How Much Do the Four Clocks Know About the Dimension They Are In: putting the exponent formula of the Yamagishi patch to the Einstein-Yamagishi Dual-Clock Framework, the tool halts between dt and dtau.
Paper 116 gave an exponent formula writing any quantity under the Yamagishi patch as X = hbar^p G_d^q c^N, and showed that the spatial dimension d sits only in the denominator n = d-1. Papers 118 and 119 showed that the single word time covers four logical types: the coordinate dt, the functional of a curve dtau, the functional of a measure dS, and the parameter of a flow (Ricci and renormalisation-group, with dimension L squared). This paper puts the former tool to the latter census. What it obtains is not a positive result but the place where the tool halts. NO NEW PHYSICAL LAW AND NO NEW THEOREM.
First, the four clocks have three different natural units. In d dimensions the routes are (p,q,N) = (1/n, 1/n, -1-3/n) for dt and dtau, and (2/n, 2/n, -6/n) for the flow parameter (verified symbolically). At d = 3, dt and dtau give the Planck time 5.3912e-44 s, while the flow parameter gives hbar G over c cubed, the Planck area 2.6123e-70 square metres — not a second but an area. The implementation of the formula was checked by feeding it the constants themselves: hbar returns (1,0,0), c returns (0,0,1), and G_d returns (0,1,0).
Second, dS does not enter this formula. Entropy has dimension M L squared T to the minus two Theta to the minus one, containing the temperature Theta, and no combination whatever of hbar, G_d and c produces Theta. The formula therefore cannot be applied until either k_B is spent or k_B = 1 is declared (a convention; Paper 95). Writing k_B explicitly as X = hbar^p G^q c^N k_B^s forces s = 1, and the remainder must then be dimensionless, giving p = q = N = 0. So the natural unit of dS is k_B itself, not a Planck quantity but a different currency. (A recorded self-correction: in the early working of this paper it was written that dS has no scale. That was wrong. It has a scale; the scale simply lies outside hbar, G and c. Having none and being elsewhere are different.)
Third, and this is the core: the formula cannot distinguish dt from dtau. Their (p,q,N) are identical, and remain so with d left as a letter (verified symbolically). Moreover the formula itself is a bijection, with inverse lambda = N + dq + 2p, mu = p – q, tau = -N – p – 2q; it discards nothing. What is lost is lost one step earlier, where a coordinate and a functional of a curve, two objects of different logical type, are assigned the same dimension T. Paper 118’s recourse to a discriminant of logical type was therefore forced, not a matter of taste: a tool that reads dimensions alone cannot separate these two in principle. This is not an endorsement of Paper 118 but a proof of its necessity. The corpus holds two tools, and where one halts the other begins.
Fourth, this is neither of the two corridors counted in Paper 120. That paper separated two ways in which dimensional analysis breaks: the beta corridor, where a quantity absent from the input is born (dimensional transmutation), and the gamma corridor, where the exponent itself shifts (anomalous dimensions). In both, dimensional analysis returns a wrong answer. Here it returns a correct answer for dt and for dtau alike; the natural unit really is the Planck time in both cases. The trouble is that the answers coincide, not that they are wrong. So this is not a breakdown but a limit of resolution, and the only one of the three in which the tool is blameless. The beta and gamma corridors mark where dimensional analysis must not be trusted; what this paper adds is where it may be trusted and is still not enough.
Fifth, the sensitivities to dimension are 1, 1, 0, 2, being q times n in units of 1/n. The flow parameter feels the dimension exactly twice as strongly as ordinary time, and the entropy clock does not feel it at all — living outside the world of hbar, G and c, it inherits none of that world’s dependence on d. Yet the dimension L squared follows from g being dimensionless and the Ricci tensor having dimension L to the minus two, and is the same in every spatial dimension: the dimension itself is blind to d. The route by which that quantity reaches a Planck value, q = 2/n, is not. The dimension of a quantity and how it is built from the fundamental constants are different matters, and one may be blind to d while the other is not. This is the same cut Paper 116 made for c cubed, where the three was separated out as a weight of time rather than a spatial dimension.
Sixth, origin is not type. A calendar, a caesium clock, a Renko bar on a currency chart (Paper 82) are all clocks made by people, but each has dimension T and the map returns the same image as for dt. So being made by people is not a property this census can see. What the census sorts is logical type, not origin, and the two are orthogonal axes. A human-made clock is therefore not a fifth type but an instance of the first, the coordinate; the number of types remains four. The note is placed to avoid a collision of numbering, following the same discipline by which Paper 118 explicitly refused to call felt time the fourth, on the ground that the number belongs to the types on the physics side. The fourth clock means the flow parameter (Paper 119) and nothing else.
IMPORTANT (scope): an application of existing tools and a self-audit (Layer 2). Both tools are already published in this corpus (the exponent formula of Paper 116, the logical types of Papers 118 and 119), and all external physics is standard (Planck units, the dimensions of the Ricci flow, Boltzmann’s constant). NO NEW PHYSICAL LAW AND NO NEW THEOREM. What is not claimed: any new physical law (there is none) / that four logical types exhaust the list (closure of the census is not demonstrated; just as Paper 119 reopened Paper 118’s census, a fifth is not excluded) / that agreement of natural units implies physical identity (dt and dtau share a unit and remain different objects, which is the point) / the sorting three belong to nature and one to us (origin is not a property this census can see) / the legitimacy of calling the fourth clock a time (reading the Planck area as a time requires dividing by c squared, and that step spends a convention). What remains: a tool has a reach, and writing down where it does not reach strengthens the corpus exactly as much as writing down where it does.
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). How Much Do the Four Clocks Know About the Dimension They Are In [Paper 122]. Zenodo.
DOI: 10.5281/zenodo.22054748
ORCID: 0009-0006-8587-3450