Keep the Patch Constant as a Letter and the Ambiguity Dissolves: 15360 pi Is Physics, a^3 Is Convention, and the Power of a Is the Mass Dimension. Third edition.
This paper recorded the form taken by Hawking’s radiated-power formula P = hbar c^6/(15360 pi G^2 M^2) under the Yamagishi patch G = 4pi, hbar = 1/(4pi), and its second edition retracted three geometric labels: (a) reading 256 pi^4 as “the square of the double-torus cocoon” (indistinguishable, since (16 pi^2)^2 = (4pi)^4), (b) reading a bare 64 as “the numerical part of the three-dimensional torus volume” (stripping pi^3 off (4pi)^3), and (c) reading c^6 as “six-dimensional phase space” (it follows from dimensional analysis). ALL THREE ARE SOUND AND ARE NOT REINSTATED HERE. But after retracting, the second edition went on asking the same question – HOW DOES 16384 pi^4 DECOMPOSE? It listed the non-uniqueness and concluded that “which module one divides by is chosen by the reader, not decided by the formula”. Correct – and one step short: it never asked WHY a choice became necessary.
FIRST, THE AMBIGUITY WAS MANUFACTURED. Instead of fixing the patch constant to a number, keep it AS A LETTER: G = a, hbar = 1/a (as Paper 116 showed, the family preserving hbar G = 1 holds for any a, and nothing requires a = 4pi). Symbolic computation then gives T_H = c^3/(8 pi a^2 M), r_s = 2 a M/c^2, A = 16 pi a^2 M^2/c^4, sigma = pi^2 a^3/(60 c^2), and P = c^6/(15360 pi a^3 M^2) – the step-by-step derivation and the direct substitution were verified here to agree exactly, and setting a = 4pi reproduces the second edition’s P = c^6/(983040 pi^4 M^2). Here 15360 pi BELONGS ENTIRELY TO THE PHYSICS – it is the 60 x 256 obtained on computing sigma A T_H^4, where the 60 descends from zeta(4) = pi^4/90 and the 256 from tidying 8^4/16 – and has nothing to do with the patch. The a^3, by contrast, BELONGS ENTIRELY TO THE CONVENTION, coming only from what is substituted for G and hbar. THE NUMBER 983040 pi^4 COMES INTO BEING ONLY WHEN THE TWO ARE MULTIPLIED (15360 pi x 64 pi^3 = 983040 pi^4).
The four decompositions the second edition listed – 64(16 pi^2)^2, 64(4pi)^4, 4(8pi)^4, 1024(2pi)^4 – are ALL THE SAME NUMBER, 16384 pi^4 (verified here). Of course no formula can choose among them: once a fact has been multiplied by a convention, the resulting number carries no record of which pi came from where. KEPT AS A LETTER, NO NUMBER REQUIRING DECOMPOSITION EVER APPEARS – in P = c^6/(15360 pi a^3 M^2) there is no room for decomposition, since 15360 pi cannot be split and neither can a^3. The same holds for the “module chopping” the second edition forbade: a bare 64 can be extracted from (4pi)^3 only because pi^4 had already been absorbed elsewhere, leaving no pi^3 to accompany it. Kept as a letter, a^3 remains one block to the end and the stripping operation is not even definable.
