r/LLMmathematics Monthly Conjectures: Open Problems, Theorems, and Exact Evidence

This maintained working-paper series is the citable companion to u/dForga's r/LLMmathematics post Monthly conjectures 1 (Start?), published on 25 July 2026, and to the monthly conjecture discussions that follow it. The nine-page front reader opens with the centered Hardy-Littlewood maximal-variation conjecture and the sharp Gaussian logarithmic-Sobolev stability problem from that post. It then records the general conformal-factor holomorphic-embedding problem and four programmes generated from exact source patterns: arithmetic reconstruction from iterated zero layers and its shifted-prime character resonance, Collatz record-tail genericity, winding-one Gaussian-prime circuits, and the nonzero D4 radial defect channel. For centered maximal variation, a human proof covers three consecutive sites and exact linear-real-arithmetic certificates cover every nonnegative real profile on at most ten consecutive sites. Fresh replay solves all seven SMT-LIB queries; the serialized Z3 proof terms await checking by an independent small proof kernel. The general variation conjecture remains open. For Gaussian logarithmic-Sobolev stability, a 640-term entropy minorant and outward-rounded 768-bit Arb calculation prove q_N < 0.577215 and Q_N < 1.15443 pi in every dimension. The sharp constant remains open. Iterated zero adjunction gives Con(G^n(A)) = Con(A) x B_n, and a bounded distributive layer lattice gives Con(G_L(A)) = Con(A) x Con(L). For number rings, finite quotient counts are a binomial shift of Dedekind-zeta coefficients. The reconstruction conjecture is proved when both fields are quadratic. Its general analytic obstruction is the fixed-shift ideal-mass conjecture sum_(p<=x) a_K(p-h) ~ C_(K,h)Li(x) with C_(K,h)>0. For abelian fields, the proposed constant is an explicit principal local product times a finite character-resonance multiplier. The ramified convolution-face factor is derived exactly, and the multiplier is proved strictly positive for every finite abelian field and nonzero shift by a conductor-filtration and finite Fourier argument. The analytic asymptotic remains open beyond two character factors. Exact replays cover 48 original and 32 multiquadratic field/shift pairs through x=2,000,000, 160 local identities, 380 general-abelian Fourier checks, and 11,620 independent Walsh/direct identities. Lean 4.32.0 checks the local inequalities and positive-convolution core. The Collatz candidate concerns a support-normalized statistic on the shortened map. On a stopping-time record's private tail, its positive variation has an exact formula whose 2-adic Haar prediction is 1/18 per private-tail step. The conjecture states that record gaps are unbounded and that the normalized private-tail contribution tends to 1/18 along gaps tending to infinity. Exhaustive search through 63,728,127 certifies a record gap 126 with exact ratio 445/8064 = 1/18 - 1/2688. A separate rational replay gives V^+_(8,447,300)=6.093638201316768... UmbrellaCorp_HR's thirteen-term arctangent formula is resolved by a 9 by 13 Gaussian-prime flow matrix of rank nine. Its nonnegative kernel has eight positive circuits, each winds once, and the supplied vector is an exact convex combination of four circuit identities. Exact multiplication gives zero imaginary part and a positive 336-digit real part. D4 triality splits the four discriminant channels into invariant and standard two-dimensional blocks; radial data yield a proved rank-at-most-one result after removal of the common scalar direction. Nonvanishing of the remaining channel is open. Closed results remain in plainly named standalone papers: the sharp circle L1 stability theorem, random monomial-unitary equidistribution, the flat holomorphic-isometric embedding classification, and the corrected gradient-descent history. The release comprises the open reader, directly visible Reddit-ready Markdown and series note, eleven standalone mathematical PDFs, eleven result-specific source or verification ZIPs, complete source and verification archives, DOI lineage, manifests, and checksums. The shifted-prime result archive includes the TeX, Lean sources, every exact replay, expected outputs, an independent proof audit, and the six primary papers used to delimit the literature. The Reddit permalink identifies the discussion; the version DOI identifies one immutable edition; the concept DOI identifies the evolving collection. Independent checking and priority information are welcome.