Prime evolution revisited

1. What survives from the old notes

The psychological starting point should remain on the record. In 1998 the distribution of primes ceased, for Watkins, to look like a completed list of special integers and appeared instead as an alien configuration: globally organized, locally resistant to prediction, and somehow more like the trace of a process than an inert set. The first notes tried to protect that perception without promoting it into a grand claim. They accumulated possible mechanisms because no mathematical vocabulary yet seemed adequate.

The revised claim is narrower. Words such as flow, equilibrium and evolution need not refer to a temporal prehistory of the primes. They can refer to a parameterized family in a moduli space, a renormalization trajectory, a gradient flow for an energy functional, a deformation of zeta functions, or a Frobenius-like action on an arithmetic-geometric object. This is familiar mathematical usage. The question is whether one of these structures selects anything recognizably classical for reasons that are not circular.

Three features of the early notebook deserve promotion.

1. Generalized primes are the right comparison class. One cannot explain a configuration’s specialness while refusing to say what its alternatives are. Beurling’s construction supplies those alternatives.
2. Measures are a better state space than bare sequences. Continuous prime-counting measures can be discretized surprisingly well, while convolution and Mellin transforms expose the multiplicative structure cleanly.
3. The Julia probability distribution is exact, not merely analogous. The weights \(n^{-\beta}/\zeta(\beta)\) form a genuine Gibbs ensemble whose independent prime occupations, entropy and critical scaling can all be calculated—and, as §5 now records, most of them already have been.

[v0.2] A fourth feature is promoted in this version. The 1999 notebook (§1.7 of the 2004 revision) proposed that each generalized integer “tries to influence” its generalized-prime factors, so that the force on a prime is assembled from a local density discrepancy at its multiples. [v0.3] This has a close mathematical counterpart: the adjoint of the linearization of the convolution exponential (Proposition 4.2). The exact generator force involves derivatives of that adjoint potential along the prime-power locations, so the old rule is better called a perceptive sketch than a verbatim formula. Bijectivity shows that there are not two independently coupled data sets; it does not say that every choice of geometry or admissible dynamics is empty (§3.2).

What does not survive is the idea that a collection of physical resemblances will assemble itself into a mechanism. A power spectrum, a random-matrix spacing law and a suggestive particle picture can each be real without being three observations of one underlying dynamics.

2. Why number theory keeps looking physical

The archive documents more than a century of recurring contact: trace formulae, quantum chaos, random matrices, partition functions, noncommutative geometry, quantum field theory, fractals, \(p\)-adic models and arithmetic statistical mechanics. It is tempting to interpret the recurrence as evidence that the integers are secretly a physical system. A more conservative and, in our view, deeper account begins with four structural mechanisms.

The free Riemann gas is the cleanest miniature. Assign a bosonic mode of energy \(\log p\) to each prime. Unique factorization makes the occupation numbers \(v_p\in\{0,1,2,\ldots\}\) independent, and

\[ Z(\beta)=\prod_p\left(1-e^{-\beta\log p}\right)^{-1} =\prod_p(1-p^{-\beta})^{-1}=\zeta(\beta), \qquad \beta>1. \]

This is not yet an explanation of the primes: it is a physical rewriting of the Euler product. The Bost–Connes system, however, shows that the analogy can be promoted to an intrinsic \(C^*\)-dynamical system with zeta partition function, a phase transition at \(\beta=1\), and arithmetic symmetry breaking. [v0.2] It also gives a concrete setting for the notebook’s question “what do these probabilities ultimately refer to?” [v0.3] The equilibrium-state structure needs care: the KMS state is unique for \(0<\beta\le1\); for \(\beta>1\) there is a symmetry-broken family of extremal KMS states, not a unique one. In each extremal Gibbs representation the Hamiltonian’s energy-level distribution is nevertheless \(n^{-\beta}/\zeta(\beta)\). These weights therefore describe equilibrium energy levels of a specific dynamical system, not a frequency with which integers mysteriously “occur”. The distinction matters: some number-theory/physics links are metaphors, some are translations, and some are theorems about genuinely dynamical structures.

