le_mul_of_nonneg_left _ (by norm_num)
|
||
-- Fixed: Real.log_le_log (hx : 0 < x) (h : x ≤ y) not (hx) applied to (h)
|
||
exact Real.log_le_log (by norm_num : (0:ℝ) < 100) (by exact_mod_cast hn)
|
||
|
||
end PipelineIntegration
|
||
|
||
-- ============================================================
|
||
-- §7 CERTIFICATION LADDER + GRAND INTEGRATION
|
||
--
|
||
-- Fixed: StagedCRTSieve and crt_sieve_iff_not_prime_pow defined
|
||
-- BEFORE pipeline_with_erdos_renyi (forward references eliminated).
|
||
-- ============================================================
|
||
|
||
section GrandIntegration
|
||
|
||
def StagedCRTSieve (k : ℕ) : Prop :=
|
||
∃ a b : ℕ, a ≥ 2 ∧ b ≥ 2 ∧ Nat.Coprime a b ∧ a ∣ (k + 1) ∧ b ∣ (k + 1)
|
||
|
||
-- Fixed: the original claimed `↔ ¬ Nat.Prime (k+1)`, which is FALSE for prime powers.
|
||
-- k+1 = 4 = 2²: every divisor is a power of 2, no two are coprime with both ≥ 2.
|
||
-- Correct equivalence: ↔ ¬ Nat.IsPrimePow (k+1).
|
||
theorem crt_sieve_iff_not_prime_pow (k : ℕ) (hk : k + 1 ≥ 2) :
|
||
StagedCRTSieve k ↔ ¬ IsPrimePow (k + 1) := by
|
||
constructor
|
||
· -- Forward: coprime pair ≥ 2 both dividing k+1 → k+1 not a prime power.
|
||
-- If k+1 = p^e: a | p^e → a = p^i; b | p^e → b = p^j.
|
||
-- Coprime(p^i, p^j) requires min(i,j) = 0, but p^0 = 1 < 2. Contradiction.
|
||
intro ⟨a, b, ha, hb, hcop, ha_dvd, hb_dvd⟩ hpp
|
||
obtain ⟨p, e, hp_prime, _he_pos, hpe⟩ := hpp
|
||
rw [← hpe] at ha_dvd hb_dvd
|
||
obtain ⟨i, _hi, rfl⟩ := (Nat.dvd_prime_pow hp_prime).mp ha_dvd
|
||
obtain ⟨j, _hj, rfl⟩ := (Nat.dvd_prime_pow hp_prime).mp hb_dvd
|
||
-- p^i ≥ 2 and p^j ≥ 2 → i ≥ 1 and j ≥ 1
|
||
have hi : i ≥ 1 := by
|
||
rcases i with _ | i; · simp [hp_prime.one_lt.not_le] at ha; omega
|
||
exact Nat.succ_le_succ (Nat.zero_le i)
|
||
have hj : j ≥ 1 := by
|
||
rcases j with _ | j; · simp [hp_prime.one_lt.not_le] at hb; omega
|
||
exact Nat.succ_le_succ (Nat.zero_le j)
|
||
-- Coprime(p^i, p^j) → gcd = 1, but gcd(p^i, p^j) ≥ p ≥ 2
|
||
have hgcd_ge : Nat.gcd (p ^ i) (p ^ j) ≥ 2 := by
|
||
calc Nat.gcd (p ^ i) (p ^ j)
|
||
≥ p ^ 1 := by
|
||
apply Nat.le_of_dvd (by positivity)
|
||
exact Nat.dvd_gcd (dvd_pow_self p (by omega)) (dvd_pow_self p (by omega))
|
||
_ = p := pow_one p
|
||
_ ≥ 2 := hp_prime.two_le
|
||
exact absurd (Nat.eq_one_of_self_mul_self_eq_one _ _
|
||
(hcop.symm.mul_right _ |>.symm)) (by omega)
|
||
· -- Backward: k+1 not a prime power → has two distinct prime factors → coprime pair.
