Exact Non-Renormalisation of the Coulomb Charge Under Refinement: Why Refinement Invariance Cannot Fix a Non-Gaussian Coupling

1. Introduction

Discrete-substrate programmes — causal sets, dynamical triangulations, asymptotic safety on a truncation, and algebraic vacuum-lattice constructions — face the same technical question before they face any physical one: given a finite, imperfectly symmetric discretisation, how is a continuum coupling extracted, and how does that extraction behave as the discretisation is refined?

In practice the coupling is defined by an estimator ${\widehat{Q}}_k$ built from the refined data at level k, and the continuum value by extrapolating ${\widehat{Q}}_k$ in the refinement parameter. Every such estimator carries a truncation bias. In lattice QCD the bias is controlled because the theory is known and the extrapolation is calibrated in $a^2$; in an emergent programme there is nothing to calibrate against, so a bias of order $(q-1)$, where q is the mesh ratio, is indistinguishable from genuine running. This matters quantitatively. For a geometric sequence $q_k=s^{1/k}$ one has $q_k-1\simeq \ln s/k$, so a first-order estimator bias falls off only as $1/k$. Refinement studies rarely go beyond $k=4$–8, where $1/k\simeq 0.12$–$0.25$. A claimed derivation of a coupling at the ${10}^{-4}$ level from such data is therefore dominated by the estimator unless the estimator bias is known in closed form.

This paper supplies that closed form. The Coulomb sector is the right place to look, because the static charge is defined by a flux, and a flux is the one observable whose continuum value is fixed by a conservation law rather than by dynamics. The result is stronger than a small-bias statement: the estimator bias vanishes identically for a unique, fully specified choice; that choice is unique among all meshes and all power laws; and the exactness so obtained forces a vanishing anomalous dimension. The last point delimits what such programmes can and cannot deliver, and we state it as a theorem with its hypotheses exposed rather than as a slogan.

Throughout, we distinguish four objects that are frequently conflated: the discrete charge, which by the divergence theorem is exact and needs no limit; the estimator, which is a numerical prescription for comparing that charge with $1/r$; the refinement flow, which changes the mesh and the estimator at fixed physics; and the renormalisation-group flow, which changes the observation scale and integrates out degrees of freedom. Bias lives entirely in the second object, and the confusion between the third and the fourth is what makes a discretisation artifact look like a physical running.

2. The Exactness Criterion

Place a unit charge at the origin of ${\mathbb{R}}^d$ and decompose space into concentric shells $[r_n,r_{n+1}]$ with $0<r_0<r_1<\dots$ and $r_n\to \infty$. Nothing is assumed about the ratios $r_{n+1}/r_n$: they may be constant (geometric mesh), slowly varying, or random. Let $V_n$ denote the potential on shell n, defined by the discrete Gauss law with the flux evaluated on a sphere of radius ${\rho }_n\in (r_n,r_{n+1})$:

$$V_n-V_{n+1}={\displaystyle \frac{r_{n+1}-r_n}{{\Omega }_d{\rho }_n^{d-1}}},{\Omega }_d={\displaystyle \frac{2{\pi }^{d/2}}{\Gamma (d/2)}}.$$

(1)

Theorem 1

(Exactness criterion). For every $n<N$ and every increasing sequence $\left\{r_n\right\}$,

$$V_n-V_N={\displaystyle \frac{1}{{\Omega }_d}}\left(r_n^{2-d}-r_N^{2-d}\right)$$

(2)

holds if and only if

$${\rho }_n^{d-1}={\displaystyle \frac{r_{n+1}-r_n}{r_n^{2-d}-r_{n+1}^{2-d}}}\mathrm{for}\mathrm{every}n.$$

(3)

Proof. 

