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T13-8: Formal Specification for φ-Field Quantization

Type Definitions​

(* Core Types *)
Inductive Binary : Type :=
  | b0 : Binary
  | b1 : Binary.

Inductive ZeckSeq : Type :=
  | empty : ZeckSeq
  | cons : Binary -> ZeckSeq -> ZeckSeq.

(* Validity Predicate *)
Fixpoint valid_zeck (s : ZeckSeq) : Prop :=
  match s with
  | empty => True
  | cons b0 rest => valid_zeck rest
  | cons b1 (cons b1 _) => False  (* No consecutive 1s *)
  | cons b1 rest => valid_zeck rest
  end.

(* Field Type *)
Record Field : Type := mkField {
  amplitude : R -> C;
  normalized : integral (norm_squared amplitude) = 1
}.

(* Quantization Map Type *)
Record QuantMap : Type := mkQuant {
  Q : ZeckSeq -> Field;
  preserves_validity : forall z, valid_zeck z -> exists f, Q z = f
}.

Axioms​

-- Unique Axiom: Self-referential complete systems increase entropy
axiom entropy_increase :
  ∀ (S : System) (t : Time),
    SelfReferentialComplete S →
    Entropy S (t + 1) > Entropy S t

-- Derived: Zeckendorf space expansion
theorem zeck_expansion :
  ∀ (n : ℕ),
    ValidSequences (n + 1) = ValidSequences n + ValidSequences (n - 1)
  := by
    intro n
    -- Proof follows from no-11 constraint
    sorry

-- Golden ratio emergence
def φ : ℝ := (1 + Real.sqrt 5) / 2

theorem phi_limit :
  (fun n => ValidSequences (n + 1) / ValidSequences n) ⟶ φ
  := by
    -- Follows from Fibonacci recurrence
    sorry

Core Theorems​

(* Quantization preserves φ-structure *)
Theorem quant_phi_structure :
  forall (z1 z2 : ZeckSeq) (Q : QuantMap),
    valid_zeck z1 -> valid_zeck z2 ->
    field_add (Q (zeck_add z1 z2)) = 
      field_scale phi (Q z1) + field_scale (1/phi) (Q z2).
Proof.
  intros z1 z2 Q H1 H2.
  (* By entropy axiom, field combination must maximize information *)
  apply entropy_maximization.
  (* φ-scaling emerges from Fibonacci basis *)
  apply fibonacci_scaling.
  (* No-11 constraint forces this unique decomposition *)
  apply no_consecutive_ones_constraint.
Qed.

(* Field operators satisfy φ-commutation *)
Theorem field_commutation :
  forall (a a_dag : Operator),
    is_annihilation a -> is_creation a_dag ->
    commutator a a_dag = scale_identity phi.
Proof.
  intros a a_dag Ha Hdag.
  (* From Zeckendorf structure of number operators *)
  unfold commutator.
  rewrite zeck_number_operator.
  (* φ emerges from Fibonacci spacing *)
  apply golden_ratio_spacing.
Qed.

Machine-Verifiable Properties​

-- Property 1: No consecutive ones
def no_consecutive_ones (s : List Bool) : Bool :=
  match s with
  | [] => true
  | [_] => true
  | true :: true :: _ => false
  | _ :: rest => no_consecutive_ones rest

-- Property 2: φ-scaling verification
def verify_phi_scaling (op : Operator) : Bool :=
  ∀ n, eigenvalue op n = φ ^ (zeck_weight n)

-- Property 3: Entropy monotonicity
def entropy_monotonic (evolution : Time → State) : Prop :=
  ∀ t, entropy (evolution (t + dt)) > entropy (evolution t)

-- Property 4: Commutator closure
def commutator_closed (ops : Set Operator) : Prop :=
  ∀ a b ∈ ops, ∃ c ∈ ops, ∃ k : ℝ,
    commutator a b = k • c ∧ k = φ ^ (some_integer)

-- Property 5: Fixed point existence
def has_fixed_point (Q : QuantMap) : Prop :=
  ∃ ψ : Field, ψ = Q (zeck_encode ψ)

Consistency Proofs​

(* System consistency *)
Theorem system_consistent :
  ~ (exists (P : Prop), provable P /\ provable (~ P)).
Proof.
  intro H.
  destruct H as [P [Hp Hnp]].
  (* Cannot have both valid and invalid Zeckendorf sequence *)
  apply (no_11_decidable P).
  (* Entropy only increases, never decreases *)
  apply entropy_irreversibility.
Qed.

(* Completeness for quantization *)
Theorem quantization_complete :
  forall (n : nat),
    exists (z : ZeckSeq), 
      valid_zeck z /\ zeck_value z = n.
Proof.
  intro n.
  (* Every natural has unique Zeckendorf representation *)
  apply zeckendorf_theorem.
  (* Representation avoids consecutive ones *)
  apply fibonacci_greedy_algorithm.
Qed.

Formal Derivation Chain​

-- Complete derivation from entropy axiom to field quantization
theorem complete_derivation :
  EntropyAxiom →
  ZeckendorfConstraint →
  FibonacciRecurrence →
  GoldenRatioEmergence →
  FieldQuantization →
  PhiCommutationRelations
  := by
    intro entropy
    -- Step 1: Entropy requires avoiding repeated patterns
    have no_11 := entropy_avoids_repetition entropy
    -- Step 2: No-11 generates Fibonacci counting
    have fib := no_consecutive_ones_fibonacci no_11
    -- Step 3: Fibonacci ratio converges to φ
    have phi := fibonacci_ratio_limit fib
    -- Step 4: Quantization inherits φ-structure
    have quant := structure_preservation phi
    -- Step 5: Operators satisfy φ-commutation
    exact field_operator_commutation quant

Verification Specification​

Record VerificationSuite : Type := {
  (* Input validation *)
  check_zeckendorf : ZeckSeq -> bool;
  
  (* Quantization verification *)
  verify_quantization : QuantMap -> bool;
  
  (* Field operator checks *)
  verify_commutation : Operator -> Operator -> bool;
  
  (* Entropy validation *)
  check_entropy_increase : Evolution -> bool;
  
  (* Complete system check *)
  verify_complete : System -> bool
}.

Definition complete_verification (suite : VerificationSuite) : bool :=
  check_zeckendorf suite empty &&
  verify_quantization suite standard_quant &&
  verify_commutation suite a_op a_dag_op &&
  check_entropy_increase suite field_evolution &&
  verify_complete suite phi_field_system.

This formal specification provides machine-verifiable proofs that φ-field quantization emerges necessarily from the entropy axiom through Zeckendorf encoding constraints.