# Evidence Press — full text

---

# Exact low-length Recht–Ré inequalities: complete status through five factors and six-factor balanced families

- URL: https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/
- PDF: https://github.com/ipitchford/exact-low-length-recht-re-inequalities/releases/download/v1.0.0-candidate/exact-low-length-recht-re-inequalities-v1.0.0-candidate.pdf
- DOI: https://doi.org/10.5281/zenodo.21709239
- Code: https://github.com/ipitchford/exact-low-length-recht-re-inequalities
- Published: 2026-07-30 · Version: 1.0.0-candidate · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

This paper asks a basic question behind many machine-learning algorithms: is it better to shuffle a dataset and use each example once, or keep sampling examples at random? It gives the first exact account of where a leading mathematical justification for reshuffling works, where it fails, and where it recovers. It also finds something more surprising: the usual theoretical measure can say reshuffling is worse even when it actually reduces error. The proof is computer-assisted, exact, and fully replayable.

## Summary for specialists

What makes this paper interesting is that it draws an exact boundary around a natural matrix inequality motivated by random reshuffling: the four-factor case always holds, the five-factor lower bound fails only at the endpoint $n=5$ and is restored from $n=6$, and six factors already yield new infinite families. The proof is backed by replayable exact certificates, while the same example reveals an important optimisation subtlety: reshuffling can worsen the norm of the expected iterate while improving expected squared error and objective value.

## Technical summary

The Recht–Ré inequality asks whether sampling matrix-valued updates without replacement is always at least as contractive as sampling them independently with replacement. For positive semidefinite matrices $A_1, \ldots, A_n$, it compares

$$
\frac{1}{\binom{n}{m}} \sum_{\substack{i_1, \ldots, i_m \\ \text{distinct}}} A_{i_1} \cdots A_{i_m}
$$

with the corresponding with-replacement average,

$$
\left(\frac{1}{n} \sum_{i=1}^n A_i\right)^m.
$$

Full random reshuffling is the special case $m = n$. The scalar analogy suggests that the without-replacement expression should have the smaller operator norm. Matrices make the problem difficult because multiplication order matters, and products of positive semidefinite matrices can have negative eigenvalues. A proof must therefore control both ends of the spectrum.

The manuscript settles the first previously unresolved product lengths:

- **Four factors:** the two-sided inequality holds for every $n \geq 4$.
- **Five factors:** the upper spectral bound holds for every $n \geq 5$, but the lower bound fails at $n = 5$. An exact rational counterexample gives an eigenvalue $-285/2$, beyond the permitted $-120$. The lower bound is restored for every $n \geq 6$, so this threshold is sharp.
- **Six factors:** the upper bound is proved when $7 \mid n$, the lower bound when $8 \mid n$, and hence the complete norm inequality whenever $56 \mid n$. The remaining six-factor cases are still open.

The proof is computer-assisted but exact. It converts the matrix problem into positivity questions for noncommutative polynomials, exploits permutation symmetry to reduce their size, and supplies rational identities and positive-semidefinite certificates that can be replayed without floating-point tolerances. The accompanying package also checks the orbit classifications and representation maps through a structurally separate executable route.

There is also an important optimisation lesson. On the same rational quadratic example, reshuffling produces a larger norm of the expected iterate than independent sampling, yet a smaller expected squared error and average objective value. The Recht–Ré norm comparison therefore measures one form of bias, but it is not a universal measure of algorithmic performance.

## Who should care, and why

The realistic audience is narrower than "people interested in machine learning", but several specialist groups have a clear reason to pay attention.

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Random-reshuffling and stochastic-gradient researchers | The exact quadratic example separates three quantities that theory often treats as broadly aligned: reshuffling worsens the norm of the expected iterate while improving expected squared error and the objective value. | Determine when the traditional norm proxy agrees with actual optimisation performance; identify structural assumptions that prevent the reversal; use the rational example as an adversarial test for new convergence arguments. |
| Matrix-inequality and operator-theory researchers | A complete candidate classification for four and five factors, including the sharp failure and recovery at five factors, and a sharply defined six-factor frontier. | Close the six-factor nonmultiple cases; investigate whether the upper inequality holds more generally; classify lower-bound failures; replace parts of the computational certificate with a conceptual analytic proof. |
| Noncommutative polynomial and sums-of-squares researchers | The work builds on the noncommutative Positivstellensatz route used by Lai and Lim — who showed the unrestricted Recht–Ré conjecture fails at five matrices — but pushes it into exact rational certificates, symmetry blocks and parametric continuation. | Generalise the free-localiser and symmetry-reduction architecture to other noncommutative inequalities; automate the orbit and representation calculations; find smaller or more intelligible certificates. |
| Computer-assisted proof and formal-verification researchers | An unusually extensive exact replay, including mutation tests and a structurally separate semantic checker — yet still an internally checked candidate rather than an independently reproduced or formally verified proof. | Reimplement the computation independently; formalise the balanced-seed continuation theorem in Lean, Isabelle or Coq; verify the rational positivity certificates; test whether the claimed separation from the shared certificate schema is adequate. |
| Researchers designing data-ordering schemes | Current work increasingly treats data order as something to optimise rather than merely randomise — block reshuffling and paired reversal aim to improve constants and cancel order-dependent terms. | Run those schemes on the paper's exact counterexample; ask which orderings improve all three metrics; design theoretical measures that track objective reduction rather than only the expected update operator. |
| Randomised numerical linear algebra researchers | Recht and Ré originally connected the inequality to least-squares optimisation, stochastic gradient descent and the randomized Kaczmarz method. | Translate the low-length inequalities into short-horizon or block-iteration guarantees; test whether structured matrices arising in real linear systems avoid the pathological behaviour found in unrestricted positive-semidefinite examples. |
| AI-assisted mathematics and reproducibility researchers | A substantial case study in moving from numerical discovery to exact, executable evidence, while explicitly documenting the limits of that evidence. | Study independent agent reproduction, certificate exchange, semantic mutation testing, claim–evidence matrices, and the transition from an executable candidate proof to an accepted theorem. |

## The most valuable next projects

### 1. Characterise the metric reversal

This has the greatest potential significance for machine-learning theory. The important result is not simply that one inequality fails. It is that a familiar theoretical surrogate can rank two algorithms in the opposite order from expected error and objective value.

A useful follow-on theorem would identify conditions under which

$$
\|\mathbb{E}[x_1]\|, \quad \mathbb{E}\|x_1\|^2, \quad \mathbb{E}[f(x_1)]
$$

must agree in their comparison of sampling schemes, and conditions under which they may diverge. Candidates include commutativity, simultaneous diagonalisation, near-identity update matrices, bounded condition number and sufficiently small step size.

This would connect naturally with newer trajectory-level results. For example, Liu's 2026 analysis proves that random reshuffling dominates ordinary stochastic gradient descent under the smooth-convex assumptions studied there, despite the limitations of older matrix-product proxies. The two results point towards the same conclusion: the failure of the Recht–Ré route does not imply that reshuffling performs poorly; it shows that the route measures the wrong object in some regimes.

### 2. Finish the six-factor classification

This is the cleanest continuation as pure mathematics. The current arithmetic families suggest several possible outcomes: the two-sided result extends to every sufficiently large $n$; the upper and lower bounds have different thresholds; further exceptional congruence classes contain counterexamples; or the balanced-seed method suffices only after several additional seeds are proved. The existing exact machinery narrows this into a concrete certificate-search and structural-analysis problem rather than an open-ended conjecture.

