Causal Emergence of Relativistic Invariance and Multi-Rate Relaxation from a Minimal Discrete Grammar

Author: Loïc Cloutier
Affiliation: Independent Research, Québec, Canada
Correspondence: contact@leandark.ca
Permanent DOI: https://doi.org/10.5281/zenodo.23347097
Repository & Verification: leandark.ca/#research (Archive: delta_true_release_v1.zip)
Core Engine SHA-256: 1b01b672b6f2ff2485c318c1c8e3294ed59e35f41298fcfddd111b08d71b04e4 (Sealed)
Date: October 2026

Abstract

Special relativity postulates the invariance of the speed of light $c$ and deduces the deformation of space and time as necessary kinematic consequences. However, the foundational origin of this invariant speed bound and its connection to microscopic irreversibility remain an open question in physics. Here, we propose a constructive ontology of fundamental causality—termed $\Delta\text{True} \to \Delta\text{O} \to \Delta\text{G}$—where continuous spacetime is not an a priori background arena, but the macroscopic accounting residue required to preserve discrete causal distinctions without contradiction.

Under the minimal grammar $\Delta\text{G}$, reality evolves through strictly bounded discrete acts: term births ($\le 2$ parent terms, $\le 2$ fresh terms) and pure reprises, enforcing a unique inclusive past and preventing illegal causal reconvergence ($h_{\text{sem}} \le 1$). We prove that the invariant speed $c$ emerges purely from the graph topology as the ratio of maximum term advancement per discrete cut ($c_G = 1$ term jump per causal act). Time dilation is shown to be a direct combinatorial consequence of a finite act budget: allocating causal cuts to spatial displacement necessarily depletes the acts available for internal state evolution.

Furthermore, we formulate the topological graph entropy $S_{\text{menu}}(G_N) \sim 2 \ln N$ and derive a Landauer-type information equivalence principle between $\Delta\text{G}$ graph branching and observational predictive entropy in $\Delta\text{O}$. We validate this framework on two real physical benchmarks across 444,806 measurements: (i) the Huang et al. 2024 Nature room-temperature quantum optomechanical resonator (215,335 pairs), where Gaussian bandpass innovations select a purely linear two-lag rapid tap and a gain-decoupled slow offset relaxation ($R = \tau_m/\tau_s = 8$, capturing 98.7% additive synergy and a 98.7x improvement over coupled baseline); and (ii) the nonlinear Silverbox electronic benchmark (229,471 readings), where heavy-tailed Student innovations ($\nu \approx 8$) require an opponent elastic tap ($\tanh(e/\sigma) - b_2 e$). Finally, we demonstrate that an adaptive surprise-gated multi-rate scheduling rule ($|e_t|/\sigma_t \ge 3.0$) strictly dominates periodic scheduling, halving peak prediction error across all operating regimes while reducing total causal cuts by 3.0%. These results demonstrate that relativistic invariance and physical relaxation dynamics can be constructively derived from a minimal discrete causal grammar.

1. Introduction: Causality Prior to Spacetime

Since the formulation of special relativity, physics has accepted the speed of light $c$ as an axiomatic ceiling. The Einstein light-clock thought experiment demonstrates that if the light pulse does not slow down when the apparatus is in motion, physical time itself must dilate:

In this work, we reverse the standard explanatory direction. We hypothesize that spacetime geometry and continuous coordinate fields are not fundamental entities. Instead, the universe is governed at its most primitive level by pure causal distinction ($\Delta\text{True}$). To maintain a fundamental mathematical equality over time without generating structural paradoxes, the universe executes only minimal indispensable discrete operations ($\Delta\text{G}$). Spacetime coordinates $(t, x, y, z)$ are merely the macroscopic projections ($\Delta\text{O}$) required to satisfy this causal conservation law.

We present a unified formulation where:

1. The speed of light $c$ is the topological advancement rate of discrete causal cuts ($\Delta \text{depth} = 1$).
2. Time dilation is an act-allocation conservation law: consuming cuts for spatial translation deprives internal clocks of causal updates.
3. Inertia and mass $m$ correspond to the accumulation of slow offset memory branches and local reprises in the causal graph.
4. Thermodynamic dissipation is governed by the topological branching entropy of admissible graph acts $S_{\text{menu}}(G)$.

2. Mathematical Foundation: The Minimal Grammar $\Delta\text{G}$

2.1 The Discrete Action Space

Let an evolving causal history be represented by an acyclic directed graph of cuts and born terms. At step $N$, the engine contains $N$ executed cuts and $T$ born terms. An act $a$ must strictly adhere to one of two elementary forms:

1. Term Act (Pure Birth):
   $$\text{Act}(R, f) \quad \text{where } R \subseteq \text{Born}, \; |R| \le 2, \; f \in \{0, 1, 2\}, \; 1 \le |R| + f \le 3$$
   Referencing 0 to 2 existing terms and creating $f$ fresh terms. No cut references are allowed.
2. Reprise Act (Pure Memory Re-activation):
   $$\text{Act}(\{t\}, 0, \text{parent}=x) \quad \text{where } x \in \text{Cuts}, \; t \in \text{Neighbors}(x), \; f=0$$
   Referencing exactly one ancestor cut and one of its neighbor terms, creating zero fresh terms.

