blackroad-os-archive/ip/trinary-encoding-spec.md

Trinary Encoding Specification (Base27, Base81, Base729)

Category: IP | Protocol Source Repo: BlackRoad-OS/blackroad-os-archive Source Path: ip/trinary-encoding-spec.md Source Commit: N/A (initial archival in this repository) Archived On: 2025-11-28 Archived By: agent:gpt-5.1-codex-max Status: Living Document

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Purpose

Define a balanced-ternary encoding stack that bridges binary data to higher-density alphabets suitable for RoadChain and related Lucidia components. The spec emphasizes deterministic mapping, collision-free symbol choices, and graceful handling of paraconsistent states (−1, 0, +1) used in the coherence formula:

[ C(t) = \frac{\Psi'(M_t) + \delta_t}{1 + |\delta_t|} ]

Balanced Ternary Model

• Digits (trits): −1, 0, +1
• Semantics:
  • −1 → Negation / contradiction
  • 0 → Null / unknown / superposition
  • +1 → Affirmation / confirmation
• Information density: log₂(3) ≈ 1.585 bits per trit.
• Normalization: The absolute value in the coherence formula collapses sign while preserving contradiction magnitude, making balanced ternary a native fit.

Alphabet Tiers

Base27 (3³)

• Symbols: A–Z, _ (27 total)
• Indexing: 0–25 map to A–Z; 26 maps to _.
• Use Cases: Compact labels, namespace prefixes, checksum syllables.

Base81 (3⁴)

• Symbols:
  • 26 uppercase: A–Z
  • 26 lowercase: a–z
  • 10 digits: 0–9
  • 19 specials: ! @ # $ % ^ & * ( ) - _ = + [ ] { } |
• Ordering (value → symbol):
  • 0–25: A–Z
  • 26–51: a–z
  • 52–61: 0–9
  • 62 → !, 63 → @, 64 → #, 65 → $, 66 → %, 67 → ^, 68 → &, 69 → *, 70 → (, 71 → ), 72 → -, 73 → _, 74 → =, 75 → +, 76 → [, 77 → ], 78 → {, 79 → }, 80 → |
• Rationale for specials: All ASCII, visually distinct, shell-safe, and avoid whitespace/quote ambiguity.
• Use Cases: URI-safe payloads, human-facing keys, structured labels where case sensitivity is acceptable.

Base729 (3⁶)

• Symbolization options:
  1. Base27 pairs (recommended): Encode each 6-trit block as two Base27 symbols (27 × 27 = 729). Example mapping: high-order trits → first symbol; low-order trits → second symbol.
  2. Base81 digraphs: Use two Base81 symbols but restrict to the first 729 combinations for backward compatibility; reject/flag overflow combos >=729.
  3. Unicode glyph set: Reserve a contiguous Unicode block (e.g., Mathematical Operators) and map indices directly for dense, single-symbol rendering.
• Use Cases: High-density archival strings, PS-SHA∞ representations, inter-agent coherence payloads.

Encoding Pipeline

1. Binary → Integer: Treat the byte string as a big-endian integer.
2. Integer → Balanced Ternary: Repeatedly divide by 3; remap remainders 0,1,2 to ternary digits 0,+1,−1 using carry to keep digits in {−1,0,+1}.
3. Group Trits: Chunk from most significant to least into:
   • 3-trit groups → Base27
   • 4-trit groups → Base81 (pad with leading zeros as needed)
   • 6-trit groups → Base729
4. Map to Alphabet: Use the ordering tables above to convert each group to its symbol.
5. Padding/Alignment: When input length is not divisible by the group size, left-pad with 0 trits; record original bit length in framing metadata when round-tripping exact byte counts matters.
6. Error Handling: Reject symbols outside the defined alphabets; for Base81 digraphs used as Base729 surrogates, enforce the 0–728 range and emit a contradiction flag (−1) for out-of-range pairs.

Binary Compatibility Notes

• Lossless decode: The balanced-ternary conversion is reversible; carries are deterministic and do not alter numeric value.
• Flag-friendly trits: The −1/0/+1 space can embed contradiction or integrity bits without leaving ternary space.
• Interchange with Base64: For interoperability, consider a shim where Base64 sextets are reinterpreted as 0–63 integers and lifted into balanced ternary blocks before symbol mapping, avoiding floating carries across block boundaries.

Example Mapping (6 Trits → Base729 via Base27 pairs)

1. Trits: [+1, −1, 0, +1, +1, −1]
2. Value: ((+1)·3^5 + (−1)·3^4 + 0·3^3 + (+1)·3^2 + (+1)·3 + (−1) = 162 − 81 + 0 + 9 + 3 − 1 = 92)
3. High digit: 92 // 27 = 3 (symbol D), low digit: 92 % 27 = 11 (symbol L)
4. Encoded pair: DL

Integration with Coherence Formula

• Contradiction channel: Represent δ_t directly as a trit (−1, 0, +1); downstream modules can attach this trit alongside payload trits without widening the alphabet.
• Normalization: Because |δ_t| collapses sign, a single trit can both mark contradiction and participate in normalization without leaving ternary space.
• Hash embedding: PS-SHA∞ outputs can be rendered in Base729 for density while preserving explicit contradiction markers during aggregation.

Implementation Checklist

• Reference encoder/decoder covering Base27/Base81/Base729
• Test vectors for binary round-trips and contradiction flag propagation
• Service-level framing that records bit-length for exact byte recovery
• Optional Unicode glyph map for Base729 when single-symbol density is required