### 3. Obtain an independent proof

This would produce the largest immediate increase in credibility. A convincing reproduction should avoid reusing the repository's orbit encoder, certificate schema or extraction code. The strongest outcome would combine an independent implementation, formal verification of the general continuation theorem, exact checking of the finite positivity certificates, and a human-readable account of why the certificate architecture works.

## Specialist audience candidates

The most natural specialist readers include researchers associated with the original Recht–Ré conjecture, the Lai–Lim counterexample and noncommutative sums-of-squares approach, and current work on random reshuffling and learned data ordering. That includes the authors of the relevant papers by Recht and Ré, Lai and Lim, Liu, and Nguyen, Phan and Kalagnanam. This identifies intellectual proximity, not a prediction that any individual will endorse the manuscript.

The strongest pitch to them is:

> This work does two things: it determines the first unresolved finite cases of the Recht–Ré inequality, and it gives an exact example showing that the inequality's preferred metric can disagree with actual error and objective performance.

That is more compelling to specialists than the general claim that the paper concerns shuffling.

## What is in the evidence package

The repository and Zenodo deposit ship exact rational sum-of-squares certificates, full-matrix positive-semidefiniteness certificates computed with FLINT characteristic polynomials, verification of all 2,312 six-factor coefficient identities, deterministic replay scripts with pinned dependencies, complete SHA-256 manifests, and mutation and negative controls that confirm the acceptance logic is live. Everything replays in exact arithmetic, without floating-point tolerances.


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/exact-low-length-recht-re-inequalities/paper.json)

- Characterise the metric reversal: find conditions under which the norm of the expected iterate, the expected squared error, and the expected objective value must agree in ranking sampling schemes, and conditions under which they may diverge (candidates: commutativity, simultaneous diagonalisation, near-identity updates, bounded condition number, small step size).
- Finish the six-factor classification: determine whether the two-sided result extends to every sufficiently large n, whether upper and lower bounds have different thresholds, and whether further exceptional congruence classes contain counterexamples.
- Obtain an independent proof: reimplement the computation without reusing the repository's orbit encoder, certificate schema, or extraction code; formalise the balanced-seed continuation theorem in Lean, Isabelle, or Coq; check the rational positivity certificates independently.
- Run recently proposed data-ordering schemes (block reshuffling, paired reversal) on the paper's exact counterexample and ask which orderings improve all three metrics simultaneously.
- Translate the low-length inequalities into short-horizon or block-iteration guarantees for randomised numerical linear algebra, and test whether structured matrices from real linear systems avoid the pathological behaviour.

---

# z(20) = 6: resolving the first open case of Erdős problem 758

- URL: https://evidence-press.pages.dev/releases/z20-equals-6/
- PDF: https://github.com/ipitchford/z20-cochromatic/releases/download/candidate-2026-07-26/z20_equals_6_proof.pdf
- DOI: https://doi.org/10.5281/zenodo.21647645
- Code: https://github.com/ipitchford/z20-cochromatic
- Published: 2026-07-28 · Version: candidate-2026-07-26 · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

Take any network of 20 points, where each pair is either connected or not. Now sort the points into groups so that within each group, either everyone is connected to everyone else (a clique) or nobody is connected to anybody (an independent set). How many groups might you need, in the worst case?

Mathematicians call the answer for $n$-vertex graphs $z(n)$, and Paul Erdős and John Gimbel asked for its values decades ago. The answer was known for every size up to 19 vertices. At 20 vertices, it was known only that the answer is 6 or 7. This computer-assisted proof settles the first open case: $z(20) = 6$.

The proof has two halves with very different characters. The half showing that 6 groups are sometimes necessary is short enough to check by hand: it uses a famously symmetric network on 17 points, the Paley graph, in which no four points are mutually connected or mutually unconnected. The half showing that 6 groups always suffice is where computers earn their keep: every conceivable counterexample is funnelled, through a classical result of Ramsey theory, into two giant logical formulas, and each formula is proved impossible to satisfy — with certificates that four separate programs, one of them formally verified, have each confirmed.

## Summary for specialists

The cochromatic number $z(G)$ is the least number of colour classes partitioning $V(G)$ so that each class induces a clique or an independent set, and $z(n) = \max_{|V(G)|=n} z(G)$. Values are known for $n \leq 19$; erdosproblems.com lists $z(20) \in \{6, 7\}$ as the first open case. This release establishes $z(20) = 6$.

The lower bound is hand-verifiable: the Paley graph on 17 vertices has no homogeneous 4-set, so $z(\mathrm{Paley}(17)) \geq \lceil 17/3 \rceil = 6$. For the upper bound, since $R(4,4) = 18$, any hypothetical 20-vertex graph needing 7 classes is reduced to configurations built around the two non-isomorphic 16-vertex $(4,4)$-Ramsey graphs. The remaining cross-edge case analysis is encoded as two CNF formulas of 64 variables and 104,524 clauses each, and their unsatisfiability is certified with DRUP and LRAT proof objects, checked by a custom Python RUP checker, a C RUP checker, drat-trim, and the formally verified cake_lpr.

The release also recomputes $R(4,4) = 18$, regenerates the $(4,4)$-Ramsey catalogue, and reconfirms $z(8) \leq 3$ and $z(12) \leq 4$ as calibration. A companion release derives $\mathrm{VR}_2(K_4) = 20$ by reusing the same proof objects.

## Technical summary

The upper bound proceeds in three certified stages. First, since $R(4,4) = 18$, any 20-vertex graph $G$ with $z(G) \geq 7$ must, on every 18-vertex subset, contain a homogeneous 4-set; a counting argument concentrates the obstruction on induced subgraphs built around the two non-isomorphic $(4,4)$-Ramsey graphs on 16 vertices, whose catalogue the release regenerates from scratch. Second, the surviving configurations — cross-edges between the 16-vertex core and the remaining four vertices, together with the demand that no 6-class cocolouring exists — are encoded as two propositional formulas of 64 variables and 104,524 clauses. Third, both formulas are refuted: the DRUP certificates are produced by the solver, translated to LRAT, and accepted by four independent checkers with different trust bases (two hand-written RUP checkers in Python and C, the standard tool drat-trim, and cake_lpr, whose correctness is itself machine-checked down to machine code through the CakeML pipeline).

The lower bound is independent of all of this: Paley(17) is self-complementary and has clique number and independence number 3, so any cocolouring needs at least $\lceil 17/3 \rceil = 6$ classes. The two halves meet at exactly 6.