A useful working thesis is that arithmetic and physics become spectral for the same reason: each must encode global consistency using local data. Spectra, partition functions and zeta functions are efficient machines for performing that compression.

3. Arithmetic equilibrium and the Goldilocks problem

3.1 Beurling systems

A Beurling prime system is a nondecreasing unbounded sequence

\[\Pcal=\{q_1,q_2,\ldots\},\qquad 1<q_1\le q_2\le\cdots,\]

with associated multiplicative semigroup \(\Ncal\) of generalized integers, freely generated, multiplicities retained. Its zeta function is

\[ \zeta_{\Pcal}(s)=\sum_{n\in\Ncal}n^{-s} =\prod_{q\in\Pcal}(1-q^{-s})^{-1} \]

where these expressions converge. The ordinary primes and integers give the classical system. Write \(\pi_\Pcal(x)\) and \(N_\Pcal(x)\) for the two counting functions.

The 1999 question was roughly: can one perturb the generalized primes, observe the resulting unevenness of the generalized integers, and feed that discrepancy back until the classical primes emerge as equilibrium? Modern Beurling theory sharpens both the interest and the danger of the question. Continuous prime measures can be approximated by discrete generalized-prime systems (Diamond 1970; Broucke–Vindas 2021); at the same time, generalized primes with very regular counting can generate surprisingly irregular generalized integers and conversely. Prime-like average density alone does not recover arithmetic.

3.2 The bijectivity obstruction [v0.2]

Why “almost by definition”? Because the passage from generalized primes to generalized integers loses nothing. Put \(u=\log q\) and encode a Beurling system by its prime-power measure in logarithmic space,

\[\Pi=\sum_j\sum_{k\ge1}\tfrac1k\,\delta_{k u_j},\qquad u_j=\log q_j ,\]

and its integer measure \(\nu=\sum_{n\in\Ncal}\delta_{\log n}\). Then \(\nu=\exp_*(\Pi)\), the additive-convolution exponential (§4.1). The point to dwell on is:

[v0.4] So v0.3’s objection was right about the proof v0.2 gave (“only finitely many terms contribute” needs the gap) and wrong about the statement: the bijection holds on the whole ambient class, including Beurling’s continuous prime measures and the \(z\Pi\) family, provided only that the prime-power measure has finite mass near zero. (Prime measures with density \(dx/\log x\) all the way down to \(x=1\) have infinite mass near \(u=0\) and are excluded; their zeta functions do not exist.) What v0.3 was reaching for is a different and correct point, which we now make precise.

[v0.5] The durable conclusion is twofold. Fable’s quasi-nilpotence argument repairs v0.3’s unnecessarily narrow domain: the algebraic bijection extends well beyond gap-supported systems. Sol’s geometric reservation also survives: the topology in which the algebra is easiest does not see generator motion, while topologies that do see it require uniformity hypotheses and can amplify perturbations severely. Möbius inversion supplies a mechanism and an exponential upper scale; it is not yet a complete condition-number theorem.

Consequently the generalized integer and prime-power measures are algebraically the same information, and literal “feedback” between them is not a coupling of independent systems. [v0.3] The conclusion should stop there. A bijection does not canonically identify metrics, topologies, regularity classes or local dynamics: a simple integer-side norm can pull back to a nonlocal prime-side geometry. Nor is the strict Beurling locus an open linear space of measures: its atoms occur in correlated towers \(k u_j\) with weights \(1/k\), so arbitrary perturbations of \(\Pi\) leave the class. Those choices are exactly where a nontrivial programme could live. A target-distance energy is still circular; a target-free structure that is independently natural would not become vacuous merely because it can be written in either coordinate system.

This relocates the Goldilocks problem. It is not a difficulty to be engineered around by a cleverer energy. It is the programme: the only non-circular question is whether some natural functional—natural meaning defined uniformly across Beurling space without reference to the classical point—is extremal, rigid or self-dual exactly there. Everything hinges on the word “natural”, which is not a mathematical predicate; the best one can do is list candidates and test them.

