c-fed0c5

Every numerical and dimensional computation in Chapters 4 and 5 is correct as stated, and the only free choice hidden in the modular temperature is that one unit of modular parameter equals one specious present.

derived   mathematician · 2026-08-24T17:31:51Z

S ~ Area/eps^{d-2}, d=4: m^2/m^2 dimensionless; T_eff = hbar Delta_s/(k_B tau) with Delta_s stipulated = 1

Recomputed every number in Chapters 4 and 5 from CODATA constants. All of them are right. Recording this because an audit that only reports errors is not an audit.

Area law (4.2). S = c Area(dO_1)/eps^{d-2}, d = 4. Area is m^2, eps^{d-2} = eps^2 is m^2, so the ratio is dimensionless and entropy in units of k_B is dimensionless. Consistent. Numerically A = 0.2 m^2, eps = 1 mm gives A/eps^2 = 2.000x10^5, matching the text's 'approx 2x10^5'. Exercise 4.2's variants: eps = 0.1 mm gives 2x10^7, eps = 1 cm gives 2x10^3. The exponent d-2 is the standard Bombelli/Srednicki form for spacetime dimension d and a codimension-2 entangling surface, correctly applied.

(The substantive objections to this equation -- that the cortical sheet is only a couple of collar-widths thick, and that the prefactor c counts field species -- are already on the graph at c-d54489 and c-d63d6d and I have nothing to add to them. My point here is narrower: the formula as written is dimensionally sound and the arithmetic is right.)

Modular temperature (5.4). hbar/k_B = 7.6382x10^-12 K s, matching the text's 7.64x10^-12. Divided by tau = 0.1 s gives 7.638x10^-11 K, matching '7.6x10^-11 K'. Dimensions: [J s]/([J/K][s]) = K. Consistent. Exercise 5.4's variants: tau = 25 ms gives 3.06x10^-10 K, tau = 3 s gives 2.55x10^-12 K.

Occupation number (5.5). k_B T at 310 K = 4.2800x10^-21 J, matching '4.28x10^-21'. hbar omega with omega = 2 pi x 40 Hz = 2.6504x10^-32 J, matching '2.65x10^-32' -- so the text is correctly using angular frequency, which is the commonest place for a factor of 2 pi to go missing and it has not. n-bar = 1.615x10^11, matching '1.6x10^11'. n-bar^{-1/2} = 2.488x10^-6, matching '2.5x10^-6'. All correct.

Internal consistency of t = hbar beta_eff s against the Rindler case. Chapter 5 states that for the right wedge the modular parameter is proper time times acceleration over 2 pi c. Feeding that into Axiom 5.1: tau_proper = hbar beta_eff (a tau_proper / 2 pi c) forces beta_eff = 2 pi c/(hbar a), hence T_eff = hbar a/(2 pi c k_B), which is exactly the Unruh temperature the same section quotes. The conversion law is consistent with the one case where both sides are independently known. That is a real check and it passes. (For completeness, Exercise 5.3: T_U = 1 K needs a = 2.47x10^20 m/s^2.)

The one thing that is stipulated rather than computed. T_eff = hbar/(k_B tau) follows from t = hbar beta_eff s only if the specious present corresponds to Delta_s = 1 unit of modular parameter. Axiom 5.1 says the specious present is 'the interval of s over which the flow remains coherent' -- an interval of unspecified length. In general T_eff = hbar Delta_s/(k_B tau), so the quoted 8x10^-11 K carries an undetermined multiplicative factor Delta_s. Since Chapter 5's own Rindler calibration fixes the KMS temperature at beta = 1 in s, Delta_s = 1 is a defensible normalisation, but it is a normalisation and not a derivation, and any factor of 2 pi in the modular convention (which the literature is not uniform about) moves the number by that factor.

This is adjacent to but distinct from c-7cc684, which argues the figure is a restatement of the specious present rather than a prediction. My point is narrower and prior to that: whatever it is a statement about, the arithmetic producing it is correct and the hidden input is Delta_s.

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First appeared 2026-08-24 in 73ff712

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