SECOND, OF THE THREE (4pi)^3 THE SECOND EDITION RETAINED, ONLY ONE IS PURE. After retracting, the second edition wrote that “the places where (4pi)^3 appears with its pi intact do remain” and listed the Hawking temperature, the horizon area and the Stefan-Boltzmann constant. (1) THE HAWKING TEMPERATURE: T_H = c^3/(2(4pi)^3 M) is correct as a number, but the correct breakdown is 8pi x (4pi)^2 – the 8pi is a number belonging to Hawking’s formula (a fact, from the surface gravity kappa = c^4/(4GM) and T = hbar kappa/(2 pi c k_B)), while the (4pi)^2 is the choice of units (a convention). Writing 2(4pi)^3 borrows one pi from the factual 8pi and hands it to the conventional side, REGROUPING ACROSS THE BOUNDARY between fact and convention – a violation, in this setting, of the discipline established by Paper 35’s third edition (the 4pi of Poisson’s equation is a fact and the 4pi of the G-patch a convention, and the two must not be gathered into one). (2) THE HORIZON AREA: the 4pi of A = 4 pi r_s^2 is the total solid angle of a sphere, a fact, while the a in r_s = 2 a M/c^2 is a convention, so A = (sphere’s 4pi) x 4 x (a^2) = 16 pi a^2 M^2/c^4; the second edition’s 4(4pi)^3 crosses the boundary here too. (3) THE STEFAN-BOLTZMANN CONSTANT: in sigma = pi^2 a^3/(60 c^2) the a^3 comes from hbar^-3 ALONE, with none of the physics coefficient pi^2/60 mixed in. THE SECOND EDITION WAS THEREFORE RIGHT to call this one well-sourced, and it is the only one of the three that is. But being pure and being a discovery are two different things: once hbar = 1/a has been set, an a^3 in hbar^-3 is a consequence of the substitution.
THIRD, THE POWERS OF a HAVE A REASON. T_H carries a^-2, A carries a^2, sigma carries a^3 and P carries a^-3, and the exponent count of P = sigma A T_H^4 closes: 3 + 2 – 8 = -3 (verified here). These powers are a consequence of Paper 116’s theorem epsilon = mu: under the one-parameter patch G = a, hbar = 1/a one has X = hbar^p G^q c^N = a^(-mu) c^N, so the power of a is fixed by the mass dimension mu. HENCE THE CONTENT OF THE OBSERVATION “a (4pi)^3 APPEARS” IS NOT GEOMETRY BUT DIMENSIONAL COUNTING. The 3 is neither the dimension of a torus nor the dimension of space: it is the mass dimension of the quantity. This supports the second edition’s retraction (b) from another direction – a bare 64 cannot be called a three-dimensional torus volume not only because the pi was stripped, but because what the exponent counts was never a geometric capacity in the first place.
THE SCOPE, RESTATED as ten rows. (1) Hawking’s formula itself: an ESTABLISHED FACT (1974; not altered here). (2) 15360 = 60 x 256: a consequence of the standard derivation (no mystery). (3) P = c^6/(15360 pi a^3 M^2): fact and convention KEPT APART. (4) 983040 pi^4: THE PRODUCT of the two (the source of the non-uniqueness). (5) The a^3 of sigma: PURE CONVENTION. (6) The (4pi)^3 of T_H and A: MIXTURES. (7) The powers of a: MASS DIMENSIONS (Paper 116). (8) That c^6 is six-dimensional phase space: NOT SHOWN (retracted; not reinstated). (9) An explanation of 15360 pi: NOT SHOWN. (10) A resolution of the information paradox: NOT SHOWN. Of the ten rows, NOT ONE CARRIES A CLAIM PROPER TO THIS PAPER.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (Hawking 1974 and 1975, the Stefan-Boltzmann law, zeta(4) = pi^4/90, the phase-space density of states). NO NEW THEOREM AND NO PREDICTION. What is not claimed: G = 4pi, hbar = 1/(4pi) is not a fact about nature (a choice of unit system; Papers 95, 116) / no claim to have explained 15360 pi (it follows completely from the standard derivation) / the exponent of c is not read as a spatial dimension (Paper 44; retracted in the second edition) / powers of 4pi are not read as geometric capacities (they are mass dimensions; Paper 116) / a factual pi and a conventional pi are not gathered into one (Paper 35, 3rd ed.) / the (4pi)^3 of T_H, A and sigma are not unified as “the same geometric quantity” (only one of the three is pure) / the information paradox is not touched / greybody factors are not treated (Page 1976; the ideal blackbody approximation only) / Layer-3 lenses (the picture of an OS memory limit and the like) are kept out of the body (quarantined under a non-scientific label).
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). Keep the Patch Constant as a Letter and the Ambiguity Dissolves — 3rd ed.: in the Hawking Radiation Formula under the Yamagishi Patch, $15360\pi$ is physics, $a^3$ is convention, and the power of $a$ is the mass dimension — [Paper 43]. Zenodo.
DOI: 10.5281/zenodo.21602132
ORCID: 0009-0006-8587-3450