Dimensional Analysis Fails in Two Ways: beta (Dimensional Transmutation) and gamma (Anomalous Dimension) Are Independent.
Paper 116 surveyed the wall, writing that the exponent formula is an identity INSIDE dimensional analysis and gives no independent route to the value of N. Paper 117 recorded that beyond that wall runs A KNOWN PASSAGE (the anomalous dimension) – “written on the map, not walked”. Paper 119 determined the type of the “time” standing in it, and wrote “this paper has not walked it either”. THREE PAPERS RUNNING, HALTED BEFORE THE SAME PLACE. This paper does not walk it either – instead it GOES AS FAR AS THE FORK AND READS THE SIGNS. THE RESULT: THERE IS NOT ONE CORRIDOR BUT TWO.
CORRIDOR ONE (the beta-root, DIMENSIONAL TRANSMUTATION). Classical massless Yang-Mills is scale invariant: its action contains NO parameter of mass dimension at all, so DIMENSIONAL ANALYSIS PREDICTS THAT NO SCALE CAN EXIST. Yet integrating the one-loop RGE mu dg/dmu = -b0 g^3/(16 pi^2) gives 1/g^2 = (b0/8pi^2) ln(mu/Lambda), hence Lambda = mu exp(-8 pi^2/(b0 g^2)). We confirm mu dLambda/dmu = 0 symbolically, and verify numerically that with alpha_s(M_Z) = 0.1180 and n_f = 5, running mu over 91.19, 30, 10 and 5 GeV leaves Lambda = 87.827 MeV UNCHANGED TO EVERY DIGIT. THE PREDICTION IS FALSIFIED – A QUANTITY THAT SHOULD NOT EXIST COMES INTO BEING. The escape route is the ESSENTIAL SINGULARITY exp(-8pi^2/(b0 g^2)), all of whose derivatives vanish as g -> 0, so it is invisible at EVERY ORDER of perturbation theory (Paper 05).
CORRIDOR TWO (the gamma-root, ANOMALOUS DIMENSION). Scaling dimensions depart from classical counting: Delta = Delta_classical + gamma. At a critical point G(r) ~ r^-(d-2+eta), and while naive dimension counting gives eta = 0, three-dimensional Ising has eta ~ 0.0363, so the true exponent is 1.0363. AGAIN THE PREDICTION IS FALSIFIED – BUT DIFFERENTLY. In corridor one, “the quantity does not exist” was overturned; in corridor two, the quantity exists and THE VALUE OF ITS EXPONENT is overturned.
THIRD, THE DECISIVE EVIDENCE THAT THE TWO ARE INDEPENDENT. At ANY fixed point beta = 0 BY DEFINITION (a fixed point being where the coupling stops running). Were beta and gamma one mechanism, gamma would have to vanish wherever beta does. YET: the Gaussian fixed point (d>4) has eta = 0; Wilson-Fisher (3D Ising) has eta ~ 0.036; two-dimensional Ising (Onsager, exact) has eta = 1/4. BETA = 0 AT ALL THREE, AND YET ETA TAKES THREE DIFFERENT VALUES. THEREFORE GAMMA IS NOT DETERMINED BY BETA. Root D is not one but two.
FOURTH, THERE IS NONETHELESS A COMMON ANCESTOR: both descend from the fact that REGULARISATION REQUIRES A SCALE – classical scale invariance is anomalous, and the trace of the energy-momentum tensor fails to vanish (T^mu_mu = (beta/2g) F^2 + …). BUT A COMMON ANCESTOR IS NOT A COMMON ROOT (Papers 86, 100): beta governs the running of the COUPLING (a scale is born), gamma the running of the NORMALISATION (an exponent shifts), and the fixed-point argument proves their logical independence. IT IS PRECISELY ON FINDING A COMMON ORIGIN THAT ONE WANTS TO MERGE; STOPPING THERE IS THE DISCIPLINE OF THIS SERIES (110, 119).
FIFTH, THE SELF-AUDIT: THE CENSUS LINE OF PAPER 117 IS CORRECTED. Paper 117 wrote that the series lives in Roots A and B alone and that Roots C and D are empty. THAT IS RIGHT ABOUT A RESULT BUT INACCURATE ABOUT CONTACT. PAPER 05 (Yang-Mills) WAS ALREADY TREATING DIMENSIONAL TRANSMUTATION HEAD-ON: it verified the mu-independence of Lambda numerically, showed that Lambda is an INVARIANT ASSEMBLED OUT OF TWO CONVENTIONS (mu and g(mu)), and had even RUN A CENSUS OF ITS OWN – a scale gets fixed in at least three ways: (a) dimensional analysis, (b) a choice of units, (c) DIMENSIONAL TRANSMUTATION, distinct roots not to be merged. THAT (c) IS THE BETA-ROOT OF THIS PAPER. THE SERIES WAS ALREADY STANDING IN THE BETA-CORRIDOR, INSIDE PAPER 05, long before Paper 117 wrote the cell as empty. WHAT WENT UNCOUNTED WAS NOT SOMEONE ELSE’S PAPER BUT MY OWN, FIVE PAPERS BACK. The gamma side, by contrast, is still ENTIRELY EMPTY: no paper of the series treats anomalous dimensions, and Paper 117’s counterexample (the eta of 3D Ising) is borrowed from another field.
SIXTH, WHAT FILLING THE GAMMA CELL WOULD REQUIRE: (1) a space of couplings in which the series’ quantities sit as coordinates; (2) a flow on it under coarse-graining; (3) a fixed point of that flow; (4) a linearisation whose eigenvalues are the exponents. THE SERIES HAS NONE OF THESE FOUR. 4pi is a solid angle (02, 13), n=2 is the exponent of a density profile (01), the Planck quantities are monomials in three constants (116); NO FLOW HAS EVER BEEN DEFINED ON ANY OF THEM. Defining some flow by force is technically possible, but BEING ABLE TO DEFINE SOMETHING IS NOT THE SAME AS ITS MEANING ANYTHING (Eddington’s caution, 101): the number emerging from a meaningless flow is the exponent of nothing. WRITING AN EMPTY CELL AS EMPTY IS THE MORE ACCURATE MAP.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts at Layer 2 (dimensional transmutation, the RG-invariance of Lambda, anomalous dimensions and critical exponents, Wilson-Fisher, Onsager’s exact solution, the trace anomaly). NO NEW THEOREM IS PROVED and no prediction is made. THIS PAPER DOES NOT WALK THE CORRIDOR: not one flow, fixed point, critical exponent or anomalous dimension is computed for the series’ own quantities. beta and gamma are NOT merged (fixed points are the counterexample). Lambda_QCD is NOT confused with a measured quantity – it is scheme-, n_f- and loop-order-dependent bookkeeping, whereas what is measurable is the mass gap (Paper 05). NOTHING IS CLAIMED ABOUT THE EXISTENCE OF YANG-MILLS AND THE MASS GAP, an unsolved Millennium Prize Problem, nor about the rigorous construction of critical phenomena. No meaning is read into the values 0.036 or 1/4 (94, 101) – only the fact that they do not vanish where beta does. PAPER 117 IS NOT RETRACTED: one line of its roll-call is corrected, and its claims stand. 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). Dimensional Analysis Fails in Two Ways — $\beta$ (Dimensional Transmutation) and $\gamma$ (Anomalous Dimension) Are Independent, and This Series Was Already Standing in One of the Corridors — [Paper 120]. Zenodo.
DOI: 10.5281/zenodo.22036433
ORCID: 0009-0006-8587-3450