--- title: "Strict interlacing and exact intersections for quartic inverse coefficients" date: 2026-09-12 version: "0.1.0-candidate" doi: 10.5281/zenodo.22727740 pdf: https://github.com/ipitchford/quartic-inverse-coefficients/releases/download/v0.1.0-candidate/manuscript.pdf repository: https://github.com/ipitchford/quartic-inverse-coefficients archive: https://zenodo.org/records/22727740 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Strict interlacing and exact intersections for quartic inverse coefficients ## Summary Two neighbouring polynomials can each have well-behaved roots without their roots fitting together in any useful order. This candidate proves the stronger statement for a family arising from the inverse of a quartic polynomial: the roots strictly alternate, and the first belongs to the even-indexed polynomial. That ordering rules out shared boundary roots. Combined with an explicitly cited external rigidity theorem, it gives a precise structural description of every corresponding affine intersection: all points are simple, their number is known exactly, and two important quantities never vanish there. ## Summary for specialists Write $H(t)=t+at^2+xt^3+yt^4$ and $H^{-1}(w)=w+\sum_{n\ge1}g_n(a,x,y)w^{n+1}$. Let $G_n(X,Y)=g_n(1,X,Y)$ and let $P_n$ be the univariate factor of the boundary specialization $g_n(0,X,Y)$. For every $d\ge2$, adjacent $P_d,P_{d+1}$ are coprime and have the stated strict positive-root interlacing. This theorem is independent of composition rigidity. Using the R3 consequence of liqsweep's pinned Corollary 5.1, the algebra $$\mathbb Q[X,Y]/(G_d,G_{d+1})$$ is finite étale of degree $\lfloor d(d+1)/6\rfloor$. The Jacobian and $G_{d+2}$ are units. This classifies reducedness, geometric cardinality and nonvanishing—not explicit coordinates or residue fields. ## Technical account Both parity pairings are represented through a common Jacobi-polynomial factor under multiplicative finite free convolution. A degree-preserving openness argument supplies strictness. Differential transforms give one pairing directly; a positive symbolic determinant excludes root collisions along a deformation for the other. The proof handles the three residue classes and the zeros introduced by common-degree reversal. For the affine conclusion, the credited quasismoothness and raising identities combine with external R3 to establish transversality. The Hilbert series then counts the reduced points. When $d\equiv1\pmod3$, the weight-three boundary orbit contributes $1/3$; subtracting it gives the floor formula. At $d=4$, the calculation is $20/6=3+1/3$. ## Evidence, assurance and limitations The uniform conclusions rest on written proofs. Exact symbolic checks verify three determinant identities and positivity certificates. Finite checks cover 96 coefficient identifications, adjacent root diagnostics through $d=60$, and affine lengths and units through $d=10$. A reversed-order control detects the orientation error that alternation alone would miss. The affine theorem explicitly imports R3 from a public research manuscript: source parameters $m=3,n=d-1$ give the required block $g_d,g_{d+1},g_{d+2}$. The Hilbert-series calculation does not remove this dependence, because reducedness was established using R3. The boundary theorem survives independently of that input. Status is unrefereed candidate. Producer-coordinated AI editorial review and local replay are recorded; unaffiliated specialist review, independent reproduction, formal verification and exhaustive priority clearance are not established. Media communicate the result and add no mathematical evidence. ## Relationship to earlier work Perry and Lewis–Perry–Straub supply the lower-degree hypergeometric precedent and existential coefficient criterion. The earlier smooth-point and structural candidates supply the motivating conjecture and differential identities. The full-$e=3$ candidate constructs a suitable point for containment; it does not provide this uniform boundary comparison or classification of every affine solution. Work from the same programme is not independent corroboration. The proposed contribution is the parameter-specific adjacent quartic comparison and its structural affine synthesis, not the invention of convolution, a new proof of universal rigidity, or completion of a higher-degree orbit-classification programme. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Polynomial-inversion researchers | Uniform adjacent-root structure and boundary coprimality | Check the common-factor and degree-drop arguments. | | Algebraic geometers | Exact reduced geometric intersection count | The affine conclusion depends on external R3. | | Symbolic-computation researchers | Small exact identities and rejection controls | Finite replay is not a universal proof. | ## Why the problem matters Existence of a useful specialization leaves open what happens at all the other points. The structural result supplies that missing information for this family. Its potential significance is within mathematics; practical applications or broad influence have not been demonstrated. ## How to inspect or reproduce the recorded checks Read the manuscript's two main theorems, then the claim and source records. With Python and the pinned SymPy dependency installed, run `python3 run_checks.py` and `python3 -O run_checks.py`. Run `python3 verify_manifest.py` to check file identity. Checks require no network; dependency installation does. The package records exact ranges and the difference between symbolic identities, finite diagnostics and prose arguments. The PDF can be rebuilt with `python3 build.py` and an installed TeX distribution. ## The most valuable next projects The most useful assurance step is independent scrutiny of the strict-convolution and no-collision arguments, together with the exact R3 specialization. Separate research targets include arithmetic descriptions of the affine points and higher-degree analogues. Neither follows merely by repeating the finite checks at larger indices. ## What is in the evidence package The archive contains the paper and editable sources, an accessible text rendering, exact scripts and semantic negative controls, claim/dependency records, the response to the supplied review, internal editorial reports, licences, replay records and checksums. GitHub and Zenodo identify the same immutable release bytes. Publication makes those objects inspectable; it does not upgrade their mathematical assurance. ## Open directions for follow-up research - Independent specialist scrutiny and reproduction of the proof and source bridge. - Arithmetic structure and residue fields of affine points. - Higher-degree analogues require new arguments. - Formalization of the strictness and determinant proof. