El universo (que otros llaman la Biblioteca) se compone de un número indefinido, y tal vez infinito, de galerías hexagonales, con vastos pozos de ventilación en el medio, cercados por barandas bajísimas.
Ficciones — Jorge Luis Borges (1941)
We isolate an abstract strategy-transfer principle for Cops and Robber: a coarse graph projection with bounded fibers and bounded distance distortion lets cops occupy a lifted macro-ball before the robber escapes, giving cop number O(sqrt N) up to polylogarithmic factors. Applied to the Hosseini-Mohar-Gonzalez Hermosillo de la Maza degree-reduction construction, this shows the known hard family for Meyniel’s conjecture already meets the square-root exponent, sharper than the usual notation suggests. A counting argument then proves a sharp limit on the strategy class itself: Cartesian tori of cycles have bounded doubling and constant cop number but linear occupation cost at every radius, so weak expansion alone cannot certify a universal robustness theorem.
A complete determination of the bounded annealed critical window for growing-radius domination in random regular graphs, resolving the open problem left by an earlier near-critical bound. Writing L_h = log B_h for the radius-h tree-ball volume, the annealed exponent obeys a universal scaling law (B_h/L_h^2) Psi_{d,h}((L_h - 2 log L_h + s)/B_h) -> 1 - e^{-s}, and its lower zero is pinned to within B_h^{-1/7+o(1)} of the scalar coupon-collector root H(C/B_h) = (1-C/B_h)^{B_h}, giving a complete fixed-order inverse-logarithmic expansion. The proof adds a quantitative reverse-transfer error analysis and an explicit capped-free profile whose only entropy loss is the cost of one terminal nonemptiness event, sharpening the earlier annealed lower bound into a two-sided window theorem. The result again strengthens to internally two-path domination, a graph-general parameter motivated by exhaustive static tube coverage in pursuit-evasion games. Quenched matching and the direct two-branch leading constant remain open.
Superseded by a proof of the full bounded critical window and its universal scaling function; retitled and restructured to match the new preprint.
A first-person, code-heavy companion to the preprint Near-Critical First-Moment Lower Bounds for Growing-Radius Domination in Random Regular Graphs. Rather than reproducing its theorem-and-proof form, this page traces where the problem came from — a cops-and-robbers hypergraph question that collapsed into a domination bound — why the answer carries an unnecessary coupon-collector logarithm, and how a chain of computational detours (a failed concavity conjecture, a catastrophic cancellation, an independent audit that caught a stale constant) repeatedly redirected the proof before it reached its final shape.
We analyze exhaustive static coverage by path tubes indexed by length- nonbacktracking robber paths from in a finite-horizon local chase on a -regular graph. An endpoint-sensitive geodesic lemma shows that, among possibly infinite -regular graphs, radius is sharp for arbitrary pairs in , while synchronized witnesses along a prescribed path require only a radius- tree-ball. If every tube is occupied, each surviving round along every path in this class ends in capture, blockage, or branch-load support on at least two branches. The tubes partition the outer ball, so deterministic coverage has minimum cost ; conditional on a specified root, i.i.d. uniform coverage has threshold for fixed . An augmented prefix-depth profile that retains the complete shallow configuration still need not determine later support. The result is local and root-dependent: it treats neither arbitrary robber walks nor a robber-independent cop strategy, and it gives no cop-number bound.
Curricula in mathematics and the sciences optimize for procedural fluency — the half of what they teach that decays once the student stops being a student. What survives twenty years on is conceptual residue, generated only as an accidental byproduct of the curriculum’s intended work. The asymmetry compounds across generations of teachers and produces a population unable to do the work that civic life requires of it.
A compendium of years of informal, empirical experiments aimed at extending the efficacy of Anki beyond rote memorization to more intricate levels of learning.
Notes from Underground is widely admired as a cornerstone of literature, culture, and philosophy. This paper develops the argument that the primary philosophical undercurrent is a rejection of logic and science as end-alls in modern life, traced through comparison with the more explicitly articulated works of Dostoevsky’s contemporaries and successors: Nietzsche, Heidegger, Shestov, Ellul, Sartre, Camus, Husserl, and Arendt.
I met a traveller from an antique land, / Who said — “Two vast and trunkless legs of stone / Stand in the desert.”
Like as the waves make towards the pebbled shore, / So do our minutes hasten to their end.
As we approach AGI, the increase in the ability of Artificial Intelligence models to infer a robust specification from a sparse prompt will lead to a devastating trend of homogeneity. We argue that this is the primary concern regarding the interaction of AI and human intelligence, rather than blanket claims that “AI reduces human cognitive ability.”
AI labs are likely deliberately reluctant to scale because they are aware that any imminient shift to locally run models as the norm would render their compute redundant. We take Anthropic as a principal case study to validate this hypothesis.
Full Unix-like kernel in ~7,000 lines of C, written for Brown CS 169 (Operating Systems with Lab): virtual memory, VFS, system calls, threading, device drivers and interrupt handlers, and file systems. Custom linker support for running userspace x86-64 ELF binaries.
TCP/IP, RIP, UDP, and DNS implementations in Go, supporting file transmission of up to 1 GB across networks of up to 8 virtual machines. Extended with a fully RFC-compliant SSH implementation (2,000+ additional lines) supporting sustained sessions of arbitrary length.