The Past Hypothesis: Boltzmann’s H-Theorem, and Honestly Admitting the Arrow of Time Is Not Yet Solved.
Why does time have a direction? A cup shatters, but shards never reassemble on their own. Ludwig Boltzmann captured this arrow statistically: entropy S = k log W (W = number of microstates for a macrostate) increases because a system moves toward macrostates with more microstates (commoner states). But Loschmidt’s REVERSIBILITY PARADOX strikes: the microscopic equations are time-reversal invariant (reversible), so why does an irreversible arrow appear macroscopically? The corpus holds half the answer — the arrow LIVES IN COARSE-GRAINING (Papers 75/77/78: micro is reversible, but the arrow points where information leaks into unresolvable degrees of freedom). But honestly, this only PUSHES the origin of the arrow back to “why the early universe was so extraordinarily low in entropy.” This PAST HYPOTHESIS — the input that the universe began in an extremely low-entropy state — is an OPEN PROBLEM, not yet explained (proposals like Penrose’s Weyl curvature hypothesis exist but are not established). The point is twofold: (1) do not conflate the coarse-graining arrow (77/78) with the past hypothesis, and honestly record that the latter is unsolved; (2) sort the thermodynamic arrow, the cosmological arrow, and (Penrose’s) gravitational entropy as DISTINCT ROOTS even as they resonate (58/48, 88). Boltzmann (in the author’s lineage) held to atomism against Mach/Ostwald’s positivism — intuition made science by royal-road math. IMPORTANT (scope): a Layer-2 structural interpretation of established statistical mechanics; no new mathematics and no new prediction; it does NOT claim to solve the past hypothesis. The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Past Hypothesis [Paper 98]. Zenodo.
DOI: 10.5281/zenodo.21758739
ORCID: 0009-0006-8587-3450