import Mathlib
import OAI.Computability.MaxCut.Games.AddressTupleLoaded
import OAI.Computability.MaxCut.Machines.MachineRhsLoad
import OAI.Computability.MaxCut.Games.Weighted

namespace OAI

/-! Executable direct addresses for the actual canonical game. Every original
edge occurrence and its translation table is retained, in the same order.
Addresses not representing realized bodies are isolated vertices. No vertex
list, duplicate removal, or inverse enumeration is evaluated by the output. -/

namespace MaxCutGames.Reduction.AddressGame

open ActualSource
open Foundations.Target
open Integration.VertexEmbedding

abbrev bodyCapacity (S : Source) (k s d : Nat) : Nat :=
  CanonicalAddress.capacity S.«variables» S.occurrences k s d

def vertexCount (S : Source) (k s d : Nat) : Nat := 2 * bodyCapacity S k s d

/-- Both bipartite sides remain distinct address blocks. -/
def vertexAddress (S : Source) (k s d : Nat) (v : ActualGame.TwoSidedVertex S k s d) :
    Fin (vertexCount S k s d) :=
  Encoding.sideEquiv (bodyCapacity S k s d) (v.1, CanonicalAddress.bodyAddress v.2.val)

theorem vertexAddress_injective (S : Source) (k s d : Nat) :
    Function.Injective (vertexAddress S k s d) := by
  intro v w h
  have hp := (Encoding.sideEquiv (bodyCapacity S k s d)).injective h
  apply Prod.ext
  · exact congrArg (fun p : Bool × Fin (bodyCapacity S k s d) => p.1) hp
  · apply Subtype.ext
    exact CanonicalAddress.bodyAddress_injective
      (congrArg (fun p : Bool × Fin (bodyCapacity S k s d) => p.2) hp)

/-- This is the endpoint routine used at runtime: canonicalize, serialize,
apply Horner radix arithmetic, and add the side block. -/
def queryAddress (S : Source) (k s d : Nat) (side : Bool) (q : ActualGame.Query S k s d) :
    Fin (vertexCount S k s d) :=
  Encoding.sideEquiv (bodyCapacity S k s d)
    (side, CanonicalAddress.bodyAddress (ActualOrbit.body (ActualGame.canonical S k s d) q))

@[simp] theorem vertexAddress_query (S : Source) (k s d : Nat) (side : Bool)
    (q : ActualGame.Query S k s d) :
    vertexAddress S k s d (side, ActualGame.vertex S k s d q) =
      queryAddress S k s d side q := rfl

/-- Proof adapter only: the output constructor does not call this inverse. -/
def explicitToAddress (S : Source) (k s d : Nat) :
    Fin (ActualGame.explicitVertexCount S k s d) → Fin (vertexCount S k s d) :=
  fun i => vertexAddress S k s d ((ActualGame.explicitVertexEncoding S k s d).symm i)

theorem explicitToAddress_injective (S : Source) (k s d : Nat) :
    Function.Injective (explicitToAddress S k s d) :=
  (vertexAddress_injective S k s d).comp (ActualGame.explicitVertexEncoding S k s d).symm.injective

@[simp] theorem explicitToAddress_apply (S : Source) (k s d : Nat)
    (v : ActualGame.TwoSidedVertex S k s d) :
    explicitToAddress S k s d (ActualGame.explicitVertexEncoding S k s d v) =
      vertexAddress S k s d v := by
  simp only [explicitToAddress, Equiv.symm_apply_apply]

/-- The corresponding injection from the original semantic vertex space. -/
noncomputable def semanticToAddress (S : Source) (k s d : Nat) :
    Fin (ActualGame.vertexCount S k s d) → Fin (vertexCount S k s d) :=
  fun i => vertexAddress S k s d ((ActualGame.vertexEncoding S k s d).symm i)

theorem semanticToAddress_injective (S : Source) (k s d : Nat) :
    Function.Injective (semanticToAddress S k s d) :=
  (vertexAddress_injective S k s d).comp (ActualGame.vertexEncoding S k s d).symm.injective

@[simp] theorem semanticToAddress_apply (S : Source) (k s d : Nat)
    (v : ActualGame.TwoSidedVertex S k s d) :
    semanticToAddress S k s d (ActualGame.vertexEncoding S k s d v) =
      vertexAddress S k s d v := by
  simp only [semanticToAddress, Equiv.symm_apply_apply]

def addressEdge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (ω : ActualGame.Outcome S k g) : Constraint (vertexCount S k s d) (2^s) where
  source := queryAddress S k s d false (ActualGame.leftQuery S k g ω)
  target := queryAddress S k s d true (ActualGame.rightQuery S k g ω)
  permutation := Encoding.translationTable
    (ActualGame.offset S k s d (ActualGame.leftQuery S k g ω))
    (ActualGame.offset S k s d (ActualGame.rightQuery S k g ω))

theorem addressEdge_eq_embed (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : ActualGame.Outcome S k g) :
    addressEdge S k g ω = embedConstraint (explicitToAddress S k s d)
      (ActualGame.explicitEdge S k g ω) := by
  simp only [addressEdge, embedConstraint, ActualGame.explicitEdge,
    explicitToAddress_apply, vertexAddress_query]

theorem addressEdge_eq_semantic_embed (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : ActualGame.Outcome S k g) :
    addressEdge S k g ω = embedConstraint (semanticToAddress S k s d)
      (ActualGame.edge S k g ω) := by
  simp only [addressEdge, embedConstraint, ActualGame.edge,
    semanticToAddress_apply, vertexAddress_query]

@[simp] theorem addressEdge_permutation (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : ActualGame.Outcome S k g) :
    (addressEdge S k g ω).permutation = (ActualGame.explicitEdge S k g ω).permutation := rfl

/-- Only canonical words, bounded radix arithmetic, and the explicit outcome
list occur in the runtime path. The correcting function is not inspected. -/
def outputInstance (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d)
    (en : NoiseEnumeration g) : Instance (2^s) where
  vertices := vertexCount S k s d
  constraints := (ActualGame.explicitOutcomes S k g en).map (addressEdge S k g)
  nonempty h := by
    have hempty := List.map_eq_nil_iff.mp h
    apply (ActualGame.outputInstanceWithEnumeration S k g en).nonempty
    change (ActualGame.explicitOutcomes S k g en).map (ActualGame.explicitEdge S k g) = []
    rw [hempty]
    rfl

