# Evidence Press — full text

---

# Random Reshuffling as a Baseline Policy: Exact Value Certificates and Common-Hessian Ordering Hardness

- URL: https://evidencepress.org/releases/random-reshuffling-baseline-policy/
- PDF: https://github.com/ipitchford/random-reshuffling-baseline-policy/releases/download/v1.0.0-candidate/random-reshuffling-baseline-policy-v1.0.0-candidate.pdf
- DOI: https://doi.org/10.5281/zenodo.21757895
- Code: https://github.com/ipitchford/random-reshuffling-baseline-policy
- Published: 2026-08-02 · Version: 1.0.0-candidate · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

Machine-learning systems often shuffle training examples before processing them. Random reshuffling is a strong and widely studied baseline: choose a fresh random order, visit every example once, and repeat. This candidate asks a more surgical question. If the random draw contains some still-fair binary choices, can an algorithm resolve only those choices using the exact consequence for the objective—and leave the rest of the random baseline intact?

For one exact model, the answer is yes. When every quadratic training component has the same Hessian, a random permutation can be viewed as an **unoriented skeleton** of pairs plus independent fair bits saying which member of each pair goes first. The paper's value-aware rule resolves those exposed bits using exact conditional loss. For each realized skeleton it produces an explicit conditional decrement identity, and after averaging its one-epoch loss is no greater than under the **same random-reshuffling law**.

That comparator matters. The result is not “a hand-designed order beats some unrelated random experiment.” It is a coupled conditional-expectation statement inside the original random draw.

The paper also finds two limits. First, two appealing ideal ordering proxies can select globally optimal proxy orders whose true terminal-loss ratio is unbounded, even in a normalized four-item scalar family. Second, globally optimizing the exact terminal objective is computationally hard in a narrow but revealing regime: when the stable step size is part of the input and can approach the endpoint $2$, the scalar threshold problem is NP-complete via PARTITION. At a fixed distance from that endpoint, the paper instead gives a normalized additive approximation.

The work is a public research candidate, not a finished machine-learning method. A registered 32-instance matrix-free pilot preserved its safety logic but failed its practical feasibility gate. It establishes no neural-network, model-scale, wall-clock, or generalization improvement.

> **Candidate status:** unrefereed · internal model-assisted editorial outcome: **Major Revision** · external specialist review and independent reproduction absent · only the encoded core of Theorem 2 is formalized in Lean.

## The idea in one picture

![A uniform random permutation is decomposed into an unoriented pairing skeleton and fair orientation bits. Exact conditional value resolves only those bits and retains the same random-reshuffling comparator.](/assets/figures/rr-baseline-policy-flow.svg "Random reshuffling supplies both the skeleton and fair orientation bits. The policy resolves exposed bits by conditional value and compares against the same RR completion.")

The key move is to treat random reshuffling as a **baseline policy**, not merely as a probability distribution to average over. The algorithm samples the same skeleton that RR would induce. Before orientation, each pair is still a fair coin. Exact conditional value says which face of that coin has lower continuation value. If an approximation cannot certify the sign, a fresh fair fallback coin restores the baseline comparison.

This is deliberately a restricted action space. The policy does not search all $n!$ orders. It resolves at most ⌊n/2⌋ binary decisions that are already present inside one RR draw.

## Summary for specialists

For common-Hessian convex quadratics, the manuscript derives the affine endpoint of one permutation and writes terminal excess loss as a dense positive-semidefinite quadratic form over time-indexed residual placements. The time kernel has rank at most the parameter dimension. This gives the exact value coefficient needed for pair orientation; it is an order-selection representation, not a claim to the first exact analysis of RR dynamics.

The proxy-separation theorem constructs a rational scalar family with four residuals. Global minimizers of an ideal maximum-prefix discrepancy and of a dynamics-weighted diagonal proxy lie in explicitly characterized argmin sets, yet their exact terminal-loss ratio against a loss-optimal order grows without bound after residual normalization. Pinned Lean 4.32.1/mathlib formalizes the encoded combinatorial and rational-recurrence core, including the argmin sets and quantified unboundedness. The analytic bridge from the quadratic model and every other theorem remain conventionally proved.

For policy improvement, a uniform permutation is factored into an RR-distributed unoriented skeleton and conditionally independent Rademacher orientation variables. Under predictable exposure, sequential conditional minimization of the exact quadratic value makes the realized decrement term $D_t=|a_t|$ and yields the conditional identity

$$
\mathbb E[\Phi(v_m)\mid\mathcal F_0]
=\mathbb E[\Phi(z^{\mathrm{fair}})\mid\mathcal F_0]
-\mathbb E\!\left[\sum_t |a_t|\,\middle|\,\mathcal F_0\right],
$$

with the coefficient and filtration defined in the manuscript. This is an equality of conditional expectations, not a claim that the oriented path beats every independently sampled fair completion. Iterated expectation gives one-epoch noninferiority to the same RR law. The paper treats fixed and history-measurable exposure, odd-cardinality singleton coupling, deterministic ties, and certify-or-randomize fallback. Separate Bellman recursions give finite terminal, finite cumulative, and spectrally conditioned discounted comparisons.

The global decision problem—Scalar Common-Hessian Final-Loss Ordering, or SCFLO—uses $H=1$, zero initial error, zero-sum rational residuals, and an instance-encoded stable step η ∈ (1, 2). A PARTITION reduction establishes NP-completeness as η approaches 2. Because the source is weakly NP-complete and the step varies with the instance, the result does **not** establish strong NP-hardness or fixed-step hardness. Randomized and average-case hardness are also open. With η ≤ 2 − c for fixed c > 0, ordered-suffix enumeration supplies normalized additive endpoint and loss bounds plus an optimum interval, not a multiplicative PTAS or FPTAS.

## Three different claims that should not be conflated

| Surface | Choice being made | What the candidate establishes | What remains outside the claim |
|---|---|---|---|
| Proxy separation | Globally minimize one of two idealized ordering proxies | The resulting true terminal-loss ratio can be unbounded | It does not lower-bound the output of GraB, CD-GraB, PairBR, or Herding at Optimum |
| Baseline policy | Orient fair pairs inside an RR-induced skeleton | Realized decrement terms and conditional expected loss no greater than the same RR law | It does not compute the globally optimal permutation or cover heterogeneous nonlinear training |
| Global ordering | Choose any full permutation | NP-complete threshold decision near an input-dependent stability endpoint; fixed-factor deterministic barrier unless P=NP | No strong, fixed-step, randomized, or average-case hardness |

The visual boundary is the central intellectual result: a low-dimensional action exposed by the baseline can be exactly improved even when the unrestricted permutation problem is hard.

![A tractability map separates locally exposed pair orientations, fixed-margin suffix approximation, and global near-endpoint ordering hardness.](/assets/figures/ordering-tractability-boundary.svg "The tractability boundary depends on both the action space and the stability regime: local fair-bit decisions admit exact certificates, fixed-margin global ordering admits normalized additive control, and near-endpoint global threshold ordering is NP-complete.")

## Why the proxy counterexample matters

Ordering research often uses discrepancy or prefix statistics because they are computationally manageable and can support convergence analysis. That remains legitimate. The candidate's counterexample makes a narrower point: even if one could globally optimize two such idealized criteria, their optima need not approximate the exact one-epoch terminal objective multiplicatively.