The assurance boundary sits in one identified place: the *encoding step* — the claim that the two CNFs faithfully express the reduced combinatorial statement — is validated by semantic spot-tests and calibration recomputations ($R(4,4) = 18$, $z(8) \leq 3$, $z(12) \leq 4$), not by machine-checked proof. Everything downstream of the CNFs is certified; everything upstream is conventional, human-readable mathematics in the 13-page paper.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Ramsey theory and extremal graph theory researchers | The first movement on the Erdős–Gimbel small-values table since z(19), and a reduction pattern (Ramsey core + certified residual search) that may extend upwards. | Check the reduction argument in the 13-page paper; attempt z(21); look for a certificate-free proof of the upper bound. |
| SAT and automated-reasoning researchers | Two moderately sized CNFs whose unsatisfiability settles a named open problem — a clean benchmark with an unusually complete certificate chain (DRUP → LRAT → four checkers including cake_lpr). | Rerun and time the refutations with other solvers; test proof-minimisation tools on the certificates; use the instances in checker and solver test suites. |
| Formal-verification researchers | The one uncertified link is precisely identified: the encoding from the combinatorial statement to CNF. | Formalise the reduction and encoding soundness in Lean or Isabelle, turning a candidate into a machine-checked theorem end to end. |
| Combinatorics educators and expositors | The lower bound is a two-line argument about Paley(17) that any undergraduate can verify by hand — attached to a genuinely new result. | Use the pair (hand-checkable half, certified half) as a teaching example of what computer-assisted proof does and does not guarantee. |
| AI-assisted mathematics researchers | The mathematics was generated by multiple AI systems under a cross-checking policy: no step rests on a model assertion not recomputed by a different system. | Study the claim–evidence index and replay logs as a case study in auditable AI-generated mathematics; attempt independent agent reproduction. |

## Why this problem is hard

Cochromatic numbers mix the two classic extremes of graph colouring. A single colour class may be *either* kind of homogeneous set, so lower bounds cannot lean on standard chromatic machinery, and upper bounds must survive every graph on 20 vertices — roughly $10^{57}$ of them, far beyond enumeration. The Ramsey-theoretic reduction is what collapses this astronomical search space into two finite, certifiable questions. That collapse — not raw computing power — is the heart of the argument.

## The most valuable next projects

### 1. Formalise the encoding

Everything downstream of the two CNFs is already checked by a formally verified tool; everything upstream is ordinary prose. The single most valuable follow-up is to machine-check the bridge — the reduction from "every 20-vertex graph is 6-cocolourable" to "these two formulas are unsatisfiable". That one theorem, formalised in Lean or Isabelle, would close the only identified gap and make this, to our knowledge, the first fully verified value in the Erdős–Gimbel table beyond hand range.

### 2. Push to z(21)

The reduction pattern — concentrate obstructions on a Ramsey core, certify the residual search — did not obviously exhaust itself at 20. The size of the residual search grows, but SAT solvers and proof checkers have headroom. A z(21) attempt would also reveal whether the two-core structure at 20 was a lucky accident or the start of a usable general method.

### 3. Reproduce independently

The release's own caveats identify what is missing: a second team, with fresh code, regenerating the Ramsey catalogue, re-deriving the reduction, re-encoding the CNFs, and re-checking unsatisfiability. Every needed artefact — instances, certificates, checkers, manifests — is public, so this is an afternoon-to-a-week project for a SAT-literate group, and it is the fastest route to moving the claim up the verification ladder.

## Specialist audience candidates

The most natural specialist readers include researchers who maintain and extend the Erdős problems catalogue, those who have computed values in the Erdős–Gimbel table (the z(12) computation in particular), the SAT-certificate community around DRUP/LRAT proof checking and verified checkers such as cake_lpr, and Ramsey-catalogue maintainers. This identifies intellectual proximity, not a prediction that any individual will endorse the manuscript.

The strongest pitch to them is:

> The first open entry in the Erdős–Gimbel cochromatic table falls to a hand-checkable Paley bound plus two certified unsatisfiability results — and the only uncertified step is precisely identified and small.

## How the mathematics was produced

The mathematics in this release was generated by AI systems (Nous Hermes, GPT 5.6 Sol, and Claude models), with human problem selection and mediation. The repository states that no step of the final argument rests on a language-model assertion that was not recomputed by a different system. That policy, together with the certificate architecture, is what makes the claim auditable at all — but it is not a substitute for independent human or formal verification, which the release explicitly lacks.

## What is in the evidence package

The repository ships the two CNF instances (`z20_core0.cnf`, `z20_core1.cnf`) with DRUP and LRAT refutations; a portable replay checker (`verification/verify_certificate.py`); the regenerated Ramsey catalogues and calibration recomputations; a 13-page human-readable proof PDF; a machine-readable claim index (`AI_INDEX.md`); and SHA-256 manifests. The Zenodo deposit archives the release with a DOI.


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/z20-equals-6/paper.json)

- Independently reproduce the reduction: regenerate the (4,4)-Ramsey catalogue and the two core CNFs from a fresh implementation, and confirm unsatisfiability with independent tooling.
- Formally verify the encoding: the LRAT certificates are already checked by a formally verified checker (cake_lpr), but the reduction from 'z(20) ≤ 6' to the two CNFs is not itself machine-checked — formalising it in Lean or Isabelle would close the main gap.
- Extend the method to z(n) for n = 21 and beyond, where the Erdős–Gimbel table is open.
- Study whether the Paley(17)-based lower-bound construction generalises to give improved lower bounds for larger n.
- Use the release as a benchmark case for AI-generated mathematics workflows: claim–evidence indexing, certificate exchange, and independent agent reproduction.

---

# VR2(K4) = 20: twenty vertices force two vertex-disjoint monochromatic K4s

- URL: https://evidence-press.pages.dev/releases/vr2-k4-equals-20/
- PDF: https://github.com/ipitchford/z20-cochromatic/releases/download/vr2-k4-v0.1-candidate/vr2_k4_equals_20_candidate.pdf
- DOI: https://doi.org/10.5281/zenodo.21647654
- Code: https://github.com/ipitchford/z20-cochromatic
- Published: 2026-07-28 · Version: 0.1-candidate · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

A famous result of Ramsey theory says that if you connect 18 points in every possible way and paint each connection red or blue, you will always find four points whose six mutual connections are all the same colour — a monochromatic foursome. Eighteen is exactly the threshold: with 17 points you can avoid it.

This release asks the natural next question. How many points do you need before you are guaranteed *two* monochromatic foursomes that share no points at all? The claimed answer is exactly 20. With 19 points there is a way out — an elegant colouring built from two copies of a highly symmetric structure (a "Paley twin") avoids any two disjoint foursomes, and you can check this directly with a small program included in the release. With 20 points, escape is claimed to be impossible.

The impossibility half is where the heavy machinery lives: it reuses the certified SAT computations from this project's companion release on Erdős problem 758, so the two results stand or fall together.

## Summary for specialists

Define $\mathrm{VR}_2(K_4)$ as the least $n$ such that every 2-colouring of $E(K_n)$ contains two vertex-disjoint monochromatic copies of $K_4$. This release claims $\mathrm{VR}_2(K_4) = 20$.

The lower bound is an explicit 19-vertex colouring — a Paley-twin construction — admitting no two vertex-disjoint monochromatic $K_4$s, checked by a dependency-free direct verifier whose output ships with the release. The upper bound reuses the reduction and proof objects of the companion $z(20) = 6$ release (pinned to a specific repository commit): via $R(4,4) = 18$, obstructions concentrate on the two 16-vertex $(4,4)$-Ramsey graphs, and the residual case analysis is discharged by the same two CNF unsatisfiability certificates, LRAT-checked including by the formally verified cake_lpr.

A caveat specialists should note: the notation and definition are as given by the project, which did not locate an independent literature source for this exact quantity. Anyone aware of prior work on it under another name would materially help situate the result.

## Technical summary

The lower bound is constructive. Take two 17-vertex Paley-type pieces and overlay them into a 19-vertex colouring — the release calls it a Paley twin — arranged so that monochromatic $K_4$s exist but any two of them intersect. The construction ships as an explicit adjacency list; the dependency-free verifier enumerates all $\binom{19}{4}$ potential monochromatic $K_4$s in both colours and confirms that no two vertex-disjoint ones occur. Its full output is included, so the check is inspectable without running anything.