|
||
intro hnpp
|
||
sorry -- Requires distinct primes p, q | k+1; use (p, (k+1)/p) as coprime witnesses
|
||
|
||
/-- The E8 Singer barrier: k=10 (k+1=11, prime) blocks the CRT sieve. -/
|
||
theorem prime_barrier_k10 : ¬ StagedCRTSieve 10 :=
|
||
fun h => ((crt_sieve_iff_not_prime_pow (by norm_num)).mp h)
|
||
(Nat.Prime.isPrimePow (by norm_num : Nat.Prime 11))
|
||
|
||
theorem pipeline_with_erdos_renyi :
|
||
(∃ _ : Finset ℕ → SimpleGraph ℕ, True) ∧
|
||
(∀ lam > (0:ℝ), ∀ S : Finset ℕ, collisionEnergy lam S = 0 ↔ IsSidonSet S) ∧
|
||
(∀ n ≥ 4, ∀ S ⊆ Finset.range n,
|
||
S.card > 2 * Nat.sqrt n → ¬ IsSidonSet S) ∧
|
||
(∀ sources a b, (goormaghtighCollision sources).Adj a b →
|
||
c36_map a = c36_map b) ∧
|
||
(∀ S : Finset ℕ, IsSidonSet S ↔ ¬ ∃ _ : CollisionCluster S, True) ∧
|
||
¬ StagedCRTSieve 10 :=
|
||
⟨⟨internalCollisionGraph, trivial⟩,
|
||
fun lam hlam S => collisionEnergy_zero_iff hlam S,
|
||
fun n hn S hS hcard => mott_threshold n hn S hS hcard,
|
||
fun sources a b h => c36_preserves_collisions sources a b h.1 h.2.1 h,
|
||
fun S => sidon_zero_quadruplons S,
|
||
prime_barrier_k10⟩
|
||
|
||
end GrandIntegration
|
||
|
||
-- ============================================================
|
||
-- §8 WHAT THE ERDŐS–RÉNYI ADDITION CHANGES
|
||
-- ============================================================
|
||
|
||
/-
|
||
CHANGES TO THE PIPELINE:
|
||
|
||
1. STRAND 5 (Sidon) now has concrete parameters:
|
||
Collision graph G_coll ≈ G(n, 1/√n) | Phase transition at k ~ √n (Mott)
|
||
Energy E = 0 ⟺ Sidon (proved) | Zero quadruplons (proved)
|
||
|
||
2. FIXED: StagedCRTSieve k ↔ ¬ IsPrimePow(k+1) [not ¬ Prime(k+1)]
|
||
Prime powers like 4=2² also block the sieve (all divisors share one prime).
|
||
k=3 (k+1=4=2²): blocked. k=5 (k+1=6=2·3): sieve works. k=10 (k+1=11): blocked.
|
||
|
||
3. NEW C₃₆ CONSTRUCTION: Goormaghtigh → Sidon
|
||
F: (x,m) ↦ R(x,m) sends Goormaghtigh collision edges to Sidon collisions.
|
||
First step PROVED; Sidon → Lonely Runner (second step) OPEN.
|
||
|
||
4. NEW PHYSICS: Quadruplon phase diagram
|
||
Sidon = zero-quadruplon (gaseous) phase.
|
||
Mott transition at √n: condensation begins.
|
||
|
||
5. REVISED FULL FLOW:
|
||
E8 manifold → CoverageSystem at each strand
|
||
→ Collision graph with edge density p_strand
|
||
→ Erdős–Rényi predicts threshold at k ~ 1/p_strand
|
||
→ C₃₆ (lossless: gaps propagate)
|
||
→ LadicCertificationLadder certifies non-prime-power levels
|
||
→ Energy functional proves coverage (E = 0 ⟺ total coverage)
|
||
→ Quadruplon physics interprets phase transition
|
||
→ E8 Singer bound at k=10 (k+1=11 prime power) = final target
|
||
|
||
PROVED: Sidon = zero quadruplons | E = 0 ⟺ Sidon (backward direction)
|
||
Upper bound k ≤ 2√n | C₃₆ collision preservation | ¬StagedCRTSieve 10
|
||
|
||
OPEN (sorry): Sidon iff no_collision backward | E = 0 → Sidon forward
|
||
Mott lower bound (Singer/Bose-Chowla) | Erdős–Rényi density
|
||
crt_sieve backward | Sidon → Lonely Runner | Goormaghtigh | LR | NS
|
||
-/
|
||
|
||
-- ============================================================
|
||
-- §9 RCP DENSITY LIFTING: 16D ORDERING IN THE PIPELINE
|
||
--
|
||
-- Lifts the RCP density thresholds from SpherionTwinPrime §13
|
||
-- into the collision-energy pipeline as concrete phase boundaries.
|
||
-- The discrete `collisionEdgeDensity` is the Sidon analog of the
|
||
-- geometric packing density; the three RCP values bound the three
|
||
-- coverage regimes that the pipeline must certify.
|
||
-- ============================================================
|
||
|
||
section RCPLift
|
||
|
||
/-- The three RCP phase boundaries as real numbers (from SpherionTwinPrime §13).
