The Yamagishi Absolute-Zero Standby and Hawking Phase-Transition Theory ── 3rd ed.: The Critical Radius Contains No G, and the 8π Is Not a Solid Angle ── [Paper 52]

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Original abstract

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. ----- 山岸・絶対零度待機とホーキング相転移理論——臨界半径に G は現れず、8π は立体角ではない。第3版。 ホーキング温度が宇宙マイクロ波背景放射の温度に等しくなる臨界質量 M_c は、ブラックホールが正味で成長する相と蒸発する相を分ける。T_CMB = 2.725 K に対し M_c = ħc³/(8πG k_B T_CMB) = 4.50×10²² kg(月の約0.61倍)、対応するシュヴァルツシルト半径は 66.9 μm である。第2版はこれを定量化し、11個の数値を検算した。第3版はその11個をすべて再確認したうえで、これらの数のどこに何が入っているのかを分解する。 第一に、これが本版の中心である——臨界半径に G は現れない。r_s(M_c) = 2GM_c/c² に M_c = ħc³/(8πG k_B T_CMB) を代入すると G は約分されて消え、r_s(M_c) = ħc/(4π k_B T_CMB) となる(記号計算で確認済:結果は G を含まない)。すなわち臨界「質量」は重力定数を G⁻¹ に比例して担うのに対し、臨界「大きさ」はまったく担わない——それは純粋に熱的な長さ ħc/(k_B T_CMB) = 8.403×10⁻⁴ m の 1/4π である(数値確認済:その値 66.87 μm が定理3.1 の r_s と一致)。この消失は偶然ではない。M_c は T_BH ∝ 1/(GM) を T_CMB に等置して得られるから M_c ∝ 1/G であり、r_s ∝ GM は G に比例するから、合成すれば必ず打ち消し合う。重力が強いか弱いかは臨界質量を上下させるが、臨界の大きさは動かさない。 第二に、そこに残る 4π は立体角ではない。体系は 8π を含む式を複数扱っている(論文17・35)。ホーキング温度の 8π とアインシュタイン方程式の 8π は、値が等しく根が異なる——前者は T_BH = ħκ/(2πck_B)、κ = c⁴/(4GM) より 8π = 4×2π であり、2π は「ユークリッド断面の周期」、すなわちウンルー効果の 2π と同じ根である。後者はニュートン極限 ∇²Φ = 4πGρ に合わせて決まるから 8π = 2×4π であり、こちらの 4π は「立体角」(論文35)、2 はトレース反転である。どちらも 25.13 だが、一方は周期性を、他方は立体角を含む。同じ数が二度現れているのではなく、別の場所から来た二つの数がたまたま等しい。そして r_s(M_c) に残る 4π は 8π を r_s = 2GM/c² の 2 で割った残りであり、住所は(κ = c²/2r_s の 2)×(ユークリッド周期 2π)である。論文17第3版が求めたのは「別根と言うこと」ではなく「各根の住所を書くこと」であった。本稿はその住所を書く。 第三に、蒸発時間の 5120π は独立な数ではない。ホーキング出力 P = ħc⁶/(15360πG²M²)(論文43)から dM/dt = −ħc⁴/(15360πG²M²) となり、これを積分すると M³/3 が現れて 15360/3 = 5120 となる(記号計算で確認済)。5120π と 15360π は二つの数ではなく、一つの根を M の冪 3 で割った二つの姿である。 第四に、「最小待機電力」という呼び名を取り下げる。理由は物理ではなく、量が量になっていないことにある。E_ZPE = (1/2)ħω は ω の関数であり、本稿はどの ω を採るかを述べていない。単一振動子について ω を下げれば (1/2)ħω → 0 であり、モードにわたる下限は存在しない(数値確認済:ω = 2π·1 Hz で 3.3×10⁻³⁴ J、2π·10³ Hz で 3.3×10⁻³¹ J、2π·5×10¹⁴ Hz で 1.7×10⁻¹⁹ J)。場の全モードについて和をとれば発散し、切断 k_max を入れれば k_max⁴ で増大する(数値確認済)。したがって「最小待機電力」は、下限としても総和としても数を持たない。論文15 は「規約は数であって、配置の関数ではない」と述べた。零点エネルギーが冷却で除去できないことは標準的な事実であり、この訂正はそれに触れない。訂正するのは、それを「最小」と呼び一つの数として扱った点だけである。 第五に、dM/dt = κ(T_CMB⁴ − T_BH⁴) の κ は定数ではない。物理的には κ = σA/c² であり、地平線面積 A = 4πr_s² ∝ M² を通じて M に依存する(さらにグレイボディ因子が加わる)。ただしこれは結論を変えない——κ は正であって括弧の符号に掛かるだけであり、dM/dt の零点の位置も、その両側での流れの向きも、括弧の符号だけで決まる。変わるのは速さであって、どちらへ動くかではない。 第2版の訂正2点は据え置く——(i) 第2節が引用していた論文20の温度関係 kT ~ (4/3)πv³ は論文20初版の撤回済の値であり、正しくは kT = (4/3)πv²(v の巾が1つ多かった)。(ii) 系3.4 を「臨界質量のブラックホールの蒸発時間」としていたが、M_c では定義により dM/dt = 0 であり蒸発しない。しかもこの平衡は不安定であり、M_c は安定点ではなく分水嶺である(第3版で数値確認:M = 0.99M_c で T_BH = 2.7525 K、M = 1.01M_c で 2.6980 K)。2.4×10⁴⁴ 年は真空中を仮定した参考値である。さらに第3版は、拡張ピタゴラス方程式 E² = (m₀c²)² + (pc)² + (kT)² + E_ZPE² が本枠組み固有の仮定であり本稿のどの結論にも使われていないことを明示し、第4節(待機モード・ガベージコレクションの読み)を第3層として隔離した。 11個の数値をすべて再検算し、いずれも第2版の表示と一致した——M_c = 4.5024×10²² kg、M_c/M_Moon = 0.6132、r_s = 66.87 μm、直径 0.134 mm、T_BH(10 M_sun) = 6.169×10⁻⁹ K、T_BH(Sgr A*) = 1.435×10⁻¹⁴ K、比 2.264×10⁻⁹、スケール因子 4.417×10⁸ 倍、t_evap(M_c) = 2.433×10⁴⁴ 年、t_evap(10 M_sun) = 2.097×10⁷⁰ 年、宇宙年齢比 1.763×10³⁴。 重要(射程):確立された標準的事実の構造的解釈と自己点検(第2層)。用いた物理はすべて標準的である(ホーキング 1975、ペイジ 1976、ウンルー 1976、ギボ

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