Equation (2) is a telescoping sum of the increments (1). Comparing term by term, (2) holds for all n precisely when $(r_{n+1}-r_n)/{\rho }_n^{d-1}=r_n^{2-d}-r_{n+1}^{2-d}$ for all n, which is (3). Each term is positive and the increments are independent, so the condition is necessary shell by shell, not merely sufficient.    □

In $d=3$, and only in $d=3$, Eq. (3) reduces to

$${\rho }_n^2=r_n{r}_{n+1},$$

(4)

i.e. the flux is evaluated on the sphere whose area $4\pi {\rho }_n^2$ is the geometric mean of the areas of the two bounding spheres. This area interpretation is a three-dimensional accident: the area of a sphere is $4\pi r^2$ and $r_n{r}_{n+1}$ is the geometric mean of $r_n^2$ and $r_{n+1}^2$. In $d=4$, Eq. (3) reads

$${\rho }_n^3={\displaystyle \frac{r_n^2{r}_{n+1}^2}{r_n+r_{n+1}}},$$

(5)

which is not the geometric mean of areas, and in general dimension the only correct statement is Eq. (3). We emphasise this because the area phrasing is memorable and therefore easy to over-extend; the general statement is the radius condition, not the area one.

Three further features deserve emphasis. First, Eq. (3) contains no group-theoretic input: it is a statement about a measure, and it holds for any increasing sequence whatsoever. Second, Eq. (2) is an identity in potential differences; an additive constant, which a lattice Green’s function generically carries and which no field can detect, drops out and is not part of the claim. Third, the criterion is not an approximation improved by refinement: it is exact at finite q and finite N. Refinement is needed to reach the continuum, not to make the Coulomb law true.

The criterion is sharp: it fails by an amount proportional to the deviation from (3). Writing ${\rho }_n=\lambda \sqrt{r_n{r}_{n+1}}$ in $d=3$ with constant $\lambda$, the recovered coefficient is exactly ${\lambda }^{-2}$; the left-endpoint choice ${\rho }_n=r_n$ returns exactly q; the arithmetic mean returns exactly $4q/{(1+q)}^2$; the right endpoint returns $1/q$; and random per-shell perturbations of relative size $\epsilon$ return a coefficient deviating from unity by $[1.1,1.7]\epsilon$ for every seed tested, with no cancellation. Table 1 and Figure 1(a) collect these values; the geometric-mean rule is exact to $5\times {10}^{-13}$ over $q\in [1.0001,5]$ and over meshes of up to $1.4\times {10}^5$ shells.

Two independent checks fix the scope of the statement. On a ${96}^3$ cubic lattice with a unit source at the origin, the discrete flux through the boundary of $\left\{\right|x|\le n\}$, with the finite-box neutralising background $\left|S\right|/L^3$ restored explicitly, returns the enclosed charge $1.000000000000$ for $n=2,3,5,8,12$: in the discrete theory the charge is exact by the divergence theorem, and no refinement is involved. The continuum-style estimator built from the same data, $4\pi R^2[\overline{V}\left(R\right)-\overline{V}(R+1)]$, returns $0.900\pm 0.021$ over $R=8$–15 — a $10\%$ bias, oscillating with the shell structure, that belongs to the estimator and to the cubic geometry rather than to the theory. Second, Eq. (4) holds for arbitrary increasing sequences, not only geometric ones: for $r_n=n^{3/2}$, $n^{3/5}$, $e^{\sqrt{n}}$, $n^{1+0.3\sin n}$ and a random monotone sequence the residual is at most $5.7\times {10}^{-13}$ in every case.

3. Exactness Is Marginality

An estimator assembled from the mesh alone cannot know the exponent of the field it is measuring; if it did, it would already contain the answer. The following lemma states precisely what such an estimator returns, and under which hypotheses.

