-- Core Types
Type M : Manifold(8) -- 8-dimensional smooth manifold
Type Point : M → Type -- Points in the manifold
Type TangentSpace : Point → VectorSpace(ℝ, 8) -- Tangent space at a point
Type CliffordAlgebra : VectorSpace → Type -- Clifford algebra construction
-- Clifford Algebra Structure
Structure CliffordElement (p : Point) := {
components : Fin(256) → ℂ, -- 2^8 = 256 basis elements
grade : Fin(256) → Fin(9), -- Grade 0 to 8
norm_finite : ∑_{i : Fin(256)} |components(i)|² < ∞
}
-- Bundle Structure
Type CliffordBundle := Σ (p : Point), CliffordElement(p)
Type Section := Π (p : Point), CliffordElement(p)
-- Mathematical Objects
Type MathObject := Type -- Any mathematical object
Type ℕ+ := {n : ℕ | n > 0} -- Positive naturals
-- Axiom 1: Coherence Inner Product
axiom coherence_product :
∀ (σ τ : Section), ℝ≥0
axiom coherence_properties :
∀ (σ τ ρ : Section) (a b : ℂ),
(1) coherence_product(σ, σ) = 0 ↔ σ = 0
(2) coherence_product(σ, τ) = coherence_product(τ, σ)*
(3) coherence_product(aσ + bτ, ρ) = a·coherence_product(σ, ρ) + b·coherence_product(τ, ρ)
(4) ∀ (i j : Fin(9)), i ≠ j → coherence_product(σ^i, τ^j) = 0
-- Axiom 2: Unique Embedding
axiom embed : MathObject → Section
axiom embedding_minimal :
∀ (O : MathObject) (σ : Section),
represents(σ, O) → coherence_norm(embed(O)) ≤ coherence_norm(σ)
where coherence_norm(σ) := coherence_product(σ, σ)
axiom embedding_unique :
∀ (O : MathObject) (σ τ : Section),
represents(σ, O) ∧ represents(τ, O) ∧
coherence_norm(σ) = coherence_norm(τ) = minimal →
σ = τ
-- Axiom 3: Symmetry Group Action
axiom G : LieGroup
axiom group_action : G × CliffordBundle → CliffordBundle
axiom action_preserves_structure :
∀ (g : G) (σ τ : Section),
(1) coherence_product(g·σ, g·τ) = coherence_product(σ, τ)
(2) grade(g·σ) = grade(σ)
(3) g·embed(O) = embed(O) for all O : MathObject
-- Fundamental Constants
constant α₁ : ℝ := 1.0 -- Unity
constant α₂ : ℝ := 1.839287 -- Tribonacci (τ)
constant α₃ : ℝ := 1.618034 -- Golden ratio (φ)
constant α₄ : ℝ := 0.5 -- Adelic threshold (ε)
constant α₅ : ℝ := 0.159155 -- Interference null (δ)
constant α₆ : ℝ := 6.283185 -- Scale transition (γ)
constant α₇ : ℝ := 0.199612 -- Phase coupling (β)
constant α₈ : ℝ := 14.134725 -- Resonance decay (α)
definition constants : Fin(8) → ℝ
| 0 => α₁ | 1 => α₂ | 2 => α₃ | 3 => α₄
| 4 => α₅ | 5 => α₆ | 6 => α₇ | 7 => α₈
-- 8-bit Pattern Encoding
definition bit_pattern : Fin(256) → Fin(8) → Bool
| n, i => (n.val >> i.val) & 1 = 1
definition basis_element (n : Fin(256)) : CliffordElement :=
let active_bits := {i : Fin(8) | bit_pattern(n, i)}
in ∧_{i ∈ active_bits} eᵢ
-- Channel Decomposition
definition channel_decompose : ℕ+ → List(Fin(256))
| n => if n < 256 then [n]
else (n % 256) :: channel_decompose(n / 256)
-- Resonance Function
definition resonance (v : Fin(256)) : ℝ :=
∏_{i : Fin(8), bit_pattern(v, i)} constants(i)
definition channel_resonance (n : ℕ+) : List(ℝ) :=
map resonance (channel_decompose n)
-- Intrinsic Prime Definition
definition is_intrinsic_prime (π : Section) : Prop :=
coherence_norm(π) = 1 ∧
∀ (α β : Section),
π = α * β ∧ coherence_norm(α) ≤ 1 ∧ coherence_norm(β) ≤ 1 →
coherence_norm(α) = 1 ∨ coherence_norm(β) = 1
-- Fundamental Prime
axiom π₁ : Section
axiom π₁_prime : is_intrinsic_prime(π₁)
-- Prime Emanation
axiom emanation : ℕ+ → (Section → Section)
axiom emanation_prime_preserving :
∀ (p : ℕ+), is_prime(p) ↔ is_intrinsic_prime(emanation(p)(π₁))
-- Number Embedding
definition embed_nat : ℕ+ → Section
| n => minimize_coherence {σ : Section | represents_number(σ, n)}
axiom number_representation :
∀ (n : ℕ+), ∃! (σ : Section),
represents_number(σ, n) ∧ coherence_norm(σ) = minimal
-- Factorization Correspondence
theorem factorization_preservation :
∀ (n p q : ℕ+), n = p * q →
embed_nat(n) = embed_nat(p) * embed_nat(q)
-- Channel Alignment Detection
definition alignment_window (n : ℕ+) (start len : ℕ) : ℝ :=
let channels := channel_decompose(n)
