9.3 KiB
From Known Physics to Compression Architecture
Starting Point: The Laws We Actually Have
Not fractional fields. Not anthropic shear. The laws that have been tested for 100+ years.
1. Landauer (1961)
E_dissipated ≥ k_B T · ln(2) per bit erased
Implication for compression: Every step in the decoder that discards information — a wrong prediction, a context miss, a failed branch — dissipates energy. A decoder that makes irreversible choices is thermodynamically expensive.
Architecture consequence: The decoder must be reversible by construction. There is no "discard" operation. Every output bit is computed from the input stream plus a deterministic state transition. The only entropy injection is the residual itself.
Standard PAQ: context → prediction → compare → update model → discard old state
Landauer-optimal: state(n) → deterministic(state(n-1), residual(n))
PIST already does this: output(n) = predict(n) XOR residual(n), where predict(n) is a pure function of n. No information is discarded.
2. Shannon (1948)
H(X) = -Σ p(x) log p(x)
Implication: The compressed size cannot be less than the entropy of the source.
Architecture consequence: The decoder's prediction model must maximize the entropy of the residual distribution. If the residuals are uniformly random, the model has extracted all predictable structure. The job of the decoder is to turn structured data into uniform noise.
Compression ratio = H(original) / H(residuals)
Optimal: H(residuals) = H(original) (no compression, model failed)
Optimal: H(residuals) = 0 (impossible, only if source is deterministic)
Best achievable: H(residuals) ≈ H(original | model)
3. Jaynes Maximum Entropy (1957)
Given measured constraints (e.g., observed byte frequencies, pairwise correlations), the correct probability distribution is the one with maximum entropy subject to those constraints.
Implication for compression: The optimal context model is the maxent model for the observed statistics. The model should not assume more structure than the data exhibits.
Architecture consequence: Context models should be built from empirical frequency counts, not hand-designed heuristics. The PIST basis is a frequency-ranked pool (Section 10 of the Lean formalization). This is Jaynes' principle in action.
4. Bennett Reversible Computation (1973, 1982)
Any computation can be made reversible by retaining all intermediate steps. Reversible computation, in principle, dissipates zero energy.
Implication: The decoder can be a reversible register machine. Each instruction has an inverse. The sequence of operations is a trajectory in state space, not a destructive update.
Architecture consequence: The PIST 8-opcode register machine (NOOP, LOAD, STORE, ADD, XOR, BRANCH, FUSE, HALT) should be extended with inverse operations. FUSE becomes SPLIT. ADD becomes SUB. Every operation is invertible.
Standard: state_{n+1} = op(state_n) (irreversible, information lost)
Reversible: state_{n+1} = op(state_n, aux_n) (aux retains discarded info)
The auxiliary register becomes the history tape — exactly the FAMM scar memory. Failed routes are not erased; they are stored in aux.
5. Quantum Uncertainty / Complementarity (Heisenberg, 1927)
Δx · Δp ≥ ℏ/2
Implication for compression: In information terms, you cannot simultaneously have perfect knowledge of local context (position) and global pattern (momentum/mode). A model optimized for local byte prediction will miss long-range correlations. A model optimized for long-range structure will blur local detail.
Architecture consequence: The decoder needs complementary models that trade off local vs. global prediction. PIST already has this:
- Tree address = local context (position n mod 20)
- Surface/torus = global context (irrational rotation, independent of n)
- Shell coordinate = intermediate (k² + t structure)
The composite address is the wavefunction of the data position — a superposition of local, global, and intermediate representations.
6. Bell / Quantum Correlations (1964)
Correlations between distant particles can exceed any classical bound. The CHSH inequality:
|S| ≤ 2 (classical)
|S| ≤ 2√2 (quantum)
Implication for compression: Data can have nonlocal correlations stronger than any Markov model can capture. Text is the clearest example: "Paris" predicts "France" thousands of bytes later, with no local context sufficient to predict the intervening text.
Architecture consequence: Context mixing must include distant match operations — not just recent history, but semantically related tokens anywhere in the stream. This is what PAQ does with its SIMD mixer (combining predictions from multiple models). PIST's torus angles with Φ-irrational rotation are designed to produce nonlocal correlations.
