The Lunarpunk Vector Space Axiom

A Formalization of Teleoplexic Isomorphism

Further to

Lunarpunk Quant Signal Remix Theorem

which is true only in so much as the Lunarpunk Hyperstition is true, a further mathematical exploration in terms of vector spaces as described in chapter 28 of Pinter’s Abstract Algebra, a DarkFi core text. Created with Deepseek.

AXIOM (Lunarpunk DLWE Isomorphism)

Let Ω be the set of all teleoplexic digital systems. For any S ∈ Ω, define three transformations:

1. T₁(S) = Markov_Boundary_Problem(S) = (E, B) where S ⊥ E | B (conditional independence)

2. T₂(S) = ZKP_Problem(S) = (statement, witness) with zero-knowledge property

3. T₃(S) = DLWE_Problem(S) = (A, A·s + e, random) where (A, A·s + e) ≈ₛ random

AXIOM: There exists a vector space V over field F (char ≠ 2) such that:

1. Ω forms a subspace of V

2. T₁, T₂, T₃ are linear isomorphisms Ω → W₁, W₂, W₃ ⊆ V

3. W₁ ≅ W₂ ≅ W₃ (isomorphic as vector spaces)

4. The isomorphism preserves three invariant functionals:

   • M(S) = 1 - I(S; E | B) = ⟨S, e₁⟩ for some basis vector e₁

   • Z(S) = Completeness/Complexity = ⟨S, e₂⟩

   • S(S) = 1 - Adv(distinguish) = ⟨S, e₃⟩

5. {e₁, e₂, e₃} forms an orthogonal basis for a 3D subspace of V

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1. VECTOR SPACE STRUCTURE

Let V = Span{privacy, control, complexity, verifiability, ...} with dim(V) = n.

Define coordinates for any system S:

S = α₁·e₁ + α₂·e₂ + α₃·e₃ + ... + αₙ·eₙ

where:

• α₁ = M_score(S)

• α₂ = Z_score(S)

• α₃ = S_score(S)

• The remaining coordinates capture narrative, regulation, etc.

The isomorphism means:

T₁⁻¹ ∘ T₂ : W₂ → W₁ is linear and invertible
T₂⁻¹ ∘ T₃ : W₃ → W₂ is linear and invertible
T₃⁻¹ ∘ T₁ : W₁ → W₃ is linear and invertible

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2. CONSEQUENCES FOR ZERO-KNOWLEDGE PROOFS

Theorem 1 (ZKP Completeness-Complexity Tradeoff):
If Z_score = ⟨S, e₂⟩ and the isomorphism holds, then for any ZKP system:

Completeness(proof) = k·⟨S, e₂⟩·Complexity(implementation)

where k is an isomorphism constant.

Proof sketch:
By the isomorphism, Z_score corresponds to a linear functional in the DLWE space:

⟨S, e₂⟩_ZKP ≡ ⟨T₃(S), f₂⟩_DLWE

where f₂ is the image of e₂ under the isomorphism. In DLWE space, verification complexity corresponds to lattice dimension n, and completeness corresponds to distinguishing advantage 1/Adv. The linear relation follows from the isomorphism preserving the ratio.

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Theorem 2 (Witness Subspace Dimension):
The witness space W for a ZKP statement has dimension:

dim(W) = n - rank(A)

where A is the constraint matrix, and this equals 1 - M_score in the Markov boundary representation.

Proof sketch:
In Markov form: S ⊥ E | B. The conditional independence means the joint distribution factorizes. The dimension of the sufficient statistic is dim(B). Under isomorphism:

dim(B) ≡ dim(witness_space) ≡ security_parameter(κ) in DLWE

Thus M_score = 1 - dim(B)/n = 1 - dim(W)/n.

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3. TRADING AS LINEAR OPTIMIZATION

Define the narrative vector N ∈ V representing market perception.

Theorem 3 (Convergence Premium):
The convergence premium is the norm of the projection error:

Premium(P) = ||Proj_{span(e₁,e₂,e₃)^⟂}(N - Π(P))||

where Π(P) = (M(P), Z(P), S(P)) in the 3D subspace.

Proof:
By the axiom, Π(P) represents mathematical reality in the invariant subspace. Market narrative N has components outside this subspace. The premium is the distance between narrative and reality in the orthogonal complement.

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Theorem 4 (Alpha as Linear Functional):
Alpha generation is a linear functional α : V → ℝ:

α(S) = ⟨w, S⟩

where the weight vector w = λ₁·e₁ + λ₂·e₂ + λ₃·e₃ with:

λ₁ = -∂U/∂M (negative: boundary erosion creates alpha)
λ₂ = -1/Z² (negative: obfuscation creates alpha)  
λ₃ = +∂AA/∂S (positive: surveillance growth creates alpha)

Proof:
This follows directly from the trading rules in the document, reinterpreted as linear functionals in the vector space.

