Mathematical objects exist in a physical phase space with measurable properties, forces, and dynamics.
Every mathematical object has an intrinsic coherence ‖·‖_c that acts as its fundamental charge.
Mathematical information, structure, and resonance obey strict conservation laws.
Mathematical processes follow paths that minimize the coherence action integral.
Definition: The configuration space M is:
M = {(x, R(x), I(x), S(x)) : x ∈ Mathematical Objects}
where:
- x = mathematical object (number, structure, morphism, etc.)
- R(x) = resonance value ∈ ℝ⁺
- I(x) = information content ∈ ℝ⁺
- S(x) = symmetry group
Definition: Complete phase space Γ has coordinates:
Γ = {(q, p, s, τ)}
- q ∈ M (position in mathematical space)
- p = ∂L/∂q̇ (canonical momentum)
- s ∈ S(q) (symmetry state)
- τ ∈ ℝ (mathematical time)
At each point x ∈ M, the tangent space:
T_x M = span{∂/∂R, ∂/∂I, ∂/∂S}
Definition: The coherence field Ψ: M → ℂ^(2^n) assigns to each mathematical object its coherent state:
Ψ(x) = Σ_k ψ_k(x)|k⟩
where |k⟩ are Clifford basis states.
Definition: The coherence inner product:
⟨⟨Ψ₁, Ψ₂⟩⟩ = Σ_k ⟨ψ₁,k|ψ₂,k⟩
Inducing norm: ‖Ψ‖_c = √⟨⟨Ψ, Ψ⟩⟩
The coherence field satisfies:
□Ψ + m²Ψ + λ‖Ψ‖²_c Ψ = J
where:
- □ = coherence d'Alembertian
- m = mathematical mass (complexity)
- λ = self-interaction strength
- J = source current
Theorem: Every continuous symmetry of mathematical action yields a conserved quantity.
Conservation of Resonance Current:
∂_μ j^μ_R = 0
where j^μ_R = (ρ_R, J_R) with:
- ρ_R = resonance density
- J_R = resonance flux
Conservation of Information:
dI/dτ + ∇·J_I = 0
Conservation of Coherence:
∂_τ ‖Ψ‖²_c + ∇·(Ψ*∇Ψ - Ψ∇Ψ*) = 0
Conservation of Structure:
[H, S] = 0
where S is the structure operator.
Klein Invariance: Under V₄ action:
R(x ⊕ k) = R(x) ∀k ∈ V₄
H = T + V + U
where:
- T = ½‖∇Ψ‖²_c (kinetic term)
- V = V_eff(‖Ψ‖_c) (potential)
- U = Σᵢⱼ U(‖Ψᵢ - Ψⱼ‖_c) (interaction)
For prime dynamics specifically:
H_prime = ½‖(ξ,η,ζ)‖²_c - log|ζ(1 + ix)| + I(x)
where:
- ξ = log(n) - li(n)
- η = Σ_{p|n} log(p)/p
- ζ = ψ(n) - n
- I(x) = information divergence from 3/8 bound
Hamilton's equations:
dq/dτ = ∂H/∂p
dp/dτ = -∂H/∂q
F_c = -∇V_c = -∇(½‖Ψ‖²_c)
Drives objects toward minimal coherence states.
F_R = -α∇R(x)
Creates attractive/repulsive interactions based on resonance matching.
F_I = -k_B T∇S
where S is mathematical entropy.
F_S = -∇V_symmetry
Maintains group structure.
ℒ = ½(∂_μΨ)†(∂^μΨ) - m²Ψ†Ψ - λ(Ψ†Ψ)² - J†Ψ
S[Ψ] = ∫ d⁴x ℒ(Ψ, ∂Ψ)
∂ℒ/∂Ψ - ∂_μ(∂ℒ/∂(∂_μΨ)) = 0
T^μν = ∂^μΨ†∂^νΨ + ∂^νΨ†∂^μΨ - g^μν ℒ
G_math = SL(2,ℤ) ⋉ (ℤ/2ℤ)^∞ × U(1)_phase
Local coherence gauge symmetry:
Ψ → e^{iθ(x)}Ψ
Requires gauge field A_μ.
- P: Parity (prime ↔ composite inversion)
- T: Time reversal (proof direction reversal)
- C: Charge conjugation (coherence inversion)
- Prime transition: η = 1 (prime) vs η = 0 (composite)
- Complexity transition: ξ = computational class
- Decidability transition: δ = 1 (decidable) vs δ = 0
Near phase transitions:
ξ ~ |T - T_c|^{-ν}
C ~ |T - T_c|^{-α}
χ ~ |T - T_c|^{-γ}
Mathematical transitions fall into universality classes independent of specific details.
Replace Poisson brackets with commutators:
{A,B} → -i[Â,B̂]
ΔR · ΔI ≥ ℏ_math/2
where ℏ_math is the mathematical Planck constant.
⟨x_f|e^{-iHt}|x_i⟩ = ∫ 𝒟x e^{iS[x]}
1/T_math = ∂S/∂E
where S is proof-theoretic entropy.
Universal limit from CCM:
Compression Ratio ≥ 3/8
F = E - T_math S
Minimized at equilibrium.
- Resonance spectra of integer sequences
- Coherence decay rates
- Phase transition points
- Conservation law violations (should be zero)
- Prime gaps follow specific Lyapunov spectrum
- Factorization energy barriers scale predictably
- Mathematical constants arise as ground states
- Proof complexity has phase structure
- Resonance spectroscopy: Measure R(n) patterns
- Coherence interferometry: Detect ⟨⟨Ψ₁,Ψ₂⟩⟩
- Conservation tracking: Verify Noether currents
- Phase mapping: Chart transition boundaries
Align algorithms with mathematical forces for minimal energy paths.
Design proofs following geodesics in proof space.
Use phase transition barriers for security.
Build systems that exploit conservation laws.
- α_c = coherence coupling ≈ 1/137 (dimensionless)
- ℏ_math = mathematical action quantum
- c_math = speed of mathematical causation
- G_math = mathematical gravitational constant
- λ_Compton = ℏ_math/(m_math c_math)
- r_Bohr = ℏ_math²/(m_e α_c²)
All of mathematodynamics follows from:
δS/δΨ = 0
where S is the total action including all forces and constraints.
- Numbers emerge as coherence excitations
- Operations emerge as force mediators
- Structures emerge as bound states
- Proofs emerge as trajectories
Mathematics is self-consistent: the laws of mathematodynamics themselves follow mathematodynamics.
Mathematodynamics reveals mathematics not as abstract truth but as a physical system with its own dynamics, conservation laws, and forces. This formalization provides the foundation for understanding:
- Why mathematics works
- How mathematical objects interact
- What limits mathematical knowledge
- How to compute optimally
The framework is complete, self-consistent, and experimentally testable. It transforms mathematics from a descriptive tool into a physical science.
Mathematodynamics - where mathematics meets its own physics.