Research
The current work follows two mathematical questions: what happens in the spaces between packed spheres, and when a positivity inequality holds with the best possible constant. The earlier network work asks how information and cargo move when the route is available only on a schedule.
The papers below distinguish proofs, exact computational results, and open questions. Their deposits carry the supporting material described in each record. TIN remains part of that record; its main development is on pause while the mathematics takes the foreground.
Lattices / voids
Place equal balls at a periodic set of sites and study what is left over. How far can a point lie from every site? What does rounding to the nearest site cost? When do the gaps form a connected passage? This series combines geometric proofs with exact enumeration. For odd dimensions, Dd+ is a periodic set with two cosets, rather than a lattice.
The covering radius is the farthest you can get from a point set. For Dd+ — the even-coordinate-sum integer lattice together with its half-shift — it is proved here in every dimension: R² = max(1, d/8) for even d ≥ 4, and 3/4, 1, 1, (2d−1)/16 for odd d = 3, 5, 7 and d ≥ 9. These are the values Agrell and Eriksson recorded without proof in 1998. Dimension two, absent from their list, is 5/8.
The normalized second moment measures the mean squared error of nearest-site rounding, after accounting for dimension and volume. There is now an exact rational formula for it in every dimension d ≥ 2, four divided differences of a truncated power. It reproduces every value previously known, including G₈ = 929/12960 for E₈, and it proves that d = 12 uniquely minimizes over the whole family.
D9+ has two inequivalent deepest holes, both at squared distance 17/16 from the sites. As hiding places they are equal; as places to escape from they are not. With tangency allowed, decrease the squared clearance t from 17/16. One hole joins an infinite network of passages at t = 33/32; the other remains a sealed pocket until t = 213/224. The whole void becomes one region exactly at t ≤ 8/9, and the last pockets to join are neither the deepest nor the shallowest.
How large can a ball be and still travel through a packing? The covering radius answers only half of that question: it records the largest ball that fits at one point. For odd d ≥ 9 the closed void percolates exactly for squared clearance t ≤ (4d−3)/32, a fixed 1/32 below the covering radius squared. Explicit tunnel segments attain it; a matching contact-hull barrier shows nothing deeper can escape.
Normalize the minimum distance between sites to √2 and allow tangency at the void boundary. For each degree 0 ≤ q ≤ d−1, the void on the quotient torus has the torus’s integral homology in that degree whenever the squared radius is at most (d−1−q)/(d−q). In degree zero, (d−1)/d is a universal connectivity floor. FCC and E₈ percolate exactly there; D9+ separates percolation from complete connectivity.
Follow the empty part of the torus as the balls shrink from the covering radius to nothing. Its persistent homology has 37 critical levels and 55 classes of bars, identical over the rationals and over every prime field, so there is no torsion anywhere in the filtration. Most of the table is explained: cube equators are regular Voronoi hexagons that kill exactly the tunnel loops which do not wind around the torus, and the essential classes enter as a torus ladder. Version 1.1 proves why the bars in higher degrees die as promptly as they do: the two cosets split the ball side into a double complex, whose squares and zigzags make every finite bar die one vertex up, except three walls that end on the next rung. In higher odd dimensions the mixed simplices below level 1 are counted exactly, and in dimension thirteen their effect on the integral homology is determined.
Four extremal even unimodular lattices of rank 48 have theta series that agree through degree three — a published theorem says they must. Degree four separates all four, and the separating coefficient counts ordered regular four-simplices with an empty circumsphere. The deposit carries the Gram matrices, orbit data for all 44,850 orbits, and the programs that aggregate them into the four rows.
Positivity / odd cycles
When do the minors of a table stay positive, and what is the sharp constant in the inequality that controls them? These papers develop proofs, exact-arithmetic certificates, and the limits of the methods. Replay code checks the finite claims stated in each deposit; an all-parameter theorem also needs its proof.
Continue the Delannoy numbers to a real first argument and lay them out as a multiplication table. Two factorizations of the same product give two products of entries, and the more lopsided pairing always wins — exactly when x ≥ 1/2, and not one step sooner. Necessity below the threshold is explicit; sufficiency above it closes the two strips the general theory leaves open.
Draw a permutation of 2r−1 letters uniformly and count its odd cycles. The Turán differences of the resulting polynomial obey a square-tail inequality, and the sharp uniform constant in it is 2.029152329129741484…, with equality only at r = 129, t = 2. The replay recomputes every certificate in exact arithmetic under separately gated chains.
An earlier theorem gave the bridge between two odd-cycle Turán quantities a factor of 2. Two is not the right number. The best possible constant is 2.655018313913361199726406175676…, attained exactly once, at r = 1350 and t = 2. The proof is one exceptional stripe, certified by an exact prefix and a positive-coefficient half-line certificate, and one uniform barrier.