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4. CONTROL CENTROID AS FIXED POINT

Theorem 5 (Control Centroid Existence):
There exists a fixed point C* ∈ V (the Control Centroid) where:

[Privacy_Asymmetry, Evidence_Integration, Platform_Capturability] = [1, 0, 1]

and this is an attractor in the dynamical system:

dS/dt = A·S + b

where A is a linear operator with eigenvalues having negative real parts for deviations from C*.

Proof sketch:
The coordinates define a 3D subspace. The condition ∂U_capture/∂t = 0 defines a hyperplane. Their intersection gives C*. The capitalist equilibrium equations:

Builders: max E[Return | Platform_Capturability, Regulatory_Alignment]
Investors: max E[Return | Narrative_Strength, Convergence_Premium]  
Users: max E[Utility | Convenience, Privacy_Theater]

are all linear or affine in the coordinates, so their simultaneous solution is a linear system.

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5. EMPIRICAL PATTERNS AS GEOMETRIC PROPERTIES

Corollary 1 (Privacy Tech Lifecycle):
The observed lifecycle Phase 1 → Phase 2 → Phase 3 is a straight-line trajectory in V:

S(t) = S₀ + t·v

where direction vector v = (-0.7, -0.5, -0.55, +0.7)/T (for M, Z, S, AA respectively).

Corollary 2 (Regulatory Capture):
Hiring former regulators applies a linear transformation:

ΔS = R·S

where R is a shear matrix reducing M_score while increasing AA_growth.

Corollary 3 (Complexity Shield):
High VAM (verification abstraction multiplicity) corresponds to:

Z_score = k₁/exp(k₂·VAM)

an exponential decay in the e₂ direction.

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6. PORTFOLIO CONSTRUCTION AS BASIS SELECTION

The trading framework chooses a basis for V:

B_trading = {e_boundary, e_verification, e_surveillance, e_narrative}

and constructs portfolios as:

Portfolio = Σ_i β_i·proj_{e_i}(S)

where:

• Long positions: β_i > 0 for control infrastructure directions

• Short positions: β_i < 0 for privacy narrative directions

The 60/40 allocation is:

Portfolio = 0.6·proj_{e_surveillance} - 0.4·proj_{e_narrative}

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7. VOLATILITY AS NORM FLUCTUATIONS

Theorem 6 (Risk Adjustment):
The volatility scaling σ_adj = 1/Historical_Volatility is equivalent to:

σ_adj = 1/||ΔS||

where ΔS is the vector of past changes in system state.

The adjusted position size is:

Position = (Base/||ΔS||)·|⟨w, S⟩|

a linear functional scaled by inverse norm of recent movements.

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8. THE ULTIMATE LINEAR MODEL

Combining all, we get the complete system:

dS/dt = A·S + B·u(t) + noise
y(t) = C·S(t)  (observable metrics)
α(t) = wᵀ·S(t)  (alpha generation)
Premium(t) = ||N(t) - Π·S(t)||

where:

• A = dynamics matrix (control infrastructure growth)

• B = input matrix (regulatory shocks, tech breakthroughs)

• C = observation matrix (extracts M,Z,S,AA scores)

• Π = projection to invariant subspace

• w = alpha weight vector

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9. WHAT FOLLOWS (SUMMARY)

1. All privacy/control systems live in one vector space V

2. Isomorphism = change of basis between problem representations

3. Market dynamics = linear transformations in this space

4. Alpha = linear functional detecting mispricing between narrative and reality

5. Convergence = attraction to fixed point C* (Control Centroid)

6. Trading = constructing portfolios along principal axes of V

7. Risk management = norm calculations in this space

The axiom reduces the complex interdisciplinary framework to linear algebra over an abstract vector space, where:

• Cryptographic primitives are basis vectors

• Market dynamics are linear operators

• Trading signals are linear functionals

• Convergence is eigenvalue analysis

This formalization makes testable predictions:

• The 3D invariant subspace (M,Z,S) should have stable geometric relationships

• System trajectories should be approximately linear in V

• The control centroid C* should be a universal attractor

• Alpha generation should be linearly separable in feature space

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Thus, assuming the axiom true: The entire Lunarpunk framework becomes an application of finite-dimensional linear algebra to crypto-economic systems, where the deep mathematics is not in the proofs but in the vector space structure itself—a structure that allegedly emerges from first principles of information, computation, and capital.

Until next time, TTFN.