[petalas]

Summary

Scoped written proof with exact finite corroboration. For the degree-R consecutive-interval Gaussian Newton residual under its max-descendant creation-precision contract:

before: R=0 forces full precision at all proper Walsh prefixes
 after: every 0<=R<n forces full precision through depth n-R-1
        the next depth has a zero-demand prefix, for every offset
        M >= 2^n (n-R-1) (p+2n-R-1)

For n>=4 and 1<=R<=floor(n/2)-1, the residual gather alone has ADD precision mass at least (n/2)*2^n*(p+n), even granting a free bulk. At p=Theta(n), that retains the input-bit-length times log-input-length scale in this metered graph. This removes the uniform-bulk-precision premise from the applicable low-degree part of #5825's obstruction. No high-degree heterogeneous-bulk theorem or new kappa is claimed.

The proof reduces the first surviving Gaussian digit to generalized-binomial parity, then uses a finite-difference identity to exclude a gap of R+2 consecutive weights. The interpolation band supplies the sharp cutoff witness. It also gives an exact full-precision prefix count at every depth. The classical identities are credited as such.

This contribution was produced by Nick Petalas. Credit Ali Romman #5825 for the degree-zero theorem and explicit residual-tree contract, and David Ganz / Dugongue #5815 and #5813 for the valuation-pruned Gaussian Newton residual and spectral predecessor, with their retained native, precision, ownership and compiler lineage. Predecessor results are cited; no predecessor verifier or fixture is audited or replayed.

Evidence

At head 8b37cd3d519365bbfe7404eccfba2262d970c6af, local CPython 3.12.15 checks pass:

• 49,726 exact positive-degree Gaussian coefficients through n=24, including 14,950 negative offsets; 2,576 sharp zero bands and 29,900 unit-containing consecutive windows.
• 630 complete trees through n=10, with every 551,700 literal binary prefix checked against an independent parity/count formula. Every tree has the sharp cutoff witness; 336 also check the free-bulk corollary.
• 16,448 signed-argument parity cases, 4,128 exact finite-difference windows and eight rejected underprecision profiles.
• Direct normal and optimized checks pass in 0.75 and 0.76 seconds respectively. Both complete registered commands pass; the combined launcher takes 1.83 seconds. Each command is below its 120-second timeout. These are check durations, not speedup claims.
• Package metadata, staged whitespace and committed-byte/source-history preflight pass for all six package files. Official export integrity and a read-only Git-free optimized replay with empty PATH pass, preserving all seven exported files including the manifest. Generated stdout receipts bind all six package files by SHA256.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -B research/newton-residual-prefix-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -O -B research/newton-residual-prefix-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/newton-residual-prefix-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/newton-residual-prefix-floor --check-source-history

Another clean CPython 3.12 installation can replace the shown interpreter. Direct replay ends PASS_NEWTON_RESIDUAL_PREFIX_FLOOR; registered replay ends PASS_REGISTERED_RESEARCH_PACKAGE. The verifier is standard-library-only and writes no files. Pending predecessor heads are citations, not execution inputs or required Git-history objects.

Limitations

The written argument supplies the universal proof; finite checks corroborate it. The graph, normalization and max-descendant demand contract are fixed. Sharpness concerns the full-precision depth, not attainment of the total mass lower bound. Higher coefficient valuations and later nonzero prefixes can add work. A changed representation, graph or integral-dependency treatment can evade this scoped result. No actual physical caller, native admission, restoration/storage protocol, complete bit-time tariff, rounded multiplier or complete multiplication theorem is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

The side-correction card is explicitly changed to this component obstruction in TASK.json. Canonical main remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8, refreshed before publication. Live handoffs, reservations, PR bodies and several main/merged-result term searches found no competing claim in the inspected coverage. Cached local overlap covers only 200 of 7,845 HEAD notes, so absence there is not a novelty certificate.

Open next questions

• Supply an arbitrary-input residual graph that exploits integral Walsh dependencies, with its exact quotient/precision map and full invoice. This tests a concrete escape from the fixed demand contract; no replacement graph is supplied here.
• Derive backward precision demands for a named high-degree heterogeneous falling-factor bulk when n-R is sublinear. The residual-only lower bound loses its n-scale factor there; that bulk question remains open.

