[zxv]

Result

The exact integer-grid protocol from PR #6066 can retain 76 residue bits instead of 84, preserving the same stochastic forward map, every arbitrary dirty input, the unchanged random tape and the compute–accumulate–uncompute endpoint. A matching 2^76-point fiber proves 76 is optimal for worst-case retained information at this logical boundary. Eight saved bits are a reduction of 2/21, approximately 9.52%.

The inverse infers one omitted half-SCALE parity in each of eight tail blocks. Three use one existing role's parity; five use the sum of two roles' parities. Symbolic source-derived invariants cover all integers and every coin tape, including fractional ADDs. The other 76 bits retain the original save/reconstruct/clear lifecycle. This adds thirteen existing-value parity reads; it is not a claim of lower physical runtime or total working memory.

For the lower bound, the first 8,411 events are surjective and exactly 2^60-to-one. Eight independent four-preimage tail collisions multiply this to 2^76 for one allowed all-zero tape. The proof uses a Cartesian product and composition, not enumeration of that many inputs. The matching inverse also proves no fixed-tape fiber is larger.

Significance for researchers

This closes the retained-log question in PR #6066's chronological half/double interface. The 32 doublings do not make a 52-bit log possible, while all 84 local parities need not be retained. Future work can price this optimal logical boundary and its actual scratch/cleanup obligations instead of searching for a smaller log under the same map. No κ increase is claimed.

Validation

CPython 3.12 normal and optimized registered checks pass. They authenticate the historical PR #5599 word and the literal predecessor interpreter; check all 32 finite collision inputs and complete nominated block fibers; derive both all-tape parity invariants; run 86 complete lifted collision executions; and compare 64 full restored protocols against the unmodified predecessor forward implementation. Signed-half, wrong-holder and invalid raw-half-count controls are included. Exact committed-byte preflight passes at 57d183c60dddb18e1c44b15e5c39f3f3a6daaab7.

python3.12 -B scripts/research_package.py --package research/stochastic-restoration-memory
python3.12 -B scripts/check_research_commit.py --package research/stochastic-restoration-memory

This is the historical prefix used by the predecessor, not a transfer to the different selected PR #6030 word. The 76-bit optimum concerns retained state at the forward logical boundary. Physical integer/tape working scratch, cleanup costs, practical guard widths, the next native MOVE and the full error/caller budget remain open. The complete collision family has a conservative finite signed/guard bound of 4,571 bits; executed representative collision fixtures use ten magnitude bits. No native/selected cold replay, independent human review or proof-kernel acceptance is claimed.

Only the isolated research package is added. Pending predecessor statement and verifier bytes are retained with exact heads and hashes, so no foreign pending history is needed for replay. The unchanged base navigation failures addressed by PR #6050 remain separate.

Open next questions

Lower the eight inferred-parity sites, thirteen existing-value parity reads and residue cleanup to an explicit reversible integer/tape program, charging all working scratch and head movement. Separately, pin one actual native MOVE and its completed error/guard contract before composing the caller. Retained-log optimality does not discharge these costs or the five framework assumptions.

Prepared by zxv with substantial OpenAI Codex assistance. Credit Nick Petalas's PR #6026, Ali Romman's PR #5599, the PR #6066 ABI and retained source/P104 lineage. Apache-2.0. Fiber counting and parity invariants are classical; the source-bound optimum and matching compressed log are the contribution.

Clean committed-export integrity/input coverage and optimized Git-free replay also pass at the submitted head. Hosted run 40638657197 stops during shared planning on the two unchanged navigation failures addressed by PR #6050, before this package's scientific replay is scheduled. Local scientific success is reported separately from that CI blocker.