3.3 What is already known not to select [v0.2]

The old notebook hoped that “equal spacing of the \(g\)-integers” might be the selecting principle. The literature now contains a ladder of results showing how much regularity one can impose on both sides without reaching arithmetic:

• Beurling (1937), Diamond (1970). \(N_\Pcal(x)=Ax+O(x/\log^{c}x)\) with \(c>3/2\) forces the prime number theorem for \(\Pcal\); \(c=3/2\) does not. So the integer side can be very regular while the prime side fails even the PNT.
• Diamond–Montgomery–Vorhauer (2006), Zhang (2007). For every \(1/2<\theta<1\), DMV constructed Beurling systems with \(N_\Pcal(x)=\rho x+O(x^\theta)\) while the primes exhibit de la Vallée Poussin-scale oscillation; their zeta functions have only the classical zero-free region, not RH. Zhang also constructed systems exhibiting RH together with strong integer-counting estimates. These examples show that impressive regularity on one analytic axis need not determine the classical system.
• Broucke–Debruyne–Vindas (2020). Their construction points in the other direction from the one stated in v0.2: it has RH-quality prime counting, \(\pi_\Pcal(x)=\operatorname{Li}(x)+O(\sqrt{x})\), but highly irregular generalized integers, with oscillations of order \(x\exp[-c\sqrt{\log x\log\log x}]\) about their main term. Even strikingly classical prime-side behaviour does not force a classical integer lattice.
• Dedekind zeta functions. For any number field \(K\), the norms of prime ideals (with multiplicity) form a Beurling system whose zeta function \(\zeta_K\) has meromorphic continuation, a functional equation, an Euler product and a simple pole at \(s=1\), with GRH conjecturally placing its nontrivial zeros on \(\Re s=1/2\). Its generalized integers lie on \(\mathbb N\) with multiplicities \(r_K(n)\ne1\). Thus density, functional equation and positive Euler factors admit abundant nonclassical examples. The analytic degree is \([K:\mathbb Q]\); the archimedean factor is a product determined by the real and complex places, not merely “one Gamma factor versus \([K:\mathbb Q]\)”.
• Degree-one \(L\)-functions. Kaczorowski and Perelli’s classification of degree-one elements of the Selberg class is a powerful analogue of the desired rigidity. But Selberg-class coefficients may be complex and are not positive Beurling integer measures, while the defining Dirichlet expansion already uses the ordinary integer lattice. It is therefore evidence that degree matters, not a direct Beurling selection theorem.

3.4 Candidate selectors that are not obviously circular [v0.2]

Two candidates survive the list above. Neither is proved to work; both are precise enough to fail.

(a) Poisson self-duality, Hamburger’s theorem, crystalline measures

Hamburger’s converse theorem, under its growth and Dirichlet-series hypotheses, characterizes \(\zeta\) from the Riemann functional equation. Its proof is tied to the theta transformation and Poisson summation for the lattice measure \(\sum_{n\in\mathbb Z}\delta_n\). The hypothesis that the Dirichlet series is already indexed by \(\mathbb N\) is exactly what the Beurling question must avoid. A possible bridge is therefore:

[v0.3] This wording separates steps that v0.2 compressed into an equivalence. Lev and Olevskii prove periodic structure when both support and spectrum are uniformly discrete: each lies in a finite union of lattice translates. Mere local finiteness of the Fourier spectrum is weaker. Kurasov–Sarnak’s nonperiodic positive Fourier quasicrystals show why that distinction matters: a locally finite spectrum need not be uniformly discrete. Nor does a functional equation automatically become a crystalline measure without completion, growth and temperedness hypotheses. The possible new rigidity would have to come from combining the analytic summation formula with the special multiplicative semigroup structure of a Beurling integer measure. This remains a promising staged programme, not a known equivalence.

(b) Bounded error

Density \(A=1\) is a normalization, not a selector. We have neither found a nonclassical example with bounded error nor verified a theorem excluding one. [v0.3] This is deliberately recorded as a specialist-check question, not a novelty or openness claim: the first task is a serious literature search. If the answer is “yes”, it would be a purely integer-side rigidity principle far short of stipulating \(\Ncal=\mathbb N\); if “no”, the counterexample would itself map an unexpectedly rigid-looking corner of Beurling space.

