CALC-108

the SPS thermal-forcing bound, reproduced from the source's own footnote

Supports CLM-0936 and CLM-0937. A calculation is required because SRC-029 p.100 publishes a bound and its own footnote 25 supplies every input needed to test it, which makes the bound checkable rather than citable. Producer-computed; the producer may not verify its own work, and the landing coordinator's independent reproduction is recorded at the end rather than standing in for verification.

Inputs, all taken from SRC-029 p.100 footnote 25 and none supplied by the producer. "This analysis used the Stephan-Boltzmann law relating energy emitted to temperature and assumed the Earth was in energy balance, with an estimated total solar flux intercepted by Earth of 1.746 * 10^17 watts, a nominal average terrestrial temperature of 15 degrees Celsius and an estimated average emissivity for Earth of 0.8875."

The relation. In energy balance eps * sigma * A * T^4 = P. Beamed power adds to P and nothing else on the left changes, so differentiating gives

dT / T = (1/4) * (dP / P)

Emissivity, area and the Stefan-Boltzmann constant all cancel. This is the load-bearing part of the check and it should be stated wherever the result is used: the 0.8875 emissivity and the Earth's radius never enter the answer, so the result cannot be rescued by disputing them. The only inputs that matter are the intercepted flux, the reference temperature, and dP.

T   = 15 degrees C = 288.15 K
P   = 1.746e17 W          (intercepted, as stated; not an absorbed flux, and no albedo applied)

CASE 1, THE CONTROL - a single SPS
  dP  = 1.5e9 W
  dT  = (288.15 / 4) * (1.5e9 / 1.746e17)   = 6.189e-7 K
  SRC-029 p.100 publishes "less than 0.000001 degrees C"
  6.189e-7 < 1e-6                            HOLDS

CASE 2, THE ONE UNDER TEST - several thousand SPS at total world consumption
  dP  = 1.5e13 W            ("about 15,000 GW", stated on the same page)
  dT  = (288.15 / 4) * (1.5e13 / 1.746e17)  = 6.189e-3 K
  SRC-029 p.100 publishes "less than 0.006 degrees C"
  6.189e-3 > 6e-3                            EXCEEDS THE PUBLISHED BOUND by 3.1 per cent

INVERSION - what would be needed for case 2 to hold as written
  T <= 4 * 6e-3 * 1.746e17 / 1.5e13 = 279.4 K = 6.2 degrees C
  against the 15 degrees C the footnote states

THE CONTROL IS THE POINT OF THE CALCULATION AND NOT AN ASIDE. Case 1 and case 2 differ by exactly 10^4 in dP and the relation is linear in dP, so the two reproduce from one arithmetic step. Case 1 passing is what establishes that the method is right and only the one bound is loose. Had both failed, the reading would have been that the producer had misread the model - wrong range, wrong flux, an albedo the footnote omits - and the finding would be about the producer. One passing and one failing, from the same line of arithmetic, excludes that.

The diagnosis is rounding, not physics. 6.189e-3 rounds to 0.006 at one significant figure. The source rounded correctly and then attached "less than" to the rounded value, which the rounding does not support. "less than 0.006" should read "about 0.006". Case 1 escapes because 0.000001 is a genuinely loose bound on 6.189e-7 rather than a one-figure rounding of it.

What this does NOT establish. Nothing about whether the model is the right model. The footnote applies no albedo, treats intercepted flux as if absorbed, and uses a single global mean temperature, and this calculation adopts all three because the object under test is the source's own consistency, not the climatology. A reader must not take 6.189e-3 K as this register's estimate of the thermal effect of 15,000 GW of SPS. It is what SRC-029's stated method yields when run, and the substantive claim SRC-029 is making with it - that the effect is negligible beside fossil generation for the same power - survives the correction intact, since 0.0062 and 0.006 are the same answer for that purpose.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-15, AND THE CONCLUSION IS ROBUST TO THE ONE INPUT THE FOOTNOTE COULD BE WRONG ABOUT. Both cases recomputed from the stated inputs and both agree to four figures. The landing coordinator then re-derived P rather than accepting it: the footnote's own emissivity and temperature with a modern Earth radius of 6371 km give an outgoing flux of 346.94 W/m2 over 5.1006e14 m2, so P = 1.7696e17 W against the footnote's 1.746e17, a 1.3 per cent difference. Running case 2 on that larger figure gives 6.106e-3 K, which still exceeds 6e-3. So the bound fails at 6.19e-3 on the source's own flux and at 6.11e-3 on a re-derived one - the finding does not depend on which flux figure is used, and that is worth more than either number, because the flux is the one stated input a critic could plausibly attack.

Origin: SRC-014 read, 2026-09-15. Reached by following SRC-014 p.13's citation back to the document it names. The bound had been sitting in a held source since registration and was not checked until a second source quoted it.

This calc was drafted as CALC-084 and the id moved before it landed. The producer listed existing ids with a ^### CALC- pattern; the headers in this file are ## CALC- with two hashes, so the listing returned nothing above CALC-083 while real ids ran to 107. CALC-084 is live and supports CLM-0865. A27: the probe's population was the three-hash headers, the rule it was meant to encode was every calc id in the file. Recorded here rather than in a log because the next person to wonder why this calc is numbered 108 will be reading this file.

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