/-- An exact instance equality, retaining the explicit edge-list order. -/
theorem outputInstance_eq_embed (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    outputInstance S k g en =
      embedInstance (ActualGame.outputInstanceWithEnumeration S k g en)
        (explicitToAddress S k s d) (explicitToAddress_injective S k s d) := by
  unfold outputInstance embedInstance ActualGame.outputInstanceWithEnumeration
  congr 1
  simp only [List.map_map, Function.comp_def, ← addressEdge_eq_embed]

@[simp] theorem outputInstance_length (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (outputInstance S k g en).constraints.length =
      Explicit.edgeCount S.occurrences k (s+d) en.indices.length := by
  simp only [outputInstance, List.length_map, ActualGame.explicitOutcomes,
    ActualEnumeration.length_indexedOutcomes]

theorem outputInstance_count (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g)
    (labeling : Fin (vertexCount S k s d) → Fin (2^s)) :
    countSatisfied labeling (outputInstance S k g en).constraints =
      countSatisfied (labeling ∘ explicitToAddress S k s d)
        (ActualGame.outputInstanceWithEnumeration S k g en).constraints := by
  simpa only [outputInstance, ActualGame.outputInstanceWithEnumeration, List.map_map,
    Function.comp_def, ← addressEdge_eq_embed] using
    count_map (explicitToAddress S k s d)
      ((ActualGame.explicitOutcomes S k g en).map (ActualGame.explicitEdge S k g)) labeling

theorem completeAt_outputInstance_iff_explicit (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    CompleteAt error (outputInstance S k g en) ↔
      CompleteAt error (ActualGame.outputInstanceWithEnumeration S k g en) := by
  rw [outputInstance_eq_embed]
  exact completeAt_embed_iff error _ _ _ (by positivity)

theorem soundAt_outputInstance_iff_explicit (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    SoundAt error (outputInstance S k g en) ↔
      SoundAt error (ActualGame.outputInstanceWithEnumeration S k g en) := by
  rw [outputInstance_eq_embed]
  exact soundAt_embed_iff error _ _ _ (by positivity)

theorem completeAt_outputInstance_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    CompleteAt error (outputInstance S k g en) ↔ CompleteAt error (ActualGame.outputInstance S k g) :=
  (completeAt_outputInstance_iff_explicit error S k g en).trans
    (ActualGame.completeAt_outputInstanceWithEnumeration_iff error S k g en)

theorem soundAt_outputInstance_iff (error : RationalError)
    (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) :
    SoundAt error (outputInstance S k g en) ↔ SoundAt error (ActualGame.outputInstance S k g) :=
  (soundAt_outputInstance_iff_explicit error S k g en).trans
    (ActualGame.soundAt_outputInstanceWithEnumeration_iff error S k g en)

theorem outputInstance_translations (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    Integration.TranslationTarget.IsTranslationInstance (Encoding.alphabetEquiv s).symm
      (outputInstance S k g en) := by
  rw [outputInstance_eq_embed]
  exact (isTranslationInstance_embed_iff (Encoding.alphabetEquiv s).symm
    (ActualGame.outputInstanceWithEnumeration S k g en) (explicitToAddress S k s d)
    (explicitToAddress_injective S k s d)).2
    (OutputTranslations.outputInstanceWithEnumeration_coordinates S k g en)

/-- The concrete vector-table interface contains no correcting-map witness. -/
def tableOutput (S : Source) (k : Nat) {s d : Nat} (T : Integration.NoiseTables.Table s d) :
    Instance (2^s) :=
  outputInstance S k (Integration.TableReduction.tableSkeleton T)
    (Integration.TableReduction.tableEnumeration T Prod.fst (fun _ _ => rfl))

theorem completeAt_tableOutput_iff (error : RationalError) (S : Source) (k : Nat)
    {s d : Nat} (T : Integration.NoiseTables.Table s d) :
    CompleteAt error (tableOutput S k T) ↔ CompleteAt error (Integration.TableReduction.output S k T) :=
  completeAt_outputInstance_iff_explicit error S k _ _

theorem soundAt_tableOutput_iff (error : RationalError) (S : Source) (k : Nat)
    {s d : Nat} (T : Integration.NoiseTables.Table s d) :
    SoundAt error (tableOutput S k T) ↔ SoundAt error (Integration.TableReduction.output S k T) :=
  soundAt_outputInstance_iff_explicit error S k _ _

theorem tableOutput_translations (S : Source) (k : Nat) {s d : Nat}
    (T : Integration.NoiseTables.Table s d) :
    Integration.TranslationTarget.IsTranslationInstance (Encoding.alphabetEquiv s).symm (tableOutput S k T) :=
  outputInstance_translations S k _ _

/-- Exact polynomial bound for the complete two-sided address space. -/
theorem vertexCount_eq_polynomial (S : Source) (k s d : Nat) :
    vertexCount S k s d = 2 * (CanonicalAddress.capacityPolynomial k s d).eval
      (S.«variables» + S.occurrences) := by
  rw [CanonicalAddress.capacityPolynomial_eval]
  rfl

end MaxCutGames.Reduction.AddressGame

namespace MaxCutGames.Reduction.FixedOutcomes

open Integration.BinaryLinear ActualHomogeneous ActualSource
open ActualEnumeration _root_.OAI.MaxCutGames.Reduction.Explicit

variable {N α β γ R : Type*}

abbrev Parameter (k s d : Nat) (N : Type*) :=
  (E k →ₗ[F2] ActualEnumeration.Ambient s d) × (N × (E k →ₗ[F2] F2))

def occurrenceTuples (m k : Nat) : List (Fin k → Fin m) :=
  functions (List.finRange m) k

/-- Ambient linear map, noise occurrence index, scalar linear functional,
in exactly the original nested enumeration order. -/
def fixedParams (k s d : Nat) (indices : List N) : List (Parameter k s d N) :=
  pairs (allLinearMaps k _ (ambientVectors s d))
    (pairs indices (allLinearMaps k F2 [0, 1]))

def assemble {m k s d : Nat} (U : Fin k → Fin m) (p : Parameter k s d N) :
    Query m k s d × (N × (E k →ₗ[F2] F2)) :=
  ((U, p.1), p.2)

