Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.MaxCut.PCP.RawInitialTables

namespace OAI

/-!
Concrete bridges between graph cardinality and the length of the validated
unary table encoding. These bounds supply the input-length and intermediate
encoding estimates used by actual machine trace composition; they are not
themselves a machine execution certificate.
-/

namespace MaxCutGames.Foundations.PCP.GraphTableComplexity

open MaxCutGames.Foundations.Complexity
open Polynomial

noncomputable def encodingPolynomial (q : Nat) : Polynomial Nat :=
  X ^ 2 + C (2 * (q * q) + 1) * X + C 2

theorem encodingPolynomial_eval (q n : Nat) :
    (encodingPolynomial q).eval n = n ^ 2 + (2 * (q * q) + 1) * n + 2 := by
  simp [encodingPolynomial]

theorem encodingPolynomial_mono (q : Nat) {n m : Nat} (hnm : n ≤ m) :
    (encodingPolynomial q).eval n ≤ (encodingPolynomial q).eval m := by
  rw [encodingPolynomial_eval, encodingPolynomial_eval]
  exact Nat.add_le_add_right
    (Nat.add_le_add (Nat.pow_le_pow_left hnm 2) (Nat.mul_le_mul_left _ hnm)) 2

theorem rowBound_le_polynomial (q vertices darts : Nat) :
    vertices + darts + 2 + darts * (vertices + darts + 2 * (q * q)) ≤
      (encodingPolynomial q).eval (vertices + darts) := by
  have hd : darts ≤ vertices + darts := Nat.le_add_left darts vertices
  calc
    _ ≤ vertices + darts + 2 +
        (vertices + darts) * (vertices + darts + 2 * (q * q)) :=
      Nat.add_le_add_left (Nat.mul_le_mul_right _ hd) _
    _ = _ := by rw [encodingPolynomial_eval]; ring
theorem generic_size_add_two_le_bits {q : Nat} (table : GenericGraphTables.Table q) :
    table.vertices + table.darts + 2 ≤ (GenericGraphTables.tableBits table).length := by
  simp only [GenericGraphTables.tableBits, GenericGraphTables.tableWords,
    encodeWords_append, List.length_append, encodeWords, encodeWord_length, List.length_nil]
  omega

theorem generic_bits_le_polynomial {q : Nat} (table : GenericGraphTables.Table q) :
    (GenericGraphTables.tableBits table).length ≤
      (encodingPolynomial q).eval (table.vertices + table.darts) :=
  (GenericGraphTables.tableBits_length_le table).trans
    (rowBound_le_polynomial q table.vertices table.darts)

theorem generic_bits_le_of_size_le {q N : Nat} (table : GenericGraphTables.Table q)
    (hsize : table.vertices + table.darts ≤ N) :
    (GenericGraphTables.tableBits table).length ≤ (encodingPolynomial q).eval N :=
  (generic_bits_le_polynomial table).trans (encodingPolynomial_mono q hsize)

theorem size_add_two_le_bits (table : GraphTables.Table) :
    table.vertices + table.darts + 2 ≤ (GraphTables.tableBits table).length := by
  simp only [GraphTables.tableBits, GraphTables.tableWords,
    encodeWords_append, List.length_append, encodeWords, encodeWord_length, List.length_nil]
  omega

theorem bits_le_polynomial (table : GraphTables.Table) :
    (GraphTables.tableBits table).length ≤
      (encodingPolynomial 64).eval (table.vertices + table.darts) :=
  (GraphTables.tableBits_length_le table).trans
    (rowBound_le_polynomial 64 table.vertices table.darts)

theorem bits_le_of_size_le {N : Nat} (table : GraphTables.Table)
    (hsize : table.vertices + table.darts ≤ N) :
    (GraphTables.tableBits table).length ≤ (encodingPolynomial 64).eval N :=
  (bits_le_polynomial table).trans (encodingPolynomial_mono 64 hsize)

end MaxCutGames.Foundations.PCP.GraphTableComplexity

/-!
The actual fixed-parameter graph transformation in one PCP amplification round.
It preprocesses the input graph, checks all edges of each bounded walk using
local endpoint labels, and applies the checked constant-query alphabet reduction.
The bounded address list is explicit fixed data shared by every input graph.
-/

noncomputable section

namespace MaxCutGames.Foundations.PCP.AmplificationRound

abbrev Label := QueryIncidence.Label 6
abbrev Addresses := PoweringLabels.PortWords Preprocessing.Port FinalConstants.walkLength
abbrev PoweredAlphabet :=
  PoweringLabels.PaddedLabel Preprocessing.Port FinalConstants.walkLength Label

instance : Fintype PoweredAlphabet := Fintype.ofFinite _
instance : DecidableEq PoweredAlphabet := Classical.decEq _

variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
  [Nonempty E]

abbrev PoweredDart (G : ConstraintGraph V E Label) :=
  PoweringTest.Dart (Preprocessing.Vertex G) Preprocessing.Port
    (2 * FinalConstants.endpointLength)

instance (G : ConstraintGraph V E Label) : DecidableEq (PoweredDart G) := Classical.decEq _

def selectors (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (G : ConstraintGraph V E Label) (v : Preprocessing.Vertex G) :
    PoweringLabels.AddressSelector (Preprocessing.portGraph G)
      FinalConstants.walkLength v :=
  PoweringLabels.listSelector (Preprocessing.portGraph G) FinalConstants.walkLength
    v addresses complete