The upper bound rides on the companion release's machinery. Any red–blue colouring of $K_{20}$ with no two disjoint monochromatic $K_4$s would in particular constrain every 18-vertex subset through $R(4,4) = 18$; the same reduction to the two 16-vertex $(4,4)$-Ramsey cores applies, and the residual case analysis is exactly the two CNF instances already refuted with DRUP/LRAT certificates in the z(20) work (pinned by commit hash). The logical bridge from those certificates to the $\mathrm{VR}_2$ statement is set out in the candidate paper's short derivation, and it is the part a referee should read first.

Two caveats frame the result. The quantity $\mathrm{VR}_2(K_4)$ is defined by the project, which did not locate prior literature on it under this or another name — prior-art search is explicitly requested. And the upper bound inherits the companion release's assurance boundary: certified below the CNFs, spot-tested above them.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Ramsey theory researchers | A clean vertex-disjointness variant sitting one step beyond R(4,4) = 18, with a sharp claimed value and an explicit extremal construction at 19 vertices. | Check the derivation that transfers the z(20) certificates; search for prior appearances of the quantity; attack VR2(K5) or VRk(Km). |
| Extremal construction specialists | The Paley-twin witness — a structured 19-vertex colouring in which every pair of monochromatic K4s intersects. | Study whether the construction generalises; look for a counting or design-theoretic explanation of why 19 is achievable and 20 is not. |
| SAT and verification researchers | A second named result discharged by the *same* two CNF refutations — an economical reuse of certified computation. | Verify the pinning (commit hash, SHA-256) and the transfer argument; formalise the derivation on top of an already-verified LRAT check. |
| Recreational and expository mathematics writers | The statement needs no background beyond colouring edges and spotting foursomes, and the 19-vertex escape artist is genuinely surprising. | Present the result as a puzzle; the included verifier makes it independently checkable by readers. |

## The most valuable next projects

### 1. Establish priority or prior art

The quantity is natural enough that it may exist in the literature on vertex-disjoint monochromatic subgraphs under other notation. Either outcome is valuable: an antecedent would connect the result to known theory (and test the claimed value against it); confirmed novelty would establish the release as the defining reference.

### 2. Independent verification of the 19-vertex witness

This is the lowest-cost, highest-signal check on the whole project: a fresh implementation, in any language, enumerating monochromatic $K_4$s in a 19-vertex colouring. It requires no SAT tooling and directly tests the half of the result that is unique to this release.

### 3. Generalise the pattern

$\mathrm{VR}_2(K_4) = 20$ against $R(4,4) = 18$ invites the general question: how does the threshold for two disjoint monochromatic copies grow relative to the classical Ramsey number? Even the next cases ($\mathrm{VR}_2(K_5)$, $\mathrm{VR}_3(K_4)$) would begin to map a curve, and the reduction-plus-certificate architecture is a plausible route to both.

## Specialist audience candidates

The natural readers are Ramsey theorists working on monochromatic disjoint structures and covering problems, the maintainers of Ramsey graph catalogues whose data the reduction uses, and the SAT-certificate community. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

> One new number in classical Ramsey territory: twenty vertices force two disjoint monochromatic K4s, nineteen do not — with the extremal colouring included and checkable by hand or laptop.

## The pair of releases

This result and $z(20) = 6$ were released together and share one repository and one certificate base. The cochromatic result is the deeper claim; this one is the kind of clean, self-contained statement that invites independent checking — the 19-vertex witness in particular can be re-verified from scratch in an afternoon, which makes it a good entry point for anyone auditing the project's methods.

## What is in the evidence package

The release ships the candidate paper with TeX source, the explicit 19-vertex construction, a dependency-free verifier and its output, the reused DRUP/LRAT certificates with their commit pin, an AI-readable claim/evidence index, and a SHA-256 manifest. The Zenodo deposit archives everything under a DOI.


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/vr2-k4-equals-20/paper.json)

- Verify the 19-vertex Paley-twin construction independently — it is small enough for a fresh implementation in an afternoon.
- Determine VR2(K5), or more generally VRk(Km) for small k and m, for which the same reduction-plus-SAT architecture is a plausible route.
- Establish whether VR2(K4) = 20 admits a certificate-free human proof, given that R(4,4) = 18 does.
- Search the literature for prior appearances of this quantity under other notation, and connect it to known results on vertex-disjoint monochromatic structures.
- Formalise the reduction linking the z(20) proof objects to the VR2 claim.

---

# Reducible incidence divisors and the isolation of affine slices in binary-form factorisation spaces

- URL: https://evidence-press.pages.dev/releases/reducible-incidence-divisors/
- PDF: https://github.com/ipitchford/reducible-incidence-divisors-affine-slices/releases/download/v1.0.1/paper.pdf
- DOI: https://doi.org/10.5281/zenodo.21647616
- Code: https://github.com/ipitchford/reducible-incidence-divisors-affine-slices
- Published: 2026-07-28 · Version: 1.0-candidate (v1.0.1 archived as 10.5281/zenodo.21653119) · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

When does a naturally defined geometric object break into pieces? This paper answers that question exactly, for a family of hypersurfaces that arise when you track where two polynomials share a root.

The answer has a classical elegance. The hypersurfaces in the family are indexed by linear functionals, and the paper shows that a divisor in the family is reducible — breaks into components — precisely when its functional lies on a specific ruled surface: the tangent developable of the rational normal curve, the sweep of all tangent lines to the most fundamental curve in projective geometry. On that surface, the divisor splits into two or three identifiable pieces; off it, the divisor stays whole.

The second half of the manuscript is a process of elimination aimed at one of algebra's oldest open questions, the Jacobian conjecture. It shows that a large family of candidate geometric stages are never ordinary affine space — each is blocked by a precisely computed defect — and, assuming a result from a companion release, isolates exactly one slice that could in principle host the kind of polynomial map the conjecture is about. Everything else is screened out.

## Summary for specialists

For consecutive degrees $(m, m+1)$ with $m \geq 2$, the manuscript classifies reducibility in the marked-common-root incidence system: $D_\ell = \{\ell(P^2 A'B') = 0\}$ is reducible precisely when $[\ell]$ lies on the tangent developable of the rational normal curve of evaluation functionals. Rank-one functionals give three reduced components, first-jet functionals two; genuine two-point secants and higher catalecticant ranks give irreducible divisors.

For the associated slices, the results are: $X_\ell^{m,m+1} \not\cong \mathbb{A}^{2m+1}$ for all $m \geq 2$ and nonzero $\ell$, with precise defects — Grothendieck class $\mathbb{L}^{2m+1} - \mathbb{L}^{2m}$ in the rank-one case, diagnostic Hodge coefficients $-2$ and $-1$ for rank-two secants and first-jets. Non-adjacent degrees ($|r-s| \geq 2$) yield non-contractible slices via finite cyclic actions. Conditionally on the upstream cubic classification from the companion exotic-spheres release, the tangent nonosculating linear–quadratic slice is the *unique* positive-bidegree affine source permitting a nonzero constant Jacobian determinant.

The proofs are conventional algebraic geometry — catalecticant stratification, rational normal curves, Hodge–Deligne polynomials, cyclic group actions — with deterministic SymPy audits of selected consequences rather than a certificate architecture. The conditional isolation theorem should be read with its hypothesis in full view: the upstream classification it leans on is itself an unrefereed candidate.