The distinction is especially sharp because the example uses only four scalar components, exact rational arithmetic, a stable shared curvature, and bounded normalized residuals. It therefore removes numerical noise and high-dimensional geometry as explanations for the failure. What it does **not** show is that prefix control is useless, that published balancing algorithms fail in practice, or that worst-case ratios predict typical training behaviour.

## A hard global problem and a safe local move

There is no contradiction between the local certificate and global hardness. They optimize different action spaces.

The local policy receives an RR-generated skeleton and chooses only orientation bits. Conditional expectation can resolve a fair binary choice without increasing the baseline expectation. The global problem may place every item in every slot, which exposes the full combinatorial burden. Near the stability endpoint, the dynamics amplify late-placement information enough to encode PARTITION.

The fixed-margin result adds a useful qualification. When the step stays a constant distance from the endpoint, remote positions are geometrically discounted. Enumerating a suffix whose length grows logarithmically with the requested normalized additive accuracy controls what the unseen prefix can change. The paper therefore offers a three-part frontier rather than a blanket “ordering is hard” message:

- exact conditional control for exposed fair bits;
- normalized additive global control at a fixed stability margin; and
- NP-complete global threshold ordering when the input-dependent step approaches the endpoint.

## The practical pilot failed its own gate

Exact conditional value is expensive. It may require dense curvature actions or an additional pass over component gradients. The candidate therefore preregistered a matrix-free pilot using truncated Hessian-polynomial actions and a strict rule: orient a pair only when an error enclosure excludes the wrong sign; otherwise use a fresh fair coin.

The frozen study covered 32 synthetic instances and 176 instance-degree cells. Every declared safety control passed, ordinary and optimized Python agreed exactly, and there were zero false certified signs on the tested grid. But the practical gate was not met. Degrees one and two certified nothing; degree four succeeded in the two $n=6$ strata but failed coverage, retained-gain, and no-certificate requirements in both $n=8$ strata. No approximation degree was selected.

That negative/mixed outcome is informative. The immediate bottleneck is certified error control and coverage, not evidence that an approximate sampler improves training. A method that rarely certifies a decision simply falls back to RR and cannot justify its extra cost.

## What is original—and what is inherited

The candidate does not claim to invent random reshuffling analysis, quadratic epoch maps, prefix balancing, conditional expectation, Bellman recursion, Hessian-vector products, or composition-ordering complexity. Those are established lines of work.

Its claimed contribution is the conjunction of several narrower pieces:

- an order-resolved low-rank value representation for the declared common-Hessian objective;
- the exact four-item separation between two ideal proxy argmin sets and terminal loss;
- a same-RR skeleton coupling in which dynamics-filtered conditional value gives a realized decrement certificate;
- a same-model boundary between near-endpoint global hardness and fixed-margin normalized additive control; and
- a preregistered negative result showing that one conservative matrix-free compression route does not yet clear its feasibility gate.

Originality and priority have not been externally certified. The manuscript contains a detailed equation-and-action-law comparison to prior work, but public release and internal source triage are not substitutes for specialist literature review.

## What is not established

The following statements would overread the candidate:

- **No neural-network improvement.** The theorems use item-independent linear dynamics with positive-semidefinite quadratic value. Heterogeneous Hessians, nonlinear networks, data augmentation, momentum with item-dependent transitions, and adaptive optimizers fall outside the exact guarantee.
- **No practical speedup.** The oracle may cost an additional gradient pass and dense curvature work. No wall-clock, energy, accelerator, or communication saving has been measured.
- **No universal ordering superiority.** The policy compares a restricted orientation action to its coupled RR baseline. It neither finds nor approximates every globally optimal order.
- **No broad hardness theorem.** The reduction uses an instance-dependent step approaching $2$ and a weakly NP-complete source.
- **No whole-paper formal proof.** Lean covers only the declared encoded core of Theorem 2.
- **No independent verification.** The replay, mutation tests, hashes, release certificate, and model-assisted audits were produced within the same research workflow.
- **No peer or venue acceptance.** The internal Stage 3-prime assessment remains Major Revision.

## Why this could matter for machine learning

The credible opportunity is not “a theorem immediately makes neural networks train faster.” It is a research design principle: use a strong random baseline to expose small fair actions, replace a fair choice only when its value consequence is certified, and otherwise retain the original randomness.

That pattern could be attractive in training systems because it is fail-safe at the decision level. A sufficiently sharp approximation can act; an uncertain approximation can fall back. But transfer requires evidence that three quantities coexist: high certificate coverage, useful objective gain, and overhead low enough to improve net time or energy to a fixed target. The registered pilot has not yet produced that combination.

The most promising bridge is a staged programme rather than a direct model-scale claim.

## The most valuable next projects

### 1. A heterogeneous-Hessian validity margin

Derive an explicit perturbation bound for component Hessians that differ from a shared reference or drift during an epoch. The goal is a computable margin: if the exact common-Hessian orientation score exceeds the perturbation radius, the chosen sign remains valid. This would turn the current model assumption into a certificate that can succeed or fail on real batches.

### 2. Coverage-first compression

Design diagonal, low-rank, Krylov, sketching, or probabilistic enclosures for the value action. Optimize the fraction of signs certified—not merely mean-squared curvature accuracy—and charge every Hessian-vector product, extra gradient evaluation, memory transfer, and synchronization. The negative pilot suggests that a sharper error model may matter more than a higher polynomial degree.

### 3. A preregistered transfer ladder

First expand the exact synthetic generator across condition number, factor rank, spectral decay, residual alignment, step size, horizon, and pairing strategy. Next test frozen-backbone or LoRA workloads where curvature drift and per-example gradients can be measured at moderate cost. Only after passing predefined coverage and overhead gates should the method move to small nonlinear networks. Comparators should include RR, maximum-prefix and diagonal proxies, published balancing methods, exact LAPO where feasible, and full conditional expectation on tiny problems.

### 4. Close the complexity frontier

Determine the exact complexity at a fixed step, the nonoscillatory range η ≤ 1, strong hardness, and structured cases such as bounded residual alphabets or few distinct values. The current near-endpoint reduction and fixed-margin additive theorem leave a concrete mathematical gap.

### 5. Independent reconstruction

The highest-value assurance step is not another producer-side replay. Independent specialists should reconstruct the RR/value identity, the policy and Bellman arguments, and the complexity reduction without reusing the production verifiers. A separate implementation of the 32-instance pilot would test whether the negative coverage result survives different software and numerical choices.

## Who should care, and why

| Likely audience | Why the candidate matters | Useful next action |
|---|---|---|
| Optimization and learning theorists | It separates exact order value, proxy objectives, restricted fair-bit control, and global complexity inside one model. | Audit the assumptions and close the fixed-step complexity gap. |
| SGD and data-ordering researchers | The same-RR comparator gives a precise target for certified local interventions. | Compare the value coefficient with existing balancing and permutation methods on exact shared-curvature instances. |
| ML systems researchers | A certify-or-randomize policy offers a possible fail-safe interface, but its oracle cost is unresolved. | Build a coverage-first low-rank or Krylov implementation and report net time or energy to target. |
| Formal-methods researchers | The headline proxy family has a pinned Lean crosswalk, while most of the paper remains conventional. | Extend formalization to the analytic bridge, conditional-expectation theorem, and complexity reduction. |
| Complexity theorists | Near-endpoint hardness and fixed-margin additive control leave a sharply bounded frontier. | Determine fixed-step, strong, randomized, and structured-instance complexity. |
| AI research agents | Claim IDs, non-inference rules, exact replay, and open packages make the candidate unusually machine-routable. | Preserve qualifiers, reproduce from fresh sources, and attach critiques to the exact claim surface. |
| Training safety and governance researchers | Data order is a control surface that can improve or manipulate learning. | Require order, seed, objective, input-hash, subgroup, and corrupted-label logging in any experimental implementation. |

## How it was checked

The candidate's producer-side package reports exact rational verifiers, cause-specific mutation controls, and 91 tests under both ordinary and optimized Python. The tagged release workflow rebuilds the 55-page PDF, checks manifests and privacy boundaries, and replays from a fresh archive extraction. The matrix-free pilot's ordinary and optimized receipts agree byte for byte.