/-- The reassociation changes only parentheses, not enumeration order. -/
def outcomeEquiv (m k s d : Nat) (N : Type*) :
    (Query m k s d × (N × (E k →ₗ[F2] F2))) ≃
      ((Fin k → Fin m) × Parameter k s d N) :=
  Equiv.prodAssoc _ _ _

private theorem pairs_associate_inline_FixedOutcomes (xs : List α) (ys : List β) (zs : List γ) :
    pairs (pairs xs ys) zs =
      xs.flatMap (fun x => (pairs ys zs).map (fun p => ((x, p.1), p.2))) := by
  simp only [pairs, List.flatMap_assoc, List.flatMap_map, List.map_flatMap,
    List.map_map, Function.comp_def]

theorem indexedOutcomes_factor (m k s d : Nat) (indices : List N) :
    ActualEnumeration.indexedOutcomes m k s d indices =
      (occurrenceTuples m k).flatMap (fun U => (fixedParams k s d indices).map (assemble U)) := by
  exact pairs_associate_inline_FixedOutcomes (functions (List.finRange m) k)
    (allLinearMaps k _ (ambientVectors s d))
    (pairs indices (allLinearMaps k F2 [0, 1]))

theorem indexedOutcomes_reassociate (m k s d : Nat) (indices : List N) :
    (ActualEnumeration.indexedOutcomes m k s d indices).map (outcomeEquiv m k s d N) =
      pairs (occurrenceTuples m k) (fixedParams k s d indices) := by
  rw [indexedOutcomes_factor]
  simp only [pairs, List.map_flatMap, List.map_map, Function.comp_def,
    assemble, outcomeEquiv, Equiv.prodAssoc_apply]

theorem occurrenceTuples_length (m k : Nat) : (occurrenceTuples m k).length = m^k := by
  simp only [occurrenceTuples, length_functions, List.length_finRange]

theorem fixedParams_length (k s d : Nat) (indices : List N) :
    (fixedParams k s d indices).length = 2^((2*k+1)*(s+d+1)) * indices.length := by
  simp only [fixedParams, length_pairs, length_allLinearMaps, length_ambientVectors,
    List.length_cons, List.length_nil, ← pow_mul]
  change 2^((s+d)*(2*k+1)) * (indices.length * 2^(2*k+1)) =
    2^((2*k+1)*(s+d+1)) * indices.length
  rw [show (2*k+1)*(s+d+1) = (s+d)*(2*k+1)+(2*k+1) by ring, pow_add]
  ring

theorem fixedParams_nonempty (k s d : Nat) {indices : List N} (hne : indices ≠ []) :
    fixedParams k s d indices ≠ [] := by
  apply List.length_pos_iff.mp
  rw [fixedParams_length]
  exact Nat.mul_pos (Nat.two_pow_pos _) (List.length_pos_iff.mpr hne)

/-- A coefficient-only representation of each fixed linear-map parameter. -/
abbrev CoefficientParameter (k s d : Nat) (N : Type*) :=
  Coefficients k (ActualEnumeration.Ambient s d) × (N × Coefficients k F2)

def coefficientValues (k : Nat) (values : List R) : List (Coefficients k R) :=
  pairs values (functions (pairs values values) k)

def fixedCoefficientParams (k s d : Nat) (indices : List N) :
    List (CoefficientParameter k s d N) :=
  pairs (coefficientValues k (ambientVectors s d))
    (pairs indices (coefficientValues k ([0, 1] : List F2)))

def fromCoefficientParameter {k s d : Nat} (p : CoefficientParameter k s d N) :
    Parameter k s d N :=
  (coefficientEquiv k _ p.1, (p.2.1, coefficientEquiv k F2 p.2.2))

/-- The concrete coefficient enumeration produces precisely the fixed parameter
list. No choice of maps or representatives is made by this constructor. -/
theorem fixedCoefficientParams_map (k s d : Nat) (indices : List N) :
    (fixedCoefficientParams k s d indices).map fromCoefficientParameter =
      fixedParams k s d indices := by
  simp only [fixedCoefficientParams, fixedParams, coefficientValues, allLinearMaps,
    pairs, List.map_flatMap, List.flatMap_assoc, List.flatMap_map, List.map_map, Function.comp_def,
    fromCoefficientParameter]

theorem fixedCoefficientParams_length (k s d : Nat) (indices : List N) :
    (fixedCoefficientParams k s d indices).length =
      2^((2*k+1)*(s+d+1)) * indices.length := by
  rw [← fixedParams_length k s d indices, ← fixedCoefficientParams_map, List.length_map]

def tableParams (k : Nat) {s d : Nat} (T : Integration.NoiseTables.Table s d) :
    List (Parameter k s d (Fin T.vectors.length)) :=
  fixedParams k s d (List.finRange T.vectors.length)

theorem tableParams_length (k : Nat) {s d : Nat} (T : Integration.NoiseTables.Table s d) :
    (tableParams k T).length = 2^((2*k+1)*(s+d+1)) * T.vectors.length := by
  simp only [tableParams, fixedParams_length, List.length_finRange]

theorem outputInstance_constraints (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    (AddressGame.outputInstance S k g en).constraints =
      (occurrenceTuples S.occurrences k).flatMap (fun U =>
        (fixedParams k s d en.indices).map (fun p =>
          AddressGame.addressEdge S k g (assemble U p))) := by
  change (ActualEnumeration.indexedOutcomes S.occurrences k s d en.indices).map
    (AddressGame.addressEdge S k g) = _
  rw [indexedOutcomes_factor]
  simp only [List.map_flatMap, List.map_map, Function.comp_def]

theorem tableOutput_constraints (S : Source) (k : Nat) {s d : Nat}
    (T : Integration.NoiseTables.Table s d) :
    (AddressGame.tableOutput S k T).constraints =
      (occurrenceTuples S.occurrences k).flatMap (fun U =>
        (tableParams k T).map (fun p => AddressGame.addressEdge S k
          (Integration.TableReduction.tableSkeleton T) (assemble U p))) :=
  outputInstance_constraints S k (Integration.TableReduction.tableSkeleton T)
    (Integration.TableReduction.tableEnumeration T Prod.fst (fun _ _ => rfl))

end MaxCutGames.Reduction.FixedOutcomes

namespace MaxCutGames.Reduction.AddressByteSemantics

open Integration.BinaryLinear Foundations.Complexity ActualSource

def queryWords {k s d : Nat} (fields : Fin k → Fin 4 → Nat)
    (X : ActualGame.Map k s d) (b : Fin k → F2) : List Nat :=
  CanonicalBodyTemplate.evalTemplate fields
    (CanonicalBodyTemplate.template (X (ActualHomogeneous.hBasis k))
      (ActualCanonical.standardTriple X) b)

def queryRank {k s d : Nat} (base : Nat) (fields : Fin k → Fin 4 → Nat)
    (X : ActualGame.Map k s d) (b : Fin k → F2) : Nat :=
  CanonicalAddress.radix base (queryWords fields X b)