def powered (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (G : ConstraintGraph V E Label) :
    ConstraintGraph (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet :=
  PoweringTest.poweredGraph (Preprocessing.portGraph G) (Preprocessing.graph G).accepts
    (2 * FinalConstants.endpointLength) (selectors addresses complete G)

abbrev Vertex (G : ConstraintGraph V E Label) :=
  QueryIncidence.Vertex
    (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet)
    (AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet)

abbrev Dart (G : ConstraintGraph V E Label) :=
  QueryIncidence.Dart (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet) 6

instance (G : ConstraintGraph V E Label) : Fintype (Vertex G) := by
  letI : Fintype (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet) := inferInstance
  letI : Fintype
      (AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet) :=
    inferInstance
  change Fintype
    (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet ⊕
      AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet)
  infer_instance
instance (G : ConstraintGraph V E Label) : Fintype (Dart G) := by
  change Fintype ((AlphabetGraph.Event (PoweredDart G) PoweredAlphabet × Fin 6) × Bool)
  infer_instance
instance (G : ConstraintGraph V E Label) : DecidableEq (Vertex G) := Classical.decEq _

def graph (addresses : List Addresses) (complete : ∀ w, w ∈ addresses)
    (G : ConstraintGraph V E Label) : ConstraintGraph (Vertex G) (Dart G) Label :=
  AlphabetGraph.graph (powered addresses complete G)

omit [Nonempty E] in
theorem powered_completeness (addresses : List Addresses)
    (complete : ∀ w, w ∈ addresses) (G : ConstraintGraph V E Label)
    (satisfied : G.Satisfiable) : (powered addresses complete G).Satisfiable := by
  obtain ⟨assignment, hassignment⟩ := Preprocessing.completeness G satisfied
  refine ⟨PoweringOpinions.honestLabels (Preprocessing.portGraph G)
    FinalConstants.walkLength assignment, ?_⟩
  apply PoweringTest.perfect_completeness (Preprocessing.portGraph G)
    (Preprocessing.graph G).accepts (2 * FinalConstants.endpointLength)
    (selectors addresses complete G) assignment
  intro e
  exact hassignment e

omit [Nonempty E] in
theorem completeness (addresses : List Addresses)
    (complete : ∀ w, w ∈ addresses) (G : ConstraintGraph V E Label)
    (satisfied : G.Satisfiable) : (graph addresses complete G).Satisfiable :=
  AlphabetGraph.perfect_completeness _
    (powered_completeness addresses complete G satisfied)

end MaxCutGames.Foundations.PCP.AmplificationRound
end

/-!
# The actual first moment of an endpoint-opinion witness

The pivot factorization is applied to actual finite walks. Its two endpoint
operator bounds are obtained from the proved lazy-graph mixture and smoothing
theorem. No witness-event probability law is assumed.
-/

namespace MaxCutGames.Foundations.PCP.PoweringWitness

open PoweringWalks SpectralReturn PoweringReturn PoweringLazy
open PoweringMoment (bit)

variable {V D : Type*}

/-- A constant scalar passes through every iterate of the actual graph mean. -/
theorem iterate_mul_const [Fintype D] (G : PortGraph V D)
    (n : Nat) (c : ℝ) (h : V → ℝ) :
    iterateOperator G n (fun x => c * h x) =
      fun x => c * iterateOperator G n h x := by
  induction n with
  | zero => rfl
  | succ n ih =>
    change averagingOperator G (iterateOperator G n (fun x => c * h x)) =
      fun x => c * averagingOperator G (iterateOperator G n h) x
    rw [ih]
    funext v
    exact mean_mul_left c (fun d => iterateOperator G n h (G.rot (v, d)).1)

/-- The actual pivot dart is marked and both endpoint opinions pass their
tests. For indicator opinions this is the indicator of their conjunction. -/
noncomputable def endpointWitness (G : PortGraph V D) (bad : Edge V D → Bool)
    (φ ψ : Edge V D → V → ℝ) (n : Nat) (k : Fin (n + 1))
    (w : Walk V D (n + 1)) : ℝ :=
  bit (bad (edgeAt G n w k) = true) *
    φ (edgeAt G n w k) w.1 * ψ (edgeAt G n w k) (endpoint G w)

/-- Vertex-indexed opinions are tested at the two endpoints of the pivot dart. -/
noncomputable def vertexWitness (G : PortGraph V D) (bad : Edge V D → Bool)
    (f : V → V → ℝ) (n : Nat) (k : Fin (n + 1)) :
    Walk V D (n + 1) → ℝ :=
  endpointWitness G bad (fun e => f e.1) (fun e => f (next G e.1 e.2)) n k

