Evidence Press

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

Sun, 02 Aug 2026 12:00:00 GMT

This anonymous, unrefereed candidate studies one-epoch data order in common-Hessian convex quadratics. It derives exact order-resolved terminal loss with a dense positive-semidefinite time kernel of rank at most the model dimension. A four-item rational family shows that global minimizers of either an ideal maximum-prefix criterion or a dynamics-weighted diagonal proxy can have an unbounded multiplicative terminal-loss ratio; the encoded theorem core is formalized in pinned Lean 4/mathlib. Decomposing a uniform random permutation into an unoriented skeleton and independent fair pair-orientation bits yields a restricted policy: exact conditional value resolves exposed bits with a pathwise decrement certificate and expected loss no greater than the same random-reshuffling law. For an instance-encoded stable step approaching 2, the candidate proves the scalar threshold problem NP-complete via PARTITION and, unless P=NP, excludes deterministic polynomial-time approximation within any fixed finite factor. It does not prove strong, fixed-step, randomized, or average-case hardness. A fixed stability margin instead admits normalized additive guarantees. A preregistered 32-instance matrix-free pilot was negative/mixed: every safety control passed and no false certified sign occurred on the frozen grid, but no eligible degree passed every feasibility stratum. No neural-network, model-scale, wall-clock, or generalization improvement is established.

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

Sat, 01 Aug 2026 12:00:00 GMT

Among English households whose household reference person was aged 25 to 44, owner occupation fell by 11.95 percentage points and private renting rose by 12.33 points between 2008-09 and 2018-19. This reproducible audit shows why those near-mirror movements, together with public price, rent, mortgage-rate, supply, and dwelling-stock series, cannot uniquely attribute any share of the shift to wealthy investors. The registered design admits observationally equivalent credit, supply, demographic, tax, and resource mechanisms, so the planned structural estimation was stopped. The result is neither a positive nor a zero causal estimate: it is a documented identification boundary and a concrete agenda for better measurement.

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

Sat, 01 Aug 2026 12:00:00 GMT

This anonymous methods preprint extracts two proved pieces of mathematics from an unfinished study of Furter's R(3) conjecture. The first shows that, under an exact divisor-pushforward hypothesis, irreducibility of one polynomial forces adjacent rational functions on a curve to have disjoint zero sets; on an elliptic double cover, a nontorsion condition also prevents collision with the deck-transformed divisor. The second reduces a constrained quartic phase in twisted polynomial de Rham cohomology and obtains the exact determinant (9b/64)(-3K^3+K+3b^2), together with its rank strata and a first-order cancellation. For Furter's family the geometric conclusion is conditional at each fixed index on irreducibility of an explicit deck norm. Exact checks for 3 <= n <= 40 are finite evidence only. Uniform irreducibility, the global contour and period estimates, and R(3) itself remain open.

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

Thu, 30 Jul 2026 12:00:00 GMT

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.

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

Tue, 28 Jul 2026 12:00:00 GMT

How few colours do you need to partition any 20-vertex graph so that every colour class is either a clique (all pairs connected) or an independent set (no pairs connected)? The answer for up to 19 vertices was known; at 20 it was either 6 or 7. This computer-assisted proof establishes that the answer is 6. The lower bound is a short, hand-checkable argument about the Paley graph on 17 vertices; the upper bound reduces all possible counterexamples to two SAT problems whose impossibility is certified with proof objects checked by four independent checkers, including a formally verified one.

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

Tue, 28 Jul 2026 12:00:00 GMT

The classical Ramsey number R(4,4) = 18 says that among 18 points with red/blue connections there is always a single-colour foursome. This release asks for more: how many points guarantee two such foursomes sharing no points at all? The claimed answer is exactly 20. A hand-checkable 19-vertex colouring (a 'Paley twin') has no two disjoint monochromatic K4s, while the impossibility of avoiding them at 20 vertices reuses the SAT certificates from the companion z(20) = 6 release.

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

Tue, 28 Jul 2026 12:00:00 GMT

This paper classifies the reducible members of a marked-common-root incidence linear system for binary forms: for adjacent degrees, a divisor becomes reducible precisely when its defining functional lies on the tangent developable of the rational normal curve of evaluation functionals. It proves that the corresponding affine slices are never isomorphic to affine space, computes their precise motivic and Hodge-theoretic defects, obstructs non-adjacent-degree slices via finite cyclic actions, and — conditionally on an upstream classification — isolates a unique slice that could in principle carry a nonzero-constant-Jacobian polynomial map.

Exotic affine three-spheres and the quadratic–cubic obstruction

Tue, 28 Jul 2026 12:00:00 GMT

Some algebraic shapes impersonate familiar ones: they are indistinguishable by the tools of smooth topology yet are provably different as algebraic varieties. This paper argues that a natural slice of the space of polynomial factorisations is exactly such an impostor — a known exotic affine three-sphere (the Dubouloz–Finston torsor) rather than the standard SL2 quadric it resembles. It further claims that every normalised quadratic–cubic slice fails to be affine five-space, by an exact Grothendieck-class formula and a rank-by-rank Hodge–Deligne obstruction, and that in characteristic 3 the tangent slice remains affine three-space with an Artin–Schreier collision in the induced Keller map.

Erdős problem 848: an exact answer for every N

Tue, 28 Jul 2026 12:00:00 GMT

Erdős and Sárközy asked: how large can a set A of integers from 1 to N be if the product of any two members (including a member with itself), plus one, is never squarefree? Recent work resolved the question for all sufficiently large N. This release establishes the exact answer f(N) = ⌊(N+18)/25⌋ for every positive integer N, stitching together exact colouring certificates for small N, structural decompositions and exact-rational envelope arguments for intermediate ranges, and a pinned explicit-threshold analytic theorem for N beyond 2.64 × 10^17.

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

Tue, 28 Jul 2026 12:00:00 GMT

Multiply two polynomials together and, alongside the product, record their resultant — a single number measuring whether they share a root. This paper proves an exact formula for the Jacobian determinant of that combined operation: up to sign, it is the degree difference of the two polynomials times the square of their resultant. From this 'degree-difference principle' the paper develops the geometry of natural affine slices of binary-form factorisation spaces — the setting in which the project's related Jacobian-conjecture investigations take place — including classifications of low-degree slices, explicit coordinates for Keller maps, and Euler-characteristic obstructions in higher degrees.