def permutationWords {k s d : Nat} (X Y : ActualGame.Map k s d)
    (b : Fin k → F2) : List Nat :=
  tableWords (Encoding.translationTable
    (CanonicalBodyTemplate.alphabetOffset (X (ActualHomogeneous.hBasis k))
      (ActualCanonical.standardTriple X) b)
    (CanonicalBodyTemplate.alphabetOffset (Y (ActualHomogeneous.hBasis k))
      (ActualCanonical.standardTriple Y) b))

def edgeWords {k s d : Nat} (base capacity : Nat) (fields : Fin k → Fin 4 → Nat)
    (X Y : ActualGame.Map k s d) (b : Fin k → F2) : List Nat :=
  [queryRank base fields X b, capacity + queryRank base fields Y b] ++
    permutationWords X Y b

def edgeBits {k s d : Nat} (base capacity : Nat) (fields : Fin k → Fin 4 → Nat)
    (X Y : ActualGame.Map k s d) (b : Fin k → F2) : List Bool :=
  encodeWords (edgeWords base capacity fields X Y b)

@[simp] theorem queryWords_length {k s d : Nat} (fields : Fin k → Fin 4 → Nat)
    (X : ActualGame.Map k s d) (b : Fin k → F2) :
    (queryWords fields X b).length = 1 + 9*k := by
  simp only [queryWords, CanonicalBodyTemplate.evalTemplate, List.length_map,
    CanonicalBodyTemplate.template_length]

@[simp] theorem permutationWords_length {k s d : Nat} (X Y : ActualGame.Map k s d)
    (b : Fin k → F2) : (permutationWords X Y b).length = 2^s :=
  tableWords_length _

theorem queryWords_actual (S : Source) {k s d : Nat}
    (q : ActualGame.Query S k s d) :
    queryWords (CanonicalBodyTemplate.sourceFields q.1 (ActualGame.names S)) q.2
        (fun j => ActualGame.rhs S (q.1 j)) =
      CanonicalEncoding.bodyWords (ActualOrbit.body (ActualGame.canonical S k s d) q) :=
  CanonicalBodyTemplate.eval_actualQuery S q

theorem queryAddress_false (S : Source) {k s d : Nat}
    (q : ActualGame.Query S k s d) :
    (AddressGame.queryAddress S k s d false q).val =
      queryRank (CanonicalAddress.base S.«variables» S.occurrences s d)
        (CanonicalBodyTemplate.sourceFields q.1 (ActualGame.names S)) q.2
        (fun j => ActualGame.rhs S (q.1 j)) := by
  rw [queryRank, queryWords_actual]
  exact Encoding.sideEquiv_false_val _

theorem queryAddress_true (S : Source) {k s d : Nat}
    (q : ActualGame.Query S k s d) :
    (AddressGame.queryAddress S k s d true q).val =
      AddressGame.bodyCapacity S k s d +
      queryRank (CanonicalAddress.base S.«variables» S.occurrences s d)
        (CanonicalBodyTemplate.sourceFields q.1 (ActualGame.names S)) q.2
        (fun j => ActualGame.rhs S (q.1 j)) := by
  rw [queryRank, queryWords_actual]
  exact Encoding.sideEquiv_true_val _

/-- Exact full serialized edge, including every forward permutation image. -/
theorem addressEdge_words (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (omega : ActualGame.Outcome S k g) :
    constraintWords (AddressGame.addressEdge S k g omega) =
      edgeWords (CanonicalAddress.base S.«variables» S.occurrences s d)
        (AddressGame.bodyCapacity S k s d)
        (CanonicalBodyTemplate.sourceFields omega.1.1 (ActualGame.names S))
        omega.1.2 (ActualGame.rightQuery S k g omega).2
        (fun j => ActualGame.rhs S (omega.1.1 j)) := by
  unfold constraintWords
  change [ (AddressGame.queryAddress S k s d false (ActualGame.leftQuery S k g omega)).val,
      (AddressGame.queryAddress S k s d true (ActualGame.rightQuery S k g omega)).val ] ++ _ = _
  rw [queryAddress_false, queryAddress_true]
  rfl

theorem addressEdge_bits (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (omega : ActualGame.Outcome S k g) :
    encodeWords (constraintWords (AddressGame.addressEdge S k g omega)) =
      edgeBits (CanonicalAddress.base S.«variables» S.occurrences s d)
        (AddressGame.bodyCapacity S k s d)
        (CanonicalBodyTemplate.sourceFields omega.1.1 (ActualGame.names S))
        omega.1.2 (ActualGame.rightQuery S k g omega).2
        (fun j => ActualGame.rhs S (omega.1.1 j)) := by
  rw [addressEdge_words]
  rfl

/-- The actual edge emitter's three payloads concatenate to the codec exactly. -/
theorem edgeBits_split {k s d : Nat} (base capacity : Nat)
    (fields : Fin k → Fin 4 → Nat) (X Y : ActualGame.Map k s d) (b : Fin k → F2) :
    edgeBits base capacity fields X Y b =
      encodeWords [queryRank base fields X b, capacity + queryRank base fields Y b] ++
        encodeWords (permutationWords X Y b) := by
  exact encodeWords_append _ _

end MaxCutGames.Reduction.AddressByteSemantics

namespace MaxCutGames.Reduction.AddressOutcomeSpecs

open Integration.BinaryLinear Foundations.Complexity ActualSource
open Integration

variable {s d : Nat} {α : Type*}

abbrev Spec (k : Nat) := MachineOutcomeRows.Spec (4*k) (1+9*k)
abbrev Values (k : Nat) := MachineOutcomeRows.Values (1+9*k)
abbrev Parameter (k : Nat) (T : NoiseTables.Table s d) :=
  FixedOutcomes.Parameter k s d (Fin T.vectors.length)

def bitVector {k : Nat} (b : Fin k → Bool) : Fin k → F2 := fun j => ofBit (b j)

def rightMap {k s d : Nat} (T : NoiseTables.Table s d) (p : Parameter k T) :
    ActualGame.Map k s d := p.1 + p.2.2.smulRight (T.vectors.get p.2.1)

def row {k s d : Nat} (T : NoiseTables.Table s d) (p : Parameter k T)
    (b : Fin k → Bool) : Spec k where
  left := CanonicalBodyMachine.template (p.1 (ActualHomogeneous.hBasis k))
    (ActualCanonical.standardTriple p.1) (bitVector b)
  right := CanonicalBodyMachine.template (rightMap T p (ActualHomogeneous.hBasis k))
    (ActualCanonical.standardTriple (rightMap T p)) (bitVector b)
  permutation := encodeWords (AddressByteSemantics.permutationWords p.1 (rightMap T p) (bitVector b))

def rows (k : Nat) {s d : Nat} (T : NoiseTables.Table s d) :
    List (MachineOutcomeRows.Row k (4*k) (1+9*k)) :=
  (FixedOutcomes.tableParams k T).map (row T)