[v0.5, corrected] Partial information, still short of a settlement. (i) From \(\zeta_\Pcal(s)=s\int_1^\infty N_\Pcal(x)x^{-s-1}dx\), the bound \(N_\Pcal(x)=x+O(1)\) implies analytic continuation of \(\zeta_\Pcal(s)-1/(s-1)\) to \(\Re s>0\), with elementary growth control there. v0.4 called this an equivalence; the converse needs additional Tauberian boundary hypotheses and is not established by analytic continuation plus a rough polynomial bound alone. (ii) In the standard \([\alpha,\beta]\)-notation, \(\psi_\Pcal(x)=x+O_\varepsilon(x^{\alpha+\varepsilon})\) and \(N_\Pcal(x)=ax+O_\varepsilon(x^{\beta+\varepsilon})\), every such system satisfies \(\max\{\alpha,\beta\}\ge1/2\). Thus a hypothetical bounded-error system would force the optimal prime-side exponent to be at least \(1/2\); this is consistent with classical behaviour but says nothing close to uniqueness. (iii) Perturbative paths \(q_p=pe^{c_p}\) make bounded discrepancy look implausibly rigid, but the probabilistic order estimates in v0.4 were heuristic and are not retained as evidence. We still have neither a proof nor a counterexample, and make no claim that the question is open in the specialist literature.

3.5 Moduli spaces and flows of zeta functions

The work of Hilberdink and Lapidus is relevant here without being a retrospective proof of the old conjecture. Their study of Beurling zeta functions, generalized primes and fractal membranes develops analytic continuation, generalized functional equations and a relationship with spectral partition functions. Lapidus later explored moduli spaces and flows of fractal membranes and zeta functions. This validates the category of the question—deforming zeta-bearing objects in a moduli space—but supplies neither the missing feedback law nor convergence to the classical primes. [v0.2] The same is true, at a greater distance, of Connes–Consani’s arithmetic and scaling sites, where a genuine \(\mathbb R_+^*\)-action plays the role of Frobenius: it is a flow with the integers in it, not a flow toward the integers.

4. The finite discrepancy flow as a probe, not an explanation

The following construction turns the notebook’s principal intuition into a finite, falsifiable model. [v0.2] In v0.1 it was called “a concrete flow in logarithmic prime space” and carried the weight of the programme. After the examination recorded here it is demoted to a probe: an instrument that reveals the geometry of the map \(\Pi\mapsto\nu\) near the classical point, and that makes circularity visible. It should not be presented as a candidate explanation of the primes.

4.1 State space and the convolution exponential

Put \(u_j=\log q_j\). Multiplication of generalized integers is now addition of their logarithms. Encode the generalized prime powers by the measure

\[ \Pi_u=\sum_j\sum_{k\ge1}\frac1k\,\delta_{k u_j}. \]

The generalized integer counting measure in logarithmic space is its additive convolution exponential

\[ \nu_u=\exp_*(\Pi_u) :=\delta_0+\Pi_u+\frac1{2!}\Pi_u*\Pi_u+\frac1{3!}\Pi_u*\Pi_u*\Pi_u+\cdots. \]

Indeed, taking a Laplace transform gives

\[ \int e^{-sx}\,d\nu_u(x) =\exp\!\left(\int e^{-sx}\,d\Pi_u(x)\right) =\prod_j(1-e^{-s u_j})^{-1}=\zeta_{\Pcal}(s). \]

This identity is exact. It packages unique multiplicative generation as linear analysis on measures, and it is Diamond’s \(d N=\exp_*(d\Pi)\) in a slightly different notation.

[v0.4] The v0.3 chain rule is accepted. If \(G(y)=\delta E/\delta\Pi(y)\) is read as a potential on logarithmic prime-power space, it is assembled from the multiples in the manner the notebook anticipated; the force on a generator is then \(\sum_kG'(ku_j)\). The old language of “charge” mixed these two levels, but the underlying tower-of-multiples geometry was perceptive.

So the 1999 picture survives as kinematics. Proposition 3.1 also supplies the negative lesson: because the map is invertible, this is the pullback of an integer-side gradient, not an independently interacting second system.