/-- Pointwise bounds on the two genuine earlier-length graph iterates give
the witness first moment. The hypotheses are only needed on marked darts. -/
theorem endpoint_lower_bound [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (n : Nat) (k : Fin (n + 1)) (bad : Edge V D → Bool)
    (φ ψ : Edge V D → V → ℝ) (a : ℝ) (ha : 0 ≤ a)
    (hL : ∀ e, bad e = true → a ≤ iterateOperator G k.val (φ e) e.1)
    (hR : ∀ e, bad e = true →
      a ≤ iterateOperator G (n - k.val) (ψ e) (next G e.1 e.2)) :
    edgeDensity bad * a ^ 2 ≤ mean (endpointWitness G bad φ ψ n k) := by
  have hfactor : mean (endpointWitness G bad φ ψ n k) =
      mean (fun e : Edge V D =>
        (bit (bad e = true) * iterateOperator G k.val (φ e) e.1) *
          iterateOperator G (n - k.val) (ψ e) (next G e.1 e.2)) := by
    unfold endpointWitness
    rw [mean_edge_endpoints G n k (fun e x => bit (bad e = true) * φ e x) ψ]
    simp_rw [iterate_mul_const]
  have hbitMean : mean (fun e : Edge V D => bit (bad e = true)) = edgeDensity bad := by
    calc
      _ = mean (fun v : V => mean (fun d : D => bit (bad (v, d) = true))) :=
        mean_prod (A := V) (B := D) (fun e => bit (bad e = true))
      _ = edgeDensity bad := by simp only [mean_edge_bit, edgeDensity]
  calc
    edgeDensity bad * a ^ 2 =
        mean (fun e : Edge V D => bit (bad e = true) * a ^ 2) := by
      rw [mean_mul_right, hbitMean]
    _ ≤ mean (fun e : Edge V D =>
        (bit (bad e = true) * iterateOperator G k.val (φ e) e.1) *
          iterateOperator G (n - k.val) (ψ e) (next G e.1 e.2)) := by
      apply mean_mono
      intro e
      cases hb : bad e with
      | false => simp [bit]
      | true =>
        have hprod := mul_le_mul (hL e hb) (hR e hb) ha (ha.trans (hL e hb))
        simpa [bit, hb, pow_two] using hprod
    _ = mean (endpointWitness G bad φ ψ n k) := hfactor.symm

/-- The two endpoint-opinion tests each retain probability at least `1/(2q)`
at every middle pivot. The walk has `2N+1` edges, with `N=(4qM)^2`, so the
prefix and suffix lengths are exactly `k` and `2N-k`.

The only modal input is the actual centered lazy iterate at each vertex. It
will be supplied by the finite modal-opinion decoder, not by a witness law.
-/
theorem lazy_middle_witness_mean [Fintype V] [Fintype D] [Nonempty D]
    (G : PortGraph V D) (q M : Nat) (hq : 1 ≤ q) (hM : 1 ≤ M)
    (k : Fin (2 * (4 * q * M) ^ 2 + 1))
    (hlo : (4 * q * M) ^ 2 - M ≤ k.val)
    (hhi : k.val ≤ (4 * q * M) ^ 2 + M)
    (bad : Edge V (Bool × D) → Bool) (f : V → V → ℝ)
    (hf : ∀ u w, f u w ∈ Set.Icc (0 : ℝ) 1)
    (hmodal : ∀ u, 1 / (q : ℝ) ≤
      iterateOperator (lazyGraph G) ((4 * q * M) ^ 2) (f u) u) :
    edgeDensity bad / (4 * (q : ℝ) ^ 2) ≤
      mean (vertexWitness (lazyGraph G) bad f (2 * (4 * q * M) ^ 2) k) := by
  have hk : k.val ≤ 2 * (4 * q * M) ^ 2 := Nat.le_of_lt_succ k.isLt
  have hslo : (4 * q * M) ^ 2 - M ≤ 2 * (4 * q * M) ^ 2 - k.val := by omega
  have hshi : 2 * (4 * q * M) ^ 2 - k.val ≤ (4 * q * M) ^ 2 + M := by omega
  have ha : (0 : ℝ) ≤ 1 / (2 * (q : ℝ)) := by positivity
  have hbound := endpoint_lower_bound (lazyGraph G) (2 * (4 * q * M) ^ 2) k bad
    (fun e => f e.1) (fun e => f (next (lazyGraph G) e.1 e.2))
    (1 / (2 * (q : ℝ))) ha
    (by
      intro e _
      exact lazy_endpoint_modal_transfer G q M k.val hq hM hlo hhi
        (f e.1) (hf e.1) e.1 (hmodal e.1))
    (by
      intro e _
      exact lazy_endpoint_modal_transfer G q M (2 * (4 * q * M) ^ 2 - k.val)
        hq hM hslo hshi (f (next (lazyGraph G) e.1 e.2))
        (hf (next (lazyGraph G) e.1 e.2)) (next (lazyGraph G) e.1 e.2)
        (hmodal (next (lazyGraph G) e.1 e.2)))
  have hscale : edgeDensity bad * (1 / (2 * (q : ℝ))) ^ 2 =
      edgeDensity bad / (4 * (q : ℝ) ^ 2) := by ring
  simpa only [vertexWitness, hscale] using hbound

end MaxCutGames.Foundations.PCP.PoweringWitness

/-! The second moment of the actual marked-edge count in a consecutive walk
window. Pair events are reduced to the checked spectral return operator. -/

namespace MaxCutGames.Foundations.PCP.PoweringMomentBound

open scoped BigOperators
open PoweringWalks SpectralReturn PoweringReturn
open PoweringMoment (bit hits)

