Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · theorem · 39b16f67

Theorem: Conditional inverse-square power and cutoff lifetime

Stage 3: energy, erasure cost, and a conditional mass law › The luminosity and clock are additional inputs · lines 1423–1453 · proof included or linked

Hawk04-D@1#thm:lifetime, Hawk04-D@1#eq:kappa, Hawk04-D@1#eq:lifetime

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Under Assumption ass:luminosity in its fixed massless-species specialization,

P (M) = \frac{κ}{M ^{2}},

where

κ = α \sum \int_{0}^{\infty} \frac{d x}{2 π} \frac{x γ _{α} ( x )}{exp ( 8 π x ) - η _{α}} .

If $κ > 0$ and $M ˙ = - κ / M^{2}$ , then until the validity cutoff,

M (t)^{3} = M_{0}^{3} - 3 κ t, t_{c} = \frac{M _{0} ^{3} - M _{c} ^{3}}{3 κ} .

For fixed $M_{c}$ and $κ$ , this cutoff lifetime has cubic large- $M_{0}$ scaling.

Proof. Set $x = M ω$ in (eq:power). The measure times the energy factor contributes $M^{- 2}$ , and the thermal denominator becomes $exp (8 π x) - η_{α}$ . Then $d (M^{3}) / d t = 3 M^{2} M ˙ = - 3 κ$ . Integrate and solve for the first crossing of $M_{c}$ .

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Hawk04-D@1#eq:kappa · line 1431
$κ = α \sum \int_{0}^{\infty} \frac{d x}{2 π} \frac{x γ _{α} ( x )}{exp ( 8 π x ) - η _{α}} .$
Hawk04-D@1#eq:lifetime · line 1438
$M (t)^{3} = M_{0}^{3} - 3 κ t, t_{c} = \frac{M _{0} ^{3} - M _{c} ^{3}}{3 κ} .$
unlabeled · line 1427
$P (M) = \frac{κ}{M ^{2}},$

Theorem: Conditional inverse-square power and cutoff lifetime

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