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:quartic-inverse-coefficients - Attempt and metric receipts: ep-attempt:quartic-inverse-coefficients-assurance-publication: published / positive; scope assurance-through-publication; target Complete review revisions and publish GitHub/Zenodo assets and guarded Evidence Press page, media and readback.; active forecast 150 minutes (90-240); Fermi components Source repairs and internal editorial gate: 1 x 30/50/80 minutes low/central/high (Established publication route; not an empirical speed comparison.); Immutable archives and communication assets: 1 x 30/50/80 minutes low/central/high (Established publication route; not an empirical speed comparison.); Composite CI and two deployment cycles: 1 x 30/50/80 minutes low/central/high (Established publication route; not an empirical speed comparison.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 22/unknown/unknown/2/0/1; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -128 minutes; ratio 0.14666666666666667; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort was not instrumented.; computeMinutes: Hardware inference time was not instrumented.; deduplicatedModelTokens: No task-local fork-aware cumulative counter available.; uncachedInputTokens: No task-local uncached-input counter available.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: metrics.outcome.agentRuns=6. Agent runs include coordinator and five reviewers. Timing values are instrumented lower bounds, not complete effort estimates. (reason: The intake snapshot remains unchanged; the terminal receipt records the observed total or explicitly limited measurement.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: metrics.outcome.reworkMinutes=1. Agent runs include coordinator and five reviewers. Timing values are instrumented lower bounds, not complete effort estimates. (reason: The intake snapshot remains unchanged; the terminal receipt records the observed total or explicitly limited measurement.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, communication - Decision object: reusable-method — Adjacent-boundary strict interlacing and structural affine count. Scope: Normalized quartic inverse coefficients in characteristic zero, d>=2; affine theorem uses R3. - Reusable methods: Structural compression (structural-compression); Exact regime stitching (regime-stitching); Explicit research-lineage reuse (research-lineage-reuse); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Common Jacobi convolution factors and a no-collision determinant prove boundary ordering; R3 and differential rank justify reducedness before Hilbert-series counting. Remaining risks: External rigidity validity is inherited, not established by replay.; Finite diagnostics do not prove universal statements.; Bounded prior-work comparison is not exhaustive priority.. - Human judgement gates: Audit the proof and exact source-to-R3 mapping.; Assess novelty and priority separately.; Retain rights, dependency and assurance boundaries. - Next assurance action: Independent proof scrutiny and reproduction; preserve the external rigidity dependency. - Claim ceiling: Unrefereed candidate. Uniform boundary interlacing is independent of rigidity; affine reducedness, exact geometric count and units use external R3. No coordinate/residue-field classification, external peer review, formal verification, exhaustive priority or impact claim. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable quartic boundary theorem and conditional structural affine classification in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: extends-result smooth-point-certificates-polydegree-containments; inherited claim: Boundary family and motivating affine conjecture; earlier structural interpretation.; inherited ceiling: Unrefereed parent; reuse is not independent corroboration.; reuses-method ep-work:lps-structural-reductions; inherited claim: Quasismoothness and raising identities, reproduced with attribution.; inherited ceiling: Unrefereed structural parent; no external validation inherited.; depends-on-claim https://github.com/blueberryvertigo/polynomial-composition-rigidity/blob/b17b6b9f7440b12fd85df2c2db9c98209ecc174d/rigidity.typ; inherited claim: R3 excludes a three-coefficient zero block, source m=3,n=d-1.; inherited ceiling: Public research manuscript; not re-proved or represented as journal-refereed. ## Verification status Unrefereed candidate. Uniform boundary interlacing is independent of rigidity; affine reducedness, exact geometric count and units use external R3. No coordinate/residue-field classification, external peer review, formal verification, exhaustive priority or impact claim. ## References 1. liqsweep (2026), Polynomial composition rigidity via critical values. External R3 input, pinned research manuscript. 2. Martinez-Finkelshtein, Morales and Perales (2024), Real roots of hypergeometric polynomials via finite free convolution. 3. Lewis, Perry and Straub (2019), An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms. 4. Anonymous (2026), Smooth-point certificates for Polydegree containments. Motivating affine conjecture. 5. Anonymous (2026), Complementary-minor duality and fixed-width reductions for the Polydegree Ideal Conjecture. Credited differential identities. 6. Anonymous (2026), A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture. Existential containment candidate, not all-point classification.