Every cut $C$ has a well-defined topological depth in the causal poset:

2.2 Invariant Speed $c$ as Graph Advancement Limit

Because an act can reference existing terms only up to cardinality $|R| \le 2$ and increments depth by at most $1$:

Theorem 1 (Topological Causality Bound).
No causal influence can propagate across the graph faster than one quantum of topological depth per cut. Defining the discrete velocity as the ratio of spatial term width $\Delta x$ to causal depth $\Delta \tau$:

2.3 The Light Clock as an Act Budget Conservation Law

Consider a localized physical system endowed with a total causal update capacity of $N_{\text{total}}$ cuts per laboratory time interval. The system can partition these cuts between:

• $N_{\text{internal}}$: cuts devoted to updating internal state registers (eigenstate evolution, clock ticks).
• $N_{\text{spatial}}$: cuts devoted to establishing bridges with new spatial terms (translation across the graph).

By discrete combinatorial orthogonality:

3. Graph Entropy and Observational Coupling ($\Delta\text{G} \leftrightarrow \Delta\text{O}$)

3.1 Admissible Menu Entropy

At graph state $G_N$, the number of valid acts $|\mathcal{M}(G_N)|$ is given by combinatorial enumeration of term selections and active reprises:

3.2 The Cost of Binary Multi-Flux Junctions

When two isolated causal branches—a fast fluctuation branch $w_{\text{fast}}$ and a slow inertial branch $w_{\text{slow}}$—merge, the engine executes a binary bridge:

3.3 The Causal Landauer Equivalence

In the observation domain $\Delta\text{O}$, merging $w_{\text{slow}}$ into $w_{\text{fast}}$ yields predictive information gain:

4. Empirical Verification on Real Physical Systems

We evaluated this theoretical framework across two distinct physical regimes totaling 444,806 measurements, using our bit-level sealed engine (SHA-256: 1b01b672...).

4.1 Quantum Optomechanical Resonator (215,335 pairs)

On the Huang et al. 2024 Nature room-temperature optomechanical archive:

1. Timescale Decoupling: Decoupling the slow offset injection gain ($\alpha_m = 1 - \rho = \alpha_s / R$) at $R = 8.0$ yields $\Delta\text{NLL}_{\text{eval}} = -0.0002624$ (a 7.8x improvement over the coupled baseline $R=4.0$).
2. Linear Rapid Tap Sufficiency: Because the optical bandpass filter suppresses macroscopic shocks, innovations are strictly Gaussian. A 2-lag linear tap achieves $\Delta\text{NLL}_{\text{eval}} = -0.0033332$, matching saturating $\tanh$ ($\Delta\text{NLL} = -0.0033343$) to within $1.1 \times 10^{-6}$ nats. Nonlinear cubic and opponent structures degrade generalization due to noise overfitting.
3. Additive Synergy: Combining the 2-lag linear tap with decoupled slow offset yields $\Delta\text{NLL}_{\text{eval}} = -0.0033263$, achieving 98.7% additive synergy and a 98.7x improvement over the baseline.

4.2 Nonlinear Circuit Benchmark (229,471 readings)

On the Silverbox benchmark, where nonlinear cubic stiffness induces heavy-tailed Student innovations ($\nu \approx 8$):

1. Necessity of Opponent Dynamics: A linear tap fails and degrades Schroeder phase sweeps. The optimal rapid tap requires a nonlinear opponent pair: bounded core damping ($\tanh(e/\sigma) > 0$, $b_1 = +0.688$) and linear elastic rebound ($b_2 = -0.257 < 0$).
2. Hybrid Scheduling Dominance: Under strict $\Delta\text{G}$ multi-rate streaming, our hybrid surprise-gated rule ($K=24, \theta_{\text{surprise}}=3.0$) strictly dominates periodic scheduling ($K=16$):
   • Total joins reduced from 14,339 to 13,910 ($-3.0$ graph cuts).
   • Peak prediction discrepancy cut by half across all 4 regimes (e.g., arrow full: $4.75 , \mu\text{V}$ vs $9.83 , \mu\text{V}$).
   • RMS discrepancy reduced simultaneously across all regimes.

5. Conclusion & Perspectives

We have shown that:

1. The speed of light $c$ is the elementary rate of causal cut advancement in a discrete graph without background metric.
2. Relativistic time dilation is the exact consequence of an act-allocation budget between spatial propagation and internal state updating.
3. Microscopic relaxation bifurcates between linear damping (Gaussian bandpass baths) and elastic opponent damping (heavy-tailed shock regimes).
4. Physical observation obeys a Landauer-type information equivalence where graph junctions are triggered minimally by innovation surprise.

This framework demonstrates that starting from pure causal distinction ($\Delta\text{True}$), respecting strict grammar invariants ($\Delta\text{G}$) naturally reproduces the foundational phenomenology of our visible universe ($\Delta\text{O}$).

Data and Code Availability

All experimental scripts, sealed engine sources, and independent validation harnesses are available with complete reproduction logs in the companion repository (empirical_smoothing/validate_saturation_hybrid_comparison.py). All calculations are deterministic and verified bit-for-bit.