A positive sequence carries a flag of ratios, and that flag has two kernel representations differing by a rank-one term. The pinned one — a Brownian bridge on the Wallis time grid — is the better object: determinantal, with an exact gap law and a fast fixed-cardinality sampler. This paper measures what the other representation loses. The correction is a strict contraction: every flag coordinate is multiplied by a survival ratio below one, on any increasing grid.
Neville chamber determinants are positive at widths two, three and four, with all-row proofs. What does not follow is any bound uniform in the width, and the paper is as explicit about that as about the theorems: the fixed-width asymptotics cancel, two natural combinatorial models fail at a stated finite row, and a proposed fixed survival factor is false from order twenty on. The uniform question is left standing.
Count the permutations of 2n+1 objects by their odd cycles and normalize. Are the consecutive ratios positive in every forward difference at the endpoint? Yes, for every n. The row is the integer trace of one meromorphic quotient, and the zeros of its carrier are real for every n, with an exact census. With Hadamard’s theorem that census makes every logarithmic derivative along the row a sum of positive terms, one for each zero. What the spacing of the zeros adds is a representation: zeros at least one apart is equivalent to the quotient being a Pick function, proved for every n ≤ 64 and stated in general as the Watson–Herglotz conjecture. The census alone already gives the Pick property at exponent one half.
Divide the odd-cycle row by a Laguerre polynomial in the finite free sense. The quotient has positive simple roots in every degree, consecutive degrees strictly interlace, and the scaled roots converge to an explicit law with a logarithmic singularity at zero and a square-root edge. The roots come from biorthogonality to one Chebyshev family, and the coefficients are moments of sums of gamma and uniform-scaled exponential variables. Moving one step along the row raises the moment order and removes two summands, and exact counterexamples show why familiar moment inequalities cannot settle that trade on their own.
Invert sinh(2√A)/(2√A) = 1/(1−z) on the branch through the origin. The result maps the upper half-plane into itself, with a strictly positive density on [1, 5.6033…] in closed parametric form, and it is univalent. The proof is one conformal map, z = 1 − u/sinh u, walked once around its boundary. It is where the bulk of the odd-cycle row converges.
Put one independent fair bit at every event of an n×n causal lattice, let each future link carry one bit computed from its causal past, and ask every 3×3 bulk diamond to reconstruct its nine interior bits from the eight crossing its future cut. Counting caps each diamond’s success at 1/2. The worst-case gap to that cap is zero for n ≤ 7 and exactly 51/512 at n = 8 — an exact value, not a bound.
The L-ensemble matrix is dense; the computation is a path. Exact rational algorithms for finite determinantal point processes whose base matrix is built from a single decreasing sequence, with a fixed-cardinality sampler, closed one-point marginals, and replay gates that mutation controls have to fail.
Corridor / TIN
Replay a frozen contact plan and check that delivered outcomes stay within preregistered statistical bounds. Certificates with explicit radii, not point estimates.
Sorts temporal contact graphs by morphology and confinement, and maps the routing-efficiency frontier across the resulting families.
Routing / SABR
A regime-indexed characterization of CCSDS 734.3-B-1 under a machine-checked reference and differential measurement against ION 4.1.4. The certification question dissolves into five enumerated places where two conformant implementations can still differ; volume numbers are under letter semantics and ship with their report artifacts. Cite the concept DOI (resolves to the current version).
Program watch
The first quantified ground-segment allocation for a crewed cislunar mission: 267.1 scheduled DSN antenna-hours at Canberra over nine days, half a 70-meter antenna’s calendar — measured from public feeds we archive hourly, scored against the public record, with evidence-class tags and a falsifier table. REV E sources the DSS-14 outage cause to the public DSE downtime feed (approved 2026-03-02 through 2028-05-05). The full archive and every derived artifact ship with it.
A first-principles audit of NASA’s post–Artemis II architecture: what the Gateway pause actually means, which modules physically exist, the surface-first Moon Base plan, and how China’s ILRS track compares. The load-bearing claims carry evidence-class tags and explicit falsifiers; every schedule claim is dated.
Logistics
Same result, two treatments. The essay is the long form — covered wagons and scheduled networks: why adding a route can make a rail system worse, and why space route topology has to be right the first time. The brief is short and decision-oriented — cryogenic supply-chain feasibility for the outer system; under the brief’s stated transport and boil-off assumptions, fewer than 3 of every 100 tonnes dispatched arrive.
Software
Contact-plan routing code: scenario configuration, routing over scheduled links, curated result artifacts, and a verifier that reports which published claim families are checkable from what ships. The repository documents its verification checks and the limits of the public release.