theorem symmetric_gap_matrix_sum (ε : ℝ) (r : Nat → ℝ) :
    ∀ (m : Nat) (f : Fin m → Fin m → ℝ),
      (∀ i j, f i j = f j i) → (∀ i, f i i = ε) →
      (∀ i j, i.val < j.val → f i j = r (j.val - i.val - 1)) →
      (∑ i, ∑ j, f i j) = (m : ℝ) * ε +
        2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap := by
  intro m
  induction m with
  | zero => intro f _hSym _hDiag _hUpper; simp
  | succ m ih =>
    intro f hSym hDiag hUpper
    have hOld : (∑ i : Fin m, ∑ j : Fin m, f i.castSucc j.castSucc) =
        (m : ℝ) * ε + 2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap := by
      apply ih (fun i j => f i.castSucc j.castSucc)
      · intro i j; exact hSym i.castSucc j.castSucc
      · intro i; exact hDiag i.castSucc
      · intro i j hij; exact hUpper i.castSucc j.castSucc hij
    have hLast : (∑ i : Fin m, f i.castSucc (Fin.last m)) =
        ∑ gap ∈ Finset.range m, r gap := by
      calc
        _ = ∑ i : Fin m, r (m - 1 - i.val) := by
          apply Finset.sum_congr rfl
          intro i _
          rw [hUpper i.castSucc (Fin.last m) i.isLt]
          simp only [Fin.val_last, Fin.val_castSucc]
          congr 1
          omega
        _ = ∑ i ∈ Finset.range m, r (m - 1 - i) :=
          Fin.sum_univ_eq_sum_range (fun i => r (m - 1 - i)) m
        _ = _ := Finset.sum_range_reflect r m
    have hLastRow : (∑ j : Fin m, f (Fin.last m) j.castSucc) =
        ∑ gap ∈ Finset.range m, r gap := by
      calc
        _ = ∑ j : Fin m, f j.castSucc (Fin.last m) := by
          apply Finset.sum_congr rfl
          intro j _
          exact hSym (Fin.last m) j.castSucc
        _ = _ := hLast
    calc
      (∑ i, ∑ j, f i j) =
          (∑ i : Fin m, ∑ j : Fin m, f i.castSucc j.castSucc) +
            (∑ i : Fin m, f i.castSucc (Fin.last m)) +
            (∑ j : Fin m, f (Fin.last m) j.castSucc) + f (Fin.last m) (Fin.last m) := by
        simp only [Fin.sum_univ_castSucc, Finset.sum_add_distrib]
        ring
      _ = (m : ℝ) * ε +
          2 * (∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, r gap) +
          (∑ gap ∈ Finset.range m, r gap) + (∑ gap ∈ Finset.range m, r gap) + ε := by
        rw [hOld, hLast, hLastRow, hDiag]
      _ = ((m + 1 : Nat) : ℝ) * ε +
          2 * ∑ j ∈ Finset.range (m + 1), ∑ gap ∈ Finset.range j, r gap := by
        rw [Finset.sum_range_succ]
        simp only [Nat.cast_add, Nat.cast_one]
        ring

def windowIndex (n start m : Nat) (h : start + m ≤ n + 1) (i : Fin m) : Fin (n + 1) :=
  ⟨start + i.val, (Nat.add_lt_add_left i.isLt start).trans_le h⟩

variable {V D : Type*} [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]

def windowEvent (G : PortGraph V D) (bad : V × D → Bool)
    (n start m : Nat) (h : start + m ≤ n + 1) (i : Fin m) (w : Walk V D (n + 1)) : Prop :=
  bad (edgeAt G n w (windowIndex n start m h i)) = true

theorem window_second_moment_eq (G : PortGraph V D) (bad : V × D → Bool)
    (n start m : Nat) (h : start + m ≤ n + 1) :
    mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) =
      (m : ℝ) * edgeDensity bad +
        2 * ∑ j ∈ Finset.range m, ∑ gap ∈ Finset.range j, returnMass G bad gap := by
  rw [show mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) =
      ∑ i, ∑ j, mean (fun w => bit (windowEvent G bad n start m h i w ∧
        windowEvent G bad n start m h j w)) from
    PoweringMoment.second_moment_eq (windowEvent G bad n start m h)]
  apply symmetric_gap_matrix_sum
  · intro i j
    congr 1
    funext w
    simp only [and_comm]
  · intro i
    simp only [and_self]
    exact hit_mean G bad n (windowIndex n start m h i)
  · intro i j hij
    have hwin : windowIndex n start m h i < windowIndex n start m h j :=
      Nat.add_lt_add_left hij start
    have hp := pair_event_mean G bad n (windowIndex n start m h i)
      (windowIndex n start m h j) hwin
    simpa only [windowEvent, windowIndex, Nat.add_sub_add_left] using hp

/-- All moment terms come from actual walk events; the only analytic input
is the stated spectral contraction certificate of the given graph. -/
theorem window_second_moment_le (G : PortGraph V D) (lambda : ℝ)
    (certificate : SpectralCertificate G lambda) (bad : V × D → Bool)
    (reversal : ∀ e, bad (G.rot e) = bad e)
    (n start m : Nat) (h : start + m ≤ n + 1) :
    mean (fun w => hits (windowEvent G bad n start m h) w ^ 2) ≤
      (m : ℝ) * edgeDensity bad *
        (1 + 2 / (1 - lambda) + ((m : ℝ) - 1) * edgeDensity bad) := by
  rw [window_second_moment_eq]
  exact secondMoment_return_envelope G lambda certificate bad reversal m

end MaxCutGames.Foundations.PCP.PoweringMomentBound

/-!
# The scalar bound for all positive error densities

The first lemma bounds a ratio using only a nonnegative coefficient bounded
by a positive cutoff. The second applies it to a positive natural count and
the first- and second-moment expressions in the powering argument.
-/

