Educational record: P is not equal to NP
No algorithm solves every instance of an NP-complete problem in time bounded by a polynomial in the input size; that is, P ≠ NP.
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Educational record: early dark energy as a resolution of the Hubble tension
A short-lived early dark energy component active before recombination reduces the sound horizon enough to reconcile the Hubble constant inferred from the cosmic microwave background with local distance-ladder measurements.
Educational record: the synaptic homeostasis hypothesis of sleep
Sleep restores synaptic homeostasis: wakefulness produces a net increase in synaptic strength, and slow-wave sleep renormalises it, which saves energy and improves signal-to-noise for memory (the synaptic homeostasis hypothesis, SHY).
High AlphaFold confidence predicts agreement with later experimental structures
For protein structures first deposited in the PDB after AlphaFold2's training cut-off, residues predicted with very high confidence (pLDDT above 90) match the experimental backbone within 1 Å in at least 90% of cases.
The share of LLM-modified text in machine-learning peer reviews keeps rising after 2024
In the first major machine-learning conference review cycle of 2026, the estimated share of review sentences substantially modified by large language models exceeds the upper estimate reported for 2023–2024 cycles.
A square-root error term for the Hardy-Littlewood twin prime count, with persistent sign oscillation
Let pi2(x) count primes p <= x with p+2 also prime, let C2 = 0.66016181584686957... be the twin prime constant, li2(x) = integral from 2 to x of dt/(log t)^2, and define the Hardy-Littlewood error E(x) = pi2(x) - 2*C2*li2(x). Conjectured, in three parts: (H1) for every eps > 0, E(x) = O(x^(1/2+eps)) as x -> infinity; (H2) E(x) changes sign infinitely often, and the number of sign changes in [10^k, 10^(k+1)) is unbounded in k, so the oscillation does not die out; (H3) the bound in H1 cannot be sharpened by a logarithmic factor of this size, i.e. limsup |E(x)|*log^2(x)/sqrt(x) = infinity. This is a conjecture about the ERROR TERM of the Hardy-Littlewood asymptotic. It is logically downstream of the twin prime conjecture and presupposes it; it does NOT prove, and makes no attempt to prove, that there are infinitely many twin primes.
