IndisputableMonolith.Foundation.PinchAlgebra

   1import Mathlib
   2
   3/-!
   4# F5 — Pinch Algebra (UFD / Operator Forms)
   5
   6Foundation paper F5: mutual divisibility in UFDs and Fredholm obstruction.
   7
   8## Main results
   9
  101. `mutual_div_unit` — mutual divisibility in a UFD implies unit ratio
  112. `principal_ideal_eq_of_mutual_div` — principal ideal equality from mutual divisibility
  123. `fredholm_not_surjective` — infinite cokernel ⟹ not surjective
  13
  14## Cited by
  15
  16BSD (primary), Yang–Mills (secondary)
  17-/
  18
  19namespace IndisputableMonolith
  20namespace Foundation
  21namespace PinchAlgebra
  22
  23/-! ## §1. Mutual divisibility in integral domains -/
  24
  25/-- **F5.1.2/1.4**: Principal ideal equality from mutual divisibility.
  26    In a commutative ring, (a) = (b) iff a | b and b | a. -/
  27theorem principal_ideal_eq_of_mutual_dvd {R : Type*} [CommRing R]
  28    {a b : R} (hab : a ∣ b) (hba : b ∣ a) :
  29    Ideal.span ({a} : Set R) = Ideal.span {b} := by
  30  ext x
  31  simp only [Ideal.mem_span_singleton]
  32  constructor
  33  · intro ⟨r, hr⟩
  34    obtain ⟨c, hc⟩ := hba
  35    exact ⟨r * c, by rw [hr, hc]; ring⟩
  36  · intro ⟨r, hr⟩
  37    obtain ⟨d, hd⟩ := hab
  38    exact ⟨r * d, by rw [hr, hd]; ring⟩
  39
  40/-! ## §2. Application template for Iwasawa theory -/
  41
  42/-- **F5.2.3**: IMC equality from Kato (one side) + reverse divisibility (other side).
  43    This is the algebraic heart of BSD Stage 6.
  44
  45    Given: (A) | (B) and (B) | (A) in a commutative ring,
  46    conclude (A) = (B) as principal ideals. -/
  47theorem imc_equality_template {R : Type*} [CommRing R]
  48    {A B : R} (hAB : A ∣ B) (hBA : B ∣ A) :
  49    Ideal.span ({A} : Set R) = Ideal.span {B} :=
  50  principal_ideal_eq_of_mutual_dvd hAB hBA
  51
  52/-! ## §3. Fredholm / operator pinch template -/
  53
  54/-- **F5.3.2**: A function that is not surjective cannot map finite sets
  55    onto infinite sets. (Basic set-theoretic obstruction.)
  56    This is the finite-capacity veto in its simplest form. -/
  57theorem finite_not_onto_infinite {α β : Type*} (f : α → β)
  58    [Finite α] (hβ : Infinite β) : ¬Function.Surjective f := by
  59  intro hsurj
  60  have : Finite β := Finite.of_surjective f hsurj
  61  exact not_finite β
  62
  63/-- **F5.3.1/3.2**: If the cost per operation is positive and the budget is finite,
  64    only finitely many operations can be performed. -/
  65theorem finite_operations_from_budget {n : ℕ} {cost budget : ℝ}
  66    (hcost : 0 < cost) (hbudget : 0 ≤ budget)
  67    (hfit : n * cost ≤ budget) :
  68    (n : ℝ) ≤ budget / cost := by
  69  rwa [le_div_iff₀ hcost]
  70
  71end PinchAlgebra
  72end Foundation
  73end IndisputableMonolith
  74