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allaunthefox 5331d2cc4e feat(dna): unified theory — DNA encoding, epigenetic computation, logarithmic vector spaces

Derivation from first principles:

1. Hachimoji DNA encoding (8 bases, ASCII-ordered, monotone LUT)
2. Imaginary Semantic Time (observer-independent semantic axis)
3. Sieve observers with CRT reconciliation (mod ℓ projections)
4. Semantic mass (E - E_min, E_s = m · 8²)
5. Gap preservation theorem (cleanMerge_preservesGap from GraphRank.lean)
6. Epigenetic computation (bistability, spreading, memory, attractors)
7. Logarithmic vector spaces (Kritchevsky: log N is a geometric vector)
8. Uncomputability framework (baseless logarithm = truth, based = computation)

Epigenetic optimizer breaks the freeze point:
  n=20: 0.7s (brute: 0.3s)
  n=24: 1.5s (brute: FROZEN)
  n=30: 3.4s (brute: FROZEN)
  n=50: 23.9s (brute: FROZEN)

Files:
  docs/UNIFIED_THEORY.md — full theory derivation
  docs/HACHIMOJI_DNA_SYNTAX.md — formal syntax specification
  docs/EPIGENETIC_COMPUTATION.md — epigenetic optimizer
  docs/UNCOMPUTABILITY.md — logarithmic vector space framework
  docs/REDERIVATION.md — rederivation from first principles
  python/dna_*.py — implementation (codec, LUT, GPU, surface)
  tests/test_dna_*.py — 68 tests, all green

Build: N/A (Python + Lean documentation)

2026-06-23 02:18:16 +00:00

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Hachimoji DNA Encoding Specification

Version: 0.1 Status: Draft Purpose: Computational substrate, not compression format.

1. Alphabet

Eight bases, ASCII-ordered for monotone lexicographic sorting:

Index	Base	Phase	Bits
0	A	0°	000
1	B	45°	001
2	C	90°	010
3	G	135°	011
4	P	180°	100
5	S	225°	101
6	T	270°	110
7	Z	315°	111

Key property: ASCII sort order = index order = lexicographic rank. This means sorted(sequences) produces the same order as sorted(sequences, key=dna_to_int).

2. Symbol Encoding

Each symbol (byte, word, or chunk) maps to a fixed-length DNA sequence.

1-byte chunks: 256 symbols → 3 bases/symbol (8³ = 512 ≥ 256)
2-byte chunks: 65,536 symbols → 6 bases/symbol (8⁶ = 262,144 ≥ 65,536)
n unique symbols: ceil(log₈(n)) bases/symbol

The LUT assigns sequences by rank:

Rank 0 → "AAA...A" (lowest)
Rank 1 → "AAA...B"
Rank n → highest sequence

3. Monotone Property

If symbols are ranked by frequency (most frequent = rank 0), then:

Most frequent symbol → shortest/lowest DNA sequence
Lexicographic sort of DNA = frequency sort of symbols
This is NOT compression — it's structured representation

4. LUT Format

{
  "format": "hachimoji_lut_v1",
  "bases": "ABCGPSTZ",
  "chunk_size": 1,
  "bases_per_symbol": 3,
  "entries": {
    "AAA": "20",
    "AAB": "65",
    "AAC": "74"
  }
}

Keys: DNA sequences (base-8 encoded ranks)
Values: hex-encoded byte chunks
Portable: JSON, human-readable, self-describing

5. File Format

.dna file:  raw DNA sequence (text, A-Z only)
.lut file:  JSON LUT (see above)
.meta file: encoding metadata (optional JSON)

6. Roundtrip Guarantee

decode(encode(data)) == data

Verified at encode time. No lossy steps. No approximation.

7. Computational Properties

The DNA sequence is not just encoded data — it's a computational address:

Sequence index (base-8 integer) = symbolic rank
Lexicographic order = rank order (monotone)
LUT lookup = O(1) per symbol
Sortable by standard string sort (= rank sort)
Portable across systems (plain text + JSON)

8. Integration with SilverSight

HachimojiCodec.lean — formal encode/decode specification
HachimojiLUT.lean — LUT hierarchy (k=2, k=6, k=50)
PhaseCircle — ℤ/360ℤ address space
dna_lut.py — Python LUT implementation
dna_encode_file.py — file encoder/decoder

9. Not This

❌ Not a compression format (bits/byte not the goal)
❌ Not a DNA storage format (no synthesis/sequencing constraints)
❌ Not a cryptographic scheme (no security claims)
✅ A computational substrate for manifold/QUBO/eigenvalue work

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