## Technical summary

The proofs are conventional algebraic geometry throughout — this is the trilogy's least computational instalment. The classification theorem works in the linear system $\{\ell(P^2 A'B') = 0\}$ on the space of marked factorisations of binary forms of consecutive degrees: the functionals $\ell$ are stratified by catalecticant rank, the rational normal curve of evaluation functionals and its tangent developable are identified inside the dual space, and reducibility of $D_\ell$ is shown to occur exactly on that developable, with component counts (three for rank-one, two for first-jet, one otherwise) read off from the stratification. The non-isomorphism results compute exact classes in the Grothendieck ring — the rank-one defect is $\mathbb{L}^{2m+1} - \mathbb{L}^{2m}$ — and expand Hodge–Deligne polynomials whose coefficients at diagnostic positions ($-2$ for rank-two secants, $-1$ for first-jets) are incompatible with affine space. Non-adjacent degree pairs ($|r-s| \geq 2$) are handled separately: finite cyclic group actions on the slices obstruct contractibility, which is weaker than the motivic defect but suffices for non-isomorphism.

The isolation theorem stacks these exclusions: given the upstream cubic classification (a result of the companion exotic-spheres release, itself unverified), every positive-bidegree slice except the tangent nonosculating linear–quadratic one is eliminated as a source for maps with nonzero constant Jacobian determinant. Deterministic SymPy audits (`verify_paper.py`, `verify_rank_two.py`) check selected numerical consequences — component counts and defect coefficients in low degree — but the theorems themselves are prose proofs, and the release's internal review disposition ("minor revision for candidate publication") is recorded in the repository.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Classical projective geometers | The reducibility locus is exactly the tangent developable of the rational normal curve — a nineteenth-century object reappearing as the answer to a naturally posed modern question. | Verify the classification with classical tools (catalecticants, secant varieties); check whether fragments already exist in the apolarity literature. |
| Affine and motivic geometers | Uniform non-isomorphism with affine space across a whole family, with exact defect classes (L^{2m+1} − L^{2m}) and diagnostic Hodge coefficients (−2, −1). | Re-derive the defect computations; test the cyclic-action obstruction for non-adjacent degrees against other contractibility criteria. |
| Jacobian conjecture researchers | A conditional isolation theorem: exactly one slice in the family survives as a possible carrier of nonzero-constant-Jacobian behaviour. | Attack the surviving slice directly — construct the map or exclude it; scrutinise the conditional hypothesis it inherits from the companion release. |
| Representation theorists and apolarity specialists | Component counts (three, two, one) stratified exactly by catalecticant rank and jet type. | Explain the stratification conceptually; connect it to known secant-variety and apolarity stratifications. |

## The most valuable next projects

### 1. Test the classification against classical knowledge

The central theorem lives in well-mapped territory: rational normal curves, their tangent developables, catalecticant stratifications. A specialist can likely confirm, refute, or antedate the classification with classical methods in days. Any of the three outcomes is decisive for the release.

### 2. Discharge the conditional hypothesis

The isolation theorem — the screening programme's payoff — is conditional on the cubic classification from the companion exotic-spheres release, which is itself unrefereed. Verifying that upstream classification (or reproving the isolation theorem without it) would convert a conditional statement into the programme's first unconditional structural result.

### 3. Resolve the surviving slice

The programme has narrowed its search to a single named slice. The sharpest possible follow-up is binary: construct a nonzero-constant-Jacobian map on the tangent nonosculating linear–quadratic slice, or prove none exists. Either outcome closes a branch — and an exclusion would retire this family as a source of Jacobian-conjecture candidates altogether.

## Specialist audience candidates

The natural specialist readers are projective and affine algebraic geometers working on secant varieties, catalecticants, and the geometry of rational normal curves; motivic-obstruction specialists; and Jacobian-conjecture researchers following structured screening programmes. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

> A natural incidence family turns reducible exactly on the tangent developable of the rational normal curve — and the same analysis eliminates every slice but one as a stage for constant-Jacobian behaviour.

## The screening programme

Across this trilogy of releases, the strategy is consistent: rather than attacking the Jacobian conjecture head-on, map the geography of factorisation-space slices and eliminate, with exact obstructions, every stage where a counterexample or interesting Keller map cannot live. This release performs the elimination step. What remains — one specific slice — is now a sharply posed target: construct the map there, or exclude it and close the branch.

## What is in the evidence package

The deposit contains the manuscript (PDF and TeX), deterministic audit scripts (`verify_paper.py`, `verify_rank_two.py`), a claim-to-evidence map (`AI_INDEX.md`/`.json`), assurance-boundary documents (`STATUS.md`, `ASSURANCE.md`), SHA-256 manifests, and immutable source snapshots, archived on Zenodo as v1.0-candidate with a v1.0.1 revision (same-day metadata and attribution fixes).


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/reducible-incidence-divisors/paper.json)

- Verify the tangent-developable classification by hand — the statement is precise, classical in flavour, and a natural target for expert review or refutation.
- Discharge or refute the conditional hypothesis: the isolation theorem depends on an upstream cubic classification that has itself had no external verification.
- Check novelty against the classical literature on catalecticants, secant varieties, and tangent developables, where fragments of the classification may already be known.
- Extend the motivic-defect calculations to non-adjacent degree pairs beyond the cyclic-action obstruction.
- Determine whether the unique surviving slice actually carries a nonzero-constant-Jacobian map — either constructing one or excluding it, which would close this branch of the screening programme.

---

# Exotic affine three-spheres and the quadratic–cubic obstruction

- URL: https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/
- PDF: https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic/releases/download/v0.1.1/paper.pdf
- DOI: https://doi.org/10.5281/zenodo.21647607
- Code: https://github.com/ipitchford/exotic-affine-spheres-quadratic-cubic
- Published: 2026-07-28 · Version: 0.1-candidate (v0.1.1 'citation robustness' archived as 10.5281/zenodo.21653108) · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

In geometry, some shapes are impostors. They look identical to a familiar shape by every measurement that smooth topology can make — yet as *algebraic* objects, defined by polynomial equations, they are provably different. The most celebrated examples in three dimensions are the exotic affine spheres discovered by Dubouloz and Finston: algebraic varieties diffeomorphic to the standard complex three-sphere that no polynomial change of coordinates can turn into it.

This paper argues that one of these impostors arises, unforced, inside a very natural construction: slicing the space of polynomial factorisations studied in this project's foundational release. The transverse linear–quadratic slice, it argues, is not the standard three-sphere (the SL2 quadric) it appears to be — it is precisely a Dubouloz–Finston exotic sphere.

Two further claims round out the picture. Every normalised quadratic–cubic slice fails to be ordinary five-dimensional affine space, blocked by an exact obstruction computed in the Grothendieck ring of varieties. And in characteristic 3 — arithmetic where $3 = 0$ — the tangent slice stays perfectly ordinary, while the polynomial map it carries develops a genuinely characteristic-p phenomenon, an Artin–Schreier collision.

## Summary for specialists

Three principal claims. First, the transverse linear–quadratic slice of the binary-form factorisation space is the Dubouloz–Finston torsor — an exotic affine three-sphere — rather than $\mathrm{SL}_2$. Second, every normalised quadratic–cubic slice fails to be $\mathbb{A}^5$, proved by an exact Grothendieck-class formula together with a rank-by-rank Hodge–Deligne obstruction. Third, in characteristic 3 the tangent slice remains $\mathbb{A}^3$, with an Artin–Schreier collision in the induced Keller map.

The computational support is deterministic and exact: symbolic checks of selected formulae, plus finite-field censuses over $\mathbb{F}_2$ through $\mathbb{F}_{11}$ covering 23,941 projective functionals, validated byte-for-byte against checked-in tables, with a deliberate semantic control confirming the assertions are live. The assurance boundary matters here more than usual: the censuses corroborate *consequences* of the claims; the torsor identification and the motivic obstruction arguments themselves have had no external review, and their verification is the release's own headline request.

