The Yamagishi Absolute-Zero Standby and Hawking Phase-Transition Theory: the critical radius contains no G, and the 8π is not a solid angle. Third edition.
The critical mass M_c at which the Hawking temperature equals the temperature of the cosmic microwave background separates the phase in which a black hole grows on net from the phase in which it evaporates. For T_CMB = 2.725 K one finds M_c = hbar c³/(8π G k_B T_CMB) = 4.50×10²² kg (about 0.61 lunar masses), with a Schwarzschild radius of 66.9 μm. The second edition quantified this and checked eleven numbers. The third edition re-verifies all eleven and then decomposes what is inside those numbers.
First, and this is the centre of the edition: the critical radius contains no G. Substituting M_c = hbar c³/(8π G k_B T_CMB) into r_s(M_c) = 2GM_c/c² makes G cancel and disappear, leaving r_s(M_c) = hbar c/(4π k_B T_CMB) (verified symbolically: the result contains no G). The critical MASS carries the gravitational constant as G⁻¹; the critical SIZE carries it not at all, being 1/4π of the purely thermal length hbar c/(k_B T_CMB) = 8.403×10⁻⁴ m (verified numerically: that value, 66.87 μm, matches r_s of Theorem 3.1). The cancellation is no accident. M_c is obtained by equating T_BH proportional to 1/(GM) with T_CMB, hence M_c proportional to 1/G; and r_s proportional to GM is proportional to G; composing the two, G must cancel. How strong gravity is moves the critical mass up or down, but does not move the critical size.
Second, the 4π that remains there is not a solid angle. The corpus handles several expressions containing 8π (Papers 17, 35). The 8π of the Hawking temperature and the 8π of Einstein’s equation are equal in value and different in root. The former follows from T_BH = hbar kappa/(2π c k_B) with kappa = c⁴/(4GM), so 8π = 4 × 2π, where the 2π is the period of the Euclidean section — the same root as the 2π of the Unruh effect. The latter is fixed by matching the Newtonian limit grad² Phi = 4π G rho, so 8π = 2 × 4π, where that 4π is the SOLID ANGLE (Paper 35) and the 2 is trace reversal. Both equal 25.13, but one contains a periodicity and the other a solid angle. It is not one number appearing twice; it is two numbers of different provenance that happen to be equal. And the 4π remaining in r_s(M_c) is what is left of 8π after division by the 2 of r_s = 2GM/c², so its address is (the 2 of kappa = c²/2r_s) × (the Euclidean period 2π). What Paper 17’s third edition asked for was not the assertion of distinct roots but the writing of each root’s address. This paper writes them.
Third, the 5120π of the evaporation time is not an independent number. From the Hawking power P = hbar c⁶/(15360π G²M²) (Paper 43) one has dM/dt = −hbar c⁴/(15360π G²M²), and integrating produces M³/3, so that 15360/3 = 5120 (verified symbolically). 5120π and 15360π are not two numbers but two faces of one root, divided by the power 3 of M.
Fourth, the name “minimum standby power” is withdrawn. The reason is not physical: the quantity is not yet a quantity. E_ZPE = (1/2) hbar omega is a function of omega, and the paper never says which omega is to be taken. For a single oscillator, lowering omega sends (1/2) hbar omega to 0, so there is no minimum over modes (verified numerically: 3.3×10⁻³⁴ J at omega = 2π·1 Hz, 3.3×10⁻³¹ J at 2π·10³ Hz, 1.7×10⁻¹⁹ J at 2π·5×10¹⁴ Hz). Summed over all field modes it diverges, growing as k_max⁴ under a cutoff (verified numerically). So “minimum standby power” has no number, neither as an infimum nor as a sum. Paper 15 states that a convention must be a number, not a function of the configuration. That the zero-point energy cannot be removed by cooling is a standard fact, and this correction does not touch it; what is corrected is only calling it “minimum” and treating it as one number.
Fifth, the kappa of dM/dt = kappa (T_CMB⁴ − T_BH⁴) is not a constant. Physically kappa = sigma A/c², and through the horizon area A = 4π r_s² proportional to M² it depends on M (greybody factors modify it further). This changes no conclusion: kappa is positive and merely multiplies the bracket, so the location of the zero and the direction of the flow on either side are decided by the sign of the bracket alone. What changes is how fast, not which way.
The two second-edition corrections are unchanged. (i) The temperature relation of Paper 20 cited in Section 2, kT ~ (4/3)πv³, was a retracted value from the first edition of Paper 20; the correct relation is kT = (4/3)πv² (the power of v was one too high). (ii) Corollary 3.4 was headed “the evaporation time of the critical black hole”, but at M_c one has dM/dt = 0 by definition and there is no evaporation; moreover the equilibrium is unstable, making M_c a watershed rather than a stable point (verified numerically in the third edition: T_BH = 2.7525 K at M = 0.99 M_c and 2.6980 K at M = 1.01 M_c). The figure 2.4×10⁴⁴ yr is a reference value assuming vacuum. The third edition further states explicitly that the extended Pythagorean equation E² = (m₀c²)² + (pc)² + (kT)² + E_ZPE² is an assumption proper to this framework and is used in none of the paper’s conclusions, and quarantines Section 4 (the standby-mode and garbage-collection readings) as Layer 3.
All eleven numbers were re-verified and all agreed with the second edition’s figures: M_c = 4.5024×10²² kg, M_c/M_Moon = 0.6132, r_s = 66.87 μm, diameter 0.134 mm, T_BH(10 M_sun) = 6.169×10⁻⁹ K, T_BH(Sgr A*) = 1.435×10⁻¹⁴ K, ratio 2.264×10⁻⁹, scale factor 4.417×10⁸, t_evap(M_c) = 2.433×10⁴⁴ yr, t_evap(10 M_sun) = 2.097×10⁷⁰ yr, ratio to the age of the universe 1.763×10³⁴.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (Hawking 1975, Page 1976, Unruh 1976, Gibbons and Hawking 1977, Fixsen 2009 for the CMB temperature, Adams and Laughlin 1997). NO NEW PHYSICAL LAW. Both the existence of the critical mass and the fact that all observed astrophysical black holes are net absorbers are well known. What is not claimed: any new physical law (there is none) / that the extended Pythagorean equation follows from standard theory (it is an assumption of this framework, is used in no conclusion here, and double-counts thermal energy under a naive reading) / that the readings “standby power” and “garbage collection” are more than metaphor (Section 4, quarantined as Layer 3) / that the critical black hole actually evaporates (at M_c, dM/dt = 0) / that the cancellation of G is unrecorded in the literature (it is two lines of algebra and this has not been checked) / that E_ZPE is a single number. What remains: equal values may have different roots. 8π arises both from a periodicity and from a solid angle. On seeing the same symbol, ask first for its address (Papers 17, 51).
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 Yamagishi Absolute-Zero Standby and Hawking Phase-Transition Theory [Paper 52]. Zenodo.
DOI: 10.5281/zenodo.21603034
ORCID: 0009-0006-8587-3450