The dimension ladder: from 0+0 to 1+3, and what the action constant does on each rung

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Record of a discussion with the user, 2026-09-27, 02:50–03:15. The user proposed the frame, first stated in Rivero 1998, arXiv:quant-ph/9803035 (the path-integral formula “as a consistency condition for the existence of certain Dirac measures over functional spaces”; see the fifth-postulate note, §7): quantization is a consistency condition that Newton’s rework of the Principia missed, so any supposedly well-posed wave equation without a quantization carries a logical issue; the ladder of dimensions runs from \(0+0\) through Newton and quantum mechanics (\(1+0\), “YM\(_0\)”) to \(1+3\); and the exponential \(e^{-1/(2b_0\hbar g^2)}\) plays a distinct role. This note records the resulting statements with their status. The two layers of §1 and the table of §2 are the parts meant for the joint paper; §§3–4 are readings, labelled as such.

1. Quantization as a consistency condition: two layers

Layer 1: every classical wave equation needs a universal action constant to be thermodynamically consistent. Theorem U of the unit-and-indeterminacy note uses Maxwell’s field, but its mechanism is general. A classical field whose normal-mode frequencies are unbounded (every wave equation on a continuum) has, in equilibrium at temperature \(T\), energy \(k_BT\) per mode by equipartition, hence infinite energy density. A finite equilibrium, which the second law and every thermometer presuppose, needs the high modes to freeze out; that compares \(k_BT\) with a frequency and so needs a constant with units of action. Two different fields in mutual equilibrium must share their occupation law, or a filter between them would run a perpetual motion of the second kind (Kirchhoff’s argument extended from colours to fields); so the constant is common to all fields. Einstein applied Planck’s constant to the vibrations of solids in exactly this way (Einstein 1907, metadata). Stated as a conjecture-level extension of Theorem U, to be proved in the same form: any classical field with unbounded mode frequencies that admits a finite thermal equilibrium requires a universal action constant, and mutual equilibrium makes it the same for all fields. A well-posed wave equation remains consistent as deterministic mathematics; as physics it is inconsistent once it can exchange energy with thermal matter. This is the kind of gap the programme attributes to Newton’s rework: the limits of the Principia are consistent, and the inconsistency appears when they are combined with thermodynamics and physical records.

Layer 2: the quantized theory must itself survive refinement. Once \(\hbar\) is present, the per-cell parameter of the lattice refinement is \(t=\hbar g_{\rm cl}^2a^{4-D}\) (zero-spacing note, §3):

• \(D<4\): \(t\to0\) and the refinement converges (superrenormalizable);
• \(D=4\): marginal; the theory survives only if its coupling runs to zero at short distances. Yang–Mills does (asymptotic freedom). Continuum \(\phi^4_4\) is trivial, a free field (Aizenman 1982; Fröhlich 1982; Aizenman and Duminil-Copin 2021; metadata), and QED has a Landau pole; on this reading both classical equations are effective theories with a cutoff;
• gravity: \(t=\hbar G/(c^3a^2)=(\ell_P/a)^2\) grows as \(a\to0\), the failure mode of the \(D>4\) row.

What “consistency” means in each layer. Layer 1 is a logical inconsistency: classical fields, the second law and finite energy are jointly contradictory without an action constant. Layer 2 is an existence condition on the continuum quantum theory, decided by estimates of the kind proved in the series/parallel and zero-spacing notes. Neither makes a classical PDE self-contradictory; both say it cannot be a complete physical theory by itself. Classical Yang–Mills in \(3+1\) Minkowski space exists globally (Eardley and Moncrief 1982, metadata), and classical \(\phi^4_4\) is well posed, although its continuum quantum theory is free: the large-quantum-number limit of that quantum theory is a free classical field. The implication “quantum theory exists \(\Rightarrow\) its classical limit exists” holds in the sense of limits; the converse fails as mathematics. The Euclidean formulation makes one version exact: the measure \(e^{-S/\hbar}\) is a classical statistical field theory in \(D\) dimensions with \(\hbar\) as temperature, so “quantum YM\(_4\) exists” (Osterwalder–Schrader) and “classical statistical YM\(_4\) at temperature \(\hbar\) exists in the continuum” are one problem, while the classical deterministic PDE is a different object.

2. The ladder, in the user’s \(1+n\) labelling

Write \(\lambda_D=\hbar g_{\rm cl}^2\) with \([\lambda_D]={\rm length}^{D-4}\), \(D=1+n\), and hold the classical coupling \(g_{\rm cl}\) fixed as \(\hbar\to0\).

(The subscripts of the series/parallel and zero-spacing notes count spacetime dimension \(D\); here YM\(_n\) is \(D=n+1\).)

\(0+0\). There is no time direction, so no Hamiltonian and no gap. What remains is one integral. It is the building block of every rung: the plaquette weight \(k_t(U)\) is a \(0+0\) theory, series moves (convolution) and parallel moves (products) assemble higher dimensions from it, and the elimination of one inserted Newtonian time (Proposition 2 of the refinement note) is the \(0+0\) piece of mechanics. A zero-dimensional integral such as \(\int d\phi\,e^{-(\phi^2/2+g\phi^4)/\hbar}\) already has an asymptotic, non-Borel-summable series in \(\hbar\) with ambiguities of order \(e^{-c/\hbar}\). At large \(N\) the one-plaquette unitary matrix integral, the building block of two-dimensional lattice Yang–Mills, has a third-order phase transition (Gross and Witten 1980, metadata): non-analyticity from a zero-dimensional integral in a limit that plays the role of \(a\to0\).

