Reusable results and explicit research frontiers

This document explains the comparative significance judgment for the five-paper selection. These are mathematical routes opened by the results, not predictions of citations or assertions that the remaining problems are solved. The selection is based on generality, sharpness, and reusable mechanisms rather than a quota of named conjectures.

P01: operators from face counts

The integral splitting replaces a large family of multiplication matrices by classical Boolean matrices. It works for all simplicial complexes, and controls all powers and characteristics simultaneously. The resulting integral cokernels can detect exactly which primes cause rank drops. Ordinary characteristic-zero Jordan type even recovers the face numbers. These are useful structural consequences beyond the named whiskered-graph conjecture.

A precise next frontier is a WLP classification for the associated Gorenstein idealizations. The manuscript proves failure in specified ranges by descending a necessary surjection to the quotient, but the converse requires controlling the mixed dual-module terms in multiplication by a general linear form. Arbitrary monomial powers and general very well-covered graphs are also outside the stated theorem; neither is a formal consequence of the square-zero change of variables as presented.

P02: exact arithmetic obstruction to simultaneous repair

The matrix theorem separates rational semisimplicity, the splitting-field Galois exponent, and local Jordan degeneration. Once the first two conditions hold, the least multiplier is an explicit finite invariant. The recurrence version accounts for initialization at index one and proves that the uniform maximum is actually attained by one integral sequence. This is a complete classification rather than a collection of sufficient constants.

The substantial remaining problem is the individual scalar sequence: given a fixed integral functional and vector, determine which local Jordan contributions cancel after observation and which conjugate contributions cancel at infinitely many primes. The entrywise theorem deliberately does not solve that problem. Minton's established trace classification answers an important unsampled case and must remain part of this frontier's baseline.

P06: arithmetic beyond the uniform bound

The Stirling paper determines exact prime support and improves the upper multiplier from a factorial to a least common multiple. More significantly, it identifies the logarithmic size of the excess over the squarefree factor to first order, at scale square root of k. This combines a uniform theorem, sharp local families, and a pointwise asymptotic. The general recurrence bound in P02 does not determine these scalar cancellations or this asymptotic.

The exact general repair factor, the proposed finite cutoff, and Miska–Ward Conjecture 4(a)–(c) remain open here. For k=p^j, the difficult new range of part (c) starts at j>=3: a proof needs higher local divisibility for every mixed index, not just a zero residue at pure prime powers. A routine refinement of the asymptotic error term would not constitute another significant paper.

P09: reconstructing addition without a regular adjoint

The finite-length theorem applies over every commutative coefficient ring and to arbitrary Lie-ring targets. It recovers addition on several intrinsic abelian Fitting images and detects the remaining defect by commuting operators. Finite-field examples show why one globally invertible action is insufficient; direct-sum examples simultaneously defeat a bounded common-centralizer shortcut. These results go beyond the already accessible associative triangular special cases.

The general centerless-Lie-ring problem remains open here. For nonmetabelian algebras, the Fitting images need not be abelian, and the first reconstruction lemma no longer applies. For infinite-dimensional or infinite-length source modules, stabilization and finite joint detection can fail. Removing either obstruction would require a new argument. No scalar semilinearity or reconstruction-complexity theorem is claimed.

P13: periods and radical theory

The period–radical inclusion applies to each individual period in every associative ring, without an identity, characteristic bound, or PI hypothesis. It upgrades the named NI conjecture to 2-primality and local nilpotence and provides immediate vanishing on semiprime rings. The proof isolates a short new reduction to a classical bounded-index nil-ideal theorem. Its brevity does not diminish its universal scope, but does limit any claim of technical depth.

The converse direction is not established: a 2-primal ring need not automatically be nilperiod, and finite classification does not classify arbitrary infinite rings. A useful next problem is to characterize which elements of the prime radical are periods and the set of exponents for each. Another is exact power-map counting in larger finite local rings beyond the square-zero twisted field construction. No general counting formula follows merely from knowing the nilradical.

Comparative conclusion

The five papers have distinct main questions and reusable outcomes. P01, P02 and P09 are broad structural classifications; P06 supplies a sharp arithmetic asymptotic and named conjecture resolutions; P13 supplies a concise universal radical theorem. This is the defensible case for a substantive specialist portfolio. It is not an assertion that every journal or expert will assign the same significance, or that these problems were historically equally difficult. All novelty comparisons and proof judgments remain documented internal AI assessments pending human scrutiny.