mirror of
https://github.com/allaunthefox/SilverSight.git
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allaun
b65ef756ca
Registering required it to actually compile (it did not). Fixes:
- capacity_bound_grid was FALSE at L₁=0 or L₂=0 (empty grid, nonempty image)
and its `norm_num : 0 < (L₁:ℤ)` could not prove positivity of a variable.
Added `0 < L₁, 0 < L₂` hypotheses; threaded through capacity_bound and
chiral_capacity_bound.
- mathlib name drift: emod_nonneg/emod_lt → Int.emod_nonneg (b ≠ 0) /
Int.emod_lt_of_pos; Finset.card_Ico → Int.card_Ico;
Finset.card_le_card_of_subset → Finset.card_le_card; Finset.image_subset →
Finset.image_subset_iff.mpr.
- replaced a `#eval` (broke on ℤ's noncomputable order instance) and its WRONG
expected value (claimed 4 over {1..6}, actually 6) with two kernel-checked
`decide` witnesses: {1,2,5,6}→4 (the Sidon set) and Ico 1 7→6.
- registered CoreFormalism.StrandCapacityBound in lakefile.
Verified: lake build OK (links into SilverSightFormal), #print axioms =
{propext, Classical.choice, Quot.sound} (no sorryAx, no custom axiom),
anti-smuggle --ci clean.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
101 lines
4.2 KiB
Text
101 lines
4.2 KiB
Text
import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Int.Basic
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import Mathlib.Data.Finset.Basic
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import Mathlib.Tactic
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open Finset
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open Nat
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/-!
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# Strand Capacity Bound
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A single chiral strand pair (L₁, L₂) maps each a ∈ A to the pair
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(a mod L₁, S−a mod L₂) on the 2-torus Z/L₁Z × Z/L₂Z.
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## Theorem
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For a single strand with moduli L₁, L₂ and any finite set A:
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|F(A)| ≤ min(|A|, L₁·L₂) × 2
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The bound has two parts:
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1. |F(A)| ≤ |A| (injectivity — F is a function from A)
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2. |F(A)| ≤ L₁·L₂ (codomain has only L₁·L₂ distinct pairs)
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-/
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variable (L₁ L₂ S : ℕ)
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/-- The strand pairing: (a mod L₁, S − a mod L₂). -/
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def strandPair (a : ℤ) : ℤ × ℤ :=
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(a % (L₁ : ℤ), ((S : ℤ) - a) % (L₂ : ℤ))
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/-- Image of A under the strand pairing. -/
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noncomputable def strandImage (A : Set ℤ) : Set (ℤ × ℤ) :=
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strandPair L₁ L₂ S '' A
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/-- Bound 1: image cardinality ≤ input cardinality (trivial, F is a function). -/
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theorem capacity_bound_input (A : Finset ℤ) :
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(A.image (strandPair L₁ L₂ S)).card ≤ A.card :=
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Finset.card_image_le
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/--
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Bound 2: the codomain grid has size L₁·L₂.
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Each residue lies in [0, L₁) resp. [0, L₂), so at most
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L₁ × L₂ distinct pairs are reachable regardless of |A|.
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-/
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theorem capacity_bound_grid (A : Finset ℤ) (hL₁ : 0 < L₁) (hL₂ : 0 < L₂) :
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(A.image (strandPair L₁ L₂ S)).card ≤ (L₁ : ℕ) * L₂ := by
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have hL₁' : (0 : ℤ) < (L₁ : ℤ) := by exact_mod_cast hL₁
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have hL₂' : (0 : ℤ) < (L₂ : ℤ) := by exact_mod_cast hL₂
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-- The grid of possible residues
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let grid : Finset (ℤ × ℤ) :=
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(Finset.Ico 0 (L₁ : ℤ)) ×ˢ (Finset.Ico 0 (L₂ : ℤ))
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have hgrid : grid.card = (L₁ : ℕ) * L₂ := by
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simp [grid, Finset.card_product, Int.card_Ico]
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-- Every strand pair lands in the grid
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have hmem : ∀ a : ℤ, strandPair L₁ L₂ S a ∈ grid := by
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intro a
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have hx : a % (L₁ : ℤ) ∈ Finset.Ico 0 (L₁ : ℤ) := by
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have hnonneg : 0 ≤ a % (L₁ : ℤ) := Int.emod_nonneg a (ne_of_gt hL₁')
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have hlt : a % (L₁ : ℤ) < (L₁ : ℤ) := Int.emod_lt_of_pos a hL₁'
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exact Finset.mem_Ico.mpr ⟨hnonneg, hlt⟩
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have hy : ((S : ℤ) - a) % (L₂ : ℤ) ∈ Finset.Ico 0 (L₂ : ℤ) := by
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have hnonneg' : 0 ≤ ((S : ℤ) - a) % (L₂ : ℤ) :=
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Int.emod_nonneg _ (ne_of_gt hL₂')
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have hlt' : ((S : ℤ) - a) % (L₂ : ℤ) < (L₂ : ℤ) :=
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Int.emod_lt_of_pos _ hL₂'
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exact Finset.mem_Ico.mpr ⟨hnonneg', hlt'⟩
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exact Finset.mem_product.mpr ⟨hx, hy⟩
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-- Image is subset of grid, so cardinality bounded by grid cardinality
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calc
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(A.image (strandPair L₁ L₂ S)).card ≤ grid.card :=
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Finset.card_le_card (Finset.image_subset_iff.mpr (fun a _ => hmem a))
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_ = (L₁ : ℕ) * L₂ := hgrid
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/--
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Combined bound: |F(A)| ≤ min(|A|, L₁·L₂).
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For a single strand with chirality (2 orientations per pair),
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the full capacity is min(|A|, L₁·L₂) × 2.
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-/
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theorem capacity_bound (A : Finset ℤ) (hL₁ : 0 < L₁) (hL₂ : 0 < L₂) :
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(A.image (strandPair L₁ L₂ S)).card ≤ min A.card ((L₁ : ℕ) * L₂) := by
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apply le_min
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· exact capacity_bound_input L₁ L₂ S A
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· exact capacity_bound_grid L₁ L₂ S A hL₁ hL₂
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/--
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With chirality: each pair can be read in 2 orders
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(identity, reflection) or (reflection, identity),
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corresponding to σᵢ vs σᵢ⁻¹ in the braid group.
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-/
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theorem chiral_capacity_bound (A : Finset ℤ) (hL₁ : 0 < L₁) (hL₂ : 0 < L₂) :
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(A.image (strandPair L₁ L₂ S)).card * 2 ≤ min A.card ((L₁ : ℕ) * L₂) * 2 := by
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nlinarith [capacity_bound L₁ L₂ S A hL₁ hL₂]
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-- Sidon example A = {1,2,5,6} with moduli 3,4, S=7 → image has 4 distinct pairs:
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-- 1↦(1,2), 2↦(2,1), 5↦(2,2), 6↦(0,1). Kernel-checked (`decide`, not
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-- `#eval`/`native_decide`), sidestepping ℤ's noncomputable order instance.
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example : (({1, 2, 5, 6} : Finset ℤ).image (strandPair 3 4 7)).card = 4 := by decide
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-- Full range Ico 1 7 = {1,…,6} instead gives 6 distinct pairs (all injective here):
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example : ((Finset.Ico (1 : ℤ) 7).image (strandPair 3 4 7)).card = 6 := by decide
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