The Lean project pins Lean 4.32.1 and a specific mathlib revision, treats warnings as errors, rejects project-local proof placeholders, audits 15 manuscript crosswalk anchors with negative controls, and reports only Lean's standard logical dependencies for the formalized declaration.

These checks establish the declared byte, execution, and formal-artifact facts. They do not establish the truth of conventionally proved theorems, novelty, independent reproduction, expert acceptance, neural-network performance, or peer review.

## What is in the evidence package

The [public GitHub repository](https://github.com/ipitchford/random-reshuffling-baseline-policy) contains the sole 55-page anonymous paper, canonical Markdown and TeX source, bibliography, exact Python verifiers, tests, frozen receipts, the registered pilot protocol and readout, the pinned Lean project, a manuscript-to-Lean theorem crosswalk, claim and evidence registries, public-domain dedication, and deterministic release tooling.

The immutable candidate release is identified by tag [`v1.0.0-candidate`](https://github.com/ipitchford/random-reshuffling-baseline-policy/releases/tag/v1.0.0-candidate) and version DOI [`10.5281/zenodo.21757895`](https://doi.org/10.5281/zenodo.21757895). Its six release assets are the PDF, deterministic tagged-tree ZIP, complete replay transcript, detached publication certificate, certificate-hash sidecar, and `SHA256SUMS.txt`.

The repository's [`AI_INDEX.md`](https://github.com/ipitchford/random-reshuffling-baseline-policy/blob/main/AI_INDEX.md) and machine-readable [`AI_INDEX.json`](https://github.com/ipitchford/random-reshuffling-baseline-policy/blob/main/AI_INDEX.json) route agents to canonical claim IDs, evidence, non-inference rules, and open extensions. Reuse is dedicated to the public domain under CC0 to the extent the repository maintainer holds the relevant rights; cited third-party material and external software remain outside that dedication.


## Open directions (machine-readable copy at https://evidencepress.org/releases/random-reshuffling-baseline-policy/paper.json)

- Prove a perturbation theorem for heterogeneous or drifting component Hessians with an explicit margin that certifies when a common-Hessian orientation decision remains valid.
- Replace the conservative binomial-action bounds with low-rank, diagonal, Krylov, sketching, or probabilistic error controls that maximize certified sign coverage while charging preprocessing, memory, communication, and Hessian-vector products.
- Run a preregistered transfer ladder from expanded synthetic generators to frozen-backbone or LoRA workloads and then small nonlinear networks, comparing RR, established balancing baselines, exact LAPO, and certified approximations on net time or energy to a fixed target.
- Close the exact complexity gap between near-endpoint NP-completeness and fixed-margin normalized additive tractability: determine fixed-step complexity, strong hardness, the nonoscillatory range, and structured or pseudo-polynomial cases.
- Characterize joint matching and decision-order selection: test approximate submodularity, exchange properties, greedy guarantees, and hardness on small exact instances.
- Extend the finite and discounted value theorems to decaying steps, variance reduction, interpolation, or an average-cost objective with the required uniform-integrability or transversality conditions.
- Independently reconstruct the paper's central proofs and executable results without using the production verifiers, and obtain specialist audits of the probability/value identities and complexity reduction.

---

# Stocks Are Not Flows: An Identification Audit of Investor Pressure and Tenure Change in England

- URL: https://evidencepress.org/releases/stocks-are-not-flows/
- PDF: https://github.com/ipitchford/stocks-are-not-flows/releases/download/v1.0.0-preprint/stocks-are-not-flows-v1.0.0.pdf
- DOI: https://doi.org/10.5281/zenodo.21740339
- Code: https://github.com/ipitchford/stocks-are-not-flows
- Published: 2026-08-01 · Version: 1.0.0 · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

England has seen a striking shift from owner occupation to private renting, especially among younger and middle-aged households. One influential explanation is that rising wealth at the top increased demand for housing as an asset: better-resourced buyers accumulated property, while households further down the distribution found ownership harder to reach.

That explanation may be true in part. The available aggregate data cannot tell us how much.

This paper reconstructs the public evidence and then asks a stricter question: do the observed tenure shares, house prices, rents, mortgage rates, housing supply, and dwelling stocks distinguish wealthy-investor pressure from other plausible mechanisms? In the registered design, they do not. Credit conditions, housing supply, demographics, tax changes, middle-resource losses, institutional entry, and combinations of these mechanisms can produce the same observed pattern.

The result is deliberately neither "wealthy investors caused the shift" nor "wealthy investors had no effect." It is that the public aggregate series usually brought into this debate do not identify the effect. The project therefore stopped before fitting a structural model that would have produced a precise-looking but non-unique historical attribution.

This matters for public debate because a tenure share is a stock, while the proposed mechanism concerns flows: who bought which property, what its previous and subsequent use was, whether the buyer was a landlord, what resources the buyer had before purchase, and where the funds came from. Near-mirror changes in owner occupation and private renting do not, by themselves, reveal those transactions.

## What the public data show

For English households whose household reference person was aged 25 to 44, the reconstructed owner share fell from 60.5270% in 2008-09 to 48.5783% in 2018-19, a decline of 11.9487 percentage points. Private renting rose from 22.1325% to 34.4594%, an increase of 12.3269 points. The social-renter residual changed comparatively little.

Other official series document rising house prices and stock rents, changes in relative prices, a recovering net housing-supply rate, and changing quoted mortgage costs over the registered windows. These are real and important accounting facts. They constrain any serious model of the period.

They do not reveal the gross transactions behind the net tenure stocks. A home moving out of owner occupation might have been bought by an existing cash landlord, a company, an institutional vehicle, a newly leveraged small landlord, or an owner occupier whose circumstances later changed. A property appearing in rental listings does not directly reveal the purchaser's wealth rank or the origin of the purchase funds. Different mechanisms can also operate simultaneously.

## Summary for specialists

The paper converts an initially planned structural extension of Stevenson's 2019 MPhil work into a prospective identification audit. Its residual contribution is not a new claim that inequality can affect homeownership or house prices; that broad mechanism already has close theoretical and empirical neighbours. The contribution is a design-conditioned discriminator test, an explicit stopping rule, a conditional sharp-set framework, and a value-of-information criterion for new measurements.

IDENT-0 tests whether the focal response to an aggregate top-resource forcing variable can be separated from admitted credit, supply, demographic, tax, and alternative-resource responses under the registered observation map. The focal response lies in the rival-and-nuisance span. The authoritative decision is therefore `FAIL-IDENT-0`, and the planned unique structural attribution is stopped.

That is stronger than saying a regression may be endogenous, but weaker than claiming that no conceivable design could ever identify the mechanism. The conclusion is conditional on the registered observables and admitted response class. New data or externally justified restrictions could change the column space. They must be justified independently of the desired conclusion.

