Scale Invariance Fixes an Exponent – in Four Different Ways: Separating Identity, Dimension Matrix, Measurement and Fixed Point; the Condition That Exactly One Pi Group Exist; and the Anomalous Dimension as a Counterexample.
Paper 01 extracted n=2 uniquely from the requirement that the shell-mass growth rate be independent of the radius. The same story – DEMAND THAT THERE BE NO SCALE, AND AN EXPONENT IS FIXED – appears to recur in Kolmogorov’s 1941 -5/3 law, in the b~1 of the Gutenberg-Richter law, in Buckingham’s Pi theorem, and in the critical exponents of the renormalisation group. This paper audits that RHYME and shows that it is NOT ONE MECHANISM BUT FOUR DISTINCT ROOTS.
ROOT A (IDENTITY): Paper 01. What forces the answer is the lemma that r^(2-n) is constant ONLY IF its exponent vanishes. No dimension matrix appears anywhere: what is demanded is INDEPENDENCE OF A VARIABLE, not covariance under a change of units.
ROOT B (DIMENSION MATRIX): K41’s E(k) = C eps^(2/3) k^(-5/3), the Kolmogorov length (Paper 03), the Planck reduction (Paper 116). Here the exponent follows from the invertibility of a dimension matrix. THIS PAPER ADDS A THIRD INSTANCE OF THE SERIES’ OWN THEOREM: Paper 39 showed that (hbar, G, c) has |det| = 2, hence half-integers and square roots; Paper 03 that (nu, eps) has |det| = 4, hence fourth roots. FOR (eps, k) ONE FINDS |det| = 3 – WHICH IS WHY THE DENOMINATORS OF 2/3 AND 5/3 ARE 3 (Cramer’s rule). AND THE EXACT CONDITION UNDER WHICH ROOT B OPERATES IS WRITTEN: only when the number of independent dimensionless groups is EXACTLY ONE. K41 (E, eps, k), the Kolmogorov length (eta, nu, eps) and the Planck reduction (X, hbar, G, c) all have n – rank = 1; ADDING VISCOSITY ALONE RAISES IT TO 2 AND THE EXPONENT IS NO LONGER FIXED – only E eps^(-2/3) k^(5/3) = f(k eta) survives. HENCE THE PHYSICS RESIDES ENTIRELY IN THE CLOSURE HYPOTHESIS, NOT IN THE ALGEBRA. Kolmogorov’s contribution was the one-line hypothesis; the exponent is what linear algebra returns.
ROOT C (MEASUREMENT): b ~ 1 is A FIT TO CATALOGUES, not a derivation. It varies systematically with region, period, depth and stress state, and no theorem forces it. Gutenberg-Richter must therefore NOT be placed in the same column as n=2 and -5/3: there scale invariance is a CONCLUSION, here it is a DESCRIPTION. (No position is taken on self-organised criticality, and nothing whatever is said about earthquake prediction.)
ROOT D (FIXED POINT): critical exponents are eigenvalues of the linearised renormalisation flow, and THEY ARE IN GENERAL NOT THE VALUES DIMENSIONAL ANALYSIS RETURNS. In three dimensions the Ising correlation function goes as 1/r^(d-2+eta) with eta ~ 0.036, whereas naive dimension counting gives eta = 0. THE NAIVE DIMENSIONAL EXPONENT d-2 = 1 IS WRONG, AND THE GAP HAS A NAME – THE ANOMALOUS DIMENSION. THIS IS THE COUNTEREXAMPLE THAT FORBIDS THE RHYME: were Root B universal, no anomalous dimension could exist. Its very existence is a proof that Roots B and D are different mechanisms.
AND THE COUNTEREXAMPLE LIVES INSIDE KOLMOGOROV’S OWN SUBJECT. The structure-function exponents satisfy zeta_3 = 1 EXACTLY – the 4/5 law S_3(r) = -(4/5) eps r, a theorem derived FROM THE NAVIER-STOKES EQUATIONS and not from dimensional analysis – while zeta_p < p/3 for p > 3 (intermittency). K41 IS EXACT AT PRECISELY ONE ORDER AND DEPARTS ELSEWHERE. Roots B and D stand side by side within one person’s work, in one subject, in one year. This is exactly where Paper 03 stopped, writing that it would “not enter intermittency corrections”; this paper takes up that line not as a question of the value of the correction but of WHICH ROOT THE EXISTENCE OF A CORRECTION BELONGS TO.
SELF-AUDIT: rho(r) = r^(-n) is NOT dimensionally homogeneous unless a reference radius is chosen, and the series has silently set it to 1. The theorem survives – differentiating dm/dr = 4 pi r0^n r^(2-n) gives 4 pi r0^n (2-n) r^(1-n), which vanishes identically only for n=2, INDEPENDENTLY OF r0 – but the address is written: Paper 01 is not talking about a system of units, and that is precisely why it is Root A. AND THE SERIES HOLDS NO ROOT-D RESULT AT ALL: Paper 01 is Root A, Papers 03, 39 and 116 are Root B, and Roots C and D are empty. Root D is precisely where physics exceeds dimensional analysis – on the far side of the wall Paper 116 surveyed there runs one known passage, and it is the anomalous dimension. THIS PAPER DOES NOT WALK THROUGH IT: no renormalisation flow, no fixed point and no running coupling is defined on this series’ quantities. Recording on a map that a passage exists is not the same as walking it.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). No new theorem is proved and no prediction is made; every ingredient (K41, the 4/5 law, intermittency, the Pi theorem, Gutenberg-Richter, the renormalisation group) is a known standard result. THE FOUR ROOTS ARE NOT MERGED INTO ONE PRINCIPLE – that is the point. n=2, -5/3, b~1 and nu~0.630 differ in mechanism AND in value; no numerology (94, 101). The 4 of the upper critical dimension of phi^4 (from power counting of the coupling) is NOT identified with the 4s of Paper 114 (Hopf, division algebras, self-duality, exotic R^4): same value, different address (17, 86). The anomalous dimension is not imported into this series’ quantities. Neither global regularity for three-dimensional Navier-Stokes (an unsolved Millennium Prize Problem) nor the rigorous construction of critical phenomena is touched. “A resonance is a signature, not a proof” (100). Layer-3 lenses are quarantined with 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). Scale Invariance Fixes an Exponent — in Four Different Ways — Separating Identity, Dimension Matrix, Measurement and Fixed Point; the Condition That Exactly One $\Pi$ Group Exist; and the Anomalous Dimension as a Counterexample — [Paper 117]. Zenodo.
DOI: 10.5281/zenodo.22005515
ORCID: 0009-0006-8587-3450