\(1+1\) on a circle is exactly a \(1+0\) rotor. Eq. (10) of the refinement note, \(E_R=\hbar c\lambda_2L\,C_2(R)/2\), equals \(\hbar^2C_2(R)/(2I)\) with \(I=1/(c\,g_{\rm cl}^2L)\): a quantum rigid rotor on the gauge group whose moment of inertia is classical (\(\hbar\) cancels from \(I\)). On the infinite line there is no particle; the string tension \(\sigma=\hbar c\lambda_2C_2/2\) between static charges goes like \(\hbar^2\).

The exponents interpolate. The only energy formed from \(\hbar\), \(c\) and a fixed classical coupling is \(E\sim\hbar c\,\lambda_D^{1/(4-D)}\), which goes like \(\hbar^{(5-D)/(4-D)}\): \(\hbar^{3/2}\) for \(D=2\), \(\hbar^2\) for \(D=3\), and a diverging exponent as \(D\to4\). At \(D=4\) no power is available, and the running coupling turns the divergent power into \(e^{-1/(2b_0\hbar g_{\rm cl}^2)}\), as dimensional regularization sees it (\((g^2)^{1/\varepsilon}\) with \(\varepsilon=4-D\) becomes an exponential of \(-1/g^2\)). This is dimensional analysis; whether each rung has a positive gap is the open content (\(C_3\), and the Millennium problem).

Classical Yang–Mills has no gap in any dimension: its waves are massless and carry arbitrarily small energy. On every rung the large-quantum-number limit is the limit in which the gap disappears.

Refinement runs in opposite directions in mechanics and in field theory. For \(D<4\), \(\hbar\to0\) and \(a\to0\) push the same per-plaquette parameter \(t=\hbar g_{\rm cl}^2a^{4-D}\) to zero: per plaquette, “more classical” and “finer lattice” are one direction, while the global scale where a gap lives stays quantum. In mechanics, refining the time cells makes each cell more quantum: the classical polygon action per cell shrinks like \(\tau^3\), while the fluctuation action per cell stays of order \(\hbar\).

3. A third route to a universal action: gravity with a least length

Classical gravitation with finite light speed has no mass-independent action (\(Gm^2/c\) depends on the mass). A fixed length \(\ell\) supplies one: \(c^3\ell^2/G\). So \((G,c,\ell)\) is a third route to a universal action, beside radiation thermodynamics (Theorem U) and the Stoney unit \(k_e/c\), and the most Newtonian of the three: \(G\) is Newton’s constant, \(c\) is Rømer’s (1676), and the missing ingredient is exactly the least magnitude that the scholium closing Book I, Section I refuses by citing Euclid X. The value needs \(\ell=\ell_P\approx1.6\times10^{-35}\) m; Newton’s interval of fits (\(1/89000\) inch) gives \(c^3\Lambda^2/G\sim10^{22}\) J s, off by about 56 orders of magnitude. The route is logically Newtonian; its value is outside Newton-age data.

4. The exponential ladder (a reading)

The factor \(e^{-1/(2b_0\hbar g^2)}\) has four properties worth keeping together.

• It vanishes to all orders in \(\hbar\), so no finite order of the loop expansion sees the gap.
• In four dimensions the running converts it into a power of length, \(e^{-c/g^2(a)}=(a\Lambda)^{2b_0c}\): dimensional transmutation. This is why the large-field criterion of the zero-spacing note is a power-law condition, \(2b_0c>4\), and why small instantons go like \((a\Lambda)^{11N/3}\).
• Every scale of pure YM\(_4\) then reads as a suppression with an “action”: \(\Lambda^k\propto e^{-k/(2b_0g^2)}\). The gap behaves like a configuration of action \(1/(2b_0)\) in units of \(g^{-2}\), which is \(24\pi^2/33\approx7.2\) for \(SU(3)\), against \(8\pi^2\approx79\) for an instanton; the dislocation threshold is \(k=D=4\) times it. This is a reading, not a result.
• It lies where perturbation theory is ambiguous: the series in \(g^2\) is not Borel summable, with ambiguities of order \(e^{-{\rm const}/g^2}\), powers of \(\Lambda\) (’t Hooft’s renormalons). The gap is in the class of quantities the perturbative expansion cannot decide, which is why non-perturbative estimates of the H1 type are unavoidable.

The same exponential runs through the programme: Theorem 5’s \(e^{-\pi^2/(8\lambda_3a)}\) in the series/parallel note, the instanton \(e^{-8\pi^2/g^2}\), Wien’s tail \(e^{-h\nu/k_BT}\) (where quantization enters Theorem U, through the lumps \(h\nu\)), and on Newton’s side the action as a phase \(e^{iS/\hbar}\) or, in Euclidean form, \(e^{-S/\hbar}\). In each case the exponential is where an action constant meets a threshold. The paper could organize the comparison by what each exponential suppresses and what survives refinement once the exponent is converted into a scale.

5. Consequence for STATE

No change to the queue. The table of §2 is proposed as the spine of the joint paper’s comparison section, and Layer 1 of §1 as a theorem to prove in the form of Theorem U for a general wave field (with the cross-field universality). The gravity route of §3 belongs with the dimensional note when that note is next revised.