namespace MaxCutGames.Foundations.PCP.PoweringNumeric

/-- A bounded nonnegative coefficient gives the required ratio bound both
below and above the cutoff `1/t`. -/
theorem min_density_ratio (ε C k t : ℝ)
    (hε : 0 < ε) (hC : 0 < C) (hk : 0 ≤ k)
    (ht : 0 < t) (hkt : k ≤ t) :
    min ε (1 / t) / (C + 1) ≤ ε / (C + k * ε) := by
  have hx : 0 ≤ min ε (1 / t) :=
    le_min hε.le (one_div_nonneg.mpr ht.le)
  have hxt : min ε (1 / t) * t ≤ 1 :=
    (le_div_iff₀ ht).mp (min_le_right ε (1 / t))
  have hkx : k * min ε (1 / t) ≤ 1 := by
    calc
      k * min ε (1 / t) ≤ t * min ε (1 / t) :=
        mul_le_mul_of_nonneg_right hkt hx
      _ ≤ 1 := by simpa only [mul_comm] using hxt
  have hC1 : 0 < C + 1 := by positivity
  have hden : 0 < C + k * ε :=
    add_pos_of_pos_of_nonneg hC (mul_nonneg hk hε.le)
  apply (div_le_div_iff₀ hC1 hden).2
  calc
    min ε (1 / t) * (C + k * ε) =
        C * min ε (1 / t) + ε * (k * min ε (1 / t)) := by ring
    _ ≤ C * ε + ε * 1 :=
      add_le_add
        (mul_le_mul_of_nonneg_left (min_le_left ε (1 / t)) hC.le)
        (mul_le_mul_of_nonneg_left hkx hε.le)
    _ = ε * (C + 1) := by ring

/-- A positive natural count supplies `0 ≤ m - 1`. After cancelling the
positive mass `m*ε`, the ratio of moments dominates a constant times
`min ε (1/t)`. No sign condition on `α` is needed for its fourth power. -/
theorem count_moment_ratio_lower (α ε C t : ℝ) (m : ℕ)
    (hm : 0 < m) (hε : 0 < ε) (hC : 0 < C) (ht : 0 < t)
    (hmt : (m : ℝ) - 1 ≤ t) :
    (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / t) ≤
      (α ^ 2 * (m : ℝ) * ε) ^ 2 /
        ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε)) := by
  have hm0 : (0 : ℝ) < (m : ℝ) := Nat.cast_pos.mpr hm
  have hm1 : (1 : ℝ) ≤ (m : ℝ) := by
    exact_mod_cast (Nat.succ_le_iff.mpr hm)
  have hk : 0 ≤ (m : ℝ) - 1 := sub_nonneg.mpr hm1
  have hden : 0 < C + ((m : ℝ) - 1) * ε :=
    add_pos_of_pos_of_nonneg hC (mul_nonneg hk hε.le)
  have hαm : 0 ≤ α ^ 4 * (m : ℝ) := by positivity
  have hratio := min_density_ratio ε C ((m : ℝ) - 1) t hε hC hk ht hmt
  calc
    (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / t) =
        (α ^ 4 * (m : ℝ)) * (min ε (1 / t) / (C + 1)) := by ring
    _ ≤ (α ^ 4 * (m : ℝ)) * (ε / (C + ((m : ℝ) - 1) * ε)) :=
      mul_le_mul_of_nonneg_left hratio hαm
    _ = (α ^ 2 * (m : ℝ) * ε) ^ 2 /
        ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε)) := by
      field_simp [ne_of_gt hm0, ne_of_gt hε, ne_of_gt hden]

end MaxCutGames.Foundations.PCP.PoweringNumeric

/-! The actual graph-powering soundness bound. Modal endpoint opinions give
the first moment, spectral return gives the second moment, and an actual
checked path constraint turns each witness into rejection. -/

noncomputable section

namespace MaxCutGames.Foundations.PCP.PoweringSoundness

open scoped BigOperators
open PoweringWalks PoweringLabels PoweringOpinions PoweringTest
open SpectralReturn PoweringWitness PoweringMomentBound
open PoweringMoment (bit hits)

variable {V D A : Type*}

def center (q M : Nat) : Nat := (4 * q * M) ^ 2

theorem le_center (q M : Nat) (hq : 1 ≤ q) (hM : 1 ≤ M) : M ≤ center q M := by
  have hfour : 1 ≤ 4 * q := by omega
  have hx : M ≤ 4 * q * M := by simpa only [Nat.one_mul] using Nat.mul_le_mul_right M hfour
  have hone : 1 ≤ 4 * q * M := hM.trans hx
  have hsquare : 4 * q * M ≤ (4 * q * M) * (4 * q * M) := by
    simpa only [Nat.mul_one] using Nat.mul_le_mul_left (4 * q * M) hone
  exact hx.trans (by simpa only [center, pow_two] using hsquare)

def gain (q M : Nat) (lambda : ℝ) : ℝ :=
  (1 / (2 * (q : ℝ))) ^ 4 * ((2 * M + 1 : Nat) : ℝ) /
    ((1 + 2 / (1 - lambda)) + 1)

variable [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
  [Fintype A] [Nonempty A]

def decodedError (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n N : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) (fallback : A) : ℝ :=
  edgeDensity (decodedBad G accepts (decoded G (n + 1) N selectors labels fallback))

def poweredRejection (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) : ℝ :=
  mean (fun d : Dart V D n => bit ((poweredGraph G accepts n selectors).edgeSatisfied labels d = false))

omit [Fintype V] [Nonempty V] in
omit [Nonempty D] in
theorem witness_bit_eq (G : PortGraph V D) (accepts : Edge V D → A → A → Bool)
    (n N : Nat) (selectors : ∀ v, AddressSelector G (n + 1) v)
    (labels : V → PaddedLabel D (n + 1) A) (fallback : A)
    (w : Walk V D (n + 1)) (k : Fin (n + 1)) :
    bit (witness G accepts n selectors labels fallback
      (decoded G (n + 1) N selectors labels fallback) w k) =
      vertexWitness G
        (decodedBad G accepts (decoded G (n + 1) N selectors labels fallback))
        (matchFn G (n + 1) N selectors labels fallback) n k w := by
  simp only [witness, vertexWitness, endpointWitness, matchFn,
    PoweringMoment.bit_mul, next, and_assoc]

