[petalas]

Summary

Scoped written proof with an exact finite certificate: the fixed PR5444 rounded inverse at three relative bits has a full 4096-preimage fiber inside [-H,H]^112 exactly when the integer radius permits H>=3. A new output attains radius three; an all-output midpoint argument excludes radius two. This answers one explicit finite-range question in zxv's PR5444. No new kappa or selected-frontier change.

F(x) = 8x - M round_away(8 M^T x / 52)
full fiber = 4096 distinct integer inputs
published sufficient radius: 16
new exact minimum radius: 3

Only research/rounded-native-fiber-radius/ is added. The source matrix is copied byte for byte with Git/blob/SHA256 provenance. The new proof and literal output witness are in PROOF.md and WITNESS.json. This contribution was produced by Nick Petalas. Credit zxv/PR5444 for the question, fiber geometry and lattice/code premises, Ali Romman/PR5323, David Ganz/PR4929, Dillon Amburgey/PR4933, Brian/PR5035 and the retained native/controller/precision lineage. Classical lattice and coding facts retain their credit.

Advance CLAIM. Base canonical main 2485805a6d47f36ea83ddde6327d453ac82d44dc, priority revision 8. The historical uniform-transfer-audit card is explicitly changed to this original finite-range side theorem, not a predecessor audit or a selected-assumption discharge.

Evidence

Before, PR5444 leaves the finite-range threshold open and supplies H=16. After, the midpoint proof excludes every output at H<=2, and exact enumeration gives a full radius-three witness.

• All 65,536 quotient candidates are considered. Exactly 4,096 distinct inputs pass the congruence and literal forward map. This fiber contains 16 inputs of radius one, 1,055 of radius two and 3,025 of radius three. These are this fiber's counts, not global maxima in smaller boxes.
• All sixteen 256-point Cartesian cells, eight quotient-coordinate pairs and every 112-owner interval/midpoint are checked. Fifteen rounding-boundary, rejection and changed-scale controls pass.
• Warning-strict normal and optimized direct checks pass with identical generated hash-bound receipts on CPython 3.12.15, macOS arm64. Individually timed commands take 1.904 and 1.912 seconds, below their 120-second limits. These are replay-deadline observations, not a speedup claim.
• Both registered commands and metadata/pins pass. Committed bytes/modes and full declared source ancestry pass at d4f642424b374cdabd20355427ee903165871257 for all eight package files.
• Official committed export and declared-input integrity pass. Normal and optimized checks pass from a read-only Git-free export with empty PATH; all nine exported files, including the export manifest, retain their content and read-only modes. Actual matrix-byte, output and false-radius mutations reject for their intended reasons in both modes.

Exact reproduction from the repository root:

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

Any standard-library CPython 3.12 executable may replace the displayed local interpreter. Expected markers are PASS_ROUNDED_NATIVE_FIBER_RADIUS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The standalone verifier needs only the eight package files and writes no files.

Limitations

The result fixes the literal 112-by-8 matrix, integer input lattice, origin-centered coordinate box, three relative bits and half-away rounding. The universal lower bound is a written proof using credited predecessor fiber/lattice premises; finite enumeration alone does not prove its all-output statement. Independent human and formal proof-kernel review are unclaimed.

Smaller-fiber optima, other precisions, repeated frames and physical caller restrictions are not classified. Native realization, reversible codec cost, source access, dirty restoration, precision/tape/stock accounting and a favorable complete supplier invoice remain open. The selected conditional kappa .003125804 and all five framework assumptions retain their status. No predecessor scientific suite, selected cold replay, Lean, Docker or full multiplication theorem is claimed.

Open next questions

1. Determine a finite-range threshold for another row of PR5444's table. Begin with that row's equality geometry; no other precision is analyzed here.
2. Bind this obstruction to one actual recovery site, including its lattice, allowed values, observations and restored outputs. The missing caller restriction and complete physical bill determine applicability.

Related open work

No currently open PR studying this fixed PR5444 map or its finite-range threshold was identified in the bounded live searches. Main README and merged searches, current handoffs and local reservations were also checked; sparse notes and unpublished work remain coverage limits.

• PR6258 studies drift and history for a different 512-coordinate rounded native cycle. Its tag bound excludes fixed bounded boxes; this PR studies one fixed 112-owner map and computes none of its orbit quantities.
• PR4172 studies allocator/codec history and physical QQ16 action costs. It supplies no finite-range value for this map, and no action count is compared here.
• PR5444 and PR6182 are merged predecessor context. PR6182 studies coordinate-parity tags at four relative bits; this PR answers the separate input-radius question and does not change that tag interface.

The completion refresh inspected all twelve open account PRs created since the advance CLAIM, including this PR. None of the eleven other contributions studies this fixed PR5444 map or computes its finite-range threshold. No same-instance numerical disagreement was identified, and no additional computations checked other submissions.

CI notes

Both hosted checks, plan and replay (research/rounded-native-fiber-radius), passed at exact head d4f642424b374cdabd20355427ee903165871257 in run 41066923358. The completed checks were read once; separate run metadata confirms completed status, success and the matching head. All declared local checks and publication/export controls above also pass. Hosted replay certifies the registered finite checks and source bindings; independent human and formal proof-kernel acceptance of the written all-output argument remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Only one new research package is added; shared tools, frozen predecessors, upstream sources and the selected record are unchanged.