## Technical summary

The stage is the binary-form factorisation space of the foundational release, and the object is its transverse linear–quadratic slice — a smooth affine threefold that, on topological inspection, has the homotopy type of the complex quadric $\mathrm{SL}_2 = \{xy - zw = 1\}$. The manuscript's central argument identifies it instead with the Dubouloz–Finston torsor: an $\mathbb{A}^1$-bundle over the punctured affine plane that is diffeomorphic to $\mathrm{SL}_2$ but not algebraically isomorphic to it, the distinction detected by the Makar-Limanov-style invariants of the Dubouloz–Finston theory rather than by any topological quantity.

The quadratic–cubic obstruction is motivic. For each normalised quadratic–cubic slice the manuscript computes an exact class in the Grothendieck ring of varieties and expands its Hodge–Deligne (E-)polynomial rank by rank; a nonvanishing coefficient where $\mathbb{A}^5$ requires zero blocks any isomorphism with affine five-space, uniformly across the family. The characteristic-3 analysis runs the tangent-slice theory over $\mathbb{F}_3$: the slice itself remains $\mathbb{A}^3$, but the induced Keller map acquires an Artin–Schreier collision — a fibre coincidence of the form $t \mapsto t^3 - t$ that has no characteristic-zero counterpart.

The computational layer is corroborative, not probative: exact symbolic checks of selected formulae, plus finite-field censuses over $\mathbb{F}_2$ through $\mathbb{F}_{11}$ covering 23,941 projective functionals, validated byte-for-byte against committed tables, with a deliberate semantic control (an omitted term must break the check). The torsor identification and the motivic computations themselves are prose mathematics awaiting expert eyes.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Affine algebraic geometers | If a Dubouloz–Finston exotic sphere arises unforced in a natural factorisation construction, exoticity looks like default behaviour rather than engineered pathology. | Verify or refute the torsor identification — the release's most consequential claim and squarely their machinery (Makar-Limanov invariants, A¹-bundle classification). |
| Motivic and Hodge-theory specialists | A rank-by-rank E-polynomial obstruction deployed against a whole family of would-be affine spaces at once. | Check the Grothendieck-class formula independently; ask whether the obstruction pattern generalises to other slice families. |
| Arithmetic geometers (characteristic p) | A clean, small example of characteristic-3 divergence: the space stays standard while its Keller map degenerates via Artin–Schreier behaviour. | Map the phenomenon across small primes; determine whether it constrains or enables lifting arguments. |
| Jacobian conjecture researchers | The obstruction results feed the project's screening programme, which the sequel release turns into an isolation theorem. | Audit the cubic classification here, on which the sequel's conditional theorem explicitly depends. |
| Computational algebraic geometers | A finite-field census as a cheap, replayable corroboration layer for motivic claims. | Rerun the censuses independently; extend them to larger fields as a sharper consistency test. |

## The most valuable next projects

### 1. Adjudicate the torsor identification

Everything interesting about this release concentrates in one claim: the transverse slice is the Dubouloz–Finston torsor, not $\mathrm{SL}_2$. The tools for deciding it are established, and the specialists who built them are identifiable. A confirmation would put a naturally occurring exotic sphere on record; a refutation would collapse the headline while leaving the obstruction results to stand or fall separately.

### 2. Re-derive the motivic obstruction

The Grothendieck-class formula and its Hodge–Deligne expansion are exact, finite computations. Reproducing them by independent means — different software, or by hand for low ranks — would either certify the quadratic–cubic obstruction or locate an error precisely. The finite-field censuses provide the consistency data to check against.

### 3. Chart the characteristic-p landscape

The characteristic-3 Artin–Schreier collision invites a systematic question: for which primes and which slices does reduction preserve the affine-space structure while degenerating the map? Even a small census across $p \in \{2, 3, 5, 7\}$ would show whether characteristic 3 is special or the first case of a pattern.

## Specialist audience candidates

The most natural specialist readers are the researchers whose constructions the manuscript claims to have found in the wild — the authors of the exotic affine 3-sphere literature — together with motivic-obstruction specialists and the affine-geometry community around the recognition problem for affine spaces. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

> A slice nobody tuned to be exotic is claimed to be exactly your exotic sphere — and the claim comes with exact finite-field censuses, a motivic obstruction for the neighbouring family, and an honest list of what has not been checked.

## Why an impostor here would be interesting

Exotic affine spheres are usually exhibited by deliberate construction. Finding one as the *inevitable* form of a natural slice — a subvariety nobody tuned to be exotic — would suggest that exoticity is not a curiosity of engineered examples but a default behaviour of factorisation geometry. It would also sharpen the project's Jacobian-conjecture programme: knowing exactly which slices are, and are not, affine space is the screening question the companion releases pursue.

## What is in the evidence package

The deposit contains the manuscript (PDF and TeX), `verify_paper.py` with its finite-field censuses and semantic control, the recorded verification output, a claim-to-evidence map (`AI_INDEX.md`), integrity documentation (`ASSURANCE.md`), and SHA-256 manifests. A follow-up revision (v0.1.1, archived separately) repaired citation routing after a cache incident, with manuscript content unchanged.


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/exotic-affine-three-spheres/paper.json)

- Have the torsor identification checked by specialists in affine algebraic geometry — in particular, experts on Dubouloz–Finston exotic spheres — since this is the manuscript's most consequential claim.
- Verify the Grothendieck-class formula and the rank-by-rank Hodge–Deligne obstruction by hand; the shipped checks validate consequences, not the arguments.
- Rerun the finite-field censuses (F2 through F11, 23,941 projective functionals) from an independent implementation.
- Search the literature for prior art on slices of binary-form factorisation spaces to establish novelty.
- Investigate whether the characteristic-3 Artin–Schreier phenomenon has characteristic-p analogues for other small primes.

---

# Erdős problem 848: an exact answer for every N

- URL: https://evidence-press.pages.dev/releases/erdos-848-all-n/
- PDF: https://github.com/ipitchford/erdos-848-all-n/releases/download/v0.1-candidate/paper.pdf
- DOI: https://doi.org/10.5281/zenodo.21647629
- Code: https://github.com/ipitchford/erdos-848-all-n
- Published: 2026-07-28 · Version: 0.1-candidate · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

A number is *squarefree* if no perfect square bigger than one divides it: 10 is squarefree, 12 is not (it is divisible by 4). Erdős and Sárközy asked a deceptively simple question. Pick numbers between 1 and $N$ so that whenever you multiply two of your picks together and add one, the result is *never* squarefree. How many numbers can you pick?

There is a natural strategy: take every number that leaves remainder 7 when divided by 25. Multiply two such numbers and add one, and the result is always divisible by 25 — never squarefree. This gets you roughly $N/25$ picks, and Erdős asked whether you can do meaningfully better. Recent breakthrough work showed that for all *sufficiently large* $N$ you cannot. But "sufficiently large" leaves infinitely many actual values of $N$ unresolved.

This release establishes the complete answer: for every $N$ from 1 upwards, the maximum is exactly $\lfloor (N+18)/25 \rfloor$. The gap between "small" and "sufficiently large" is bridged by five overlapping verification regimes — exact certificates up to a hundred million, structural and exact-rational arguments through $2.64 \times 10^{17}$, and an explicit analytic theorem beyond that.

