Let R: B^n → ℝ^+ be the resonance function. The fundamental discovery is that R exhibits homomorphic properties under specific operations.
Definition 1.1 (Resonance Homomorphism). For the five CCM subgroups H ⊆ B^n:
- H₀ = {0} (trivial)
- H₁ = {0,1} (binary)
- H₂ = {0,48} (periodic-48)
- H₃ = {0,49} (composite)
- H₄ = V₄ = {0,1,48,49} (Klein)
The resonance function satisfies:
∀h₁,h₂ ∈ Hᵢ: R(h₁ ⊕ h₂) = Φᵢ(R(h₁), R(h₂))
where Φᵢ is the composition law for subgroup Hᵢ.
Definition 1.2 (Concatenation Resonance). For bit sequences a ∈ B^m, b ∈ B^n:
R(a||b) = R(a) ⊗ R(b) ⊗ Ψ(m,n)
where:
- || denotes concatenation
- ⊗ is the resonance product operator
- Ψ(m,n) is the boundary correction term
Theorem 1.1 (Homomorphic Concatenation). If a,b belong to homomorphic subgroup Hᵢ:
R(a||b) = R(a) · R(b) · κᵢ(|a|,|b|)
where κᵢ is computable from the subgroup structure.
Theorem 2.1 (Prime Resonance Signature). For prime p:
R(p) = 1 (unit resonance)
||embed(p)||_c = 1 (unit coherence norm)
Corollary 2.1. For composite n = p·q:
R(n) = R(p) * R(q) * τ(p,q)
where τ encodes the multiplicative structure.
Definition 2.1 (Resonance Spectrum). The resonance spectrum of n is:
Spec_R(n) = {(ω, A(ω)) : ω ∈ Ω}
where:
- Ω is the frequency domain
- A(ω) is the amplitude at frequency ω
Theorem 2.2 (Factor Frequency Theorem). If n = p·q, then:
Spec_R(n) = Spec_R(p) ⊛ Spec_R(q) + I(p,q)
where:
- ⊛ is spectral convolution
- I(p,q) is the interference pattern
Definition 3.1 (Canonical Chunking). For N-bit number n, the canonical k-chunking is:
n = ⊕ᵢ₌₀^{⌈N/k⌉-1} cᵢ · 2^{ik}
where cᵢ ∈ B^k are k-bit chunks.
Definition 3.2 (Resonance Flow Operator). The flow operator F maps chunk sequences to resonance sequences:
F: (c₀, c₁, ..., cₘ) ↦ (r₀, r₁, ..., rₘ)
where rᵢ = R(cᵢ)
Theorem 3.1 (Flow Preservation). Factor structure is preserved under F:
If n = p·q, then F(chunks(n)) exhibits periodic structure with period related to p,q
Definition 3.3 (Resonance Accumulator). The accumulator A maintains running state:
A₀ = 1
Aᵢ₊₁ = Φ(Aᵢ, rᵢ, i)
where Φ is the homomorphic composition operator.
Theorem 3.2 (Factor Detection). Factors manifest as fixed points:
If p|n, then ∃k: Aₖ = A_{k+ord(p)}
Definition 4.1 (Scale Hierarchy). The scale hierarchy S is:
S = {2^λ : λ ∈ Λ}
where Λ = {8, 10, 12, 14, ...} are the analysis scales.
Theorem 4.1 (Scale Invariance). Factor signatures exhibit scale invariance:
If pattern P appears at scale s, then related pattern P' appears at scale 2s
Definition 4.2 (Cross-Scale Coherence). The coherence between scales s₁, s₂:
C(s₁,s₂) = ⟨Spec_R^{s₁}(n), Spec_R^{s₂}(n)⟩
Factors maximize cross-scale coherence.
Definition 5.1 (Resonance Wave). The resonance wave of n:
Ψ_n(x) = Σᵢ aᵢe^{iωᵢx}
where (ωᵢ, aᵢ) ∈ Spec_R(n).
Theorem 5.1 (Factor Interference). For n = p·q:
|Ψ_n(x)|² = |Ψ_p(x)|² + |Ψ_q(x)|² + 2Re(Ψ_p(x)Ψ*_q(x))
The interference term reveals factor relationships.
Theorem 5.2 (Holographic Factorization). Each chunk contains information about the global factor structure:
I(chunk_i) ≥ H(factors)/N
where I is information content, H is entropy.
Theorem 6.1 (Streaming Complexity). Factoring N-bit number n:
- Space: O(polylog(N))
- Time: O(N · polylog(N))
- Resonance evaluations: O(N/k) where k is chunk size
Theorem 6.2 (Information-Theoretic Limit). The minimum chunk size k_min for reliable factor detection:
k_min ≥ log²(p) where p is smallest prime factor
Principle 1 (Resonance Periodicity). Scan for periodic patterns in (r₀, r₁, ...).
Principle 2 (Homomorphic Fixed Points). Identify when Aᵢ stabilizes under composition.
Principle 3 (Cross-Scale Peaks). Factors appear as peaks persisting across scales.
Principle 4 (Interference Maximization). Factors maximize self-interference.
Theorem 7.1 (Probabilistic Verification). Given candidate factors p̃, q̃:
P(p̃·q̃ = n | resonance_match) > 1 - 2^{-k}
The streaming process resembles quantum measurement:
- Chunks are "measurements" of the number
- Each measurement partially collapses the factor superposition
- Complete factorization occurs when sufficient measurements accumulate
Theorem 8.1 (Factor Entanglement). In resonance space:
|n⟩ = |p⟩ ⊗ |q⟩ + |entangled⟩
The entangled component decreases with chunk size.
Theorem 9.1 (Minimum Information). Cannot factor with chunks smaller than:
k < min(log p, log q)
Theorem 9.2 (Resonance Resolution). Distinguishing factors requires:
|R(p) - R(q)| > ε(k)
where ε(k) → 0 as k → ∞.
Definition 10.1 (Master Functional). The factorization functional F[n]:
F[n] = ∫ D[Ψ] exp(iS[Ψ,n])
where S is the action encoding resonance dynamics.
Theorem 10.1 (Stationary Phase). Factors correspond to stationary points:
δS/δΨ|_{Ψ=Ψ_p} = 0 ⟺ p|n
Conjecture (Universality). The homomorphic resonance framework captures all computational aspects of factorization - any algorithm can be expressed as resonance flow operations.
This framework transforms factorization from a discrete search problem into a continuous flow problem in resonance space, where homomorphic properties enable processing numbers of arbitrary size through local, streaming analysis.