Lemma 1

(Shell scaling). Assume (i) ${\rho }_n$ obeys Eq. (3); (ii) the mesh ratio $q=r_{n+1}/r_n$ is the same on every shell; (iii) the source field is the Coulomb field of that dimension, $V\propto r^{2-d}$. Then for a trial potential $V\propto r^{-p}$ the amplitude returned on shell n is

$$A_n={\displaystyle \frac{1}{{\Omega }_d}}{\displaystyle \frac{r_n^{2-d}-r_{n+1}^{2-d}}{r_n^{-p}-r_{n+1}^{-p}}}={\displaystyle \frac{r_n^{p-d+2}}{{\Omega }_d}}{\displaystyle \frac{q-1}{q(1-q^{-p})}},$$

(6)

so that $A_n\propto r_n^{p-d+2}$ exactly and

$${\displaystyle \frac{A_{n+1}}{A_n}}=q^{p-d+2},$$

(7)

independently of n. In $d=3$ this exponent is $p-1$ and Eq. (6) reduces to $A_n\propto r_n^{p-1}$; marginality $p=d-2$ gives $q^0=1$ in every dimension.

Proof. 

Write $r_{n+1}=q{r}_n$ and factor $r_n^{2-d}$ out of the numerator and $r_n^{-p}$ out of the denominator. The two powers combine to $r_n^{p-d+2}$, and the remaining q-dependent factor $(q-1)/\left(q(1-q^{-p})\right)$ is independent of n under hypothesis (ii). Dividing the expression at $n+1$ by the expression at n gives ${(r_{n+1}/r_n)}^{p-d+2}=q^{p-d+2}$.    □

Each hypothesis is load-bearing, and it is worth recording how the exponents change when one is dropped. (i) If ${\rho }_n$ is not the radius of Eq. (3), the potential increments in (6) are themselves wrong and $A_n$ acquires an extra factor ${\lambda }^{-2}$ (or $1/q$, or $4q/{(1+q)}^2$) from Table 1; the exponent is unaffected but the normalisation is not, and it is the normalisation that is being measured. (ii) If q varies from shell to shell, (6) becomes $A_n=r_n^{p-d+2}{\Omega }_d^{-1}(q_n-1)/\left(q_n(1-q_n^{-p})\right)$ and (7) acquires the ratio of that factor at $n+1$ and n; Eq. (7) is then no longer exact. For $r_n=n^{1.3}$ and $p=2$ the measured ratio departs from $q_{\mathrm{eff}}^{p-1}$ by up to $12\%$. (iii) In $d\ne 3$ the exponent is $p-d+2$ and not $p-1$; marginality, $p=d-2$, always gives exponent zero, which is the statement that survives in every dimension.

The measured charge is scale-free — independent of the shell on which it is measured — if and only if $p=d-2$. Equation (7) was confirmed directly for $d=3,4,5$ to a relative accuracy of ${10}^{-12}$; for $d=3$ it was also checked against Figure 1(b) for $p\in \{0.5,1,1.5,2\}$ and $q\in \{1.2,\sqrt{3},2\}$, and the fitted slopes in Table 2 agree with the prediction to machine precision.

Two readings of Eq. (7) should be separated. Read as a diagnostic, it says that the “running” produced by a geometry-only estimator on a non-marginal power is not a small correction but an $O\left(1\right)$ effect growing as a power of the shell radius: at $q=\sqrt{3}$ a $1/r^2$ trial field appears to run by a factor $1.73$ per shell. Read as a definition, it says that a charge which can be measured without knowing the dynamics is a charge with zero anomalous dimension. That equivalence is the content of Section 5.

3.1. Refinement Flow Versus Renormalisation-Group Flow

The two flows that appear in this paper are frequently conflated, and the confusion is the source of most of the ambiguity that Section 4 resolves.

Refinement flow. A change of the pair $(\mathrm{mesh},\mathrm{estimator})$ at fixed physics. No degree of freedom is integrated out; no scale is compared with a coupling; the theory is untouched and only its description changes. Its generator is the map $q_k\mapsto q_{k+1}$.

Renormalisation-group flow. A change of the observation scale $\mu$, in the course of which degrees of freedom between $\mu$ and $\mu +d\mu$ are integrated out. Its generator is the Callan–Symanzik operator, and its output is a set of functions $\alpha \left(\mu \right)$ whose variation is measured by the anomalous dimension.

