The Prime Structure exists in phase space Π with coordinates:
- Position: x ∈ ℤ₊ (positive integers)
- Momentum: p = d(π(x))/dx (rate of prime density change)
- Information: I(x) = -∑ᵢ pᵢ log(pᵢ) where pᵢ are prime factors of x
Unity Constraint: The foundational relation α₄α₅ = 1 anchors the entire structure, tying together:
- 48-byte page size (first non-trivial unity byte)
- V₄ Klein orbit (bit-flips in {0,4,5} leave H invariant)
- 96-value resonance spectrum (|Image R| = 96)
- 3/8 compression limit (96/256 = 3/8)
The prime-generating dynamics are governed by the CCM Hamiltonian based on the coherence norm ‖·‖_c:
H(x,p,I) = ½‖(ξ,η,ζ)‖²_c + V_eff
Where:
- The coherence norm ensures Klein flips are isometric under α₄α₅ = 1
- V_eff = -log|ζ(1 + ix)| (potential from Riemann zeta)
- I measures distance from optimal 3/8 compression bound
The information term I(x) quantifies deviation from CCM's universal compression limit, with minima at maximally coherent states (primes).
Lyapunov Exponent: λ(n) = lim_{k→∞} (1/k)log|dφᵏ(n)/dn|
Where φ(n) is the Euler totient function iteration.
Correlation Dimension: D₂ = lim_{r→0} log(C(r))/log(r)
Where C(r) measures prime pair correlations at scale r.
The Prime Structure's attractor in phase space has:
- Fractal Dimension: D_f = log(96)/log(256) ≈ 0.823 (from resonance reduction)
- Fundamental Period: 48 (first non-trivial unity byte)
- Resonance Classes: 96 (coarse-graining of 256 states via R)
- Basin of Attraction: V₄ Klein orbits preserving unity constraint
The attractor reduces the full 256-state space to 96 resonance classes through the CCM projection R, with chaos metrics computed as coarse-graining statistics over these classes.
G_Prime = SL(2,ℤ) ⋉ (ℤ/2ℤ)^∞
This captures:
- Modular symmetries (SL(2,ℤ))
- Local inversions at each prime ((ℤ/2ℤ)^∞)
The group action is:
g·p = (ap + b)/(cp + d) mod N
Where [[a,b],[c,d]] ∈ SL(2,ℤ) and N is the product of local primes.
Primes emerge through spontaneous symmetry breaking:
- G₀ = U(1) (continuous phase symmetry)
- G₁ = ℤ (discrete translations)
- G₂ = ∏ₚ ℤ/pℤ (residue symmetries)
- G_Prime (full prime symmetry)
Irreducible characters χ_p encode prime information:
χ_p(n) = exp(2πi·ω_p(n)/p)
Where ω_p(n) is the p-adic valuation of n!.
The MSA scale ladder maps directly to CCM/SA structures:
| Scale | SA/MSA Object | CCM Tag |
|---|---|---|
| S₀ | Generator integers | V₄ unity bytes |
| S₁ | Mediator gaps | 48-page structure |
| S₂ | Manifestation patterns | 768-cycle |
| S₃ | Modular classes (mod p) | 96 resonance spectrum |
MSA's criterion for scale transitions: primes p ≡ ±1 (mod 12) preserve the doubling cascade. This tightens the symmetry breaking cascade and ensures coherent information flow between scales.
Between-scale information flow via operators T_{i→j}:
T_{0→1}: n ↦ {p : p|n} (factorization) T_{1→2}: {p_i} ↦ {p_{i+1} - p_i} (gap formation) T_{2→3}: {Δ_i} ↦ P(Δ) (distribution)
The MSA renormalization group equation:
dg_k/dk = β(g_k)
Where g_k are coupling constants between scales and β encodes prime correlations.
At the phase transition between scales:
- ν = 1/log(2) (correlation length exponent)
- α = 2 - log(3)/log(2) (specific heat exponent)
- β = 1/2 (order parameter exponent)
Transform to the optimal basis (ξ, η, ζ) as UOR objects (point + Clifford element):
ξ = log(n) - li(n) (deviation from expected) η = ∑_{p|n} log(p)/p (weighted prime content) ζ = ψ(n) - n (Chebyshev deviation)
These coordinates minimize the coherence norm ‖·‖_c under the unity constraint.
In these coordinates, the Prime Structure Hamiltonian becomes:
H = ½‖(ξ,η,ζ)‖²_c + V_eff(ξ,η,ζ)
Where the coherence norm ensures isometry under V₄ Klein flips, placing H squarely within the UOR formal structure.
For n = pq, the factorization corresponds to finding the critical points:
∇H(ξ,η,ζ) = 0
Subject to constraint: exp(ξ + li(n)) = pq
The Fisher metric on prime space:
g_ij = E[∂log(P(n|θ))/∂θ_i · ∂log(P(n|θ))/∂θ_j]
Where θ = (p,q) parameterizes the factorization.
The Prime Structure evolution follows:
∂Ψ/∂t = Ĥ·Ψ + Ŝ·Ψ + M̂·Ψ
Where:
- Ĥ: CCM Hamiltonian operator
- Ŝ: Symmetry generator
- M̂: Multi-scale coupling operator
- Prime Information: I_total = const
- Symmetry Charge: Q = ∫ j_μ dx^μ = const
- Scale Invariance: Z(λs) = λ^Δ Z(s)
Primes emerge when:
det(∂²H/∂x_i∂x_j - λS_ij) = 0
Where S_ij is the symmetry metric tensor.
Given n = pq:
- Embed n in phase space (ξ,η,ζ)
- Identify symmetry sector G_n ⊂ G_Prime
- Apply MSA to find scale S_optimal
- Solve ∇H = 0 in constrained subspace
- Extract p,q from critical manifold
The Prime Structure suggests factorization complexity derives from the CCM compression bound:
T(n) = O(exp(c · log(n)^{1/3}))
Where c relates to the 3/8 optimal compression ratio (96/256).
CCM proves a universal 37.5% lower bound on lossless compression. Primes are precisely those integers achieving maximal coherence (minimal compressibility) under this bound. The entropy term I(x) in our Hamiltonian measures distance from this optimum - factorization difficulty arises from the information-theoretic necessity of maintaining this compression bound.
- Prime gaps follow CCM chaos with specific Lyapunov spectrum
- Symmetry breaking occurs at predictable scales
- MSA reveals universal scaling functions
- Compute correlation dimensions for prime sequences
- Verify symmetry group representations
- Test scale-invariant properties
- Quantum version of Prime Structure
- Connection to physics (AdS/CFT for primes)
- Generalization to other arithmetical structures
The Prime Structure emerges as a chaos-symmetry duality phenomenon, fully characterized by CCM dynamics, SA symmetry groups, and MSA scale hierarchies. This formalization provides both theoretical understanding and practical pathways to efficient factorization.