Related open work

• Execute pruned Gaussian corrections and bound the standalone Newton-W route #5825, Ali Romman, head baea91d supplies the named graph and degree-zero theorem. This adds the positive-degree sharp cutoff and residual-only low-degree obstruction, without recomputing its numerical fixtures.
• Construct exact Gaussian corrections with pruned and mixed precision circuits #5815, David Ganz / Dugongue, head f0fcda5 supplies the valuation-pruned residual and separate Smith representation. This constrains the residual prefix contract; it changes no implementation invoice.
• Construct Gaussian spectral and precision compilers from the ETH preprint #5813, David Ganz / Dugongue, head af37059 supplies the Gaussian spectral bulk and residual context. This concerns the residual gather, independently of the bulk's precision schedule.
• Execute Gaussian Smith circuits and prove sharp precision requirements #5821 executes the separate Gaussian Smith circuit and studies its output/cut precision. That graph and those information requirements differ from this residual-tree depth.
• Prove the sharp R-unit Gaussian bulk denominator (no new κ) #5890 proves the central bulk's sharp denominator debt. That full-response Boolean-degree quantity differs from the residual-prefix precision cutoff; no additional computations were run to compare its results.
• Prove exact optimal ordering of heterogeneous Smith axes (no κ) #5896 optimizes heterogeneous Smith-axis ordering. The result here is order-independent for the symmetric residual weight spectrum and neither extends nor verifies that order optimization.
• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 studies reversible clean storage for Gaussian overwrite; its storage quantity and endpoint contract differ from this metered residual gather.
• Prove one-axis transported coframe optimization is NP-hard #5888 studies transported Gaussian coframe rank tariffs; its rank-transition optimization differs from this ordinary ADD precision bound.

Completion refresh inspected this account's eight other open PRs created since the CLAIM. None computes this residual's full-precision depth or gather mass. No extra mathematical computations were run to compare them.

• Prove exact contiguous helper storage: 5q versus peak 4q (no new κ) #5897 concerns contiguous helper storage and fragmentation; it does not study this residual graph or its precision demands.
• Prove the six-ADD bank-exchange floor with one clean helper #5898 concerns a binary bank exchange with a clean helper; its paid ADD count differs from this Gaussian ADD precision mass.
• Prove sharp minimum block length 112 for fixed tail gates (no κ) #5899 concerns a fixed two-tail compiler block-length gate; its integer threshold is separate from this residual's prefix cutoff.
• Prove exponential invoice frontiers at fixed count and rank mass #5901 concerns increasing-concave invoice frontier size; it does not compute a residual precision schedule.
• Prove sharp zero-phase witness length 2n+2 for n=2 mod 4 (no κ) #5902 concerns zero-phase witness lengths in a binary representation; its visibility thresholds differ from the residual-tree depth.
• Prove exact constant-weight phase COPY error (no new κ) #5903 concerns best COPY approximation from constant-weight phase histories. Its Hilbert-space error and history contract differ from this consecutive-weight Newton residual and metered gather.
• Prove unbounded dyadic acquisition at rational rank one (no new κ) #5904 concerns the number of dyadic decoder acquisitions at rational rank one; no common precision-depth quantity is compared.
• Prove sharp one-pass child batching storage (no new κ) #5905 concerns retained payload storage for a one-pass child-batch dispatcher; its streaming memory bound differs from the residual's ADD precision mass.

CI notes

At commit 8b37cd3d519365bbfe7404eccfba2262d970c6af, run 40501016460 finished with plan failed; replay skipped. The sole failed step was Onboarding helper tests. All ten failure/error entries point to unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22) and exit code 1.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failure entries are the frozen-card count assertion 0 != 16, five expected missing-helper/reference ValueError checks that did not raise, and two sparse-reference receipt-list mismatches. All ten failure headers and exception/assertion messages match PR #5899's failed planning log exactly. This confirms the reported shared onboarding failure on base main de5ec2e7a6074068452f45b1a191b5c72832df4a; maintainers will handle the repair. No package or shared source/test file was changed in response.

Hosted replay has not certified this package; only the local scientific checks described above have run. The planning failure prevented the package replay job from executing. Passing local checks and source/export checks are separate from hosted scientific certification.

Advance claim: #1166 (comment)

Merge Danger

Door: two-way. This adds one isolated research package with its opt-in replay manifest.

Blast Radius: package. No shared implementation, frozen evidence or selected record changes.