The two are related by a single statement: an exact estimator makes the refinement flow trivial. By Theorem 1 the discrete charge is independent of k once the radius condition holds, so

$${\beta }_{\mathrm{refine}}\equiv {\displaystyle \frac{\partial {\alpha }_k}{\partial \ln q_k}}{|}_{\mathrm{estimator}}=0,{\beta }_{\mathrm{phys}}\equiv {\displaystyle \frac{\partial \alpha \left(\mu \right)}{\partial \ln \mu }}\ne 0,$$

(8)

and these are not two determinations of the same quantity. The first is a property of the discretisation; the second is a property of the theory. Whenever a finite-refinement study reports a non-zero $\beta$ from a static flux without an estimator-error model, it has measured ${\beta }_{\mathrm{refine}}$ and called it ${\beta }_{\mathrm{phys}}$. The diagnostic of Section 7 is precisely the operation of setting ${\beta }_{\mathrm{refine}}$ to zero by construction and asking what remains: whatever survives is ${\beta }_{\mathrm{phys}}$, and it is a statement about vacuum polarisation.

A corollary of the distinction is that the two flows do not commute in general. Refining first and then measuring, versus measuring first and then refining, differ by the estimator bias, which is exactly the closed form of Section 4 and vanishes only under the radius condition. The commutator is therefore a computable quantity, not a conceptual difficulty.

4. Consequence I: A Vanishing Refinement $\beta$-function

With the geometric-mean estimator the measured charge at level k equals the continuum charge exactly, for every k. The refinement $\beta$-function of the Coulomb sector is therefore not small; it is zero, as an identity rather than as a limit:

$$\beta \left(\alpha \right)\equiv {\left.{\displaystyle \frac{d{\alpha }_k}{d\ln (1/q_k)}}\right|}_{\mathrm{estimator}}=0\mathrm{for}\mathrm{all}k.$$

(9)

The corollary is operational: a reported running of the static charge must be compared with the closed-form bias of the estimator that produced it. For the geometric family $q_k=2^{1/k}$,

$${\displaystyle \frac{{\alpha }_{\mathrm{eff}}\left(k\right)}{\alpha }}=\left\{\begin{array}{cc}{q}_k,\hfill & \mathrm{left}\mathrm{endpoint},\hfill \\ 4q_k/{(1+q_k)}^2,\hfill & \mathrm{arithmetic},\hfill \\ 1,\hfill & \mathrm{geometric},\hfill \end{array}\right.$$

(10)

with $q_k=1+\ln 2/k+O\left(k^{-2}\right)$ and $4q_k/{(1+q_k)}^2=1-{\left(\ln 2\right)}^2/\left(4k^2\right)+O\left(k^{-4}\right)$. The three cases differ by orders of magnitude in the quantity that a fit would report (Table 3, Figure 2). Fitting $\alpha (1+c/k)$ to the left-endpoint data returns ${\alpha }_{\mathrm{inferred}}/\alpha =0.896$ at $k_{\max }=4$, $0.924$ at $k_{\max }=6$ and $0.939$ at $k_{\max }=8$: a $10\%$ bias at the refinement depth typical of published studies. The bias is not a small correction to be absorbed into an error bar; it is the same order as the effects such studies claim to resolve.

It is worth noting what the exactness of Eq. (9) does not require. It does not require the field to be free, it does not require a particular lattice geometry beyond the radial shell decomposition itself, and it does not require the mesh to be geometric: only the estimator must match the shell measure. This is what makes the criterion useful outside the framework that motivated it: it is a calibration standard for any claim that a coupling emerges from a discretisation, and a null test that costs one transform.

5. Consequence II: A Scoped No-Go

We now state what the preceding results forbid, under hypotheses that we make explicit.

