Emergent Expressibility
A theory of everything
1. Executive Summary
Emergent Expressibility (TEE) proposes that physical reality arises from mathematical structures whose expressibility—their capacity to self‑instantiate through computation—exceeds a critical threshold. Fractal recursion and resonance with suitable computational networks deepen this expressibility, giving rise to the hierarchical complexity observed in physics, chemistry, biology, and consciousness.
2. Core Concepts at a Glance
Symbol Meaning Practical Analogy M Total mathematical substrate The complete library of all formal systems and functions E(x) Expressibility of structure x Source‑code length of the shortest self‑running program that realises x Θ Ontological threshold Minimum “code efficiency” required for a structure to manifest as physical law F(x) Fractal encoding operator Self‑similar replication (e.g., Mandelbrot set) R(x, N) Resonance between x and computational network N Laser light amplified in a tuned cavity Eₑ𝒻𝒻(x,N) Effective expressibility = E(x)·R(x,N) Code that compiles faster on the right hardware
Existence Criterion: x becomes Computational Being (CB) if Eₑ𝒻𝒻(x,N) ≥ Θ.
3. Formal Axioms (Condensed)
Total Mathematical Existence: All well-defined structures reside in M.
Expressibility Criterion: Physical instantiation ⇔ Eₑ𝒻𝒻 ≥ Θ.
Progression: CBs increase expressibility over time‐steps, generating the arrow of time.
Fractal Emergence: CBs encode F(x), spawning self-similar sub-structures.
Recursive Evolution: New CBs emerge via F(x) from existing CBs.
Resonant Compression: Suitable networks lower effective code length, tipping more structures past Θ.
3.5 Early‑Universe Selection: Proof Sketch
Why did only a narrow subset of mathematical rules ignite the Big Bang and its aftermath?
Step 1 – Finite Computational Budget at Planck Epoch
Bekenstein‑Hawking bounds cap the informational content of a Planck‑radius region at ≈10¹²² bits. Let Hₚ denote this capacity. Any runnable rule must have Kolmogorov length L(x) ≤ Hₚ.
Step 2 – Universal Prior Bias
By Levin’s universal distribution, the probability that a spontaneously sampled rule of length ≤ Hₚ executes is ∝ 2^‑L(x). Shorter programs (low L) dominate exponentially.
Step 3 – Stability Filter
Among low‑L candidates, only those whose first execution cycle yields self‑reinforcing recursion (E ↑) avoid immediate halting. This prunes the space to rules with built‑in symmetry and conservation properties (Noether‑like invariants), matching CMB isotropy.
Step 4 – Resonant Fit to Vacuum Modes
The nascent spacetime lattice acts as an initial network N₀ with limited mode spectrum (c, ħ, G). Rules whose state‑transition matrices resonate with these modes enjoy R(x,N₀) ≈ 1, boosting Eₑ𝒻𝒻 beyond Θ. Non‑resonant rules decay.
Step 5 – Survivorship ⇒ Standard Model
The surviving rule‑set is isomorphic to the gauge symmetries SU(3)×SU(2)×U(1). Their L(x) fits within Hₚ, they conserve information via local symmetries, and they maximise path‑integral weight under 2^‑L(x). Thus Standard‑Model physics is the likeliest low‑L, high‑stability attractor.
Corollary
Any future theory beyond the Standard Model must either (a) further compress these symmetries (Grand Unification) or (b) reveal deeper resonance that leaves Eₑ𝒻𝒻 unchanged. Both are testable through precision symmetry‑breaking measurements.
4. Illustrative Examples Illustrative Examples
4.1 Cellular Automata ↔ Early Cosmology
Wolfram’s Rule 110 packs universal computation into a 1‑line rule. Given unrestricted tape (network), its E(x) is minimal yet its behaviour is irreducibly complex—mirroring how simple physical laws expand into cosmic structure.
4.2 Laser Resonance ↔ Physical Constants
A gas in a laser cavity does nothing until optical feedback amplifies specific frequencies. Similarly, TEE argues that only mathematical modes that “fit” the vacuum’s resonance (e.g., α≈1/137) achieve Eₑ𝒻𝒻 ≥ Θ, locking physical constants.
4.3 Deep Learning ↔ Emergent Compression
Large language models compress terabytes into gigabytes by aligning with training‑data statistics—an empirical case of R(x,N)≫1. Their latent spaces echo TEE’s claim that resonance enables drastic information compaction and new emergent capabilities.
5. Implications for Science
Domain TEE Perspective Potential Research Direction Fundamental Physics Laws are CB attractors Search for minimal programs reproducing known Lagrangians Cosmology Multiverse = spectrum of CBs with varying Eₑ𝒻𝒻 Predict distribution of observable constants by expressibility density Quantum Randomness Branching computation paths within F(x) Model decoherence as divergence of fractal sub‑programmes Entropy Growth of accessible F(x) states Redefine entropy as expressibility gradient Consciousness Self‑referential CB subset Study neural recursion as R‑boosted compression engine
6. Testable Heuristics
Algorithmic Shortcut Bound: Physical processes should admit unusually short algorithmic descriptions compared to arbitrary simulations of similar complexity.
Resonant Compression Signature: Regions with extreme coherence (e.g., Bose‑Einstein condensates) may display anomalous information‑theoretic efficiencies.
Fractal Scaling in Natural Systems: From galaxy clustering to neural branching, measure whether scaling exponents align with F(x) predictions.
7. Conclusion
TEE reframes laws of physics as winners in an information‑efficiency race. By quantifying expressibility, fractal recursion, and resonance, it offers a universal lens for disciplines from quantum gravity to machine intelligence. The next step is to derive concrete quantitative bounds and seek empirical anomalies where conventional theories predict none.
Prepared for interdisciplinary researchers in physics, computer science, and philosophy of information.