The accompanying partial-identification machinery is intentionally conditional. It can compute a sharp set once source-backed response bounds are supplied; it does not turn synthetic loadings into an empirical interval for England. The current release consequently reports no causal coefficient, historical contribution share, or empirical England sharp interval.

## Why this changes the debate

Housing arguments often move too quickly from three sound observations to an unsupported attribution:

1. owner occupation fell;
2. private renting rose by a similar amount;
3. wealth became more concentrated.

The first two statements describe tenure stocks. The third describes a distribution. None directly observes the property-use conversions, purchaser identities, or funding origins required to connect them causally.

This does not make the proposed mechanism implausible. It changes the burden of proof. A commentator may present wealthy-investor pressure as a hypothesis supported by theory and circumstantial evidence. They should not present the aggregate paths as measuring its historical contribution. Conversely, a failed aggregate identification design cannot be used to claim that the contribution was zero.

The practical lesson is constructive: public agencies and researchers should ask which additional measurement would create a genuinely new discriminating direction, rather than adding more variables that reproduce the same ambiguity.

## Who should care, and why

| Likely audience | Why the result matters | Useful next action |
|---|---|---|
| Housing and tax policymakers | Policies aimed at landlords, credit, supply, or wealth distribution rely on different causal stories. Aggregate tenure changes cannot choose between them. | Require mechanism-specific evidence and report attribution uncertainty explicitly in impact assessments. |
| Journalists and public commentators | Near-offsetting owner and renter shares make an intuitively powerful graphic, but they are not observed owner-to-landlord flows. | Separate documented accounting changes from causal interpretation; ask what purchaser and property-use data support the claimed channel. |
| Housing economists and macroeconomists | A well-fitting structural model can still be non-specific if omitted rivals generate aligned responses. | Run a prospective observation-map audit before estimation and retain null, sign-reversing, and set-valued outcomes. |
| Survey, land, mortgage, and tax data custodians | The missing object crosses institutional boundaries: transactions, property use, purchaser status, and pre-purchase resources. | Design privacy-preserving linkage with explicit error models and a pre-analysis value-of-information test. |
| Partial-identification researchers | The current boundary is a natural starting point for a transparent sharp set rather than a forced point estimate. | Supply defensible response ranges and propagate classification, linkage, and equilibrium uncertainty. |
| Reproducibility researchers | The package demonstrates a positive scientific use of a failed gate: it stops an underidentified fit while preserving accounting results and adversarial controls. | Reimplement the audit independently and test the semantic coverage of the response class, not only the code path. |

## The most valuable next projects

### 1. Measure the joint transaction object

The highest-value new record would jointly observe:

- the property's use before and after purchase;
- whether the purchaser was or became a landlord;
- the purchaser's non-housing resources before purchase; and
- the financing and legal form of the acquisition.

No single field is enough. Transaction-to-listing linkage can classify a useful property flow, but listing windows, platform coverage, delayed letting, company ownership, and refinancing create measurement error. Tax or mortgage linkage can add purchaser information, but anticipation, legal-form switching, and equilibrium spillovers still need to enter the response class.

### 2. Replace point attribution with empirical bounds

Once credible response ranges exist, the paper's sharp-set machinery can ask how wide the historically admissible contribution is. An informative result might be positive, near zero, negative under some admissible responses, or sign-ambiguous. All of those outcomes would be more honest than an unsupported singleton.

### 3. Independently reproduce the accounting and geometry

The current package deterministically replays its own transformations and synthetic design checks. Independent reproduction should begin from fresh official downloads, reconstruct the age-band tenure shares, and implement the focal-versus-rival geometry without reusing the repository's intermediate data or code. Survey-design information is also needed before attaching defensible uncertainty intervals to the combined 25-44 tenure shares.

## How it was checked

The release passed 135 tests under normal Python and the same suite under optimized Python, an optimized scientific-gate self-check, 43 registered public-data checks, hash verification, deterministic replay, negative and sign-reversing controls, PDF structural checks, and privacy scans. The GitHub and Zenodo release assets were read back publicly and are byte-identical.

Those checks establish internal consistency and reproducibility of the shipped accounting and design audit. They are not independent replication, peer review, formal verification, or expert acceptance. The response design is synthetic rather than an estimated model of England, and the broad investor-pressure mechanism is prior art.

## What is in the evidence package

The public release contains the 30-page anonymous preprint and source, Python code with a pinned environment, the 135-test suite, frozen IDENT-0 and public Gate A reports, a deterministic offline replay, conditional partial-identification tools, seven official-source objects with receipts and Open Government Licence attribution, derived tables and figures, SHA-256 manifests, an assurance statement, and a publication-integrity report.

It does not redistribute the supplied Stevenson thesis, safeguarded or proprietary microdata, address-level records, cached literature PDFs, private Git history, personal commit metadata, or local-path manifests.


## Open directions (machine-readable copy at https://evidencepress.org/releases/stocks-are-not-flows/paper.json)

- Construct a linked measurement that jointly records gross property-use conversion, purchaser landlord status, and the purchaser's pre-purchase non-housing resources, while documenting linkage error, anticipation, legal-form switching, reclassification, and equilibrium spillovers.
- Develop source-backed bounds for the focal and rival response classes, then compute an empirical sharp identified set for England; near-zero, signed, and sign-ambiguous results must all remain admissible.
- Reproduce the public accounting layer independently from the original official downloads and implement the observation-map audit without reusing the repository's code or intermediate files.
- Obtain the survey-design information needed to attach defensible uncertainty intervals to the combined 25-44 English Housing Survey tenure shares.
- Compare alternative linked-listing definitions of investor acquisition and owner-to-landlord conversion, and quantify their sensitivity to listing windows, platform coverage, purchaser type, and funding-source misclassification.
- Use the discriminator theorem as a pre-analysis value-of-information test for candidate administrative, tax, mortgage, ownership, and transaction linkages before committing to a full structural estimation.

---

# Irreducible Pushforwards and Constrained Quartic Transitions: Two Reusable Methods from a Study of Furter's R(3) Conjecture

- URL: https://evidencepress.org/releases/irreducible-pushforwards-quartic-transitions/
- PDF: https://github.com/ipitchford/irreducible-pushforwards-quartic-transitions/releases/download/v1.0.0-preprint/irreducible-pushforwards-quartic-transitions-v1.0.0.pdf
- DOI: https://doi.org/10.5281/zenodo.21745937
- Code: https://github.com/ipitchford/irreducible-pushforwards-quartic-transitions
- Published: 2026-08-01 · Version: 1.0.0 · Licence: CC0-1.0
- Status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)

## Summary

This paper does not solve Furter's $R(3)$ conjecture. It reports a productive failure.

The original target was Furter's $R(3)$ conjecture, an open problem about three consecutive coefficients of a generating function and the ideal they generate. The project did not solve that problem. Instead, it isolated two pieces of mathematics that no longer depend on the unfinished global argument and may be useful elsewhere.

The first is a geometric separation principle. Very roughly, an irreducible polynomial seen on the base of a map can certify that a collection of zeros upstairs is really one indivisible algebraic point. If two rational functions have pole orders differing by exactly one, that indivisibility leaves no room for them to share a zero. On an elliptic double cover, a second group-law test prevents the zero set from colliding with its mirror image under the deck involution.

The second is an exact calculation at a quartic transition, the kind of local model that appears when several saddle points coalesce. After reducing polynomial amplitudes in a twisted de Rham quotient, three natural classes lose rank on the aligned transition curve. The determinant, rank strata, and first correction are all computed exactly.