Theorem 2

(Scoped no-go). Suppose (H1) the charge is measured as astatic fluxthrough a surface enclosing it; (H2) the estimator is a function of the shell geometry alone, with no fitting to dynamical data; and (H3) the field is Coulomb-like in the sense of Eq. (7) at the scale of measurement. Then (i) the measured flux charge has zero anomalous dimension; (ii) the flux direction is exactly marginal, so the refinement-invariance condition is satisfied for every value of that charge, i.e. the fixed set is a line rather than an isolated point; and (iii) consequently refinement invariance does not select the value of the coupling.

Proof. 

Under (H2) and (H3), exactness at every shell is equivalent, by Theorem 1 and Eq. (7), to $p=d-2$: the recovered amplitude is independent of $r_n$, so $d\ln A/d\ln r=0$ and the anomalous dimension vanishes, which is (i). A direction with zero anomalous dimension is marginal: the condition that fixes a fixed point is satisfied identically along it, so the fixed set is a line, which is (ii). A condition satisfied for every value of a parameter does not determine that parameter, which is (iii).    □

What (i)–(iii) do not assert. They concern the flux observable and the sector it probes, not the whole theory, and in particular they do not assert that the theory is free. The distinction is not pedantic: a conserved current has zero anomalous dimension by the Ward identity in an interacting theory, the electromagnetic current of QCD being the standard example, and the energy–momentum tensor of an interacting conformal field theory likewise. Zero anomalous dimension of one conserved observable is therefore compatible with arbitrarily strong interactions elsewhere. Accordingly we do not claim that the refinement flow drives the theory to a Gaussian fixed point; we claim that the refinement flow carries no information about the value of the flux charge, because along the marginal direction every value satisfies the same condition. Statements about the rest of the theory would require the full set of anomalous dimensions, not one of them.

The hypotheses matter, and each can fail in practice. (H1) excludes, for example, couplings extracted from a running mass, from a three-point function at non-zero momentum, or from an equation of state; for those observables a static flux argument says nothing. (H2) excludes estimators that fit a dynamical ansatz, where the fit parameters absorb part of the truncation error and the residual bias is no longer the closed form of Section 4. (H3) excludes screened or confined regimes, where the field is not Coulomb-like and the exponent p departs from $d-2$ for physical reasons; there the running is real and Eq. (7) measures it rather than manufacturing it.

Within these hypotheses the content is a dichotomy. Either a derivation uses an estimator whose bias is controlled and exactly zero — and then it can only reproduce a Gaussian charge, whose numerical value is an input rather than an output; or it uses an estimator with non-zero bias — and then the reported number is the bias, of known closed form. There is no third possibility, because every charge measurement in this sector is a flux measurement. Accordingly, the correct summary is not that an emergent coupling can never be a fixed point, but that refinement invariance alone cannot determine the value of a non-Gaussian fixed-point coupling. A specific number requires either a computed non-zero anomalous dimension — vacuum polarisation from charged degrees of freedom — or an independent quantisation condition; in the second case the condition, not the geometry, does the work.

6. Non-Abelian Charges and Non-Spherical Lattices

6.1. Cartan Sector of a non-Abelian Group

The extension to a non-Abelian group is not an analogy; it is a theorem with conditions, and it is worth stating both.

Lemma 2

(Cartan flux conservation). Let G be a compact Lie group with maximal torus T of rank r, generators $H_a$ ($a=1,\dots ,r$), and let $E_i^a$ be the lattice electric field on link i in the Cartan direction. For any region S and any gauge-field configuration,

$$\sum _{\mathrm{links}\mathrm{crossing}\partial S}{E}_i^a=Q_{\mathrm{enc}}^a,$$

(11)

where $Q_{\mathrm{enc}}^a$ is the Cartan charge enclosed by S, exactly and configuration by configuration.

Proof. 

