Outward digit mapping for RedTail

The requested notation is interpreted provisionally as two unbounded integer rays: (-infinity, -B] union [B, +infinity). B is a positive integer boundary; this experiment uses B=16. The open middle interval is excluded. This is a new mapping, not a mathematical inverse of a saturating clamp, which has already lost distinctions.

For every integer digit or accumulated digit value x, define:

F_B(x) = x − B       for x <= 0
         x + B − 1   for x >= 1

The inverse on the two output rays is:

F_B_inverse(y) = y + B       for y <= −B
                 y − B + 1   for y >= B

This is one-to-one: each branch is a translation, and their output ranges do not overlap. Zero is assigned to the negative endpoint. This explicit zero convention makes binary 0 map to −B and binary 1 map to +B. For ternary digits −1, 0, +1, outputs are −B−1, −B, +B; zero no longer has a neutral numerical output. An additional neutral output would require changing the two-ray alphabet.

At B=16, input −2, −1, 0, 1, 2 maps to −18, −17, −16, 16, 17. The ASCII byte A, binary 01000001, maps digit by digit to [−16, 16, −16, −16, −16, −16, −16, 16]. Digit ordering and leading zeros are retained.

Infinity denotes an unbounded range; no input is replaced by a literal infinity. Fixed binary digits alone produce only the two endpoints. If later dynamics generate arbitrary integers, the same rule places those values along the outer rays while retaining their offsets.

The trial applies this mapping after RedTail’s complete length-bearing 4+2 record. It does not pass negative numbers into the production logarithm guard or reinterpret signed integers as GF(256) elements. The established scalar clamp and previous Refracting Light remain unchanged.

Verification

The executable checks 65,536 signed integers and all 65,536 two-byte binary messages. All outputs are distinct within their respective tests, and all inverse checks pass. In comparison, ordinary saturation to [−16,16] has only 33 scalar outputs on the signed test domain. Five representative full RedTail records also round-trip, including empty input and leading-zero messages. Arbitrary-precision checks at positive and negative 10^100 pass.

The injectivity argument applies beyond those finite tests while complete integer values and ordered digit sequences are retained. The output is a representation, not a fixed-size digest. Clipping, taking magnitudes, discarding order, or reducing modulo a fixed word size can lose that property. A later 64-bit hash still requires independent collision analysis. No post-quantum security is established by this transform.

Run python3 outward_digit_clamp.py. Results are saved in outward-digit-clamp-results.json. This is a provisional implementation of the outward-branch interpretation, pending any correction to the intended boundary or treatment of zero.