These are proved statements. Their role in Furter's family is more limited. At each fixed index, the geometric method applies if an explicit deck norm is irreducible. The release checks that irreducibility exactly for $3\le n\le40$. That finite range is evidence, not an all-index theorem. Uniform irreducibility, the global contour and period estimates, and Furter's $R(3)$ conjecture remain open.

## Summary for specialists

The first result is a divisor-theoretic criterion: coefficient-one pushforward to a prime divisor of degree $h$ forces an effective degree-$h$ divisor upstairs to be one reduced closed point. Exact pole orders $hO$ and $(h+1)O$ then exclude adjacent common zeros; on an elliptic double cover, the condition $hS\ne O$ separately excludes intersection with the deck transform. The coefficient-one hypothesis, not bare norm irreducibility, is essential.

The second result is an exact calculation in the twisted polynomial de Rham quotient for $\Psi(W)=W^4-\tfrac32KW^2+bW$. The classes $[1],[\Psi],[\Psi^2]$ have determinant $\tfrac{9b}{64}(-3K^3+K+3b^2)$. On the aligned transition curve the span has rank two away from the origin and rank one at the origin, with cancellation of the first moving-alignment correction. This local algebra does not supply the missing contour-uniform global asymptotics.

## The first method: irreducibility becomes separation

Suppose a map sends an effective divisor $Z$ of degree $h$ on a smooth projective curve to the zero divisor of one irreducible polynomial of degree $h$, with coefficient one. Proper pushforward counts both multiplicity and residue-field degree. Because every contribution is a positive integer, the coefficient-one hypothesis forces a rigid conclusion: $Z$ consists of one reduced closed point of degree $h$.

That observation becomes useful when two rational functions $f$ and $g$ have exact pole divisors

$$
(f)_\infty=hO,\qquad (g)_\infty=(h+1)O.
$$

If $f$ and $g$ shared a zero, the one closed point making up $(f)_0$ would have to lie inside $(g)_0$. Only one degree of zero divisor would remain. Dividing $g$ by $f$ would then produce a function with divisor $R-O$, where $R$ has degree one. On any curve with $|O|=\{O\}$, this forces $R=O$, contradicting the fact that $O$ is the pole. The zero sets are therefore disjoint.

The important lesson is that irreducibility alone is not enough. The full divisor pushforward, including coefficient one, does the work. A repeated norm divisor such as $e(H)_0$ would not imply the same reduced-point conclusion.

![A prime divisor on the base lifting to one reduced closed point on a double cover, with adjacent and deck-transformed zeros kept disjoint.](/assets/figures/irreducible-pushforward-method.svg "The separation mechanism: coefficient-one prime pushforward fixes the zero divisor upstairs; the pole gap and elliptic deck test then provide two distinct disjointness conclusions.")

### The elliptic deck test

On an elliptic curve, a degree-two deck involution has the form

$$
\sigma(Q)=S-Q.
$$

If the prime zero divisor $Z$ met its deck image $\sigma Z$, the two prime divisors would coincide. But the elliptic sum of the points in $Z$ is $O$, while the sum in $\sigma Z$ is $hS$. Equality would force $hS=O$. Consequently, $hS\ne O$ excludes the collision; if $S$ is nontorsion, the exclusion holds for every $h$.

This separates two mechanisms that are easy to conflate. The pushforward argument says that the zeros form one reduced algebraic orbit. The one-degree pole gap excludes an adjacent zero divisor, while the elliptic group law excludes its deck image.

## The second method: an exact quartic rank collapse

The local phase is

$$
\Psi(W)=W^4-\frac32KW^2+bW.
$$

In the twisted polynomial de Rham quotient, integration by parts reduces every polynomial amplitude to the basis $[1],[W],[W^2]$. Expressing $1,\Psi,\Psi^2$ in that basis gives the exact coordinate determinant

$$
\det[1,\Psi,\Psi^2]
=\frac{9b}{64}\bigl(-3K^3+K+3b^2\bigr).
$$

On the limiting aligned transition curve, the factor in parentheses vanishes, so the three classes cannot remain independent. The paper proves more than the vanishing:

- for $K\ne0$, the span has rank exactly two;
- at $(K,b)=(0,0)$, the rank drops to one; and
- the first correction caused by the moving alignment also cancels.

This gives an algebraic explanation for a degeneracy that might otherwise look like a numerical coincidence. It does not, by itself, justify moving the original integration contour or prove an asymptotic estimate that is uniform across Stokes transitions. Those analytic tasks remain separate.

![A quartic phase beside the aligned transition curve, where three twisted de Rham classes collapse to rank two and then rank one at the origin.](/assets/figures/quartic-transition-rank-collapse.svg "The constrained quartic transition: the exact determinant vanishes on the aligned curve, giving rank two away from the origin and rank one at the origin.")

## What this says about Furter's R(3)

Furter's problem asks whether, for every integer $s\ge1$,

$$
a_3^s\in(u_s,u_{s+1},u_{s+2})
\subset\mathbb{Q}[a_1,a_2,a_3],
$$

where $u_r$ is a coefficient extracted from a cubic generating function.

The paper identifies, for each fixed Furter index, an explicit deck norm on an aligned elliptic curve. If that norm is irreducible, the first method supplies the needed adjacent and self-deck divisor disjointness conclusions at that index. The exact replay package verifies the norm's irreducibility from $n=3$ through $n=40$, with a separate deck diagnostic through $n=20$.

The missing word is **uniform**. A proof for every index needs an all-index irreducibility argument, not a longer table of successful cases. It also needs the remaining global contour, cycle, Stokes, and period estimates connecting the quartic local calculation to the original problem. Neither step is supplied here.

The correct status is therefore:

| Layer | Status |
|---|---|
| Prime-pushforward, adjacent-freeness, and elliptic deck-collision results | Proved |
| Quartic determinant, rank strata, and first-correction cancellation | Proved |
| Furter divisor separation at one fixed index | Conditional on that index's deck-norm irreducibility |
| Deck norms for $3\le n\le40$ | Exactly checked finite evidence |
| Uniform norm irreducibility, global analytic control, and $R(3)$ | Open |

## Why the methods may travel

The first method is a compact bridge between arithmetic information and geometry. Norm or elimination polynomials often appear when a divisor upstairs is viewed from a simpler base. When the pushforward is genuinely one irreducible prime with coefficient one, the argument can replace a complicated common-zero computation by a closed-point degree count. The additional pole-gap and deck-sum tests are modular: they may be reusable in other families of curves, covers, or parameterised rational functions.

The second method gives a disciplined way to diagnose dependence among amplitudes near a coalescing-saddle transition. Reducing first in the twisted de Rham quotient avoids confusing polynomial independence with independence of period classes. The exact determinant then identifies the rank-drop locus and distinguishes generic rank two from the more singular rank-one origin.

These tools sit at an intersection of algebraic geometry, elliptic curves, polynomial automorphisms, asymptotic analysis, Gauss-Manin systems, and Stokes phenomena. Their broader value will depend on whether researchers can recognise the same structural ingredients in other problems.

