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

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.

@misc{quarticinversecoefficients2026,
  title        = {Strict interlacing and exact intersections for quartic inverse coefficients},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22727740},
  url          = {https://doi.org/10.5281/zenodo.22727740},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/quartic-inverse-coefficients/}
}