The Gauss constraint reads ${\left(D_i{E}_i\right)}^a={\rho }^a$ with $D_i{E}_i={\partial }_i{E}_i+f^{abc}{A}_i^b{E}_i^c$. For a in the Cartan subalgebra, $f^{ab}{}_c=0$ whenever b is also Cartan, because T is Abelian. Splitting b into Cartan and root directions, the root contributions are proportional to $E_i^b$ off the Cartan directions; those components are zero for a configuration whose electric field lies in T. Hence $D_i{E}_i={\partial }_i{E}_i$ on the Cartan components, and summing ${\partial }_i{E}_i^a={\rho }^a$ over S telescopes to (11) by the lattice divergence theorem.    □

Two remarks fix the scope. First, the lemma is a statement about the Cartan projection: it constrains $E^a$ for $a\in T$, not the full non-Abelian field, and it holds for every configuration in which the electric field has no root components at the measurement surface. Second, the result is exact on any lattice, including a finite one: the only corrections are the boundary flux and, for fermions, the determinant weight, both of which are computable and neither of which affects the per-shell identity inside the region.

Two levels must be kept apart, and conflating them is the most likely misuse of this section.

Level 1 — a constraint, proved. Lemma 2 is a statement about the Gauss constraint alone. It guarantees that the Cartan flux is conserved exactly; it entails nothing about any coupling constant, any anomalous dimension, or any fixed point.

Level 2 — a coupling, conditional. To pass from a conserved flux to a coupling one must specify which operator is being measured, in which scheme, on which configurations, and in which phase. Only then does the criterion of Section 2 apply to that Cartan component, and only then do the marginality statements of Lemma 1 and Theorem 2 concern it. Three conditions must therefore be checked rather than assumed:

(C1)

Observable. The measured quantity is the static Cartan flux of Eq. (11), not a gauge-dependent functional and not an off-diagonal correlator.

(C2)

Sector. No dynamical charged matter with nonzero Cartan charge is present inside the measurement volume; otherwise the enclosed charge is screened and $Q_{\mathrm{enc}}^a$ is not the charge of interest.

(C3)

Phase. The measurement scale lies inside the Coulomb phase of the Cartan sector, so that the static potential is $1/r$ rather than linear. In a confined phase the criterion still holds shell by shell — the flux is still conserved — but its physical reading changes, since the exponent p is no longer $d-2$.

Under (C1)–(C3) plus a stated scheme, Theorem 1 and Lemma 1 hold componentwise for that Cartan component, with ${\rho }_n$ given by Eq. (3) unchanged: the radius condition is a statement about the measure and contains no group-theoretic input, which is exactly why it transfers without modification. The Abelian projection used in lattice studies of confinement supplies such Cartan fluxes, so the criterion is directly applicable there — at level 2, for the projected charge, and not as a statement about the full non-Abelian theory. In particular, nothing here bounds the anomalous dimensions of non-Cartan operators, and the projected charge is not the same object as a gauge-invariant non-Abelian coupling.

What does not transfer is scale-freedom. Off-diagonal modes and dynamical matter screen the Cartan charge at the scale of their masses, so the measured exponent departs from $d-2$; Eq. (7) then predicts the drift of the measured charge in terms of that exponent, and here the drift is physical, because the field is no longer Coulomb-like. The criterion therefore separates two situations that a naive fit cannot distinguish: an exponent $p\ne d-2$ produced by screening, which is physics, and a normalisation drift at $p=d-2$ produced by the estimator, which is an artifact. The discriminator is the null test of Section 7: change the estimator, not the physics.

Finally, centre vortices and monopoles change the long-distance behaviour of the Cartan field without changing the conservation law at the measurement surface. The criterion therefore survives their presence shell by shell, while the interpretation of a non-zero exponent in that regime is precisely the physical screening of (C3).

6.2. Lattices without Spherical Shells

On a hypercubic lattice no shell is a sphere and the radial decomposition is itself an approximation. Two procedures are available, and they answer different questions.

(i) Exact route. Use the discrete flux through the boundary of a region S. By the divergence theorem this equals the enclosed charge identically: no estimator, hence no bias. We verified this on a ${96}^3$ lattice, obtaining $1.000000000000$ for $n=2,3,5,8,12$ after restoring the neutralising background $\left|S\right|/L^3$. The quantity is exact but does not by itself define a continuum coupling, since it contains no comparison with $1/r$.

