Incoprea research · independent proofs · 31 October 2026

Three provable questions

Exact results for the per-bit energy floor, the invasion front and osmotic minimum work — proofs for questions 16, 10 and 17 of the notebook's twenty promising science questions.

#	Question	Verdict	Exact result
16	How closely can a link approach its signal-energy floor?	Confirmed in part	E_b/N₀ > ln 2 = −1.59 dB, approached monotonically; no finite optimum
10	Which growth and dispersal combination sets invasion speed?	Refuted	v = 2√(rD), a geometric mean; the ratio r/D predicts nothing (R² = 0.08)
17	How does water recovery change ideal separation work?	Direction confirmed	w = π_f[−ln(1−r)]/r, strictly convex; the claimed π_f/r crosses the true curve at r = 1 − 1/e

Three-Proven-Questions.pdf →14 pages · A4 · print-ready. The three claims, the three proofs, the numbers, the figures. Three-Proven-Questions.html →Same document in the browser. Math is rendered with KaTeX, so it needs one CDN fetch. README.md →What is in the folder, the verdicts, how to reproduce every number. SOURCES.md →The three questions quoted verbatim, and every citation with its DOI. verification/VERIFICATION.md →Every machine check: route, expected value, observed value.

Not peer reviewed. No theorem here is new — the results are Fisher (1937), Kolmogorov–Petrovskii–Piskunov (1937), Shannon (1948), Bramson (1983) and Wang–Violet–DuChanois–Elimelech (2020). What is new is the comparison, and the independent numerical replication in code/. Reproduce with cd code && ./run_all.sh.