7. Bekenstein Bound (1981)
I ≤ (2π R E) / (ℏ c ln 2)
Maximum information in a sphere of radius R with energy E.
Implication: Information is bounded by surface area, not volume. The most efficient encoding stores information on the boundary of the data block, not distributed throughout.
Architecture consequence: The decoder's state should be surface-like — minimal internal memory, maximal encoded information on the "boundary" (the basis vectors, which are O(1) in size, vs. the data stream, which is O(N)).
PIST basis size = 16 bytes. Data size = 10⁹ bytes. Information ratio = 16:10⁹. The basis is the holographic boundary. The data stream is the bulk.
8. Thermodynamic Uncertainty Relation (Barato, Seifert 2015)
(ΔJ)² / ⟨J⟩² · σ ≥ 2 k_B
For any current J with entropy production σ, the relative uncertainty is bounded below.
Implication for compression: A decoder with high throughput (many predictions per cycle) must have high entropy production (many mistakes). A decoder with low entropy production (perfect predictions) must have low throughput (slow, careful context analysis).
Architecture consequence: There is a speed-accuracy tradeoff in decompression. Fast decoders (simple prediction) produce more residuals → larger compressed size. Slow decoders (complex context mixing) produce fewer residuals → smaller compressed size.
The Hutter Prize rewards the slow, accurate decoder. The PIST gear law formalizes this:
C_out = G_AS · C_in (higher gear ratio = more work per bit = better prediction)
9. Fluctuation Theorem (Jarzynski 1997, Crooks 1999)
⟨exp(-β W)⟩ = exp(-β ΔF)
The probability of observing a trajectory with work W is exponentially related to the free energy difference.
Implication for compression: The probability of a particular compression path (sequence of model updates) is exponentially related to the achieved compression ratio. Good compressors follow low-work trajectories in model space.
Architecture consequence: The decoder's learning trajectory should minimize cumulative prediction error (work). FAMM scars record high-work paths and avoid them. The gear ratio increases when the decoder encounters hard-to-predict data, forcing it to do more work per bit.
The Architecture That Emerges
From these 9 known laws, the decoder architecture is:
| Physical Law | Decoder Feature |
|---|---|
| Landauer reversibility | Deterministic state transitions, no information erasure |
| Shannon entropy | Residual distribution should be maximally random |
| Jaynes maxent | Context model from empirical frequencies |
| Bennett reversibility | Inverse operations, history tape (FAMM scars) |
| Uncertainty principle | Complementary local/global models (composite address) |
| Bell correlations | Nonlocal context mixing (distant matches) |
| Bekenstein bound | O(1) basis state (holographic boundary) |
| Thermodynamic uncertainty | Speed-accuracy tradeoff (gear law) |
| Fluctuation theorem | Low-work trajectories preferred (basis fusion) |
This is PIST
Not because we assumed it. Because it is the unique architecture consistent with:
- Reversible computation (Landauer + Bennett)
- Entropy maximization (Shannon + Jaynes)
- Complementary addressing (uncertainty)
- Nonlocal correlations (Bell)
- Holographic state (Bekenstein)
- Speed-accuracy tradeoff (thermodynamic uncertainty)
- Optimal trajectories (fluctuation theorem)
What is NOT justified by known physics
- Fractional derivatives α = Φ⁻ⁿ
- Anthropic shear angle θ
- Four forces as resonant modes
- DNA as a PIST decoder
- Torsional cosmology
These are speculative extensions. They may be correct, but they are not derivable from the 9 laws above.
What IS justified
The PIST decoder, with its:
- Deterministic composite addressing
- O(1) basis state
- Reversible prediction
- FAMM scar memory (history tape)
- Gear ratio (speed-accuracy tradeoff)
- Noncommuting prediction layers (complementary models)
is a thermodynamically optimal decompression engine.
The Remaining Question
Given this architecture, what is the best basis function for enwik9?
Not the speculative physics. The empirical question: what 16-byte basis, combined with what composite address mixing, minimizes the residual entropy?
The physics gives us the structure. The data gives us the parameters.