[aliromman]

This construction makes the rounded inverse-code output exactly recoverable with eight quotient tags modulo 13. It proves that 13^8 retained states, or 30 packed bits, are necessary and sufficient per real scalar coordinate when H >= 24. A simple separate-field layout uses 32 bits. The complete κ calculation and selected frontier are unchanged.

Proof, exact receipt, source pins, and checker. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.

t = M^T x; q = round_away(t/52); y = x-Mq
retain a_j = q_j mod 13
recover q from s=M^T y, its tag, and:
ceil((-26-s)/4) <= q <= floor((26-s)/4)
round_away((s+56q)/52) = q
x = y+Mq.

The recovery proof covers every integer input and the source nearest-rounding rule with ties away from zero. At most 14 scalar candidates need inspection per mode. The zero fiber contains all x=Mv with v in {-6,...,6}^8; full column rank gives the sharp state lower bound. Tags are retained owned output.

Use CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/inverse-code-recovery-tags
python3.12 -B scripts/check_research_commit.py --package research/inverse-code-recovery-tags

Both registered modes passed under Python 3.12.13, with warnings as errors and with -O. Expected verifier marker: PASS_SHARP_INVERSE_KERNEL_TAGS. The checks cover the exact Gram identity, all valid scalar residuals for q=-200,...,200, 40 actual source-kernel round trips, zero-fiber fixtures and a collision for 12 residue classes. The large fiber count follows from the proof; it was not enumerated. The committed-byte check, export inventory check and both direct exported replay modes passed. git diff --check passed. Full-history validation was not run in the shallow checkout. No full physical caller, Lean, native/compiler or selected replay ran.

This is an injective interface, not a priced reversible circuit. Tag storage, movement and later uncomputation remain caller obligations; tags cannot be erased while retaining only y.

This packages the completed Discussion result as the same contribution with a committed proof and replay. @JoyboyBrian’s PR #5035 supplies the finite kernel, rounding rule and original collision. @dillona’s PR #4933 supplies the mode factorization and physical wrappers. The consumed files are pinned at the base in SOURCES.json. @mdumitrean and the retained native, ownership, precision and compiler authors keep their source credit. New proof and checks: Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Apache-2.0. This is author-checked evidence; independent acceptance is not claimed.