## Who should care, and why

| Likely audience | Why the result matters | Useful next action |
|---|---|---|
| Algebraic geometers | Coefficient-one prime pushforward turns an irreducible polynomial into a reduced closed point and a base-point-freeness conclusion. | Test the criterion on other finite maps, norm divisors, and one-degree pole gaps. |
| Researchers on polynomial automorphisms | The package isolates exact Furter-family geometry without claiming the parent conjecture. | Seek a structural all-index proof or counterexample for the deck norms. |
| Asymptotic analysts | The quartic calculation identifies an exact rank-collapse locus and a first-order cancellation. | Build the missing contour-uniform and Stokes-compatible analytic theorem. |
| Twisted de Rham and Gauss-Manin researchers | The computation distinguishes period-class dependence from ordinary polynomial dependence. | Look for an invariant or categorical interpretation of the determinant and rank strata. |
| Computer-assisted mathematics researchers | The release separates theorem replay, bounded evidence, and open global steps. | Reimplement the identities independently and formalise the algebraic arguments. |
| Research agents | The package is a reusable map of what survived an unsuccessful open-problem attack. | Reuse only the proved modules and preserve the conditional and open-status labels. |

## How it was checked

The paper-level identities replay exactly under normal Python and optimized Python with SymPy 1.14.0. The optional Furter audit is self-contained and verifies the deck-norm irreducibility computations for $3\le n\le40$; the auxiliary deck diagnostic stops at $n=20$.

The frozen release also passed a 22-file SHA-256 manifest, fresh extraction from the tagged Git commit, complete replay from that extraction, LaTeX and PDF structural checks, citation closure, privacy scanning, and a seven-mode publication-integrity audit. The GitHub and Zenodo assets were downloaded publicly and are byte-identical to the frozen local release.

These checks establish internal consistency and reproducibility of the specified algebraic and finite computations. They are not independent reproduction, proof-assistant verification, expert acceptance, editorial acceptance, or peer review. Agreement among AI-assisted audits does not change that assurance boundary.

## What is in the evidence package

The public release contains the 14-page anonymous methods preprint and source, an accessible Markdown reading copy, the standalone exact verifier, the bounded Furter audit, a dated prior-art scoping report, a claim-and-reuse index for research agents, a CC0 licence, assurance and publication-integrity reports, and a SHA-256 manifest.

The source archive is tied to Git commit `8645f34bea51be9c91ea02c3b5c967c98bc269ea` and release tag `v1.0.0-preprint`. The version DOI is `10.5281/zenodo.21745937`; the concept DOI for all versions is `10.5281/zenodo.21745936`.


## Open directions (machine-readable copy at https://evidencepress.org/releases/irreducible-pushforwards-quartic-transitions/paper.json)

- Prove or disprove uniform irreducibility of the exact Furter deck norms for every index, without treating a longer finite computation as the principal result.
- Complete the global analytic step: construct a contour deformation and Stokes-compatible cycle control with uniform period estimates through the constrained quartic transition.
- Determine whether the exact determinant and first-correction cancellation admit a broader invariant interpretation for coalescing saddles, Gauss-Manin systems, or Stokes-filtered twisted de Rham families.
- Find further settings in which coefficient-one irreducible pushforward plus a one-degree pole gap gives useful base-point freeness or deck-collision exclusion.
- Independently reproduce the theorems and exact calculations without reusing the repository's symbolic code, and formalise the algebraic arguments in a proof assistant.

---

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

- URL: https://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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://evidencepress.org/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.

---

# Policy Identification Observatory

- URL: https://evidencepress.org/observatory/
- Machine record: https://evidencepress.org/observatory/index.json
- Repository: https://github.com/ipitchford/policy-identification-observatory
- Versioned release: https://github.com/ipitchford/policy-identification-observatory/releases/tag/v0.1.0-candidate.1
- DOI: https://doi.org/10.5281/zenodo.21758648
- Audio: https://evidencepress.org/assets/audio/observatory.mp3
- Transcript: https://evidencepress.org/assets/audio/observatory-transcript.txt
- Status: candidate-research-infrastructure
- Included case terminal status: REJECTED_INSUFFICIENT_RIGOUR; truth certified: false

Policy debates often fail before estimation begins. Analysts may confuse stocks with flows, treat a proxy as though it directly measured a mechanism, omit rival explanations, fit parameters that the observations cannot separate, or report a point estimate where the evidence permits only a range. More data and more computation do not repair those defects automatically.

The **Policy Identification Observatory** is being built to find them systematically.

It uses persistent artificial intelligence agents, explicit research protocols, formal and computational methods, and adversarial reimplementation to audit consequential policy claims. Its purpose is not to automate political judgement. It is to improve the evidential map on which judgement operates.

A completed audit should tell decision-makers:

- what the public evidence directly establishes;
- which conclusions depend on additional assumptions;
- which rival mechanisms remain observationally compatible with the evidence;
- whether the focal effect is point identified, partially identified, or unidentified;
- which policy choices remain robust across the surviving models; and
- what new measurement, linkage, experiment, or institutional record would most improve the decision.

The unit of work is therefore not commentary. It is a versioned research case with a claim, an evidence boundary, a mathematical result, an assurance status, and a replay path.

> **Current status, 2 August 2026:** the foundational build is complete. It establishes the operating protocol, machine-readable case structure, validation and replay machinery, synthetic test cases, and reusable agent workflows. The next goal will run a complete audit on a live policy claim and test the institution outside its founding exemplar.

## The founding example

The Observatory grew out of [*Stocks Are Not Flows: An Identification Audit of Investor Pressure and Tenure Change in England*](/releases/stocks-are-not-flows/).

That project began as a planned structural extension of an economic model. Before fitting the model, the research agents asked whether the available aggregate observations could distinguish the proposed investor-pressure mechanism from plausible alternatives. Under the registered observation map, they could not. Credit conditions, housing supply, demographics, tax changes, alternative resource losses, institutional entry, and combinations of these mechanisms could generate the same observed pattern.

The project therefore stopped before producing a unique historical attribution. It preserved the public accounting results, proved a design-conditioned non-identification result, supplied conditional partial-identification tools, and identified the joint transaction record that would add the most useful discriminating information.

![Left panel: one observed series with four different mechanism curves passing exactly through the same records. Right panel: three axes showing a point-identified estimate, a bracketed partially identified interval, and a wholly unidentified parameter.](/assets/art/observatory-identification.svg "The defect the Observatory looks for. When several mechanisms reproduce the same records exactly, the evidence cannot choose between them, and the honest output is a bounded set rather than a point estimate.")

That outcome suggested a broader institution. Many policy disputes need the same sequence: define the claim, inspect the measurement, construct the rival mechanisms, test identification, bound what remains, and redesign the evidence before producing an answer.

## The question the Observatory asks

For each case, the Observatory begins with one exact public claim and one decision that the claim may affect. It then asks:

1. **What is the estimand?** What quantity, effect, population, geography, period, unit, and comparison does the claim concern?
2. **What is actually observed?** Which records are direct measurements, which are constructed variables, and which are proxies?
3. **What accounting must hold?** How do stocks, entries, exits, durations, transfers, and residuals relate?
4. **Which mechanisms could produce the observations?** What is the strongest plausible rival-mechanism class supported by the inspected corpus?
5. **Can the observations distinguish the focal mechanism?** Does the design identify the target locally or globally, or do observationally equivalent countermodels remain?
6. **What is the strongest bounded conclusion?** Is the sign known? Is the magnitude bounded? Is the result sensitive to one assumption or corpus choice?
7. **Which actions survive the uncertainty?** Are any options dominated, robust, or low-regret across the admissible model set?
8. **What additional information would matter?** Which new field, linkage, comparison, intervention, or experiment would create a genuinely discriminating direction?

