A Photocount Reading of the Interpolating Function, and a No-Go for Deriving Its Normalisation from a Clock
Abstract
A companion paper removed the MOND acceleration scale as an independent parameter and found that 155 SPARC rotation curves then select, with nothing fitted, the member n = 2 of the family mu_n(Y) = 1 - (1+Y)^(-n), which is kappa = 1/2. This paper reports what happened when we tried to DERIVE that integer, and one structural result obtained along the way. FIRST, A NO-GO. A cuscuton clock's own field equation fixes V'(tau) = -3 H U, so a vanishing potential forces a vanishing expansion rate: any cuscuton clock in an expanding universe REQUIRES a potential. A potential is precisely the free additive constant that makes the acceleration-scale normalisation a zero mode. Hence the twelve-gate relativistic construction of our earlier work provably cannot derive kappa, and the two tracks cannot be merged: they assign U = 3.68e-25 eV^4 and U = 2.52e-11 eV^4 to the same coefficient, a ratio of 6.9e13, and forcing agreement demands a margin of 2.58 where the margin cannot exceed one. SECOND, AN IDENTIFICATION. The family is exactly Mandel's n-mode photocount formula: for n independent thermal modes of mean occupancy Y, mu_n is the probability that at least one quantum is present (Monte Carlo at Y = 2, n = 2 over 400,000 trials: 0.88914 against the closed form 0.88889). Equivalently it is one minus a Tsallis q-exponential with q = 1 + 1/n, the ordinary exponential kernel being the infinite-mode limit. This explains why the exponent is an INTEGER -- it is a mode count -- without fixing its value, and replaces the question with a sharper one: why does the vacuum present exactly two modes to an acceleration? THIRD, THE CLOSURES. The modes are not Unruh modes: a fixed-frequency thermal occupancy is 3.72e-44 where the reading requires 0.01, failing by 41 orders in the only regime MOND is about. Twelve pre-registered mathematical criteria scatter over three different members, with n = 1 the modal choice (5 of 9 unique selections); only the convergence boundary selects n = 2, and as a boundary rather than an optimum. Effective-field-theory positivity is inapplicable, since the kinetic matrix degenerates where mu(0) = 0, and granting it anyway the single scale-free invariant 9(n+1)/(16n) is continuous and monotone, so no bound can select an integer; causality is saturated identically, c_par^2 tending to 2 in the deep regime for every kernel. No published entropic or holographic derivation produces the family beyond n = 1. Finally the d-dimensional generalisation mu_n^(d)(Y) = 1 - (1 + Y^(d-2))^(-n) exists and is forced, but the mode count enters the deep limit as a multiplicative prefactor and cannot change a power, so it is dimensionally inert and no dimension fixes it; four candidate counts remain degenerate. WE CONCLUDE THAT THE EXPONENT IS AN EMPIRICAL QUANTITY, and that after these closures this is the result rather than a placeholder for one. Sharpening it means beating the relation's intrinsic scatter, which needs better mass-to-light ratios and distances rather than better theory. The clock no-go is written at constant U and the tau-dependent case is not computed. The photocount identification is an identification: nothing here shows the vacuum response IS such a count. Twelve criteria is not exhaustive. This paper concerns one coefficient and is not a verdict on any construction. AI-assisted research programme; not peer reviewed.
