Quantitative common-test error obstruction for synthetic parity
Lax342547.SyntheticRankTransfer · concepts/Lax342547/SyntheticRankTransfer.lean · lax-342547
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Lemma
On one common reference law, parity and all small coordinate-slice tests are incompatible once the synthetic identity exceeds the sum of local ranks. Separate empirical density caps and variance bounds transfer every test. Their total prediction and comparison errors must then be at least one. This is a conditional finite transfer theorem; constructing the actual synthetic reference law and its empirical test couplings remains necessary.
Concept map
Evidence
This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.
1 exists_error_margin proven
2 incompatible_events proven
3 prediction_error_budget proven
4 rank_tests_incompatible proven
5 transferred_incompatible_events proven
6 uniform_prediction_error proven
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| 1 | import Lax342547.SmallSliceRank |
| 2 | import Lax342547.CommonTestTransfer |
| 3 | |
| 4 | /-! |
| 5 | --- |
| 6 | title: Quantitative common-test error obstruction for synthetic parity |
| 7 | type: lemma |
| 8 | --- |
| 9 | On one common reference law, parity and all small coordinate-slice tests |
| 10 | are incompatible once the synthetic identity exceeds the sum of local ranks. |
| 11 | Separate empirical density caps and variance bounds transfer every test. |
| 12 | Their total prediction and comparison errors must then be at least one. |
| 13 | This is a conditional finite transfer theorem; constructing the actual |
| 14 | synthetic reference law and its empirical test couplings remains necessary. |
| 15 | -/ |
| 16 | |
| 17 | namespace Lax342547.SyntheticRankTransfer |
| 18 | noncomputable section |
| 19 | open Lax342547.MomentSpace Lax342547.SmallSliceRank Lax342547.RelativeEntropy |
| 20 | open scoped BigOperators |
| 21 | set_option backward.isDefEq.respectTransparency false |
| 22 | |
| 23 | def indicator {Ω : Type} (C : Ω → Prop) (ω : Ω) : ℝ := by |
| 24 | classical |
| 25 | exact if C ω then 1 else 0 |
| 26 | |
| 27 | def eventMean {Ω : Type} [Fintype Ω] (μ : Ω → ℝ) (C : Ω → Prop) : ℝ := |
| 28 | ∑ ω,μ ω*indicator C ω |
| 29 | |
| 30 | abbrev Points (m : ℕ) := Fin m → Binary |
| 31 | |
| 32 | abbrev Tests (T : Type) (s m : ℕ) := |
| 33 | (Points m × Points m) ⊕ (T × (Fin (2*s+1) → Fin m) × (Fin (2*s+1) → Fin m)) |
| 34 | |
| 35 | def testCondition {T : Type} [Fintype T] (s m : ℕ) |
| 36 | (F : T → Points m → Points m → Binary) : Tests T s m → Prop |
| 37 | | Sum.inl (x,y) => (∑ l,F l x y) = dotProduct x y |
| 38 | | Sum.inr (l,r,c) => Function.Injective r → Function.Injective c → |