@[simp] theorem row_left_length {k s d : Nat} (T : NoiseTables.Table s d)
    (p : Parameter k T) (b : Fin k → Bool) : (row T p b).left.length = 1+9*k :=
  CanonicalBodyMachine.template_length _ _ _

@[simp] theorem row_right_length {k s d : Nat} (T : NoiseTables.Table s d)
    (p : Parameter k T) (b : Fin k → Bool) : (row T p b).right.length = 1+9*k :=
  CanonicalBodyMachine.template_length _ _ _

@[simp] theorem rows_length (k : Nat) {s d : Nat} (T : NoiseTables.Table s d) :
    (rows k T).length = 2^((2*k+1)*(s+d+1)) * T.vectors.length := by
  simp only [rows, List.length_map, FixedOutcomes.tableParams_length]

@[simp] theorem selected_rows (k : Nat) {s d : Nat} (T : NoiseTables.Table s d)
    (b : Fin k → Bool) :
    MachineOutcomeRows.selected (rows k T) b =
      (FixedOutcomes.tableParams k T).map (fun p => row T p b) := by
  simp only [MachineOutcomeRows.selected, rows, List.map_map, Function.comp_def]

/-- A total proof-side interpretation. The fallback handles arbitrary malformed
static specs; actual canonical rows always take the checked decoding branch. -/
def interpreted (width : Nat) (bits : List Bool) : {words : List Nat // words.length = width} :=
  let words := (decodeWords bits).getD []
  if h : words.length = width then ⟨words, h⟩
  else ⟨List.replicate width 0, List.length_replicate⟩

theorem interpreted_encoded (width : Nat) (words : List Nat) (hlen : words.length = width) :
    (interpreted width (encodeWords words)).val = words := by
  simp [interpreted, hlen]

def values {k : Nat} (fields : Fin k → Fin 4 → Nat) (spec : Spec k) : Values k where
  left := (interpreted (1+9*k)
    (MachineFieldTemplate.templateOutput spec.left (CanonicalBodyMachine.savedFields fields))).val
  right := (interpreted (1+9*k)
    (MachineFieldTemplate.templateOutput spec.right (CanonicalBodyMachine.savedFields fields))).val
  left_length := (interpreted (1+9*k)
    (MachineFieldTemplate.templateOutput spec.left (CanonicalBodyMachine.savedFields fields))).property
  right_length := (interpreted (1+9*k)
    (MachineFieldTemplate.templateOutput spec.right (CanonicalBodyMachine.savedFields fields))).property

theorem row_left_output {k s d : Nat} (T : NoiseTables.Table s d) (p : Parameter k T)
    (b : Fin k → Bool) (fields : Fin k → Fin 4 → Nat) :
    MachineFieldTemplate.templateOutput (row T p b).left (CanonicalBodyMachine.savedFields fields) =
      encodeWords (AddressByteSemantics.queryWords fields p.1 (bitVector b)) :=
  CanonicalBodyMachine.templateOutput_lowerTemplate fields _

theorem row_right_output {k s d : Nat} (T : NoiseTables.Table s d) (p : Parameter k T)
    (b : Fin k → Bool) (fields : Fin k → Fin 4 → Nat) :
    MachineFieldTemplate.templateOutput (row T p b).right (CanonicalBodyMachine.savedFields fields) =
      encodeWords (AddressByteSemantics.queryWords fields (rightMap T p) (bitVector b)) :=
  CanonicalBodyMachine.templateOutput_lowerTemplate fields _

@[simp] theorem values_row_left {k s d : Nat} (T : NoiseTables.Table s d)
    (p : Parameter k T) (b : Fin k → Bool) (fields : Fin k → Fin 4 → Nat) :
    (values fields (row T p b)).left =
      AddressByteSemantics.queryWords fields p.1 (bitVector b) := by
  change (interpreted (1+9*k) _).val = _
  rw [row_left_output]
  exact interpreted_encoded _ _ (AddressByteSemantics.queryWords_length _ _ _)

@[simp] theorem values_row_right {k s d : Nat} (T : NoiseTables.Table s d)
    (p : Parameter k T) (b : Fin k → Bool) (fields : Fin k → Fin 4 → Nat) :
    (values fields (row T p b)).right =
      AddressByteSemantics.queryWords fields (rightMap T p) (bitVector b) := by
  change (interpreted (1+9*k) _).val = _
  rw [row_right_output]
  exact interpreted_encoded _ _ (AddressByteSemantics.queryWords_length _ _ _)

theorem correct_rows {k s d : Nat} (T : NoiseTables.Table s d)
    (fields : Fin k → Fin 4 → Nat) (b : Fin k → Bool) :
    ∀ spec ∈ MachineOutcomeRows.selected (rows k T) b,
      MachineFieldTemplate.templateOutput spec.left (CanonicalBodyMachine.savedFields fields) =
        encodeWords (values fields spec).left ∧
      MachineFieldTemplate.templateOutput spec.right (CanonicalBodyMachine.savedFields fields) =
        encodeWords (values fields spec).right := by
  rw [selected_rows]
  intro spec hspec
  obtain ⟨p, _, rfl⟩ := List.mem_map.mp hspec
  constructor
  · rw [values_row_left]
    exact row_left_output T p b fields
  · rw [values_row_right]
    exact row_right_output T p b fields

theorem payload_row {k s d : Nat} (T : NoiseTables.Table s d)
    (p : Parameter k T) (b : Fin k → Bool) (fields : Fin k → Fin 4 → Nat)
    (B C : Nat) :
    MachineOutcomeRows.payload B C (values fields) (row T p b) =
      AddressByteSemantics.edgeBits B C fields p.1 (rightMap T p) (bitVector b) := by
  unfold MachineOutcomeRows.payload MachineAddressEdge.edgeBits
  rw [values_row_left, values_row_right, AddressByteSemantics.edgeBits_split]
  rfl

def rhsBits (S : Source) {k : Nat} (U : ActualGame.Question S k) : Fin k → Bool :=
  fun j => (S.equation (U j)).rhs