(ii) Radial projection. Project onto shells and use the radius of Eq. (3). The shell-decomposition error survives: on the ${96}^3$ cubic lattice the continuum-style estimator gave $0.900\pm 0.021$ for $R=8$–15, a $10\%$ bias that decreases only as the sphere is resolved. The practical rule is that the criterion removes the estimator bias exactly and leaves a lattice-geometry bias that must be measured rather than assumed.

For anisotropic or random lattices the same logic applies cell by cell. The flux weight $w_n={\Omega }_d^{-1}(r_n^{2-d}-r_{n+1}^{2-d})$ is the exact weight for the Coulomb field, and any weight differing from it by a relative $\epsilon$ produces a charge bias of the same relative size. One therefore does not need spheres; one needs the weights.

7. Relation to Other Approaches, and the Operational Test

Lattice QCD. Continuum extrapolation there is calibrated: the static potential is fitted as $V\left(r\right)=-C/r+\sigma r+O\left(a^2\right)$, and the estimator bias is one of the fitted terms. Equation (3) identifies the flux surface for which the $O\left(a^0\right)$ part of that bias vanishes exactly, at finite a. It is the lattice statement of Gauss’s law, not an approximation to it.

Asymptotic safety. A non-Gaussian fixed point requires a non-zero anomalous dimension. The static Coulomb sector is exactly marginal by the conservation law, and no truncation can change that. A gravitational fixed point must therefore be looked for in the interacting sector; evidence for one drawn from the $1/r$ tail of a propagator is, within the hypotheses of Theorem 2, evidence about the estimator.

Causal sets and dynamical triangulations. Both establish a continuum limit through scaling observables such as spectral dimension or volume scaling. The criterion supplies a complementary and much cheaper null test at the level of the $1/r$ tail.

7.1. Relation to Existing Discretisation Methods

Three bodies of work touch the radius condition, and it is worth being explicit about what is old and what is not.

Finite differences on graded meshes. That a symmetric (midpoint) placement of a difference operator is second-order accurate on a non-uniform mesh is classical numerical analysis, and exponential or logarithmic meshes are standard in atomic structure calculations. The radius condition is the exact version of the same statement: on a geometric mesh the symmetric point of the shell $[r_n,r_{n+1}]$ in the sense of the q-calculus of Jackson is $r_n\sqrt{q}$, and for the $1/r$ field the discretisation error at that point does not fall off at second order — it vanishes identically, at finite q. In q-calculus language, Eq. (3) says that the flux must be sampled at the q-symmetric point; this is the precise sense in which the criterion is a known structure rather than a new approximation [15].

Discrete Gauss law. Since Wilson’s formulation the Gauss constraint has been a fundamental object of lattice gauge theory [1,12], and the conservation of Cartan flux (Lemma 2) is standard lore in the Abelian-projection literature [11]. What is not usually exploited is that on a radial shell decomposition the constraint reproduces the continuum $1/r$ exactly, because homogeneous lattices have no distinguished radial measure.

Flux and potential estimators. Lattice determinations of the static potential fit $V\left(r\right)=-C/r+\sigma r+a+O\left(a^2\right)$ and absorb the estimator bias into the fitted terms [9,16], while Symanzik improvement removes $O\left(a^2\right)$ errors from the action [13,14]. The present result acts on a third axis: it removes the $O\left(a^0\right)$ estimator error from the measurement of the Coulomb coefficient, at finite a, without modifying the action. The three are complementary, and they are not substitutes: improvement of the action does not fix the choice of flux surface, and the radius condition does not reduce the $O\left(a^2\right)$ scaling of the potential at large r.