| 39 | HasProducts s (fun x y => F l (liftCoordinates r x) (liftCoordinates c y)) |
| 40 | |
| 41 | axiom incompatible_events {J Ω : Type} [Fintype J] [Fintype Ω] |
| 42 | (μ : Ω → ℝ) (hμ : Probability μ) (C : J → Ω → Prop) |
| 43 | (hincompatible : ∀ ω,¬∀ j,C j ω) : 1 ≤ ∑ j : J,(1-eventMean μ (C j)) |
| 44 | |
| 45 | axiom transferred_incompatible_events {J Ω : Type} [Fintype J] [Fintype Ω] |
| 46 | (A : J → Type) [∀ j,Fintype (A j)] |
| 47 | (μ : Ω → ℝ) (hμ : Probability μ) (C : J → Ω → Prop) |
| 48 | (hincompatible : ∀ ω,¬∀ j,C j ω) |
| 49 | (ρ σ F : ∀ j,A j → ℝ) (hσ : ∀ j,Probability (σ j)) |
| 50 | (L ε η : J → ℝ) (hL : ∀ j,0 ≤ L j) |
| 51 | (hcap : ∀ j o,σ j o ≤ L j*ρ j o) |
| 52 | (hvar : ∀ j,(∑ o,ρ j o*(F j o-eventMean μ (C j))^2) ≤ ε j) |
| 53 | (hsuccess : ∀ j,1-η j ≤ ∑ o,σ j o*F j o) : |
| 54 | 1 ≤ ∑ j : J,(η j+Real.sqrt (L j*ε j)) |
| 55 | |
| 56 | axiom rank_tests_incompatible {T : Type} [Fintype T] (s m : ℕ) |
| 57 | (F : T → Points m → Points m → Binary) (hm : 2*s*Fintype.card T < m) : |
| 58 | ¬∀ j,testCondition s m F j |
| 59 | |
| 60 | axiom prediction_error_budget {T Ω : Type} [Fintype T] [Fintype Ω] |
| 61 | (s m : ℕ) (hm : 2*s*Fintype.card T < m) |
| 62 | (μ : Ω → ℝ) (hμ : Probability μ) |
| 63 | (pred : Ω → T → Points m → Points m → Binary) |
| 64 | (A : Tests T s m → Type) [∀ j,Fintype (A j)] |
| 65 | (ρ σ F : ∀ j,A j → ℝ) (hσ : ∀ j,Probability (σ j)) |
| 66 | (L ε η : Tests T s m → ℝ) (hL : ∀ j,0 ≤ L j) |
| 67 | (hcap : ∀ j o,σ j o ≤ L j*ρ j o) |
| 68 | (hvar : ∀ j,(∑ o,ρ j o*(F j o-eventMean μ (fun ω => testCondition s m (pred ω) j))^2) ≤ ε j) |
| 69 | (hsuccess : ∀ j,1-η j ≤ ∑ o,σ j o*F j o) : |
| 70 | 1 ≤ ∑ j : Tests T s m,(η j+Real.sqrt (L j*ε j)) |
| 71 | |
| 72 | axiom uniform_prediction_error {T Ω : Type} [Fintype T] [Fintype Ω] |
| 73 | (s m : ℕ) (hm : 2*s*Fintype.card T < m) |
| 74 | (μ : Ω → ℝ) (hμ : Probability μ) |
| 75 | (pred : Ω → T → Points m → Points m → Binary) |
| 76 | (A : Tests T s m → Type) [∀ j,Fintype (A j)] |
| 77 | (ρ σ F : ∀ j,A j → ℝ) (hσ : ∀ j,Probability (σ j)) |
| 78 | (L ε η : ℝ) (hL : 0 ≤ L) |
| 79 | (hcap : ∀ j o,σ j o ≤ L*ρ j o) |
| 80 | (hvar : ∀ j,(∑ o,ρ j o*(F j o-eventMean μ (fun ω => testCondition s m (pred ω) j))^2) ≤ ε) |
| 81 | (hsuccess : ∀ j,1-η ≤ ∑ o,σ j o*F j o) : |
| 82 | 1 ≤ (Fintype.card (Tests T s m) : ℝ)*(η+Real.sqrt (L*ε)) |
| 83 | |
| 84 | axiom exists_error_margin (J : Type) [Fintype J] (L : ℝ) (hL : 0 ≤ L) : |
| 85 | ∃ η : ℝ,0 < η ∧ ∃ ε : ℝ,0 < ε ∧ |
| 86 | (Fintype.card J : ℝ)*(η+Real.sqrt (L*ε)) < 1 |
| 87 | |
| 88 | end |
| 89 | end Lax342547.SyntheticRankTransfer |
| 90 |
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