@[simp] theorem bitVector_rhsBits (S : Source) {k : Nat} (U : ActualGame.Question S k) :
    bitVector (rhsBits S U) = fun j => ActualGame.rhs S (U j) := rfl

theorem rightMap_actual (S : Source) {k s d : Nat} (T : NoiseTables.Table s d)
    (U : ActualGame.Question S k) (p : Parameter k T) :
    rightMap T p =
      (ActualGame.rightQuery S k (TableReduction.tableSkeleton T) (FixedOutcomes.assemble U p)).2 := rfl

/-- Each static row emits exactly the concrete game's constraint encoding. -/
theorem payload_row_actual (S : Source) (k : Nat) {s d : Nat}
    (T : NoiseTables.Table s d) (U : ActualGame.Question S k) (p : Parameter k T) :
    MachineOutcomeRows.payload
        (CanonicalAddress.base S.«variables» S.occurrences s d) (AddressGame.bodyCapacity S k s d)
        (values (CanonicalBodyTemplate.sourceFields U (ActualGame.names S)))
        (row T p (rhsBits S U)) =
      encodeWords (constraintWords (AddressGame.addressEdge S k (TableReduction.tableSkeleton T)
        (FixedOutcomes.assemble U p))) := by
  rw [payload_row, bitVector_rhsBits, rightMap_actual S T U p]
  exact (AddressByteSemantics.addressEdge_bits S k (TableReduction.tableSkeleton T)
    (FixedOutcomes.assemble U p)).symm

def tupleBits (S : Source) (k : Nat) {s d : Nat} (T : NoiseTables.Table s d)
    (U : ActualGame.Question S k) : List Bool :=
  (FixedOutcomes.tableParams k T).flatMap (fun p =>
    encodeWords (constraintWords (AddressGame.addressEdge S k (TableReduction.tableSkeleton T)
      (FixedOutcomes.assemble U p))))

theorem tuplePayload_actual (S : Source) (k : Nat) {s d : Nat}
    (T : NoiseTables.Table s d) (U : ActualGame.Question S k) :
    (MachineOutcomeRows.selected (rows k T) (rhsBits S U)).flatMap
        (MachineOutcomeRows.payload
          (CanonicalAddress.base S.«variables» S.occurrences s d) (AddressGame.bodyCapacity S k s d)
          (values (CanonicalBodyTemplate.sourceFields U (ActualGame.names S)))) =
      tupleBits S k T U := by
  rw [selected_rows, List.flatMap_map]
  simp only [payload_row_actual, tupleBits]
  rfl

private theorem encodeWords_flatMap_inline_AddressOutcomeSpecs (items : List α) (words : α → List Nat) :
    encodeWords (items.flatMap words) = items.flatMap (fun x => encodeWords (words x)) := by
  induction items with
  | nil => rfl
  | cons x items ih => simp only [List.flatMap_cons, encodeWords_append, ih]

/-- Complete output-byte factorization used by the actual outer loop. -/
theorem gameBits_factor (S : Source) (k : Nat) {s d : Nat} (T : NoiseTables.Table s d) :
    gameBits (AddressGame.tableOutput S k T) =
      encodeWords [(AddressGame.tableOutput S k T).vertices, 2^s,
        (AddressGame.tableOutput S k T).constraints.length] ++
      (FixedOutcomes.occurrenceTuples S.occurrences k).flatMap (tupleBits S k T) := by
  rw [gameBits, gameWords, encodeWords_append, encodeWords_flatMap_inline_AddressOutcomeSpecs]
  rw [FixedOutcomes.tableOutput_constraints]
  erw [List.flatMap_assoc]
  congr 1
  apply List.flatMap_congr
  intro U _
  erw [List.flatMap_map]
  rfl

end MaxCutGames.Reduction.AddressOutcomeSpecs

namespace MaxCutGames.Integration.AddressTupleBody

open Turing Reduction Foundations.Complexity
open AddressMachineSpace

variable {k s d noiseCount : Nat} {Λ : Type}

abbrev Arena (k s d noiseCount : Nat) := AddressMachineSpace.Tape k s d noiseCount
abbrev Rows (k : Nat) := List (MachineOutcomeRows.Row k (4*k) (1+9*k))
abbrev Slots (k s d noiseCount : Nat) := AddressMachineSpace.addressEdgeSlots k s d noiseCount
abbrev BodyState (k : Nat) := AddressMachineSpace.State k
abbrev CleanupLabel (k s d noiseCount : Nat) :=
  AddressTupleCleanup.Label k s d noiseCount
abbrev RowLabel (k s d noiseCount : Nat) (rows : Rows k) :=
  MachineOutcomeRows.Label (Slots k s d noiseCount) rows
abbrev Label (k s d noiseCount : Nat) (rows : Rows k) :=
  MachineSourceTuple.Label k ⊕ (Unit ⊕
    (RowLabel k s d noiseCount rows ⊕ (CleanupLabel k s d noiseCount ⊕ Unit)))

instance labelFintype (rows : Rows k) : Fintype (Label k s d noiseCount rows) := by
  unfold Label CleanupLabel RowLabel
  infer_instance

variable {rows : Rows k}

def sourceLabels (labels : Label k s d noiseCount rows → Λ) : MachineSourceTuple.Label k → Λ :=
  fun l => labels (.inl l)
def rhsLabel (labels : Label k s d noiseCount rows → Λ) : Λ := labels (.inr (.inl ()))
def rowLabels (labels : Label k s d noiseCount rows → Λ) : RowLabel k s d noiseCount rows → Λ :=
  fun l => labels (.inr (.inr (.inl l)))
def cleanupLabels (labels : Label k s d noiseCount rows → Λ) : CleanupLabel k s d noiseCount → Λ :=
  fun l => labels (.inr (.inr (.inr (.inl l))))
def resetLabel (labels : Label k s d noiseCount rows → Λ) : Λ :=
  labels (.inr (.inr (.inr (.inr ()))))

def main (rows : Rows k) : Label k s d noiseCount rows := .inl (MachineSourceTuple.labelAt 0)

def reset (exit : Option Λ) : TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) Λ (BodyState k) :=
  .load (fun _ => initialState k) (MachineFieldTemplate.jump exit)

