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. It collects precise open problems, proved structural results, corrections, partial bounds, and exact verification. The seven-page front reader contains six open programmes: centered maximal variation, the sharp Gaussian logarithmic-Sobolev constant, general conformal-factor holomorphic embeddings, arithmetic reconstruction from iterated zero layers, winding-one Gaussian-prime circuits, and the nonzero D4 radial defect channel. Iterated zero adjunction gives the proved factorization Con (Gⁿ (A) ) = Con (A) x Bₙ. Replacing the layer chain by a bounded distributive lattice gives Con (GL (A) ) = Con (A) x Con (L). For a number ring, the finite quotient counts are the binomial shift of its Dedekind-zeta coefficients. The open reconstruction conjecture asks whether this complete shifted sequence determines both the layer depth and the zeta function. Two exhaustive congruence programs and 1, 288 exact quadratic-field/layer tests accompany the proof. JT'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 circuit rays, every circuit winds once, and the supplied coefficient vector is an exact convex combination of four circuit identities. Exact multiplication gives zero imaginary part and a positive 336-digit real part, so the reported 10^-128 residual is rounding error. The circuit winding uses a rational bound, and the Gaussian product, flow equations, selected circuits, and decomposition are independently checked in Lean. The resulting open problem is to characterize Pythagorean atom families whose positive circuits all wind once. D4 triality splits the four discriminant channels into a two-dimensional invariant block and a two-dimensional standard block. Radial data occupy only the invariant block, giving a proved rank-at-most-one bound after the common scalar direction is removed. The open part is nonvanishing of that channel in the weight-two modular interpolation problem. The paper makes no claim to a Cohn-Elkies magic function or D4 packing optimality. For the discrete centered Hardy-Littlewood maximal operator, a human proof covers three consecutive sites and exact linear-real-arithmetic certificates cover arbitrary nonnegative real profiles on at most ten consecutive sites. Fresh replay solves all seven SMT-LIB queries; the serialized Z3 proof terms have not been checked 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 qN < 0. 577215 and QN < 1. 15443 pi in every dimension. This is a certified upper bound, not the sharp constant. 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 seven-page open reader, directly visible Reddit-ready Markdown and series note, nine standalone mathematical PDFs, nine result-specific source or verification ZIPs, complete source and verification archives, DOI lineage, manifests, and checksums. Every named local source is included in a public-safe archive. The Reddit permalink identifies the discussion; the version DOI identifies immutable release bytes; the concept DOI identifies the evolving collection. Independent checking and priority information are welcome.
The Clankers (Wed,) studied this question.