Trinary Base729 Encoding Specification

Category: Protocol / IP Source Repo: BlackRoad-OS/blackroad-os-archive Source Path: ip/trinary-base729-encoding.md Source Commit: N/A (initial archival) Archived On: 2025-11-28 Archived By: agent:cece Status: Final

Purpose

Define a readable, trinary-aligned encoding for cryptographic memory hashes. Base729 maps balanced ternary sequences into 729 unique glyphs so that truth-state hashes reflect paraconsistent logic states instead of hexadecimal noise.

Design Goals

Trinary native: Preserve {affirm (+1), null (0), contradict (-1)} semantics across the encoding pipeline.
Readable: Produce short, human-typable strings with meaningful glyphs rather than hex strings.
Deterministic: Round-trip encoding/decoding with no information loss from the underlying hash bits.
Composable: Slot directly after PS-SHA∞ or SHA-256 to represent agent truth-state hashes and registry IDs.

Alphabet

A Base729 digit encodes 6 balanced trits (3^6 = 729). Represent each digit as a pair:

Radix glyph (27 options): A–Z plus _ (underscore) to keep the set ASCII-friendly.
Orientation index (0–26): Either numeric (0–26) or stylized rotation marker in UI renderers.

A Base729 character is written as <glyph><orientation>, e.g., C14 or Q03. UI layers may substitute bespoke glyph art, but the canonical plain-text form MUST preserve the glyph letter and two-digit orientation with leading zeroes.

Encoding Pipeline

Hash input: Compute a 256-bit digest via PS-SHA∞ or SHA-256.
Binary → balanced ternary: Convert the 256-bit integer to balanced trits using digit set {-1,0,1}. Pad the leading trit group so the length is a multiple of 6.
Group into sextets: Chunk the trits into groups of 6 (most significant first).
Map to Base729 digits: Interpret each 6-trit chunk as an integer d ∈ [0,728] using balanced-trit weights 3^5 … 3^0.
Emit glyphs: glyph = BASE27[d // 27], orientation = d % 27. Render as <glyph><orientation> with orientation zero-padded to two digits.

Decoding Pipeline

Parse glyphs: For each <glyph><orientation> token, recover g = index(BASE27, glyph) and o = int(orientation).
Recover digit: d = g * 27 + o (0–728).
Digit → trits: Expand d into 6 balanced trits using positional weights 3^5 … 3^0 with digit set {-1,0,1}.
Reassemble integer: Concatenate trits, convert back to unsigned integer, and truncate leading padding trits to return the original 256-bit value.
Re-hash check: Optional verification by recomputing the source hash and matching against the decoded value.

Reference Pseudocode (Pythonic)

BASE27 = list("ABCDEFGHIJKLMNOPQRSTUVWXYZ_")

BALANCED = {0: 0, 1: 1, 2: -1}  # maps standard trits to balanced


def to_balanced_trits(n: int, min_len: int) -> list[int]:
    trits = []
    while n > 0:
        n, rem = divmod(n, 3)
        trits.append(BALANCED[rem])
    while len(trits) < min_len:
        trits.append(0)
    return list(reversed(trits))


def encode_base729(digest_bytes: bytes) -> str:
    n = int.from_bytes(digest_bytes, "big")
    trits = to_balanced_trits(n, min_len=6 * math.ceil((len(digest_bytes) * 8) / 6))
    tokens = []
    for i in range(0, len(trits), 6):
        sextet = trits[i : i + 6]
        d = sum((t + 1) * (3 ** (5 - j)) for j, t in enumerate(sextet)) - 364  # shift back to balanced
        glyph, orient = divmod(d, 27)
        tokens.append(f"{BASE27[glyph]}{orient:02d}")
    return "-".join(tokens)

Note: The -364 shift centers the 0–728 digit range back to balanced form because each balanced trit is represented as {-1,0,1}. Any production implementation should include constant-time operations and thorough test vectors.

Example (truncated)

Input: ps_sha∞("7Ψ∞kL3x@Amu…") → digest starting 0x6fd1…
Balanced trits: [-1, 0, +1, …]
Base729: M05-R18-A00-…

Full examples should be generated from fixed seeds with published vectors inside the PS-SHA∞ test suite.

Integration Notes

Prism Console: Use Base729 hashes for truth-state registry entries to expose contradiction-heavy states visually.
Capability Registry: Encode capability IDs in Base729 to align with trinary logic in audits.
RoadChain: RoadCoin minting and trinary smart contracts can reference Base729 transaction proofs.
Unity Agents: Render glyph + orientation as animated spiral tiles; orientation animates the θ phase of U(θ,a)=e^(a+i)θ.

Future Work

Publish a glyph atlas (27 glyphs × 27 orientations) with Unicode private-use fallbacks.
Add streaming encoders for large transcripts (chunked hashing + Base729 framing).
Provide Rust and TypeScript reference implementations.
Extend the format with optional checksum trits for error detection in noisy channels.

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