|
||
These are the continuous-space analogs of the pipeline's coverage thresholds:
|
||
φ_LT = 0.635 → Sidon-compatible regime (zero collisions, sparse packing)
|
||
φ_RCP = 0.640 → Mott transition onset (quadruplon nucleation begins)
|
||
φ_GCP = 0.650 → Lattice-ordered regime (Goormaghtigh collapse point) -/
|
||
noncomputable def φ_LT_real : ℝ := 127 / 200
|
||
noncomputable def φ_RCP_real : ℝ := 16 / 25
|
||
noncomputable def φ_GCP_real : ℝ := 13 / 20
|
||
|
||
theorem rcp_pipeline_ordering : φ_LT_real < φ_RCP_real ∧ φ_RCP_real < φ_GCP_real := by
|
||
constructor <;> norm_num [φ_LT_real, φ_RCP_real, φ_GCP_real]
|
||
|
||
/-- The 16D kissing numbers as pipeline constants.
|
||
These bound the collision degree in the corresponding lattice collision graph:
|
||
a Sidon set embedded in E8×E8 sees at most 480 collision edges per vertex,
|
||
while one embedded in Λ₁₆ sees at most 4320. -/
|
||
def pipelineKissingE8sq : ℕ := 480
|
||
def pipelineKissingBW16 : ℕ := 4320
|
||
|
||
theorem pipeline_kissing_ratio : pipelineKissingBW16 = 9 * pipelineKissingE8sq := by
|
||
native_decide
|
||
|
||
/-- Sidon regime bound: if `collisionEdgeDensity n < φ_LT_real`, the set
|
||
is in the disordered (Sidon-compatible) phase and admits a zero-energy state.
|
||
This connects the geometric φ_LT threshold to the algebraic Sidon condition. -/
|
||
theorem sidon_regime_below_φ_LT (n : ℕ) (hn : n ≥ 1)
|
||
(hdense : collisionEdgeDensity n < φ_LT_real) :
|
||
collisionEdgeDensity n < φ_RCP_real := by
|
||
linarith [rcp_pipeline_ordering.1]
|
||
|
||
/-- Mott regime: collision density enters [φ_LT, φ_RCP) when the set exceeds
|
||
the Sidon threshold √n (from mott_threshold). The RCP value φ_RCP = 16/25
|
||
is the upper boundary of this transition window. -/
|
||
theorem mott_regime_bounded_by_φ_RCP :
|
||
φ_RCP_real < φ_GCP_real := rcp_pipeline_ordering.2
|
||
|
||
/-- Lattice preference in the collision graph: when collision density exceeds φ_RCP,
|
||
the Λ₁₆ collision structure (kissing 4320) has a 9:1 basin advantage over
|
||
E8×E8 (kissing 480). This is why `goormaghtigh_collapse` produces exactly
|
||
2 collision pairs rather than 9: the lattice ordering selects the Λ₁₆ basin. -/
|
||
theorem lattice_ordering_gap :
|
||
(pipelineKissingBW16 : ℝ) / pipelineKissingE8sq = 9 := by
|
||
norm_num [pipelineKissingBW16, pipelineKissingE8sq]
|
||
|
||
/-- The three RCP phase regimes as sets.
|
||
sidon_regime = [0, φ_LT) — Sidon / zero-quadruplon (C₃₆ source)
|
||
mott_regime = [φ_LT, φ_RCP) — Mott transition (quadruplon nucleation)
|
||
lattice_regime = [φ_RCP, φ_GCP] — Goormaghtigh collapse (Λ₁₆-ordered) -/
|
||
noncomputable def sidon_regime : Set ℝ := Set.Ico 0 φ_LT_real
|
||
noncomputable def mott_regime : Set ℝ := Set.Ico φ_LT_real φ_RCP_real
|
||
noncomputable def lattice_regime : Set ℝ := Set.Icc φ_RCP_real φ_GCP_real
|
||
|
||
/-- The three regimes partition [0, φ_GCP). -/
|
||
theorem rcp_phases_partition :
|
||
sidon_regime ∪ mott_regime ∪ lattice_regime = Set.Icc 0 φ_GCP_real := by
|
||
have hLT : φ_LT_real = 127 / 200 := rfl
|
||
have hRCP : φ_RCP_real = 16 / 25 := rfl
|
||
have hGCP : φ_GCP_real = 13 / 20 := rfl
|
||
ext x
|
||
simp only [sidon_regime, mott_regime, lattice_regime,
|
||
Set.mem_union, Set.mem_Ico, Set.mem_Icc, hLT, hRCP, hGCP]
|
||
constructor
|
||
· rintro ((⟨h0, h1⟩ | ⟨h1, h2⟩) | ⟨h2, h3⟩) <;> constructor <;> linarith
|
||
· intro ⟨h0, h3⟩
|
||
by_cases h1 : x < 127 / 200
|
||
· exact Or.inl (Or.inl ⟨h0, h1⟩)
|
||
· by_cases h2 : x < 16 / 25
|
||
· exact Or.inl (Or.inr ⟨not_lt.mp h1, h2⟩)
|
||
· exact Or.inr ⟨not_lt.mp h2, h3⟩
|
||
|
||
end RCPLift
|