[petalas]

Summary

For a quadratic with exactly one root inside and one outside the unit circle, prove the sharp inverse bound

P(z) = c(z-a)(1-bz), |a|<1, 0<|b|<1
m = min(|a|,|b|), delta = min_{|z|=1}|P(z)|
delta ||1/P||_W <= (1+m)^2/(1+m^2) < 2

Claim type: scoped proof with exact finite corroboration. The same optimal factors bound ||P(tau Z_L^w)^(-1)||_infinity, uniformly over cyclic length, integer stride and unit phase. The radius-dependent factor is attained by a=r,b=-r; the least uniform factor is two. This replaces #803's general degree-two sufficient multiplier 81 on the stated mixed-root subclass. No new kappa is claimed.

Add only research/mixed-quadratic-circle-bound/. This contribution is produced by Nick Petalas. Credit Rafael Guevara #803 for the general bound and its nonoptimality question, Ali Romman #621 for the cyclic setting, Thiago Costa #802 for Fourier-transfer work and Ryan Pinto #692 for the retained geometric-resolvent lineage. The proof uses classical geometric-series and Fourier methods and makes no literature-priority claim for them. Predecessor results are cited premises, without importing or replaying their executables.

Evidence

Before: #803 gives (1+4n)^2/delta, hence 81/delta at degree two, without an optimal degree factor. After: PROOF.md gives the exact two-sided Laurent norm, the sharp radius-sensitive inequality, uniform factor two, and cyclic sharpness by coefficient folding as length grows.

Local CPython 3.12.15 checks passed:

• 192 exact endpoint inequalities; 768 dense-elimination versus finite-geometric inverse comparisons, covering 2,688 entries and 192 antipodal exact-gap cases; 63 rational sharpness identities; eight semantic controls.
• Registered warning-strict ordinary and optimized commands, and direct standalone execution. Each registered command took under one second locally; this is timeout evidence, not a performance claim.
• Generated source-bound receipt replay; altered count, Boolean type, source hashes and duplicate JSON keys reject. Exclusive-create rejects overwrite while preserving the receipt. An actual proof-byte mutation in the exported copy rejects against its pre-mutation receipt.
• Metadata/pins, exact committed bytes and source history, official export/integrity and Git-free optimized replay pass at 6022a32a01cf8bdc19d382fa1a657ecd4afada14.

From the repository root, using the local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/mixed-quadratic-circle-bound
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/mixed-quadratic-circle-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/mixed-quadratic-circle-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/mixed-quadratic-circle-bound --check-source-history

A reviewer can substitute their CPython 3.12 executable. Replay uses only the standard library and writes nothing by default. --emit-to NEW_EXTERNAL_FILE creates an optional generated receipt, and --check-receipt FILE recomputes the result and all six source hashes. Historical source pins are at canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Full-main, merged-PR, live PR and Discussion overlap searches were refreshed before committing; sparse notes, arbitrary diffs and unpublished work remain coverage limits.

Limitations

The universal quantifiers rest on the written proof. Finite enumeration does not exhaust complex roots. The bound concerns exact mixed quadratic operators, not the general arbitrary-degree case. Coefficient preparation, rounded execution, dirty restoration, fixed-tape costs, complete caller invoices and supplier admission remain open. No selected-frontier change or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or full selected cold replay was run, and no proof obligation was deferred solely to CI.

Open next questions

• Determine the optimal multiplier for quadratics with both roots outside the circle, starting with separated two-pole phases. Coefficient cancellation blocks the present disjoint-series argument.
• Identify one actual mixed-quadratic caller and carry this bound through its complete rounding recurrence, including preparation, every prefix and restoration.

Related open work

No open PR on the same sharp mixed-quadratic Wiener multiplier was found in the inspected coverage. Nearby cyclic-inverse work includes:

• Implement phase-preserving cyclic recovery with bounded-word rounding #4908 implements phase-preserving cyclic recovery with bounded-word rounding for a linear observation operator. This PR proves a sharp exact norm inequality for a mixed quadratic, and does not supply that physical implementation.
• Certify complete block-cascade conditioning and audit paid-word admission (no new κ) #4842 studies complete observation-cascade Euclidean conditioning. This PR studies Laurent coefficient norms and induced infinity norms for scalar quadratic cyclic polynomials; no common numerical invariant is compared.

The completion refresh read this account's nine other open PRs created since the CLAIM: #4909, #4910, #4911, #4912, #4915, #4916, #4917, #4918 and #4919. They study query capacity, tensor-error dimension, a fixed source-basis walk, square-zero call compilation, idempotent orbit counts, ring-valued fanout, a full-mask rank, signed unitriangular tariffs and shared subspace maps. None studies this mixed-quadratic Wiener multiplier or reports a competing value for it. No additional computations were run to compare their results.

CI notes

Both hosted checks passed on exact head 6022a32a01cf8bdc19d382fa1a657ecd4afada14 in run 39873686308: planning in 3m20s and the registered package replay in 1m11s. Run metadata confirms the exact head, completed status and successful conclusion. Registered replay runs both warning-strict ordinary and optimized commands, which also passed locally. Local committed-byte/source-history, export/integrity, Git-free replay and receipt controls passed. Hosted execution records the bounded package checks, not independent human or formal proof-kernel acceptance of the universal theorem.

Merge Danger

Door: two-way

One isolated proof and replay package. Retention does not admit a supplier.

Blast Radius: research