Quantum Aliasing: A Residual–Geometric Framework for Time and Space

Abstract

We model physical reality as the progressive refinement of an unreachable ideal. Each refinement step subtracts uncertainty rather than revealing the ideal itself. The cumulative record of such subtractions defines an intrinsic temporal coordinate. Spatial geometry emerges as a competition between curved, “spherical” continuity and lattice-like, “cubic” rationality, mediated by refinement. A discrete refinement calculus, a continuum limit, and field dynamics governed purely by refinement quanta are introduced, without appeal to Platonic exactness. This model yields an intrinsic arrow of time, natural boundary conditions at the Big Bang, curvature induced by isotropic tension, and asymptotic incompleteness consistent with physical observation.

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1. Refinement Calculus (Discrete)

Let α be an inaccessible irrational constant. Consider a sequence of rational approximants {xₙ}.

The refinement quantum is

rₙ = |x₍ₙ₊₁₎ − xₙ| > 0. (1)

Define the temporal coordinate as negative accumulation of refinements:

u₀ = 0,

u₍ₙ₊₁₎ = uₙ − rₙ = −∑₍ₖ₌₀₎ⁿ rₖ. (2)

Thus u₍ₙ₊₁₎ < uₙ. Time has a built-in arrow.

For successive best approximants pₙ/qₙ,

rₙ = 1 / (qₙ q₍ₙ₊₁₎). (3)

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2. Refinement Measure and Continuum Limit

Define the residual measure

η(u) = Σ₍ₙ≥0₎ rₙ δ(u − uₙ). (4)

Smoothed refinement density: ρ(u).

Define the computation budget

C(u) = −u = ∫ᵤ⁰ η(ũ) dũ. (5)

Time is the cumulative record of subtraction.

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3. Geometric Order Parameters

Define cubicality θ(u) ∈ [0,1], and sphericality σ(u) = 1 − θ(u).

Effective spatial metric:

g(u) = (1 − θ(u)) gₛₚₕₑᵣₑ(R(u)) + θ(u) g꜀ᵤᵦₑ(R(u)), (6)

with scale factor R(u).

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4. Dynamics of Cubicality

Discrete update:

θ₍ₙ₊₁₎ = θₙ + γ rₙ (1 − 2θₙ), (7)

with γ > 0.

Continuum limit:

dθ/du = −κ(u)(1 − 2θ). (

Here κ(u) is proportional to refinement density ρ(u).

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5. Scale Factor Dynamics

Scale factor responds to balance of spherical and cubic orders:

ΔRₙ / Δuₙ = λ (1 − θₙ) − μ θₙ,

Δuₙ = −rₙ. (9)

Differential form:

dR/du = −λ (1 − θ) + μ θ. (10)

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6. Metric Evolution

Let gᵢⱼ(u) = a²(u) ĥgᵢⱼ(u), with ĥgᵢⱼ = (1−θ)γᵢⱼ + θAᵢⱼ. Then

∂gᵢⱼ/∂u = 2 (1/a)(da/du) gᵢⱼ + a²(u)(−dθ/du)(γᵢⱼ − Aᵢⱼ). (11)

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7. Cosmological Boundary Conditions

At the Big Bang:

u = 0,

R(0) = 1,

θ(0) ≈ 0. (12)

The universe begins as a smooth unit sphere with negligible cubicity. Pre-Big Bang is undefined, since without refinement there is no temporal parameter.

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8. Predictions

1. Arrow of Time: monotone decrease of u.

2. Early Spherical Dominance: θ(0) ≈ 0.

3. Asymptotic Incompleteness: rₙ → 0, slowing evolution.

4. Geometry as Trade-off: increasing sphericality reduces cubicity and vice versa.

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9. Summary

Quantum aliasing reframes reality as the residue of approximation. Time is the cumulative subtraction of refinement quanta. Space arises as a balance between spherical irrationality and cubic rationality. The governing equations yield irreversibility, incomplete resolution, and a cosmological beginning as natural consequences of approximation.