[petalas]

Summary

Scoped mathematical proof with an exact finite certificate. For the actual eight-mode matrix M from Dillon Amburgey's PR4933, an explicitly feasible mode fiber has exact integer rounding radius one:

x = (e_0 + e_7 + e_20 + e_111)/2
M^T x = M^T e_48
min { ||z-x||_infinity : z in Z^112, M^T z=M^T x } = 1

An error smaller than one forces zero outside the four fractional coordinates. Two preserved modes then force z_0-z_7=0 and z_0+z_7=1, contradicting integrality. The output e_48 attains radius one. Translation and scaling give the same obstruction at every grid spacing. Arbitrary nonlinear rules and correlated randomness are allowed, with mode equality required for each returned output. Equality only in expectation is outside this contract.

This contribution was produced by Nick Petalas. Credit Dillon Amburgey/PR4933 for the mode matrix and inverse-code factorization, JoyboyBrian/PR5069 for the fixed-grid arithmetic context, Dugongue's coded producer and their retained native/controller/compiler lineage. Ali Romman's PR5838 distinguishes possible physical correlations from independently adverse defects; that word and error calculation are not replayed. Classical lattice and finite-box arguments retain their credit. No new kappa or selected-frontier change.

Evidence

• Before: ordinary unconstrained nearest rounding has half-unit coordinate error; exact preservation of these eight modes is a new requirement.
• After: the feasible fiber above excludes every output within the open unit box and attains distance one. All sixteen integer corners are enumerated. Column-oriented and separate row-oriented calculations agree.
• Warning-strict normal and optimized checks pass 91 translated/scaled attaining fixtures and nine rejection controls. Source corruption, a wrong lift and a sum-preserving corner with wrong modes reject. Each registered check takes about 0.14 seconds locally under CPython 3.12.15, well below its 30-second limit.
• Registered replay, exact committed bytes/modes and declared source ancestry pass at 4659323. The official seven-file export passes input coverage. Both direct modes pass from a read-only Git-free export; all eight exported files, including its export manifest, retain their bytes and modes. Generated receipts bind package/source hashes and are emitted to stdout.

From the repository root, with the local interpreter path shown explicitly:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/mode-preserving-rounding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/mode-preserving-rounding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/mode-preserving-rounding
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/mode-preserving-rounding --check-source-history --check-modes

Any CPython 3.12 executable may replace that machine-specific path. Expected markers include PASS_MODE_PRESERVING_ROUNDING radius=1 corners=16 fixtures=91 controls=9 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is one sharp feasible-fiber result and a uniform lower obstruction, not a global radius-one upper theorem. It does not claim that a native caller reaches the witness or must preserve modes exactly. A finer grid, approximate mode preservation or retained residue state changes the contract. Physical rounding, full propagation, restoration, precision/storage/tape costs and supplier admission remain open. All five framework assumptions remain. No predecessor theorem audit, selected scientific replay, Lean build, Docker run or complete multiplication theorem is supplied. All registered commands ran locally. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the least uniform radius over all nonempty integer fibers of this exact mode map. Start with the integer kernel lattice's fundamental domain; a global attaining construction is missing.
• Bind approximate mode preservation to one reached physical caller state and propagate its permitted mode defects. Reachability and complete physical costs remain blockers.

Related open work

• Prove the exact radius-three full recovery fiber threshold (no κ) #6269 uses the same source mode matrix in a fixed rounded inverse and proves a radius-three input-box threshold for a full recovery fiber. This result changes the program contract to arbitrary output rounding with exact mode preservation, and computes output distance on a different fiber. The two radii are different quantities and are not compared.
• Prove linear drift in a rounded native identity cycle (no κ) #6258 proves drift of a fixed rounded native identity cycle. It supplies no exact mode-preserving rounding radius; this proof neither replays nor changes that cycle.

Completion refresh checked the six other open account PRs created since the CLAIM: #6307, #6308, #6309, #6310, #6311 and #6313. None studies this mode map or its output-fiber rounding radius. The related lines above remain current; no common computed quantity disagrees.

PR4933, PR5069, PR5323, PR5444, PR5838 and PR6096 are merged contextual predecessors. Their inverse, recovery, error and residue statements are cited at their stated scopes, without extra computations to check their reported values.

CI notes

Both hosted checks passed at 46593234bfe658b63fb829f0d22b7eb216e392a2 in Registered research packages run 41107770945: plan in 4m17s and replay (research/mode-preserving-rounding) in 1m23s. The checks were read once. Separate run metadata confirms completed status, success and the matching head SHA. Hosted replay executes the registered normal and optimized checks. All local scientific, publication and portable-export checks listed above also pass; no registered command was deferred to CI. These checks do not establish independent human or formal proof-kernel acceptance.

Merge danger

Door: two-way. Blast radius: isolated. Six files are added under research/mode-preserving-rounding. No shared implementation or selected-frontier record is edited.