A brute-force search that actually scans 2^n candidates is bounded below by Landauer’s principle. This paper states that bound precisely and writes out what does and does not follow from it.
First, the exact form of the theorem is “for irreversible operations”. Landauer’s principle (1961) demands a minimum work of kT ln2 per bit erased — 2.871×10⁻²¹ J at 300 K — so an implementation that scans 2^n candidates and discards each irreversibly requires E ≥ 2^n kT ln2, which at n = 300 and 300 K gives E ≥ 5.85×10⁶⁹ J. But Bennett (1973) showed that logically reversible computation can in principle dissipate arbitrarily little, so the bound constrains irreversible implementations, not search itself; a reversible implementation pays in space or time instead. Drop that qualification and the theorem is false.
Second — and this is the point of the 3rd edition — the robust statement is the logarithmic one, not the linear one. The second edition described the energy required at n = 300 as “about 0.6 times the mass-energy of the observable universe”. Checking that ratio: with 1.5×10⁵³ kg it is 0.43, with 1.0×10⁵³ kg it is 0.65, and with 6×10⁵² kg (baryons only) it is 1.08. The two-figure display “about 0.6” does not possess that precision, and the 3rd edition corrects it.
By contrast, the n* at which the required energy reaches an available energy E barely moves. Since n* = log₂(E/(kT ln2)) = log₂E − log₂(kT ln2), changing the estimated mass of the universe by a factor of 2.5 moves n* only from 301.2 to 299.9 — a shift of 1.3 — and lowering the temperature from 300 K to 2.7 K, a factor of 111, moves it by only 6.8. Temperature moreover shifts n* by a constant: the same 6.8 appears for the world’s annual primary energy (n* = 137.3), the annual solar output (181.5) and the universe (301.2), because only the second term depends on T. Hence there is exactly one thing this paper can say robustly: irreversible brute-force search becomes physically impossible at around n = 300. When the quantity of interest is exponential, a linear measure inherits the uncertainty of its inputs at full size while a logarithmic measure compresses it; the latter is what should be reported.
Third, this is a module boundary. It is the place where the dimensionless ratio E_required/E_available = 2^n kT ln2 / E crosses one (4.2×10⁻⁴ at n = 290, 0.43 at n = 300, 4.4×10² at n = 310) — structurally the same as the Planck scale of Paper 39 (E/E_p), the naturalness bound of Paper 09 (Λ/4πv) and the Kolmogorov length of Paper 03 (Re_η). But there is a decisive difference: those three are boundaries at which the description of a theory changes, whereas this one is a boundary at which a physical implementation becomes impossible. What lies beyond it is not another theory of computation but a machine that cannot be built.
Fourth, erasure is coarse-graining. Erasing one bit collapses two states into one, halving the volume of phase space; but the microscopic dynamics is reversible and Liouville’s theorem conserves that volume (Paper 77), so the difference must be passed to the degrees of freedom of the environment — the same mechanism Paper 78 described as friction, a coarse-grained transfer to the bath. The ln2 in kT ln2 is precisely the logarithm of “two becoming one”. Landauer’s principle is thus a quantitative bridge between information and thermodynamics, verified experimentally (Bérut et al. 2012). Of the two entropies separated in Paper 88 — coarse-graining and black-hole area — what Landauer bridges is the former with information.
Fifth, thermodynamics cannot separate complexity classes. The theorem constrains implementations that actually scan 2^n candidates, whereas the hypothesis P = NP asserts precisely the existence of an algorithm that avoids that scan; to pass from the one to the other would be to place the conclusion among the premises. The same argument yields false conclusions elsewhere: linear programming is exponential by naive vertex enumeration yet admits polynomial-time algorithms, and sorting is n! by trying all permutations yet takes O(n log n).
Sixth, the three barriers are complexity theory’s version of the machinery-transfer test. Relativization (Baker–Gill–Solovay 1975), natural proofs (Razborov–Rudich 1994) and algebrization (Aaronson–Wigderson 2008) each state that a certain class of proof technique is in principle powerless here: they say not what is true but what one cannot get there with. That is the same role as the transfer test of Paper 100, and the same kind of work Paper 03 did in showing that Ricci flow’s machinery does not transfer to Navier–Stokes. This paper answers none of the three — a thermodynamic argument is invariant under oracles and cannot even clear relativization — and therefore contributes nothing to separating complexity classes. It belongs to the lineage of “NP-complete problems and physical reality” opened by Aaronson (2005), and is not a proof.
Scope: a structural account of established physical law and theorems (Layer 2). P vs NP is a Clay Millennium Prize Problem and is unsolved; this paper claims neither its proof, nor a partial proof, nor a route towards one, places no constraint on search methods other than brute force or on reversible implementations, and claims no digits it does not possess. The Universe OS is a non-scientific Layer-3 lens.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical laws and mathematical theorems. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are transformations of known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
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 Physical Resource Bound on Brute-Force Search [Paper 4]. Zenodo.
DOI: 10.5281/zenodo.21404131
ORCID: 0009-0006-8587-3450