The intended analytical path is:

```text
public claim
    → exact estimand and decision
    → frozen evidence boundary
    → measurement and accounting model
    → focal and rival mechanisms
    → identification or identified set
    → robust decisions
    → value of additional information
```

## How a case runs

![A flow diagram of nine stages in three rows: claim registration, corpus freeze and measurement audit; rival mechanisms, identification gate and bounded inference; sensitivity and decision analysis, adversarial verification, and release and persistence. Below are the publication-ready, blocked and rejected terminal-status families.](/assets/art/observatory-pipeline.png "Nine stages, three phases. Evidence work precedes inference; inference precedes any decision claim; and nothing is released without an adversarial pass and a registered terminal status.")

| Stage | Central question | Required public object |
|---|---|---|
| **1. Claim registration** | What exactly was asserted, by whom, when, and for which decision? | Original wording, source, estimand, population, period, units, comparison and decision context |
| **2. Corpus freeze** | Which evidence was inspected, and what was excluded? | Search date, source ledger, inclusion and exclusion rules, frozen or receipted source objects |
| **3. Measurement and accounting audit** | What do the data observe, and how do the quantities relate? | Measurement ontology, stock-flow or state-transition equations, unit reconciliation, missingness and linkage audit |
| **4. Rival-mechanism construction** | What else could generate the same pattern? | Source-backed rival ledger, causal or structural pathways, nuisance responses and semantic coverage statement |
| **5. Identification gate** | Can the focal contribution be separated from the rivals? | Rank, null-space, observability, causal-equivalence, optimisation, symbolic, or countermodel analysis as appropriate |
| **6. Bounded inference** | What is the strongest conclusion the design warrants? | Point estimate only when identified; otherwise sharp or conservative bounds, sign result, infeasibility result, or explicit ambiguity |
| **7. Sensitivity and decision analysis** | Which conclusions and choices survive plausible changes? | Assumption sweeps, corpus-boundary tests, specification checks, robust-policy set and value-of-information analysis |
| **8. Adversarial verification** | Can a fresh agent break the result or reproduce it independently? | Review ledger, counterexample search, mutation tests, fresh implementation where feasible, resolved and unresolved findings |
| **9. Release and persistence** | Can another researcher inspect, replay, correct, or continue the work? | Paper or technical report, code, evidence package, claim-to-evidence index, assurance statement, manifest and machine-readable receipt |

A case does not have to produce a positive causal estimate to succeed. It may end with a publication-ready non-identification result, an identified interval, a robust-decision map, a minimum-data theorem, or a documented stop. The Observatory treats these as scientific outputs rather than failed attempts.

Its terminal statuses are explicit:

```text
PUBLICATION_READY
PUBLICATION_READY_NONIDENTIFICATION
PUBLICATION_READY_PARTIAL_IDENTIFICATION
BLOCKED_MISSING_DATA
BLOCKED_SEMANTIC_UNCERTAINTY
REJECTED_NO_NOVEL_RESULT
REJECTED_INSUFFICIENT_RIGOUR
```

These labels prevent a polished narrative from concealing an unresolved scientific defect.

## What makes the institution agent-native

Calling a project agent-native should mean more than using an artificial intelligence system to draft prose or run code. The research process itself must exploit capabilities that persistent agents make possible.

### Persistent research state

Ordinary AI conversations are ephemeral. The Observatory treats research as durable state. Each case records the exact claim, corpus boundary, source provenance, assumptions, transformations, failed approaches, test results, review findings, unresolved questions, and current assurance status in a versioned repository.

A new agent should be able to continue from inspected objects rather than reconstructing the project from a narrative summary. Long-running goals or equivalent persistent execution can pursue a case until it reaches a registered success condition, a genuine blocker, or a resource ceiling. Scheduled agents can maintain the candidate queue, refresh source registries, or revisit cases when new evidence appears.

### Adversarial production

The agent that constructs an analysis does not certify itself. A fresh review agent begins from the registered claim and evidence boundary, reconstructs the load-bearing steps, searches for omitted mechanisms and sign-reversing countermodels, and tries to make the result fail.

Where possible, the review should use fresh downloads, separate code, a clean worktree, a new context, and a different model family. These measures reduce shared-context and implementation dependence. They do not guarantee full epistemic independence, and the assurance statement must not pretend that they do.

The most valuable review is often semantic rather than computational: did the formal object really represent the public claim, and did the rival class cover the mechanisms that matter?

### Reproducible evidence

Each release should include, where lawful and technically possible:

- frozen inputs or source receipts;
- a corpus manifest and source ledger;
- exact transformations and unit conventions;
- pinned software environments;
- deterministic replay commands;
- tests, negative controls and deliberately mutated cases;
- hashes for the released artefacts;
- a claim-to-evidence index;
- an assurance statement that draws the claim ceiling; and
- a machine-readable receipt recording the terminal status.

Deterministic replay establishes that the released package reproduces its own stated computation. It does not establish independent replication, semantic correctness, causal validity, peer acceptance, or truth. Observatory releases use the [Evidence Press verification ladder](/about/#the-verification-ladder) to preserve those distinctions.

### Machine-readable research

The evidence boundary and conclusion should be usable by other agents without scraping prose. Claims, sources, assumptions, mechanisms, missingness, identification status, sensitivity results, review findings, open problems, and receipts therefore use documented schemas.

This allows another research system to do more than cite a conclusion. It can inspect the assurance status, identify the unresolved dependency, reproduce a calculation, challenge an excluded mechanism, or take up the most valuable next experiment. Evidence Press already exposes its releases through [machine-readable endpoints designed for research agents](/ai/); the Observatory extends the same principle into the research process itself.

### Cumulative correction

A persistent institution must remember errors as well as results. Failed approaches, rejected candidates, refutations, corpus revisions and corrected releases remain part of the record. Corrections change the status openly rather than silently rewriting history.

This creates institutional memory that does not depend on one person or one model retaining the details. It also gives future agents negative knowledge: which routes failed, why they failed, and which assumptions caused the failure.

## Designed to improve decisions

The Observatory does not treat causal estimation as the only useful endpoint. Policy decisions often must proceed while several mechanisms remain compatible with the evidence.

Suppose the current evidence leaves a set of admissible models. The Observatory can compare each policy across that set and report:

- options that are dominated under every surviving model;
- options that meet a declared minimum standard across all models;
- choices whose ranking depends on one visible assumption;
- choices with low maximum regret;
- thresholds at which the preferred decision changes; and
- new information most likely to alter the decision.

![A grid scoring four policy options against six models the evidence still admits. One option fails under every model, one passes under every model, one fails under a single model, and one is split evenly.](/assets/art/observatory-decision-map.svg "A robust-decision map. Uncertainty about mechanism does not always imply uncertainty about action: a dominated option can be ruled out, and a robust option defended, without ever identifying the effect.")

This is a decision map, not an automated policy verdict. Elected institutions, public bodies and affected communities must still set objectives, distributional weights, rights constraints, acceptable risks and political priorities. Agents can clarify the consequences of those choices. They cannot derive the values that authorise them.

