The Internal Structure of the Exponent 7 Is Not Unique: The 5/2 + 9/2 of the Yamagishi Planck-Pressure Limit Names the Route Taken. Third edition.
This paper treats the Planck pressure P_Pl = c^7/(hbar G^2) approximately 4.63e113 Pa and stated that the exponent 7 decomposes rigorously as 7 = 5/2 + 3 x (3/2). THE THREE RETRACTIONS OF THE SECOND EDITION ARE ALL SOUND AND ARE NOT REINSTATED – the first edition’s seven-dimensional breakdown (3D geometry + 1D time + 2D complex phase + 1D memory bus, none defined in any other paper and assembled after the fact to match the exponent), the “exact” agreement of Theorem 3.1 (the proof is written throughout with order estimates and drops the 8 pi; keeping it gives a discrepancy of about 25.13), and the decomposition (16 pi^2)^2 in the citation of Paper 43. The SI numbers were also re-verified and all three agree: F_Pl = c^4/G = 1.210e44 N, P_Pl = 4.633e113 Pa, F_max = c^4/(4G) = 3.026e43 N; P_Pl = F_Pl/l_P^2 was confirmed independently. What the third edition corrects is that “rigorous decomposition”.
FIRST, THE INTERNAL STRUCTURE OF 7 IS NOT UNIQUE. There are several routes to P_Pl and which route one takes changes the breakdown (verified here): E_P/l_P^3 gives 5/2 + 9/2, F_P/l_P^2 gives 4 + 3, rho_P c^2 gives 5 + 2, P_power/(l_P^2 c) gives 5 + 3 – 1, F_P c^3 gives 4 + 3. All give 7, and all give different breakdowns. AND THE F_P, rho_P AND P_power OF THE ALTERNATIVE ROUTES ARE THE VERY QUANTITIES THIS PAPER’S OWN COROLLARY 2.3 SETS OUT – F_P = E_P/l_P = c^4/(4 pi), P_power = E_P/t_P = c^5/(4 pi), rho_P = E_P/(c^2 l_P^3) = c^5/(4 pi). The evidence that the decomposition is not unique is written by this paper itself, two sections later. HENCE “7 = 5/2 + 3 x (3/2)” IS NOT THE INTERNAL STRUCTURE OF THE EXPONENT BUT THE NAME OF THE ROUTE CHOSEN, the one passing through E_P and l_P. The algebra is right (5/2 + 9/2 = 7 is certain); what changes is only the classification – a record of a path, not a description of a structure.
SECOND, THIS IS THE FOURTH TIME THE SERIES HAS MET THIS SHAPE. Paper 43 arrived at the non-uniqueness of the decomposition of 16384 pi^4 (64(4 pi)^4 = 4(8 pi)^4 = 1024(2 pi)^4), Paper 45 at R = 2 being a choice of unit (rescaling gives R’ = 2/lambda), Paper 48 at (4 pi)^n n! being cut in the wrong place (2^n n! is one block of the Pfaffian). And this paper arrives at there being several routes. All four are the same shape – splitting one number and reading a meaning into the split – and in all four several splits existed. WHEN SEVERAL SPLITS EXIST, WHICH SPLIT TO CHOOSE IS NOT DECIDED BY THE NUMBER; IT IS DECIDED BY THE READER. This does not forbid decomposition; it only demands that a decomposition state which route was taken (Paper 17: saying “distinct roots” is only half, and the address must be written before the claim becomes checkable).
THIRD, F_max = c^4/(16 pi) IS LIKEWISE A PRODUCT OF A FACT AND A CONVENTION. The maximum-force conjecture of Gibbons (2002) and Schiller (2005), F_max = c^4/(4G) approximately 3.026e43 N, is not an established theorem and remains under discussion – the second edition’s limitation, “this paper cites it as supporting evidence and does not use it as a premise it relies on”, was right and is kept. In general, setting G = a gives F_max = c^4/(4a), where the 4 is a number belonging to Gibbons’ conjecture (the factual side) and a is the patch constant (a convention; any positive value; Paper 116). Only at a = 4 pi does 4a become 16 pi, but that is a fact multiplied by a convention and the resulting number keeps no record of which came from where (the same shape as Papers 43, 44). So the expression F_max = c^4/(16 pi) is correct, but 16 pi must not be read as a single geometric quantity. The second edition’s phrasing, “it is 1/4 of the Planck force F_P = c^4/(4 pi)”, is accurate, and that form – the one preserving the separation – is used.
FOURTH, R_max = 1/l_P^2 IS NOT A THEOREM. The proof of Theorem 3.1 used R_max as though given, speaking of “the maximum curvature corresponding to a minimum length scale (the Planck length)”, but the existence of a minimum length is not established – the Planck length is where the effective-field-theory description is expected to fail, and it has not been shown that there is a lower bound on length there. The second edition’s physical reading (“once the curvature reaches the Planck scale, the premise of treating spacetime as a continuum in an effective field theory breaks down and a quantum-gravitational description is required”; “P_Pl gives the order at which classical general relativity loses its descriptive power”) is right, and this edition states it on the side of the proof as well. The conclusion does not change – that the upper bound on curvature and the upper bound on pressure reduce to the same combination c^7/(hbar G^2) is a consequence of dimensional analysis, and the agreement of orders stands. The second edition’s caution about 8 pi is also kept: retaining the coefficient gives P_max = P_Pl/(8 pi), a discrepancy of 25.13. What agrees is the order, not the number.
For the c^7 to c^6 transition the second edition’s limitation is kept unchanged: the statement that once the local pressure reaches P_Pl an event horizon forms and subsequent emission is governed by Hawking radiation P = c^6/(983040 pi^4 M^2), and the measure of using the undecomposed form to match Paper 43’s second edition (so as not to call back a retracted reading through a citation), are unaltered. The second edition’s limitation – “this paper gives no independent proof that this connection is a physical necessity beyond numerical agreement; what is established is that both are generated from the same algebraic hierarchy” – was also right. But in the light of the preceding sections even “the same algebraic hierarchy” must be weakened: which route one takes within that hierarchy is not unique. That they can be generated is certain; that this is the only route of generation has not been shown. The Layer-3 lens (the halting of the rendering pipeline, memory attributes locked to a dump state, a write limit and a read rate) was correctly limited by the second edition as “an interpretation, not a theorem”, and this edition carries the limitation out by removing it from the body.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (the Planck units, Einstein’s equation, the maximum-force conjecture of Gibbons 2002 and Schiller 2005, the Schwarzschild metric). NO NEW THEOREM AND NO PREDICTION. What is not claimed: a decomposition of an exponent is not called an “internal structure” (it names a route) / a factual number and 4 pi are not gathered into one (Papers 43, 44, 48) / the existence of a minimum length is not assumed / the maximum-force conjecture is not treated as an established theorem (supporting evidence only) / agreement with the curvature limit is not claimed at the level of coefficients (a discrepancy of 25.13) / the exponent 7 is not read as seven dimensions (retracted in the second edition) / the c^7 to c^6 transition is not claimed to be more than a numerical agreement / uniqueness of the generating route is not claimed / the Layer-3 lens is not placed in 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). The Internal Structure of the Exponent $7$ Is Not Unique — 3rd ed.: The $\frac52+\frac92$ of the Yamagishi Planck-Pressure Limit Names the Route Taken — [Paper 50]. Zenodo.
DOI: 10.5281/zenodo.21602796
ORCID: 0009-0006-8587-3450