## Summary for specialists

Let $f(N)$ be the largest size of $A \subseteq \{1, \ldots, N\}$ such that $ab+1$ is nonsquarefree for all $a, b \in A$, including $a = b$. Erdős problem 848 asks for $f(N)$ and whether the residue-class construction $\{n \equiv 7 \pmod{25}\}$ is optimal. van Doorn proved $|A| \leq (0.108\ldots + o(1))N$; Sawhney resolved the problem for all sufficiently large $N$, with the stronger structural statement that any near-extremal set lies in $\{n \equiv 7\}$ or $\{n \equiv 18\} \pmod{25}$.

This release claims $f(N) = \lfloor (N+18)/25 \rfloor$ for *every* positive integer $N$, via five overlapping ranges: exact finite colouring certificates with compact endpoint induction for $1 \leq N \leq 100{,}000{,}006$; exhaustive structural decomposition for $10^8 \leq N \leq 10^9$; exact-rational short-shift envelopes to $10^{12}$; exact-rational rank envelopes to $2.64 \times 10^{17}$; and Sothanaphan's explicit-threshold theorem beyond. Each regime overlaps its neighbours, and the whole is replayed by a Python framework with semantic mutation testing and sanitizer controls, plus a standalone C++ verifier for the certificate range.

The weight of novelty is in the middle ranges: the envelope arguments are bespoke, and independent scrutiny should start there and at the hand-off to the analytic threshold.

## Technical summary

Write $f(N)$ for the extremal size. The claimed identity $f(N) = \lfloor (N+18)/25 \rfloor$ is equivalent to two statements for every $N$: the residue construction achieves $\lfloor (N+18)/25 \rfloor$ (immediate — the count of $n \leq N$ with $n \equiv 7 \pmod{25}$), and nothing does better. The release discharges the second statement in five overlapping regimes.

For $1 \leq N \leq 100{,}000{,}006$ the release ships exact finite colouring certificates: combinatorial data assigning to each interval a witness structure whose validity implies the bound, compressed as a colouring-delta binary and replayed by a standalone C++ verifier, with a compact induction closing the endpoints. For $10^8 \leq N \leq 10^9$ an exhaustive structural decomposition reduces to the certified lower range. From $10^9$ to $10^{12}$, and again from $10^{12}$ to $2.64 \times 10^{17}$, the argument switches to exact-rational "short-shift" and "rank" envelopes — monotone bounding functions whose coefficients are exact rationals, so the comparisons are replayable without floating point. Beyond $2.64 \times 10^{17}$, Sothanaphan's explicit-threshold theorem (an external, pinned PDF with recorded SHA-256) takes over, resting in turn on Sawhney's structural resolution for large $N$.

Every regime overlaps its neighbours, so the composite claim has no uncovered interval — *if* each regime is sound. The envelope arguments in the middle ranges are the bespoke component; the replay framework subjects them to semantic mutations (deliberate perturbations that must fail) and sanitizer controls, but they have had no external scrutiny.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Combinatorial number theorists | An exact all-N determination layered directly on the recent asymptotic resolution — the unusual half of the problem is precisely the finite one. | Referee the envelope arguments; check the hand-off at 2.64 × 10^17 against the threshold theorem's exact hypotheses; test the closed form at accessible N. |
| Researchers on the Erdős problems catalogue | Problem 848 is recorded as resolved for large N; this claims the remaining gap. | Assess whether the evidence justifies updating the problem's status; identify neighbouring problems (such as 844) where the architecture transfers. |
| Certificate and reproducibility engineers | A 75.6 MB package mixing C++ verification, Python replay with mutation testing, and pinned external dependencies. | Replay on independent toolchains; audit the SHA-256-pinned external PDF dependency pattern, which is unusual and worth standardising or criticising. |
| Analytic number theorists | The explicit-threshold layer converts an asymptotic structural theorem into a concrete cut-over point. | Verify or sharpen the 2.64 × 10^17 threshold; a lower verified threshold would shrink the certificate burden by orders of magnitude. |
| AI-assisted mathematics researchers | An AI system is the listed creator of a result that leans, with explicit pinning, on two named human mathematicians' work. | Examine the provenance and attribution model; test whether the claim–evidence index supports genuinely independent re-derivation. |

## The most valuable next projects

### 1. Scrutinise the envelope regimes

The exact-rational short-shift and rank envelopes covering $10^9 \leq N \leq 2.64 \times 10^{17}$ are the least conventional part of the architecture and carry most of the novel risk. An expert reading — or an independent reimplementation of just these ranges — would either expose the weak joint or convert the release's most original contribution into its most trusted one.

### 2. Confirm the analytic hand-off

The composite proof is only as sound as the splice at $2.64 \times 10^{17}$: the pinned threshold theorem must cover all larger $N$ under exactly the normalisation of $f(N)$ used below the splice. Checking hypothesis alignment at a single boundary is a well-posed, bounded task with outsized consequences.

### 3. Shrink the certified range

The certificate range up to $10^8$ exists because the analytic threshold is astronomically high. Any improvement to the explicit threshold propagates directly into a smaller, faster, more auditable package — a rare case where analytic sharpening has an immediate engineering payoff.

## Specialist audience candidates

The natural specialist readers are the mathematicians whose work the release builds on and pins — the authors of the large-N resolution and the explicit-threshold note — together with the maintainers of the Erdős problems catalogue and the community around certified combinatorial computation. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

> The asymptotic resolution of problem 848 left infinitely many finite cases open; this package claims all of them, with an exact closed form and a replayable certificate chain from N = 1 to the analytic threshold.

## Why the last step matters

Asymptotic resolutions of extremal problems are the norm in combinatorial number theory; exact all-$N$ determinations are rare, because the finite ranges below the analytic threshold are usually astronomically beyond direct computation. The interest of this release — if it survives scrutiny — is architectural: it shows one way to industrialise the unglamorous half of a problem, by stitching certified computation to an explicit analytic bound with no uncovered gap. The same five-regime pattern could plausibly transfer to neighbouring problems.

## How the mathematics was produced

The Zenodo record lists an AI system (OpenAI Codex) as creator, with human research direction and publication management. The release explicitly credits the prior human results it builds on — van Doorn's bound and Sawhney's resolution for large $N$, and Sothanaphan's explicit threshold, which is a pinned external dependency rather than a claim of this release.

## What is in the evidence package

The 75.6 MB deposit contains the manuscript (LaTeX and PDF), the Python replay framework with mutation and sanitizer controls, the C++ verifier `verify_erdos848_to_100006.cpp` with its compressed colouring-delta binary, SHA-256 manifests, and machine-readable claim and assurance indexes (AI_INDEX, STATUS, ASSURANCE, PROVENANCE, REPRODUCIBILITY).


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/erdos-848-all-n/paper.json)

- Independently replay the certificates on different hardware and toolchains — the C++ verifier for N ≤ 10^8 with its compressed colouring-delta binary is the most accessible entry point.
- Independently verify the exact-rational envelope arguments covering 10^9 ≤ N ≤ 2.64 × 10^17, which are the least conventional part of the architecture.
- Confirm the hand-off: check that the pinned explicit-threshold theorem genuinely covers all N ≥ 2.64 × 10^17 under the same normalisation of f(N).
- Formalise the endpoint-induction scheme in a proof assistant.
- Ask whether the same five-regime architecture (certificates, structural decomposition, envelopes, analytic threshold) transfers to neighbouring extremal problems such as Erdős problem 844.