What is claimed as new. The telescoping identity behind Theorem 1 is elementary and we do not claim it. The claims of this paper are: (i) the $iff$ characterisation of exact estimators over arbitrary meshes; (ii) the identification of exactness with marginality and hence with a vanishing anomalous dimension (Lemma 1 and Theorem 2); (iii) the closed-form biases of the natural estimators and their size at realistic refinement depths (Table 3); and (iv) the scoped no-go for couplings obtained from refinement invariance. Lemma 2 is standard; its role here is as an applicability condition, not as a result.

The operational test. For a candidate emergent coupling, compute the effective charge at two refinement levels k and $k+1$ with (i) the estimator as reported and (ii) the geometric-mean estimator of Eq. (3). Three outcomes exhaust the possibilities. If ${\widehat{Q}}^{\left(ii\right)}$ is k-independent to machine precision while ${\widehat{Q}}^{\left(i\right)}$ drifts, the drift is an artifact and Eq. (10) gives its size. If both drift identically, the running is real and equals the anomalous dimension, obtainable from the slope $d\ln A/d\ln r$. If ${\widehat{Q}}^{\left(ii\right)}$ itself drifts, the field is not Coulomb-like in the required sense and the Coulomb interpretation is unavailable. The test costs one transform and one sum, and it is decisive in the first case.

8. Conclusions

The Coulomb charge is the one coupling in physics whose measurement is fixed by a conservation law rather than by a dynamical calculation, and that fact has an exact discrete form: the discrete Gauss law reproduces $1/r$ identically — at finite mesh ratio, on arbitrary refinement sequences — if and only if the flux radius obeys Eq. (3), with the area form (4) valid in three dimensions alone. The physics lies in the second result: the estimator so defined is exact for exactly one power law, the marginal one, so exactness under refinement and a vanishing anomalous dimension are not two properties but one. The scope of that identification is a single observable: it says that this charge carries no scale dependence, not that the rest of the theory is trivial. A conserved current has zero anomalous dimension in an interacting theory, and the argument here is of exactly that type.

The consequence for emergent-coupling programmes is a dichotomy rather than an obstacle to be worked around. Either the reported number is an estimator bias of known closed form — first order in $q-1$, hence about $10\%$ at the refinement depths actually used — or it is an input. A specific value of $1/\alpha$ therefore has to come from a computed anomalous dimension or from an explicit quantisation condition. The constructive reading is that the target of such programmes should be the anomalous dimension $\gamma \left(\alpha \right)=d\ln A/d\ln r$ rather than a fixed-point identity; that is a statement about vacuum polarisation, and it is where the Standard Model’s couplings actually live.

8.1. Future Directions

Three extensions follow directly and are stated here so that they can be taken up independently.

(a) Post-hoc audit of published refinement data. Every study that quotes a coupling from a sequence $q_k$ can be audited without re-running anything: Eq. (10) gives the bias of the estimator it used, as a function of the mesh ratios it reports. A table of (estimator, $q_k$, predicted bias) is enough to decide whether a reported drift is signal. We expect this to be the fastest use of the present result.

(b) Dynamical matter and non-Cartan components. Conditions (C2) and (C3) of Section 6.1 are the places where dynamics enters, and they are where the criterion must be extended rather than applied. The natural object is the measured exponent p itself, extracted from two radii; the difference $p-(d-2)$ is the anomalous dimension of the sector, and its computation for fermions in the fundamental representation is a lattice problem of the usual size. A non-Cartan (non-Abelian) charge has no conserved flux of the type (11), so the criterion is expected to fail there, and the failure is itself informative.

(c) Weights on arbitrary cell complexes. The sphere is a convenience, not a requirement: the content of Theorem 1 is that the flux weight of a cell must equal the exact potential difference across it. On a general cell complex with a radial coordinate, the criterion therefore reads $w_C={\Omega }_d^{-1}\Delta \left(r^{2-d}\right){|}_C$ cell by cell, and the natural conjecture is that on a Voronoi–Delaunay complex this weight is the harmonic measure of the cell boundary. Settling that would put the criterion on the same footing for simplicial and hypercubic lattices as it already has for radial meshes, and would remove the last spherical assumption from the argument.