def instruction (rows : Rows k) (labels : Label k s d noiseCount rows → Λ) (exit : Option Λ) :
    Label k s d noiseCount rows →
      TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) Λ (BodyState k)
  | .inl l => MachineSubroutine.statement (sourceLabels labels) (some (rhsLabel labels))
      (MachineSourceTuple.program l)
  | .inr (.inl _) => MachineRhsLoad.statement (rhsField k s d noiseCount)
      (some (rowLabels labels (.inl ())))
  | .inr (.inr (.inl l)) => MachineOutcomeRows.instruction (Slots k s d noiseCount) rows
      (rowLabels labels)
      (AddressTupleCleanup.entry k s d noiseCount (cleanupLabels labels) (some (resetLabel labels))) l
  | .inr (.inr (.inr (.inl l))) => AddressTupleCleanup.instruction k s d noiseCount
      (cleanupLabels labels) (some (resetLabel labels)) l
  | .inr (.inr (.inr (.inr _))) => reset exit

def program (rows : Rows k) : Label k s d noiseCount rows →
    TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) (Label k s d noiseCount rows) (BodyState k) :=
  instruction rows id none

def rhs (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length) : Fin k → Bool :=
  fun j => F.equations[(tuple j).val].rhs

def loaded (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) : Arena k s d noiseCount → List Bool :=
  MachineSourceTuple.stageTapes F tuple base k

def tupleBits (rows : Rows k) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length) : List Bool :=
  (MachineOutcomeRows.selected rows (rhs F tuple)).flatMap (MachineOutcomeRows.payload B C values)

noncomputable def emitted (rows : Rows k) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) : Arena k s d noiseCount → List Bool :=
  MachineAddressEdge.appended (Slots k s d noiseCount) (loaded F tuple base)
    (tupleBits rows B C values F tuple)

noncomputable def finalTapes (rows : Rows k) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) : Arena k s d noiseCount → List Bool :=
  AddressTupleCleanup.cleared k s d noiseCount (emitted rows B C values F tuple base)

structure Ready (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) : Prop where
  source : base .source = SourceEncoding.inputBits F
  saved : ∀j, base (.savedIndex j) = encodeWord (tuple j).val
  index : base .index = []
  work : base .work = []
  scratch : base .scratch = []
  copy : base .copyScratch = []
  fields : ∀j slot, base (.field j slot) = []

noncomputable def budget (rows : Rows k) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) : Nat :=
  (k * (10 * (SourceEncoding.inputBits F).length + 20) + 1) + 1 +
    ((MachineOutcomeRows.rowCosts (Slots k s d noiseCount) B C values
      (MachineOutcomeRows.selected rows (rhs F tuple)) (loaded F tuple base)).sum + 1) +
    AddressTupleCleanup.budget k s d noiseCount (emitted rows B C values F tuple base) + 1

noncomputable def run (rows : Rows k) (labels : Label k s d noiseCount rows → Λ)
    (exit : Option Λ)
    (P : Λ → TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) Λ (BodyState k))
    (atLabels : ∀l, P (labels l) = instruction rows labels exit l)
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) (ready : Ready F tuple base)
    (oldRhs : Fin k → Bool) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (valid : MachineOutcomeRows.Invariant (Slots k s d noiseCount) B C values
      (MachineOutcomeRows.selected rows (rhs F tuple)) (loaded F tuple base)) :
    StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (labels (main rows)), (((oldRhs, ()), ()), none), base⟩
      (some ⟨exit, initialState k, finalTapes rows B C values F tuple base⟩)
      (budget rows B C values F tuple base) := by
  have sourceRun := MachineSourceTuple.placedLoadUnaryInTime
    (sourceLabels labels) (some (rhsLabel labels)) P (fun l => atLabels (.inl l))
    F tuple base ready.source ready.saved ready.index ready.work ready.scratch ready.copy (oldRhs, ())
  have rhsRun := MachineRhsLoad.loadInTime (rhsField k s d noiseCount)
    (rhsLabel labels) (some (rowLabels labels (.inl ()))) P (atLabels (.inr (.inl ())))
    (((oldRhs, ()), ()), none) (loaded F tuple base) (rhs F tuple) (fun _ => [])
    (by
      intro j
      change loaded F tuple base (MachineSourceTuple.rhsField j) = _
      rw [loaded, MachineSourceTuple.output_rhs]
      change encodeWord (if F.equations[(tuple j).val].rhs then 1 else 0) ++
        base (.field j 3) = _
      rw [ready.fields j 3]
      rfl)
  have rowsRun := MachineOutcomeRows.phaseInTime (Slots k s d noiseCount) rows
    (rowLabels labels)
    (AddressTupleCleanup.entry k s d noiseCount (cleanupLabels labels) (some (resetLabel labels)))
    P (fun l => atLabels (.inr (.inr (.inl l))))
    (loaded F tuple base) (rhs F tuple) () none B C values valid
  have cleanupRun := AddressTupleCleanup.execution k s d noiseCount
    (cleanupLabels labels) (some (resetLabel labels)) P
    (fun l => atLabels (.inr (.inr (.inr (.inl l)))))
    (emitted rows B C values F tuple base) (((rhs F tuple), ()), ()) none
  simp only [MachineDrainMany.finalRegister_none] at cleanupRun
  have resetRun : StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (resetLabel labels), ((((rhs F tuple), ()), ()), none),
        finalTapes rows B C values F tuple base⟩
      (some ⟨exit, initialState k, finalTapes rows B C values F tuple base⟩) 1 := by
    refine { steps := 1, evals_in_steps := ?_, steps_le_m := Nat.le_refl _ }
    change TM2.step P ⟨some (resetLabel labels), _, _⟩ = _
    change some (TM2.stepAux (P (resetLabel labels)) _ _) = _
    rw [show P (resetLabel labels) = reset exit from atLabels (.inr (.inr (.inr (.inr ()))))]
    cases exit <;> rfl
  have first := StateTransition.EvalsToInTime.trans (TM2.step P) _ _ _ _ _ sourceRun rhsRun
  have second := StateTransition.EvalsToInTime.trans (TM2.step P) _ _ _ _ _ first rowsRun
  have third := StateTransition.EvalsToInTime.trans (TM2.step P) _ _ _ _ _ second cleanupRun
  have last := StateTransition.EvalsToInTime.trans (TM2.step P) _ _ _ _ _ third resetRun
  refine { steps := last.steps, evals_in_steps := last.evals_in_steps, steps_le_m := ?_ }
  exact last.steps_le_m.trans (by dsimp only [budget]; omega)

def source (F : SourceEncoding.Input) : ActualSource.Source :=
  ActualSource.Source.ofList F.equations F.nonempty

theorem rhs_eq_actual (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length) :
    rhs F tuple = AddressOutcomeSpecs.rhsBits (source F) tuple := rfl

theorem canonicalValues_eq_actual (F : SourceEncoding.Input)
    (tuple : Fin k → Fin F.equations.length) :
    AddressTupleLoaded.canonicalValues F tuple =
      CanonicalBodyTemplate.sourceFields tuple (ActualGame.names (source F)) := by
  funext j
  apply congrArg (fun names => CanonicalBodyTemplate.recordFields names (tuple j).val)
  funext slot
  fin_cases slot <;> rfl

theorem tupleBits_actual (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (T : NoiseTables.Table s d) :
    tupleBits (AddressOutcomeSpecs.rows k T)
      (CanonicalAddress.base F.«variables» F.equations.length s d)
      (AddressGame.bodyCapacity (source F) k s d)
      (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple)) F tuple =
      AddressOutcomeSpecs.tupleBits (source F) k T tuple := by
  unfold tupleBits
  rw [rhs_eq_actual, canonicalValues_eq_actual]
  exact AddressOutcomeSpecs.tuplePayload_actual (source F) k T tuple