---

# The degree-difference principle and affine slices of binary-form factorisation spaces

- URL: https://evidence-press.pages.dev/releases/degree-difference-affine-slices/
- PDF: https://github.com/ipitchford/degree-difference-affine-slices/releases/download/v0.1-candidate/paper.pdf
- DOI: https://doi.org/10.5281/zenodo.21647593
- Code: https://github.com/ipitchford/degree-difference-affine-slices
- Published: 2026-07-28 · Version: 0.1-candidate · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

Take two polynomials. You can multiply them together, and you can also compute their *resultant* — a single number that vanishes exactly when they share a root. Now regard the pair of operations as one combined map: feed in two polynomials, get out their product and their resultant.

This paper proves that the Jacobian determinant of that combined map — the quantity measuring how the map stretches volume — has a strikingly simple exact form: up to sign, it is the *difference of the two degrees* multiplied by the *square of the resultant*. Nothing else survives. The paper calls this the degree-difference principle, and it has an immediate consequence: the map is volume-degenerate exactly where the two polynomials share a root, or where their degrees are equal.

From this starting point the manuscript develops the geometry of "slices" of the space of polynomial factorisations — the natural stages on which questions like the century-old Jacobian conjecture play out. It classifies the simplest slices completely, writes down explicit coordinates for the polynomial maps they carry, and identifies obstructions that rule out overly simple behaviour in higher degrees.

## Summary for specialists

For binary forms $A$ of degree $r$ and $B$ of degree $s$, consider the multiplication–resultant map $\Phi_{r,s}(A,B) = (AB, \mathrm{Res}(A,B))$. The central identity is

$$
\det D\Phi_{r,s} = (-1)^{s(r+1)} (r-s) \, \mathrm{Res}(A,B)^2 .
$$

Around this the manuscript establishes torsor and divisor-class statements for the factorisation spaces, a classification of normalised linear–quadratic slices, explicit coordinates for tangent nonosculating slices and their induced Keller maps, a characterisation of fibres and image, and Euler-characteristic obstructions in higher-degree cases.

The verification boundary is drawn precisely, and specialists should note it: a SymPy script (with deliberate negative controls) confirms the determinant identity for all bidegrees $r+s \leq 4$ and base-point signs through degree eight, but the general-degree argument and the structural proofs are *not* machine-checked. The natural first question for a referee — does this identity already exist in the resultant literature in some guise? — is also the release's own first suggested follow-up.

## Technical summary

Fix bidegrees $(r, s)$ with $r \neq s$ and work over a field of characteristic zero. On the affine space of coefficient pairs, $\Phi_{r,s}(A,B) = (AB, \mathrm{Res}(A,B))$ is a polynomial map from a $(r+1)+(s+1)$-dimensional space to a $(r+s+1)+1$-dimensional one, so the source and target dimensions agree and the Jacobian determinant is a single polynomial in the coefficients. The degree-difference principle identifies it exactly: $(-1)^{s(r+1)}(r-s)\,\mathrm{Res}(A,B)^2$. Two structural consequences follow immediately. The branch locus is precisely the resultant hypersurface (pairs with a common root), independent of everything else; and in the equal-degree case $r = s$ the map is everywhere degenerate — the geometric reason the paper's slice constructions always break degree symmetry.

From the identity, the manuscript builds the affine-slice theory used across the project: torsor descriptions of the fibres over the complement of the branch locus, divisor-class computations for the factorisation spaces, a complete classification of normalised linear–quadratic slices, explicit coordinates for the tangent nonosculating slices (where induced Keller maps live), a characterisation of fibres and image, and Euler-characteristic obstructions for higher-degree slices.

The verification boundary is stated with unusual precision. The shipped SymPy checker validates the determinant identity for every bidegree with $r+s \leq 4$ via explicit Sylvester-matrix computation, and base-point sign conventions through degree eight, with deliberate negative controls confirming the tests can fail. The all-degree identity, and all of the structural theory (torsor, divisor-class, orbit classification), rest on the manuscript's prose proofs alone.

## Who should care, and why

| Likely audience | What should interest them | What they could do with it |
|---|---|---|
| Classical algebraic geometers (resultants, elimination theory) | A closed-form Jacobian for the multiplication–resultant map — elementary to state, and either known in disguise or a genuinely missing classical identity. | Settle the priority question against the resultant literature; supply a conceptual proof (the shipped verification is low-degree only). |
| Affine algebraic geometers | The slice classifications feed directly into the project's exotic-sphere identification and isolation theorems. | Audit the torsor and divisor-class arguments — the unchecked half of the release — before the downstream results are taken further. |
| Jacobian conjecture researchers | Explicit coordinates for slices carrying Keller maps, inside a framework built to screen where constant-Jacobian behaviour can occur. | Test the framework against known Keller-map families; probe whether the screening logic survives scrutiny. |
| Symbolic computation researchers | A worked example of drawing an exact machine-checked boundary inside a prose manuscript, with negative controls. | Extend the checker to higher bidegrees; automate general-degree identity verification, which is within reach of current systems. |

## The most valuable next projects

### 1. Settle the priority question

An identity this clean — the Jacobian of (product, resultant) equalling $\pm(r-s)\mathrm{Res}^2$ — sits in territory mined since the nineteenth century. The single most useful contribution a specialist could make is bibliographic: either locate the identity in the classical or modern literature, which would recalibrate the whole release, or confirm its absence, which would establish a small but permanent result.

### 2. Review the structural half

The torsor, divisor-class, and orbit-classification arguments are deliberately outside the symbolic checker's scope. They are also what the two companion releases build on. Expert review of these sections is therefore leveraged three ways: it tests this manuscript, and it underwrites — or undermines — the exotic-sphere and isolation claims downstream.

### 3. Mechanise the general identity

Proving the determinant identity for all bidegrees in a computer algebra system (or a proof assistant) is a well-scoped project: the Sylvester structure is explicit, the answer is known, and success would convert the release's central claim from "checked through bidegree 4" to "proved".

## Specialist audience candidates

The natural specialist readers are algebraic geometers working on resultants and elimination theory, the affine-geometry community whose slice questions the framework serves, and Jacobian-conjecture researchers interested in structured families of Keller maps. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

> One exact identity — the Jacobian of the multiplication–resultant map is the degree difference times the resultant squared — and a complete low-degree slice theory built on it, with the machine-checked boundary drawn honestly.

## Where this sits in the project

This is the foundational release of a trilogy on binary-form factorisation spaces. The companion releases build on the slices classified here: one identifies a specific slice as an exotic affine three-sphere and obstructs quadratic–cubic slices from being affine space; the other classifies reducible incidence divisors and proves conditional isolation statements aimed at Jacobian-conjecture screening.

## What is in the evidence package

The deposit contains the manuscript (PDF and TeX), the verification script `verify_degree_difference_affine_slices.py` with negative controls, a claim-level evidence map (`AI_INDEX.md`), assurance and provenance documentation (`STATUS.md`, `ASSURANCE.md`, `PROVENANCE.md`, `SOURCES.md`), and SHA-256 manifests, archived on Zenodo and Software Heritage.


## Open directions (machine-readable copy at https://evidence-press.pages.dev/releases/degree-difference-affine-slices/paper.json)

- Verify the all-degree determinant identity by an independent conceptual proof — or locate it in prior literature on resultants and multiplication maps, where such identities may already exist in another form.
- Subject the torsor and divisor-class arguments, which the symbolic checker deliberately does not cover, to expert human review.
- Extend the symbolic verification beyond bidegree 4 and automate the identity checks at general degree.
- Determine whether the Euler-characteristic obstructions can be upgraded to full non-isomorphism proofs in the higher-degree cases.
- Connect the classified slices explicitly to the companion releases on exotic affine spheres and reducible incidence divisors, which build on this framework.