The Yamagishi Macroscopic Superposition Limit Theory: the density correction never reached the table of radii, and E_G=Gm²/R hides the separation. Third edition.
Against the problem of the limit of macroscopic superposition, epitomised by Schrödinger’s cat, this paper offers the picture of an automatic state commit in a universe OS and estimates its critical particle number N_crit using existing objective-collapse models. It claims no new physics whatever. Through the second edition it was shown that the first edition’s 2^N memory model, confronted with the holographic bound, gives N about 405 and so disagrees with the claimed 10^11 by eight orders of magnitude; that the CSL N-squared law holds only within the correlation length r_C; and that the effective density of the Penrose table is not a solid density. What the third edition adds is the observation that the second edition’s own correction was never carried back to the other place the same error affects.
First, the density correction never reached the table of radii. Section 3.2 of the second edition noted that the effective density of the table is about 62 kg per cubic metre, not a solid density, and corrected the collapse time accordingly. Yet the table of radii in the note to Section 3.1 was left constructed in exactly the same way — N lattice points at a spacing of 3e-10 m. If the density is wrong, the radius is wrong. Recomputed at a solid density of 2000 kg per cubic metre, every radius shrinks by the factor (2000/61.9)^(1/3) = 3.18 (verified numerically). The row at N = 10^11, which the second edition described as exceeding r_C by about nine times, in fact exceeds it by about 2.7 times. The conclusion survives; the figure does not.
And the range in which the N-squared law holds exactly moves too. Solving R(N) = r_C gives N = 1.6e8 at the second edition’s density and N = 5.0e9 at solid density (verified numerically). So where Corollary 3.3 item four said that the N-squared law holds exactly for N below about 10^8, it understated the range by the same factor of thirty-two. The correct figure is N below about 5e9. Both of these are consequences of an error the second edition itself found one subsection later: a correction was made in one place and not carried back to the other place the same error affects — a shape this corpus keeps meeting, and this paper is one instance of it.
Second, E_G approximately equal to G m squared over R hides the separation of the superposition. In Penrose’s criterion E_G is the difference in gravitational self-energy between the two superposed mass distributions, not the self-energy of one body. For a uniform sphere of radius R superposed at two positions separated by d, that difference behaves as G m squared over R when d is comparable to R or larger, but as G m squared d squared over R cubed when d is much smaller than R. Writing E_G as G m squared over R is equivalent to taking d comparable to R — the body superposed at a separation comparable to its own size, the situation in which collapse is fastest. For smaller d the collapse time is longer by the factor (R/d) squared (verified: four times at d = 0.5 R, a hundred times at 0.1 R, ten thousand times at 0.01 R). In real interference experiments the separation is usually far smaller than the size of the object, and then the collapse time is much longer than the table’s value. This does not invalidate the table — as an order-of-magnitude guide the form is the one Penrose himself used. What is corrected is the failure to say that a second variable, d, is hidden in the table. Read as a function of N alone, the table gives the fastest case.
Both corrections act in the same direction, making collapse slower, so the actual transition region is likely at higher N than the second edition wrote. Item five, the separation, was added to Corollary 3.3 and the figure in item four corrected. The conclusion that N_crit is not uniquely fixed is reinforced by both.
Third, all four files of the second edition (Japanese and English, tex and html) contained control characters (tabs). The backslash-t had been lost, so item four of the conclusion did not display the phrase about the validity conditions of the model, nor the symbol tau, correctly. The third edition repairs this and verifies that every file is free of control characters.
What this paper returns to the corpus: the second edition found two errors in the first — the effective density and the validity condition of the N-squared law. And having found the former, it did not carry the consequence back to the table of the latter. A radius is determined by a density; if the density is wrong, so is the radius. As a result the second edition printed a figure, about nine times over, that is off by more than a factor of three, and understated the validity range of the N-squared law by a factor of thirty-two. Neither changes a conclusion; both are checkable errors. The shape is this: when you find an error in one quantity, re-examine everything computed from it. A correction is complete only when it has been carried to everywhere it propagates.
Every figure was re-verified and every figure the second edition printed was reproduced (what is corrected are quantities the second edition had not computed): 405.3; four rows of the CSL table; five rows of radii; the effective density 61.9 kg per cubic metre; five rows of the Penrose table (2.3e13, 4.9e6, 49, 1.1e-5, 4.9e-19 s); the factor 3.18; and N at r_C of 1.6e8 and 5.0e9. The second edition’s limitations are all retained: the CSL N-squared law is confined to r_C; the Penrose table’s effective density is not a solid density; and the first edition’s claim that many-worlds is unnecessary was withdrawn, this mechanism being placed as one alternative among objective-collapse theories. Experiments reach N of order 10^3 to 10^4 (Fein et al. 2019), with no capacity yet to test the region in question.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). Every model used is pre-existing (GRW-CSL, Penrose 1996, the holographic bound, Fein et al. 2019). NO NEW THEOREM AND NO PREDICTION. What is not claimed: any new physical law (there is none) / that N_crit follows from the OS picture alone (it requires a specific collapse mechanism) / the 2^N memory model (not adopted, since it presupposes a classical substrate) / that many-worlds is dispensable (objective collapse is an alternative, not a proof) / that the table’s collapse times are independent of the separation (d comparable to R is taken silently) / that the N-squared law holds at N = 10^11 (it is an extrapolation). What remains: a correction is complete only when it has been carried to everywhere it propagates.
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 Yamagishi Macroscopic Superposition Limit Theory [Paper 55]. Zenodo.
DOI: 10.5281/zenodo.21603425
ORCID: 0009-0006-8587-3450