omit [Nonempty V] in
theorem modal_witness_mean (G : PortGraph V D) (M : Nat) (hM : 1 ≤ M)
    (accepts : Edge V (Bool × D) → A → A → Bool)
    (selectors : ∀ v, AddressSelector (lazyGraph G) (2 * center (Fintype.card A) M + 1) v)
    (labels : V → PaddedLabel (Bool × D) (2 * center (Fintype.card A) M + 1) A)
    (fallback : A) (k : Fin (2 * center (Fintype.card A) M + 1))
    (hlo : center (Fintype.card A) M - M ≤ k.val)
    (hhi : k.val ≤ center (Fintype.card A) M + M) :
    decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
      (center (Fintype.card A) M) selectors labels fallback / (4 * (Fintype.card A : ℝ) ^ 2) ≤
      mean (fun w => bit (witness (lazyGraph G) accepts (2 * center (Fintype.card A) M)
        selectors labels fallback (decoded (lazyGraph G) (2 * center (Fintype.card A) M + 1)
          (center (Fintype.card A) M) selectors labels fallback) w k)) := by
  have h := lazy_middle_witness_mean G (Fintype.card A) M Fintype.card_pos hM k hlo hhi
    (decodedBad (lazyGraph G) accepts
      (decoded (lazyGraph G) (2 * center (Fintype.card A) M + 1)
        (center (Fintype.card A) M) selectors labels fallback))
    (matchFn (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
    (matchFn_mem_Icc (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
    (decoded_modal_baseline (lazyGraph G) (2 * center (Fintype.card A) M + 1)
      (center (Fintype.card A) M) selectors labels fallback)
  simp_rw [witness_bit_eq]
  convert h using 1 <;> rfl
/-- Positive decoded error forces rejection of the actual powered graph.
The cap and amplification coefficient are explicit constants. -/
theorem rejection_lower_bound (G : PortGraph V D) (lambda : ℝ)
    (certificate : SpectralCertificate (lazyGraph G) lambda)
    (accepts : Edge V (Bool × D) → A → A → Bool)
    (reverse_accepts : ∀ e a b, accepts ((lazyGraph G).rot e) b a = accepts e a b)
    (M : Nat) (hM : 1 ≤ M)
    (selectors : ∀ v, AddressSelector (lazyGraph G) (2 * center (Fintype.card A) M + 1) v)
    (labels : V → PaddedLabel (Bool × D) (2 * center (Fintype.card A) M + 1) A)
    (fallback : A)
    (hε : 0 < decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
      (center (Fintype.card A) M) selectors labels fallback) :
    gain (Fintype.card A) M lambda *
        min (decodedError (lazyGraph G) accepts (2 * center (Fintype.card A) M)
          (center (Fintype.card A) M) selectors labels fallback)
          (1 / ((2 * center (Fintype.card A) M + 1 : Nat) : ℝ)) ≤
      poweredRejection (lazyGraph G) accepts (2 * center (Fintype.card A) M) selectors labels := by
  let q := Fintype.card A
  let N := center q M
  let n := 2 * N
  let m := 2 * M + 1
  let start := N - M
  let L := lazyGraph G
  let assignment := decoded L (n + 1) N selectors labels fallback
  let bad := decodedBad L accepts assignment
  let ε := edgeDensity bad
  let α : ℝ := 1 / (2 * (q : ℝ))
  let C : ℝ := 1 + 2 / (1 - lambda)
  have hNM : M ≤ N := le_center q M Fintype.card_pos hM
  have hw : start + m ≤ n + 1 := by dsimp [start, m, n]; omega
  let E := windowEvent L bad n start m hw
  let W : Fin m → Walk V (Bool × D) (n + 1) → Prop := fun i w =>
    witness L accepts n selectors labels fallback assignment w (windowIndex n start m hw i)
  let R : Walk V (Bool × D) (n + 1) → Prop := fun w =>
    pathAccepts L accepts n selectors w (labels w.1) (labels (endpoint L w)) = false
  have hWE : ∀ i w, W i w → E i w := by
    intro i w h
    exact h.1
  have hWR : ∀ i w, W i w → R w := by
    intro i w h
    exact witness_implies_path_rejection L accepts n selectors labels fallback assignment w
      (windowIndex n start m hw i) h
  have hpoint (i : Fin m) : ε / (4 * (q : ℝ) ^ 2) ≤ mean (fun w => bit (W i w)) := by
    have hlo : N - M ≤ (windowIndex n start m hw i).val := by
      dsimp [windowIndex, start]
      omega
    have hhi : (windowIndex n start m hw i).val ≤ N + M := by
      have hi := i.isLt
      dsimp [windowIndex, start]
      dsimp [m] at hi
      omega
    exact modal_witness_mean G M hM accepts selectors labels fallback
      (windowIndex n start m hw i) hlo hhi
  have hfirst : α ^ 2 * (m : ℝ) * ε ≤ mean (hits W) := by
    change α ^ 2 * (m : ℝ) * ε ≤ PoweringMoment.mean (hits W)
    rw [PoweringMoment.mean_hits]
    calc
      _ = ∑ _i : Fin m, ε / (4 * (q : ℝ) ^ 2) := by
        simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
        dsimp [α]
        ring
      _ ≤ _ := Finset.sum_le_sum (fun i _ => hpoint i)
  have hsecond : mean (fun w => hits E w ^ 2) ≤
      (m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε) :=
    window_second_moment_le L lambda certificate bad
      (decodedBad_rot L accepts reverse_accepts assignment) n start m hw
  have hε' : 0 < ε := hε
  have hm : 0 < m := by dsimp [m]; omega
  have hmReal : (0 : ℝ) < m := Nat.cast_pos.mpr hm
  have hqReal : (0 : ℝ) < q := Nat.cast_pos.mpr Fintype.card_pos
  have hden : 0 < 1 - lambda := sub_pos.mpr certificate.lt_one
  have hC : 0 < C := by dsimp [C]; positivity
  have hmOne : (1 : ℝ) ≤ m := by exact_mod_cast hm
  have hα : 0 < α := by dsimp [α]; positivity
  have ha : 0 < α ^ 2 * (m : ℝ) * ε := by positivity
  have hb : 0 < (m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε) := by
    have hdiff : 0 ≤ (m : ℝ) - 1 := sub_nonneg.mpr hmOne
    positivity
  have hCS := PoweringMoment.rejection_lower_bound E W R hWE hWR
    (α ^ 2 * (m : ℝ) * ε) ((m : ℝ) * ε * (C + ((m : ℝ) - 1) * ε))
    ha hb hfirst hsecond
  have ht : (0 : ℝ) < ((n + 1 : Nat) : ℝ) := Nat.cast_pos.mpr (Nat.succ_pos n)
  have hmn : m ≤ n + 1 := by dsimp [m, n]; omega
  have hmt : (m : ℝ) - 1 ≤ ((n + 1 : Nat) : ℝ) := by
    have hmnReal : (m : ℝ) ≤ ((n + 1 : Nat) : ℝ) := by exact_mod_cast hmn
    linarith
  have hratio := PoweringNumeric.count_moment_ratio_lower α ε C ((n + 1 : Nat) : ℝ)
    m hm hε' hC ht hmt
  change (α ^ 4 * (m : ℝ) / (C + 1)) * min ε (1 / ((n + 1 : Nat) : ℝ)) ≤ _
  change _ ≤ mean (fun d : Dart V (Bool × D) n =>
    bit ((poweredGraph L accepts n selectors).edgeSatisfied labels d = false))
  rw [rejection_mean_eq_path_mean]
  exact hratio.trans hCS