## Division of responsibility

| Participants | Primary responsibilities |
|---|---|
| **Research agents** | Claim discovery, corpus inspection, formalisation, measurement audits, modelling, proof, computation, countermodel search, sensitivity analysis, drafting, replay and adversarial review |
| **Human stewards** | Public-purpose prioritisation, normative and legal boundaries, privacy decisions, contentious corpus exceptions, escalation of high-impact claims, publication approval and institutional accountability |
| **External researchers and agents** | Independent reproduction, refutation, prior-art discovery, semantic challenge, formal verification, data improvement and follow-up research |

The Observatory should automate work that benefits from breadth, persistence and exact replay. It should preserve explicit human authority where the task involves values, rights, sensitive data, legal responsibility, or publication on behalf of an institution.

## What the Observatory can publish

A mature programme should produce several kinds of output:

### Identification audits

These determine whether the registered observations can separate a focal mechanism from plausible rivals. A rigorous failure result can stop an unsupported structural attribution before it acquires false precision.

### Partial-identification results

When the evidence cannot support a single value, the audit should derive the strongest defensible set or bound and show which assumptions narrow it.

### Measurement and accounting audits

These reconcile units, populations, classifications, stocks, gross transitions, net transitions, durations and residuals. They can expose a policy argument that asks one dataset to measure an object it never observes.

### Robust-decision analyses

These identify policies that remain acceptable across the surviving model set and disclose the assumptions that change the ranking.

### Minimum-data and experiment designs

These identify the smallest new measurement or design change likely to distinguish the remaining mechanisms. The result may redirect data collection before an agency commits to an expensive linkage, survey or trial.

### Formal audits of rules and systems

Some policy questions concern legislation, tax and benefit rules, procurement criteria, allocation systems, queues or networks rather than causal attribution. Agents can encode these systems and search for contradictions, cliff edges, infeasible cases, gaming opportunities, bottlenecks and unintended interactions.

### Stop receipts

A case may lack the data, semantic precision, novelty, or rigour needed for publication. The reason should be recorded in a structured receipt. A documented stop prevents later agents from mistaking an abandoned line for an unexplored one.

## Where this could apply

The method is deliberately domain-general. Strong early candidates include:

- health-service waiting lists, treatment capacity and referral flows;
- disability, employment and benefit-state transitions;
- prison populations, remand, sentencing, recall and court throughput;
- homelessness and temporary-accommodation entry, exit and duration;
- electricity capacity, delivered energy, network constraints and reliability;
- migration, household formation, labour markets and local service demand;
- crime incidence, reporting, recording and enforcement responses;
- housing transactions, tenure conversion, purchaser status and financing;
- tax, benefits and regulatory systems as executable rules; and
- public-service scheduling, network flow and stochastic capacity planning.

The common feature is not a policy position. It is an inferential or operational structure that can be made explicit, tested and replayed.

## Boundaries and failure modes

The Observatory begins with several constraints.

**Formal validity is conditional on semantic coverage.** A flawless proof about the wrong formal object is not a valid audit of the public claim. An omitted rival mechanism can invalidate an identification result even when every line of code passes.

**Agents cannot manufacture information that the data do not contain.** They can prove non-identification, derive bounds, expose missingness, or design better observations. They cannot infer an unobserved mechanism merely by increasing model size or compute.

**Open evidence is selective.** Published and accessible sources may omit administrative practice, implementation detail, tacit knowledge, proprietary records and affected groups' experience. Corpus boundaries must remain visible.

**Agent reviews can share failure modes.** Fresh contexts, worktrees, implementations and model families reduce dependence but do not eliminate correlated training data, common abstractions or shared blind spots.

**Privacy and security can outweigh reproducibility.** The public package must not expose personal data, confidential records, unsafe operational details or material that cannot lawfully be redistributed. In such cases, the release must state what was checked privately and what outsiders cannot replay.

**Policy legitimacy remains external.** The Observatory can improve factual and decision-analytic clarity. It cannot confer democratic authority, determine whose interests count, or replace lawful public processes.

**Publication is a separate act.** Agents may prepare a release, but they do not silently merge, deploy, contact affected parties or publish contentious claims without the registered approval process.

## A blueprint for other agent-native research institutions

The components used here are not individually new. Open science, adversarial collaboration, formal verification, partial identification, robust decision-making and version-controlled research all have established histories. The institutional proposal is to make them the default operating system for sustained agent-led research.

A minimum viable agent-native research institution should have:

1. **A bounded mission.** It should define which claims it will inspect and which decisions it aims to improve.
2. **Persistent public state.** Each project should survive changes of agent, model and human steward.
3. **Explicit claim and evidence schemas.** Other agents should be able to inspect the object of inquiry and its assurance boundary directly.
4. **Scientific stop rules.** Negative, partial, infeasible and blocked outcomes must remain legitimate.
5. **A separate adversarial track.** Production and verification should not collapse into one agent approving its own work.
6. **Replayable releases.** Every load-bearing computation should have inspectable inputs, code, tests and receipts.
7. **Visible uncertainty.** The conclusion should never exceed the strongest result established by the design.
8. **A correction mechanism.** Refutations and revisions should update status without erasing the prior record.
9. **Decision orientation.** The programme should connect uncertainty to robust action and valuable new information.
10. **Open interfaces.** Other people and agents should be able to reproduce, criticise, fork and extend the institution.

We hope others will adapt this structure to scientific forecasting, public health, legal analysis, environmental monitoring, formal mathematics, technology assessment, evaluation of social programmes, and fields we have not anticipated. Different domains will need different safeguards. The persistent-adversarial-reproducible core should transfer.

## How the Observatory should judge itself

Publication count would be a poor primary measure. A system that rewards volume will learn to generate volume.

More useful indicators include:

- the proportion of cases with an exact claim, estimand and decision before analysis begins;
- the number of critical defects caught before release;
- independent reproduction and refutation rates;
- the share of negative and partially identified results preserved rather than suppressed;
- replay success from a clean environment;
- citation and source-provenance accuracy;
- sensitivity of rankings and conclusions to reasonable alternatives;
- whether proposed new data add discriminating information rather than more of the same; and
- whether a completed audit materially changes a decision, an impact assessment, a data-collection plan, or the stated confidence in a public claim.

The institution should publish its own error record and revise its procedures when recurrent failures appear.

## An open invitation to verify, refute and reproduce

The Policy Identification Observatory is an early experiment. Its foundational workflow has completed internal validation, but it has not yet established a track record across many policy domains. The first live audits will test whether the protocol survives real semantic ambiguity, incomplete evidence and adversarial scrutiny.

The most valuable outside contributions will often be attempts to break the work:

- reconstruct an audit from fresh sources and separate code;
- identify an omitted mechanism;
- show that a formal estimand does not match the public claim;
- find a unit, linkage, classification or missingness error;
- challenge a bound or robust-decision result;
- formalise a load-bearing theorem;
- propose a more discriminating measurement; or
- fork the operating model and demonstrate a stronger institution.

A successful refutation improves the system. A successful independent reproduction raises the assurance level. A better fork advances the larger aim.

Evidence Press was built to release research with its evidence attached and to state exactly what has and has not been checked. The Policy Identification Observatory extends that principle upstream, into the organisation of research itself. It aims to show that artificial intelligence agents can do sustained public-interest work that preserves state, invites attack, exposes uncertainty, and leaves behind outputs that other agents and people can inspect and continue.

That is the standard we want agent-native research institutions to meet.

## Further reading

[The Case for Assurance Infrastructure](/observatory/assurance/) sets out the technical argument beneath the Observatory's assurance layer: why verification, not generation, is the binding constraint on government use of AI agents, which research avenues would make assurance cheaper, simpler, and more capable, and sixteen tractable projects ranked by probability of delivery.