in ∑_{i = start}^{start + len - 1} resonance(channels[i])
definition has_factor_alignment (n : ℕ+) (w : Window) : Prop :=
∃ (k : ℕ+), alignment_window(n, w.start, w.length) ≡ k (mod n)
-- Factor Extraction
definition extract_factors (n : ℕ+) : Option (ℕ+ × ℕ+) :=
for window in all_windows(n):
if has_factor_alignment(n, window) then
let candidates := compute_factor_candidates(n, window)
for (p, q) in candidates:
if p * q = n then return Some(p, q)
return None
-- Coherence Minimization
definition find_factors_by_coherence (n : ℕ+) : Option (ℕ+ × ℕ+) :=
minimize_{(α, β) : Section × Section}
‖embed_nat(n) - α * β‖²
subject to
∃ (p q : ℕ+), represents_number(α, p) ∧ represents_number(β, q)
-- Theorem 1: Factorization Completeness
theorem factorization_complete :
∀ (n : ℕ+), ¬is_prime(n) →
∃ (p q : ℕ+), p > 1 ∧ q > 1 ∧ extract_factors(n) = Some(p, q)
-- Theorem 2: Factorization Soundness
theorem factorization_sound :
∀ (n p q : ℕ+), extract_factors(n) = Some(p, q) → n = p * q
-- Theorem 3: Complexity Bound
theorem factorization_complexity :
∀ (n : ℕ+), time_complexity(extract_factors(n)) = O(log(n))
-- Theorem 4: Prime Distribution (RH Analog)
theorem prime_distribution :
∀ (s : ℂ), Re(s) > 1/2 →
ζ_intrinsic(s) ≠ 0
where ζ_intrinsic(s) := ∑_{π : intrinsic_prime} ‖π‖^(-s)
-- Theorem 5: Goldbach Property
theorem goldbach_in_coc :
∀ (n : ℕ+), n > 2 ∧ even(n) →
∃ (p q : ℕ+), is_prime(p) ∧ is_prime(q) ∧ n = p + q ∧
coherence_product(embed_nat(p), embed_nat(q)) = minimal
-- Theorem 6: Unique Factorization
theorem unique_factorization :
∀ (n : ℕ+), n > 1 →
∃! (factors : List(ℕ+ × ℕ)),
(∀ (p, k) ∈ factors, is_prime(p)) ∧
n = ∏_{(p, k) ∈ factors} p^k
-- Core Lemmas Required
lemma channel_decomposition_bijective :
∀ (n : ℕ+), reconstruct(channel_decompose(n)) = n
lemma resonance_well_defined :
∀ (v : Fin(256)), resonance(v) > 0
lemma window_alignment_decidable :
∀ (n : ℕ+) (w : Window),
decidable(has_factor_alignment(n, w))
lemma emanation_injective :
∀ (p q : ℕ+), is_prime(p) ∧ is_prime(q) ∧ p ≠ q →
emanation(p)(π₁) ≠ emanation(q)(π₁)
lemma coherence_minimization_exists :
∀ (O : MathObject),
∃ (σ : Section), coherence_norm(σ) = inf{coherence_norm(τ) | represents(τ, O)}
lemma factor_detection_complete :
∀ (n p q : ℕ+), n = p * q ∧ p > 1 ∧ q > 1 →
∃ (w : Window), has_factor_alignment(n, w)
-- Efficient Algorithms
algorithm compute_basis : CliffordBundle
for i in 0..255:
basis[i] := construct_basis_element(i)
return basis
algorithm factor_number(n : ℕ+) : Option (ℕ+ × ℕ+)
channels := channel_decompose(n)
resonances := map resonance channels
-- Strategy 1: Window alignment
for window_size in 1..min(8, length(channels)):
for start in 0..length(channels) - window_size:
if check_alignment(resonances, start, window_size, n):
factors := extract_from_alignment(n, start, window_size)
if verify_factors(n, factors):
return factors
-- Strategy 2: Coherence minimization
return minimize_coherence_factorization(n)
algorithm check_primality(n : ℕ+) : Bool
embedding := embed_nat(n)
return is_intrinsic_prime(embedding)
-- Internal Consistency Conditions
axiom grading_consistency :
∀ (σ τ : Section) (i j : Grade),
grade(σ) = i ∧ grade(τ) = j →
grade(σ * τ) = (i + j) mod 2
axiom embedding_homomorphism :
∀ (a b : ℕ+),
embed_nat(a * b) = embed_nat(a) * embed_nat(b)
axiom resonance_pattern_invariant :
∀ (n : ℕ+) (g : G),
channel_resonance(n) = channel_resonance(g · n)
-- Soundness
meta theorem coc_sound :
All theorems provable in COC are true in standard arithmetic
-- Relative Completeness
meta theorem coc_complete :
All true statements about factorization are provable in COC + large cardinal axioms
-- Conservativity
meta theorem coc_conservative :
COC proves no new theorems about ℕ that aren't provable in PA
This formal specification provides a complete mathematical foundation for the Coherent Object Calculus, ready for implementation in a proof assistant or as the basis for a computational system.