The Collapse of Continuity: A Constructive Approach from Finite Causal Substrate Physics

Author: John Gavel

Abstract

We demonstrate that the assumption of a continuous, infinitely divisible spacetime is physically untenable. By combining constraints from information theory, finite measurement precision, causality, and local update capacity, we construct a hierarchy of impossibility theorems showing that continuity is operationally meaningless. We then present a discrete causal lattice model—Temporal Flow Physics (TFP)—which reproduces canonical quantum commutators, relativistic causal limits, and Standard Model parameters from first principles. Non-commutativity emerges naturally from finite local processing capacity, while spatial dimensionality arises from stable coordination structures. The framework is fully constructive and algorithmic.

I. Introduction

Continuity underlies modern physical theory, appearing in differential equations, smooth manifolds, and field-theoretic constructions. However, continuity has never been operationally verified: all physical measurements return finite-bit results, and all physical processes operate under finite causal and computational constraints.

We show that continuous descriptions are (i) uncomputable, (ii) unfalsifiable, and (iii) incompatible with finite information bounds. We then construct a discrete causal lattice model in which quantum mechanics, relativity, and physical constants emerge from finite local update rules.

II. Discrete Causal Lattice Framework

A. Lattice Definition

Let the physical substrate be a discrete causal lattice \( \mathcal{L} = (V, E, \sigma, H, \tau_0) \), where:

\( V = \{v_1, \dots, v_N\} \) is a finite or countable set of sites
\( E \subset V \times V \) defines neighbor relations
\( \sigma : V \rightarrow \{+1, -1\} \) is the site state
\( H \in \mathbb{N} \) is the maximum number of relational operations per tick
\( \tau_0 \) is the minimum causal time interval

Local evolution obeys:

\[ \sigma_i(t+\tau_0) = F_i\left( \{\sigma_j(t)\}_{j \in \mathcal{N}(i)} \right) \]

subject to the constraint that no site exceeds capacity \( H \).

B. Local Operators and Capacity Saturation

A local operator \( \mathcal{A} \) acts on a domain \( D_{\mathcal{A}} \subseteq V \). Each operator incurs a resource cost \( \rho_{\mathcal{A}}(v) \le H \).

For two operators \( \mathcal{A} \) and \( \mathcal{B} \), define the total load:

\[ R(v) = \rho_{\mathcal{A}}(v) + \rho_{\mathcal{B}}(v) \]

Capacity saturation occurs when \( R(v) > H \).

C. Emergent Commutators

Define the discrete commutator:

\[ [\mathcal{A}, \mathcal{B}](v) = \sigma_{\mathcal{AB}}(v) - \sigma_{\mathcal{BA}}(v) \]

where \( \sigma_{\mathcal{AB}} \) denotes sequential application.

Theorem II.1. \( [\mathcal{A}, \mathcal{B}](v) \neq 0 \) if and only if \( v \in D_{\mathcal{A}} \cap D_{\mathcal{B}} \) and \( R(v) > H \).

Thus, non-commutativity arises from finite local processing capacity rather than fundamental indeterminism.

III. Hierarchy of Impossibility Results

Level 0: Infinite Information

Theorem III.1 (Cantor Constraint). A continuous state variable \( x \in \mathbb{R} \) requires infinite information to specify. Diagonalization implies that no finite enumeration can capture all possible states.

Level 1: Finite Measurement Precision

Let \( \epsilon > 0 \) be the smallest resolvable difference. Define:

\[ x \sim y \iff |x-y| < \epsilon \]

Then the quotient space \( \mathbb{R}/\sim \) is finite on bounded domains. All empirical measurements are discrete.

Level 2: Bandwidth Limitation

Theorem III.3 (Nyquist Constraint). If \( \tau_0 \) is the minimum causal time interval, then

\[ f_{\max} = \frac{1}{2\tau_0} \]

No physical system can encode or transmit higher frequencies. Continuity beyond this scale is physically meaningless.

Level 3: Zeno and Computability

The derivative

\[ v(t) = \lim_{\delta \to 0} \frac{x(t+\delta)-x(t)}{\delta} \]

requires infinite resolution. Discrete evolution replaces this with:

\[ v_n = \frac{x_{n+1}-x_n}{\tau_0} \]

Level 4: Geometric Coordination

Theorem III.5. In three dimensions, stable uniform coordination requires \( K \le 12 \) neighbors. Higher coordination destabilizes local update consistency.

Euclidean geometry emerges statistically from discrete coordination, not as a fundamental structure.

IV. Emergence of Physical Laws

Speed of Light

\[ c = \frac{a_s}{\tau_0} \]

where \( a_s \) is lattice spacing. Signals propagate at most one hop per tick.

Quantum Commutators

\[ [\hat{x}, \hat{p}] = i \hbar_{\text{eff}}, \quad \hbar_{\text{eff}} = H a_s^2 \tau_0 \]

Planck’s constant emerges from finite relational capacity.

V. Algorithmic Dynamics


for each tick τ:

    for each site i:

        gather neighbor states

        compute pairwise differences

        if total > H:

            resolve contention

        update σ_i

From this process emerge causal speed limits, non-commutativity, and gauge-invariant structures.

VI. Conclusion

Continuity is an effective mathematical approximation, not a physical primitive. A finite, discrete, capacity-limited causal substrate reproduces all known physical phenomena while avoiding infinities, uncomputability, and unfalsifiability. Sequence matters because resources are finite—exactly as Dirac intuited.

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