/-- All generic template-correctness obligations are discharged from the
actual loader's canonical field values and the checked fixed-row specs. -/
theorem actualInvariant (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (T : NoiseTables.Table s d) (base : Arena k s d noiseCount → List Bool)
    (ready : Ready F tuple base)
    (clean : MachineAddressEdge.Clean (Slots k s d noiseCount) base) (B C : Nat)
    (hB : base (headerTape k s d noiseCount .baseValue) = encodeWord B)
    (hC : base (headerTape k s d noiseCount .capacity) = encodeWord C) :
    MachineOutcomeRows.Invariant (Slots k s d noiseCount) B C
      (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple))
      (MachineOutcomeRows.selected (AddressOutcomeSpecs.rows k T) (rhs F tuple))
      (loaded F tuple base) := by
  constructor
  · exact AddressTupleLoaded.loaded_clean F tuple base clean
  · exact AddressTupleLoaded.loaded_radix F tuple base B hB
  · exact AddressTupleLoaded.loaded_capacity F tuple base C hC
  · intro spec member
    have fields := AddressTupleLoaded.loaded_savedFields F tuple base ready.saved ready.fields
    change (fun i => loaded F tuple base (MachineAddressEdge.addressSlots
      (Slots k s d noiseCount) (.field i))) = _ at fields
    rw [fields]
    exact AddressOutcomeSpecs.correct_rows T (AddressTupleLoaded.canonicalValues F tuple)
      (rhs F tuple) spec member

/-- Actual fixed-row execution. Physical readiness and stored header values
are its only data hypotheses; no template or body-execution premise remains. -/
noncomputable def run_fixed_rows (T : NoiseTables.Table s d)
    (labels : Label k s d noiseCount (AddressOutcomeSpecs.rows k T) → Λ) (exit : Option Λ)
    (P : Λ → TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) Λ (BodyState k))
    (atLabels : ∀l, P (labels l) = instruction (AddressOutcomeSpecs.rows k T) labels exit l)
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) (ready : Ready F tuple base)
    (clean : MachineAddressEdge.Clean (Slots k s d noiseCount) base)
    (oldRhs : Fin k → Bool) (B C : Nat)
    (hB : base (headerTape k s d noiseCount .baseValue) = encodeWord B)
    (hC : base (headerTape k s d noiseCount .capacity) = encodeWord C) :
    StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (labels (main (AddressOutcomeSpecs.rows k T))), (((oldRhs, ()), ()), none), base⟩
      (some ⟨exit, initialState k,
        finalTapes (AddressOutcomeSpecs.rows k T) B C
          (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple)) F tuple base⟩)
      (budget (AddressOutcomeSpecs.rows k T) B C
        (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple)) F tuple base) :=
  run (AddressOutcomeSpecs.rows k T) labels exit P atLabels F tuple base ready oldRhs B C
    (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple))
    (actualInvariant F tuple T base ready clean B C hB hC)

theorem finalTapes_eq (rows : Rows k) (B C : Nat)
    (values : MachineOutcomeRows.Spec (4*k) (1+9*k) → MachineOutcomeRows.Values (1+9*k))
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) (ready : Ready F tuple base) :
    finalTapes rows B C values F tuple base = MachineAddressEdge.appended
      (Slots k s d noiseCount) base (tupleBits rows B C values F tuple) :=
  AddressTupleFrame.cleared_appended_loaded F tuple base ready.index ready.work ready.fields _

noncomputable def run_actual (T : NoiseTables.Table s d)
    (labels : Label k s d noiseCount (AddressOutcomeSpecs.rows k T) → Λ) (exit : Option Λ)
    (P : Λ → TM2.Stmt (fun _ : Arena k s d noiseCount => Bool) Λ (BodyState k))
    (atLabels : ∀l, P (labels l) = instruction (AddressOutcomeSpecs.rows k T) labels exit l)
    (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) (ready : Ready F tuple base)
    (clean : MachineAddressEdge.Clean (Slots k s d noiseCount) base)
    (oldRhs : Fin k → Bool)
    (hB : base (headerTape k s d noiseCount .baseValue) =
      encodeWord (CanonicalAddress.base F.«variables» F.equations.length s d))
    (hC : base (headerTape k s d noiseCount .capacity) =
      encodeWord (AddressGame.bodyCapacity (source F) k s d)) :
    StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (labels (main (AddressOutcomeSpecs.rows k T))), (((oldRhs, ()), ()), none), base⟩
      (some ⟨exit, initialState k, MachineAddressEdge.appended (Slots k s d noiseCount)
        base (AddressOutcomeSpecs.tupleBits (source F) k T tuple)⟩)
      (budget (AddressOutcomeSpecs.rows k T)
        (CanonicalAddress.base F.«variables» F.equations.length s d)
        (AddressGame.bodyCapacity (source F) k s d)
        (AddressOutcomeSpecs.values (AddressTupleLoaded.canonicalValues F tuple)) F tuple base) := by
  have actual := run_fixed_rows T labels exit P atLabels F tuple base ready clean oldRhs
    (CanonicalAddress.base F.«variables» F.equations.length s d)
    (AddressGame.bodyCapacity (source F) k s d) hB hC
  rw [finalTapes_eq _ _ _ _ F tuple base ready, tupleBits_actual] at actual
  exact actual

theorem saved_of_current (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (base : Arena k s d noiseCount → List Bool)
    (currentWords : ∀j, base (current k s d noiseCount j) = encodeWord (digits j).val) :
    ∀j, base (.savedIndex j) = encodeWord (digits j.rev).val := by
  intro j
  simpa only [AddressMachineSpace.current_rev] using currentWords j.rev

end MaxCutGames.Integration.AddressTupleBody

end OAI