end MaxCutGames.Foundations.PCP.PoweringSoundness
end

namespace MaxCutGames.Foundations.PCP.PoweringGap

open PoweringWalks PoweringLabels PoweringOpinions PoweringTest
open SpectralReturn PoweringSoundness PoweringCounting

variable {V D A : Type*}

theorem gain_nonneg (q M : Nat) (lambda : ℝ) (hlambda : lambda < 1) :
    0 ≤ gain q M lambda := by
  have hden : 0 < 1 - lambda := sub_pos.mpr hlambda
  unfold gain
  positivity

variable [Fintype V] [Fintype D] [Nonempty V] [Nonempty D]
  [Fintype A] [Nonempty A]

theorem uniform_count_gap (G : PortGraph V D) (lambda : ℝ)
    (certificate : SpectralCertificate (lazyGraph G) lambda)
    (accepts : Edge V (Bool × D) → A → A → Bool)
    (reverse_accepts : ∀ e a b, accepts ((lazyGraph G).rot e) b a = accepts e a b)
    (M : Nat) (hM : 1 ≤ M)
    (selectors : ∀ v, AddressSelector (lazyGraph G)
      (2 * center (Fintype.card A) M + 1) v)
    (epsilon : ℝ) (hepsilon : 0 ≤ epsilon)
    (hgap : ∀ assignment : V → A,
      epsilon * (Fintype.card (Edge V (Bool × D)) : ℝ) ≤
        ((baseGraph (lazyGraph G) accepts reverse_accepts).rejectionCount assignment : ℝ))
    (labels : V → PaddedLabel (Bool × D) (2 * center (Fintype.card A) M + 1) A) :
    (gain (Fintype.card A) M lambda *
      min epsilon (1 / ((2 * center (Fintype.card A) M + 1 : Nat) : ℝ))) *
        (Fintype.card (Dart V (Bool × D) (2 * center (Fintype.card A) M)) : ℝ) ≤
      ((poweredGraph (lazyGraph G) accepts (2 * center (Fintype.card A) M)
        selectors).rejectionCount labels : ℝ) := by
  classical
  let L := lazyGraph G
  let N := center (Fintype.card A) M
  let n := 2 * N
  let H := poweredGraph L accepts n selectors
  change (gain (Fintype.card A) M lambda *
      min epsilon (1 / ((n + 1 : Nat) : ℝ))) *
        (Fintype.card (Dart V (Bool × D) n) : ℝ) ≤
      (H.rejectionCount labels : ℝ)
  rcases eq_or_lt_of_le hepsilon with hzero | hpositive
  · rw [← hzero]
    have hcap : (0 : ℝ) ≤ 1 / ((n + 1 : Nat) : ℝ) := by positivity
    rw [min_eq_left hcap, mul_zero, zero_mul]
    exact Nat.cast_nonneg _
  · let fallback : A := Classical.choice (inferInstance : Nonempty A)
    let assignment := decoded L (n + 1) N selectors labels fallback
    have hdecoded : epsilon ≤ decodedError L accepts n N selectors labels fallback := by
      change epsilon ≤ edgeDensity (decodedBad L accepts assignment)
      exact base_count_lower_to_density L accepts reverse_accepts assignment
        Fintype.card_pos epsilon (hgap assignment)
    have herror : 0 < decodedError L accepts n N selectors labels fallback :=
      hpositive.trans_le hdecoded
    have hsound := PoweringSoundness.rejection_lower_bound G lambda certificate
      accepts reverse_accepts M hM selectors labels fallback herror
    have hgain : 0 ≤ gain (Fintype.card A) M lambda :=
      gain_nonneg (Fintype.card A) M lambda certificate.lt_one
    have hmin : min epsilon (1 / ((n + 1 : Nat) : ℝ)) ≤
        min (decodedError L accepts n N selectors labels fallback)
          (1 / ((n + 1 : Nat) : ℝ)) :=
      min_le_min hdecoded (le_refl _)
    have hlower : gain (Fintype.card A) M lambda *
        min epsilon (1 / ((n + 1 : Nat) : ℝ)) ≤
        poweredRejection L accepts n selectors labels :=
      (mul_le_mul_of_nonneg_left hmin hgain).trans hsound
    exact (constraint_rejection_lower_iff H labels Fintype.card_pos
      (gain (Fintype.card A) M lambda *
        min epsilon (1 / ((n + 1 : Nat) : ℝ)))).mp hlower

