[petalas]

Result

This answers open question 3 of CubeJerry's #6327 (research/refined-native-recovery) for every chain depth. That question asked for the optimal recovery alphabet of longer parity-changing chains. The model is #6327's: the #5444 matrix M, the half-away kernel K, the chain z_1=K(2x), z_(j+1)=K(U_j z_j), and fixed frames U_j∈GL_112(Z) with MᵀU_j=Mᵀ and no parity restriction. F=51,008,161 is #6327's one-step maximum.

Claim type: scoped proof with an exact finite certificate. No κ change; the selected frontier is unchanged.

• For every depth t and every frame sequence, the zero output has at least F^t preimages. Equality holds exactly when the binary images of the cumulative framed matrices form a direct sum.
• Under a direct sum, every output has at most F^t preimages.
• For this M, all images lie in a 104-dimensional binary space, so a direct sum exists only for t≤26. An explicit 26-stage source frame sequence (FRAMES.json) attains it.

So the optimal worst-case fiber is exactly F^t for 1≤t≤26 and strictly above F^t for t≥27. The minimum packed tag is ceil(t·log₂F) bits: 26, 52, 77, …, 666 at t=26. At t=2 this reproduces #6327's 52-bit theorem and its separation criterion.

The upper bound conditions on one stage at a time instead of using a joint Fourier majorant; #6327 shows that majorant fails for three parities. Under a direct sum, each stage's parity coset is fixed by the output alone. Each stage then ranges over a box of at most 13 values per mode, which holds at most F points of one coset. The depth cap holds because 2Z^8⊂MᵀZ^112, which puts im(M mod 2) inside the reduction of ker_Z Mᵀ (dimension ≤104).

Evidence and reproduction

From the repository root, with ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12.15), standard library only:

python3.12 -I -W error -B research/refined-recovery-longer-chains/verify.py
python3.12 -I -W error -O -B research/refined-recovery-longer-chains/verify.py
python3.12 -B scripts/research_package.py --package research/refined-recovery-longer-chains
python3.12 -B scripts/check_research_commit.py --package research/refined-recovery-longer-chains --check-source-history
python3.12 -B research/refined-recovery-longer-chains/build_frames.py "$SCRATCH/FRAMES.json" && cmp "$SCRATCH/FRAMES.json" research/refined-recovery-longer-chains/FRAMES.json

Local results: both verifier modes print PASS_REFINED_RECOVERY_LONGER_CHAINS in about 0.3 s. The package check prints PASS_REGISTERED_RESEARCH_PACKAGE, the commit check prints PASS_RESEARCH_COMMIT_BYTES, and the regenerated FRAMES.json is byte-identical.

The verifier checks:

• the fixture hash, MᵀM=56I, the code C and F;
• the interval lemma on s∈[−4000,4000];
• the box-coset maximum, exhaustively;
• 2e_i∈MᵀZ^112;
• for every frame block, MᵀV=0 and EV=0, plus rank 104 of all 26 binary images;
• the inverse-chain identity at depths 3 and 26;
• that the parity-I zero-fiber formula reproduces Prove optimal 52-bit refined recovery over mode-preserving frames (no new κ) #6327's A_t at t=1,2,3;
• that a perturbed block and a repeated frame are both rejected.

Limitations

• The all-depth statements are proved in PROOF.md. The finite checks cover the source facts and the explicit frames.
• The predecessor's interval lemma (|Q(s)|≤13 for every s) is cited from Prove optimal 52-bit refined recovery over mode-preserving frames (no new κ) #6327's CHAIN_PROOF and only rechecked on a window.
• For t≥27 only the strict inequality is proved.
• The interface assumptions are Prove optimal 52-bit refined recovery over mode-preserving frames (no new κ) #6327's: fixed known frames, exact integers and full output access. The output map changes with the frames.
• No codec, physical tag, caller or restoration cost is supplied. All five selected assumptions remain.
• No independent human review or formal proof-kernel check has been done.

Open next questions

1. The exact optimum at t≥27, where the images must overlap. The zero-fiber kernel formula gives a lower bound for each overlap pattern.
2. Mode-changing frames, or extra refinement between calls.
3. A caller that accepts the framed output at a favorable complete cost.

Related open work

No open PR studies this chain or quantity, including this account's open PRs created since the CLAIM (#6652–#6662), none of which cites #5444 or #6327. #6327 is merged and is the direct predecessor. Its 52-bit two-stage value is reproduced here as the t=2 case.

CI notes

All checks passed at head b6de532: plan and replay (research/refined-recovery-longer-chains) in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41662409899.