Statement. For alpha(P)=N(P)/D(P) with sums of K + A·bP/(1+bP), the number of crossings of alpha=1 has the parity of the two limits alpha(0)=K_N/K_D and alpha(inf)=(K_N+A_N)/(K_D+A_D).
Status. Proved by continuity; verified in 3,000 random cases with zero violations and again with Hill exponents and decreasing channels (0/400). Refuted a stronger claim of 'at most two crossings'.
Statement. theta is locally identifiable iff the sensitivity matrix has full column rank; when K < N a null direction exists and no amount of precision recovers it.
Status. Proved (rank argument) and computed on four inverse problems: depth profiles, closure dating, mass balance, damping mechanisms. Conditioning scales as the inverse separation of weightings.
Statement. In Newtonian gravity with exact spherical symmetry and known M and R, g(R) = -GM/R^2 for every radial density profile with that mass.
Status. Proved by Gauss's law; computed that a density model 91.7% wrong differs in g by 6.4e-7 relative.
Statement. Both are frequency-flat, so eta(omega) is rank-deficient for the pair; no number of frequencies separates them.
Status. Proved; conditioning 1.7e15 from frequency alone, 9.7 once a temperature axis is added.
Statement. A single integrated rate cannot determine the history that produced it: many histories give the same marker.
Status. Computed: four histories with identical intermetallic thickness differ in damage by 3.8x.
Statement. One equation with N unknowns has rank 1; assaying more streams does not help until equations are as numerous as streams.
Status. Computed: rank 1 and infinite conditioning for every pattern of unassayed streams.
Statement. A predictor formed from the outcome inflates R^2 by a computable factor set by the variance ratio.
Status. Computed: 3.57x inflation at one parameter choice; 50/100 of the parameter grid exceeds 1.5x.
Statement. Hill and PWM tail indices disagree by a factor that does not shrink as 1/sqrt(n).
Status. Computed on 602 real events: spread 2.80x at n=50 and 2.36x at n=602 — a 12x data increase buys 1.19x. More data cannot fix it.
Statement. The number of sensors bounds the number of distinguishable modes independently of bandwidth.
Status. Computed: 8 sensors give 9 signatures for 54 modes; 46 are confounded.
Statement. If one reported column is a function of another, the pair carries one measurement, and any inference that treats them as independent is circular.
Status. Computed on 602 real events: log-log R^2 = 0.99985, residual 0.0064 dex.
Incoprea Research · the best 1,000 · 3 November 2026. Sorted from every result produced today. Computation only.