The Fringes Do Not Remember the Exponent: Flux Conservation Forces n = d-1, and the Fringe Spacing Comes from Phase – Two Distinct Roots. Third edition.
This paper held two results. (A) that the condition for the flux of a spherical wave to be conserved in r fixes the exponent uniquely at n = 2 (Section 2), and (B) that the superposition of two spherical waves reproduces the standard fringe spacing delta-y = lambda D / a (Theorem 3.1). BOTH PIECES OF ALGEBRA ARE CORRECT AND THE THIRD EDITION CHANGES NEITHER. What the third edition shows is that THE TWO DO NOT SHARE A ROOT.
FIRST, THE FRINGES DO NOT REMEMBER THE EXPONENT. Relaxing the |psi| proportional to 1/r that (A) forces to a general |psi| proportional to r^(-m), the maxima were located numerically (no approximation; a dense scan over four million points with parabolic vertex refinement). Between m = 1 (what (A) forces) and m = 3 (an entirely different value), the first maximum sits at 5.00004983966e-3 m and 5.00002450778e-3 m, a difference of 2.53e-8 m; the second maxima differ by 5.07e-8 m and the third by 7.60e-8 m. That is FIVE PARTS PER MILLION OF ONE FRINGE (5e-3 m). Sweeping m from 0 to 3 changed the mean fringe spacing by a relative 7.6e-6 only. THE EXPONENT ACTS ON THE ENVELOPE AND NOT ON THE FRINGE SPACING. The formula says why: what fixes the fringes is the phase difference k(r1 – r2) alone, into which the falloff of the amplitude does not enter. The residual 1e-8 is not of phase origin either – it is the slight imbalance of the two amplitudes arising from the two slit distances being unequal. HENCE (A) AND (B) ARE NOT TWO APPEARANCES OF ONE MECHANISM: one fixes how the brightness falls, the other fixes the spacing of the fringes.
SECOND, 4PI APPEARS IN NEITHER. The intensity of Theorem 3.1, I = (2|A|^2/R^2)[1 + cos(k a y / R)], contains no occurrence of 4pi. Setting the amplitude A to 1, to 1/sqrt(4pi) = 0.2820947918 and to 17, the maxima were found at 1.000045e-2 m in all three cases. 4PI ENTERS ONLY THE NORMALISATION OF THE FLUX AND CANCELS OUT OF THE SHAPE OF THE PATTERN. This is an instance of the general check Paper 11’s third edition wrote down: a quantity that enters a normalised probability only as an overall factor is, by definition, invisible to observation. The “4pi wavefield” of the second edition’s title therefore contributes nothing to the description of the interference. The second edition’s verdict – that the appearance of 4pi in any spherical integral is an identity and not a mechanism – is kept unchanged.
THIRD, (A) DOES NOT SELECT THREE DIMENSIONS. In d spatial dimensions with |psi|^2 proportional to r^(-n), the area of the sphere S^(d-1) of radius r scales as r^(d-1), so the flux scales as r^(d-1-n) and the conservation condition is n = d – 1 (checked numerically for d = 2, 3, 4, 5 at r = 0.5, 1, 2, 5, 100, with a spread of 0 in every case). n = 2 is merely the d = 3 case, and the condition itself holds in the same form in every dimension. One therefore CANNOT say “space is three-dimensional because the flux is conserved”. This is exactly how Paper 114 classified it – n = d – 1 stands on the side that does not select – and it is the same quantity Paper 116 addressed by noting that d enters only through the denominator n = d – 1. The Flatland analogy of the introduction must accordingly NOT be read as explaining the number of dimensions; it is a picture for the fact that the observing surface is of lower dimension than the field.
FOURTH, TWO INTERNAL CITATIONS ARE UPDATED. (i) The second edition’s Section 4 described the detector picture as “based on the Lazy Evaluation protocol of Paper 11”. Paper 11’s third edition has since failed that metaphor on the machinery-transfer test: lazy evaluation in computer science is a structure in which the value is already determined and only the computation is deferred (a thunk), and that core is precisely what the Kochen-Specker theorem forbids. Paper 11’s third edition computed the Peres-Mermin magic square itself and confirmed that the product of the rows and the product of the columns come out +1 and -1. THE METAPHOR CROSSES OVER AS A PICTURE; THE MACHINERY DOES NOT. This edition therefore removes the appeal to Paper 11. What remains is only the vocabulary of a drop in dimension, which the second edition’s Note 4.2 had already limited: P_2D is not a projection operator in the mathematical sense, but a notational convenience and not a derivation. THAT LIMITATION WAS CORRECT AND IS KEPT. This paper does not solve the measurement problem.
(ii) The second edition’s Note 4.1 deferred the Born exponent 2 to Paper 53 alone, writing that “the usable exponent is fixed at 2 by unitarity”. That is correct, but it is THE WEAKEST OF THREE. Paper 11’s third edition holds a roll-call of three addresses: GLEASON’S THEOREM (Paper 71 – in dimension three or more any probability measure is tr(rho P); a theorem, not a postulate), UNITARITY (Paper 53 – isometries of l^p for p not equal to 2 are phases and permutations; Banach-Lamperti), and THE 2 OF THE HOPF MAP (Paper 68 – same root or rhyme, undecided). Paper 53 itself states that it is weaker than Gleason’s theorem (its Note 2.3). This paper was pointing only at the weakest one. The third edition writes all three.
THE SCOPE, RESTATED as eight rows. (1) Flux conservation forces n = d – 1: a FACT. (2) That n = 2 means three dimensions: NOT SHOWN. (3) Reproduction of delta-y = lambda D / a: a FACT. (4) That (A) and (B) are one mechanism: NOT SHOWN. (5) 4pi: an IDENTITY. (6) P_2D: NOTATION. (7) A solution of the measurement problem: NOT SHOWN. (8) Any change to quantum predictions: NONE. Of the eight rows, two are facts proper to this paper – the flux condition of Section 2 and the fringe spacing of Theorem 3.1. And what the third edition adds is the single line that those two are independent of each other.
IMPORTANT (scope): a structural interpretation and self-audit of elementary established facts (Layer 2). The physics used is entirely standard (spherical waves, double-slit interference in the Fraunhofer limit, flux conservation). NO NEW THEOREM AND NO PREDICTION. What is not claimed: no claim to have solved the measurement problem (P_2D is notation) / lazy evaluation is not used as a mechanism (it fails in Paper 11, 3rd ed.) / n = d – 1 is not read as “the reason space is three-dimensional” (it holds in every dimension; Papers 114, 116) / 4pi is not treated as a mechanism (an identity of spherical integration, absent from the pattern) / (A) and (B) are not merged into one mechanism (perturbation separates them) / wave-particle duality is not denied (the terminology functions well as a summary of experimental fact) / no prediction of quantum mechanics is altered / no claim to have derived the Born rule (there are three addresses, and the strongest is Gleason’s theorem). The item retracted in the second edition (the first edition’s integral of |Psi|^2 d-Omega = 4 pi |A|^2, a confusion of integration elements) is not reinstated.
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 Fringes Do Not Remember the Exponent — 3rd ed.: in the Yamagishi Double-Slit Geometric Wavefront Restoration Theory, flux conservation forces $n=d-1$ while the fringe spacing comes from phase: two distinct roots — [Paper 41]. Zenodo.
DOI: 10.5281/zenodo.21436048
ORCID: 0009-0006-8587-3450