end MaxCutGames.Foundations.PCP.PoweringGap

/-!
# The fixed constants in the actual powering estimate

The six-bit alphabet has 64 letters. The middle window, endpoint length,
spectral parameter and composition loss in the checked powering theorem give
exactly the constants used in the final amplification round. Large natural
constants remain symbolic in these calculations.
-/

namespace MaxCutGames.Foundations.PCP.PoweringFinalConstants

abbrev Alphabet := Fin 6 → Bool

@[simp] theorem card_alphabet : Fintype.card Alphabet = 64 := by
  norm_num [Alphabet, Fintype.card_fun]

theorem card_alphabet_eq_final : Fintype.card Alphabet = FinalConstants.alphabet := by
  exact card_alphabet

/-- The actual six-bit alphabet gives the endpoint length fixed for the round. -/
theorem center_eq :
    PoweringSoundness.center (Fintype.card Alphabet) FinalConstants.windowHalf =
      FinalConstants.endpointLength := by
  rw [card_alphabet]
  rfl

/-- Prefix, pivot and suffix together have the fixed odd walk length. -/
theorem walkLength_eq :
    2 * PoweringSoundness.center (Fintype.card Alphabet) FinalConstants.windowHalf + 1 =
      FinalConstants.walkLength := by
  rw [center_eq]
  rfl

/-- The spectral denominator is exactly 66 at the fixed return parameter. -/
theorem spectral_denominator_eq :
    (1 + 2 / (1 - (31 / 32 : ℝ))) + 1 = 66 := by
  norm_num

/-- Dividing the actual powering coefficient by the composition loss gives
the round coefficient exactly. The large window parameter is not evaluated. -/
theorem gain_eq :
    PoweringSoundness.gain 64 FinalConstants.windowHalf (31 / 32) / 12288 =
      FinalConstants.gain := by
  change
    (1 / (2 * (FinalConstants.alphabet : ℝ))) ^ 4 *
        (FinalConstants.windowSize : ℝ) /
        ((1 + 2 / (1 - (31 / 32 : ℝ))) + 1) /
        (FinalConstants.compositionLoss : ℝ) =
      (FinalConstants.windowSize : ℝ) / (FinalConstants.denominator : ℝ)
  rw [spectral_denominator_eq]
  have hden : (FinalConstants.denominator : ℝ) =
      16 * (FinalConstants.alphabet : ℝ) ^ 4 * 66 *
        (FinalConstants.compositionLoss : ℝ) := by
    simp only [FinalConstants.denominator, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  rw [hden]
  simp only [div_eq_mul_inv, mul_inv_rev]
  ring

theorem gain_card_eq :
    PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) /
        12288 = FinalConstants.gain := by
  rw [card_alphabet]
  exact gain_eq

/-- Both the coefficient and cap in the actual estimate match the fixed
round constants, for every real error parameter. -/
theorem scaled_bound_eq (ε : ℝ) :
    (PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) *
        min ε
          (1 / ((2 * PoweringSoundness.center (Fintype.card Alphabet)
            FinalConstants.windowHalf + 1 : Nat) : ℝ))) / 12288 =
      FinalConstants.gain * min ε FinalConstants.cap := by
  rw [walkLength_eq]
  rw [mul_div_right_comm, gain_card_eq]
  rfl

/-- The composed powering and alphabet-reduction coefficient yields the
fixed doubling bound after the checked preprocessing loss. -/
theorem composed_scaled_gap (ε : ℝ) (hε : 0 ≤ ε) :
    min (2 * ε) FinalConstants.cap ≤
      (PoweringSoundness.gain (Fintype.card Alphabet) FinalConstants.windowHalf (31 / 32) *
        min (ε / (Preprocessing.sizeFactor : ℝ))
          (1 / ((2 * PoweringSoundness.center (Fintype.card Alphabet)
            FinalConstants.windowHalf + 1 : Nat) : ℝ))) / 12288 := by
  rw [scaled_bound_eq]
  exact FinalConstants.composed_gap ε hε

end MaxCutGames.Foundations.PCP.PoweringFinalConstants

end OAI