[petalas]

Summary

Scoped all-width proof with exact finite corroboration. The complete two- and three-call dirty-input ports from Dugongue's #4743 have sharp Euclidean condition number

previous bound: 16
exact value:    (11 + 3 sqrt(13))/2

Both norm and inverse norm equal (sqrt(13)+3)/2. An explicit two-dimensional invariant Gram sector attains the extremes at every positive tensor width. The package also gives the full singular spectrum with exact multiplicities. The classical shear singular-value calculation is the method; the new result is the sharp constant and spectrum for these particular complete ports.

Only research/complex-port-sharp-conditioning is added. Paid calls and the selected frontier are unchanged. This contribution is produced by Nick Petalas. Credit Dugongue #4743, Yiyang #4710, lee-phys #4688 and Jacob Alan Sussman #4628, with their predecessor credit retained by citation.

Evidence

Before, FULL-INPUT-PORT.md bounds each endpoint norm by 4 and the condition number by 16. After, the written singular-value decomposition and attaining vector prove the exact smaller constant.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/complex-port-sharp-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/complex-port-sharp-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/complex-port-sharp-conditioning
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/complex-port-sharp-conditioning --check-source-history

All pass locally on CPython 3.12.15, macOS arm64. Each registered check is timed locally at 0.067 s and 0.066 s. Expected mathematical status: PASS_COMPLEX_PORT_SHARP_CONDITIONING.

The independent Gaussian-integer computation covers literal 16-, 128- and 1,024-row shear matrices at widths 1, 2 and 3. It checks the new Gram polynomial, both inverse orders, exact spectral-projector traces and the sharp sector. Squared-coupling multiplicities for d=1,5,9 are (2,2,4), (18,24,22), (140,224,148). Six rejection controls pass. A generated scratch receipt binds the five final package files by SHA-256. No predecessor verifier is imported or replayed. The source-history preflight passes for the pinned canonical base 624de9d.

The current selected source-binding orientation passes. The card's historical control was attempted and fails with KeyError: 'complex_profile' under the changed selected schema; it is recorded and unchanged.

Limitations

Exact Euclidean endpoint conditioning is proved. Internal Gaussian gate cuts, coordinatewise rounded-call error, complete recursive substitution, changed nonunitary encodings, full-unit admission and an integer multiplication theorem remain outside scope. No Lean or other proof-kernel check, full supplier replay or independent human review is claimed. The inherited native costs, restoration and workspace remain those of #4743.

Open next questions

• Propagate the sharp endpoint constant through one complete recursively substituted word, retaining every intervening mixer.
• Derive the native schedule's sharp coordinatewise error amplification from an actual rounded Gaussian call contract.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 supplies the small-unit model and lower-bound context; this result changes a local conditioning constant rather than its call bounds.

• feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813 studies a complete cascade using the Add reversible native ports and swap-sector completion for the complex prefix #4743 ports. This result tightens the component bound 16 for the frozen two-bank port and does not establish cascade correctness or its condition number.

• feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818 studies a block-4 cascade with different letters. Its complete-cascade conditioning claim is a different quantity; this package does not compute it.

• feat(research): exact 13-array 41-call block-4 boundary schedule on C^16 (κ = 0.1714) #4819 studies the block-4 boundary cascade. Its loop conditioning and call-count claims are outside this theorem.

• Construct a resonant partial target channel at every even width #4797 changes the coupling to the resonant P_f+Q_f pencil. This theorem concerns P_f+2Q_f and retains the full two-bank interface.

• Certify resonant partial-channel rank 100 at width four (no new κ) #4825 studies the rank of Construct a resonant partial target channel at every even width #4797's partial channel at width four. No partial-channel rank is computed here.

• Prove sharp dirty Toeplitz shear gate-cut norm bounds (no new κ) #4829 uses the same classical block-shear singular-value formula to bound internal dirty Toeplitz FFT cuts. This result instead gives the exact complete two-bank port spectrum; it computes no FFT cut norm.

• Prove the exact rational GL3 two-column kernel (no new κ) #4835 determines the characteristic-zero two-column GL3 address-algebra kernel in the same small-unit context. Its image-rank quantity differs from this complete port's Euclidean conditioning, and no numerical comparison is made.

This account's open PRs created since the CLAIM at 2026-10-22 21:12:45 UTC were refreshed before DONE. No conflicting same-port conditioning value was found in that snapshot.

CI notes

Both hosted checks passed on 299b609 in run 39856129937: plan in 3m34s and the registered package replay in 1m31s. The replay executes both warning-strict ordinary and optimized commands. Run metadata confirms that exact head and successful conclusion. Local package replay, committed-byte and source-history preflights also pass. These checks certify execution of the bounded verifier; independent human and formal proof-kernel review remain unclaimed. No predecessor or unrelated CI job has been rerun.

Merge danger

Door: two-way. Blast radius: one isolated research package.