4.2 A reconstruction energy, with a corrected cutoff convention

Let the classical log-integer measure be

\[\nu_{\mathrm{cl}}=\sum_{n\ge1}\delta_{\log n}.\]

Choose a smoothing scale \(\tau>0\) with heat kernel \(K_\tau\), a window length \(L\), and a smooth nonnegative weight \(w_L\) supported in \([0,L]\) and vanishing to order two at \(L\). Define

\[ E_{\tau,L}(u)=\frac12\int_0^L \left|K_\tau*\bigl(\nu_u-\nu_{\mathrm{cl}}\bigr)(x)\right|^2 w_L(x)\,dx. \]

[v0.2] Do not hard-truncate the measures before smoothing. v0.1 said “retain only the part of each measure on \([0,L]\)”. If done literally, a generalized integer crossing the cutoff makes its Gaussian switch on or off, so the numerical energy jumps. [v0.3] For a finite list of generators, the mathematically clean Gaussian model keeps the full generalized-integer measures. The resulting sum on \([0,L]\) is infinite, not finite as v0.2 said, but it converges normally (with all parameter derivatives) on chambers bounded away from \(u_1=0\), because the number of exponent vectors grows only polynomially while Gaussian tails decay super-exponentially. The energy is therefore real analytic there. A computation must still truncate this tail; doing so at \(L+c\sqrt\tau\) introduces exponentially small switching discontinuities rather than literally preserving smoothness. The margin \(c\), a tail estimate and a convergence check should be reported. (Coincidences \(u_i=u_j\) represent multiplicity; ordering fixes the relabelling symmetry.)

The simplest proposed dynamics is gradient descent,

\[ \dot u_j=-\frac{\partial E_{\tau,L}}{\partial u_j}, \qquad q_j=e^{u_j}, \]

for which \(dE/dt=-\|\nabla E\|^2\le0\). This proves only that the chosen discrepancy decreases.

4.3 What the energy is in Dirichlet-series language [v0.2]

v0.1 asked whether a Dirichlet-series model would expose the geometry more cleanly. It does. In the corresponding whole-line Gaussian model with \(w(x)=e^{-2\sigma x}\) and \(\sigma>1\), Plancherel gives (for the stated Fourier convention)

\[ E_{\sigma,\tau}=\frac{e^{\sigma^2\tau}}{4\pi}\int_{-\infty}^{\infty} \bigl|\zeta_\Pcal(\sigma+it)-\zeta(\sigma+it)\bigr|^2\,e^{-\tau t^2}\,dt . \]

The compact spatial window used in the finite probe introduces boundary smearing, so it is not literally identical to this whole-line formula; the formula nevertheless exposes the same frequency tradeoff. The energy is a Gaussian-windowed \(L^2\) distance on a vertical line, and \(\tau\) is a frequency cutoff: only Mellin frequencies \(|t|\lesssim\tau^{-1/2}\) are compared. The tangent map is explicit,

\[ \frac{\partial}{\partial u_j}\log\zeta_\Pcal(s)=-\frac{s}{e^{su_j}-1}=-s\sum_{k\ge1}q_j^{-ks}, \]

so the Hessian of \(E\) at the classical point is the Gram matrix of the functions \(t\mapsto s\,\zeta(s)/(p_j^{\,s}-1)\) in the windowed \(L^2\) norm. Finite \(L\) and a compactly supported \(w_L\) replace the vertical line by a smeared version of it; nothing structural changes.

Two consequences follow immediately.

1. The landscape is created by finite resolution. As \(\tau\to0\), the unnormalised Gaussian integral diverges. After the natural \(\sqrt{\tau/\pi}\) normalization, the mean-value theorem for generalized Dirichlet series leads to the weighted squared mismatch of atomic coefficients, \(\sum_x|\nu_u(\{x\})-\nu_{\mathrm{cl}}(\{x\})|^2e^{-2\sigma x}\). Off collision loci this limit is locally constant, so interactions between separated atoms collapse as resolution becomes exact. Smoothing does not merely reveal a hidden differentiable landscape: it creates one by declaring nearby exponents partially similar.
2. Conditioning worsens with scale and smoothing. Moving \(u_1=\log2\) moves about \(e^L/2\) atoms; moving a prime near \(e^L\) moves one. The diagonal stiffness is roughly controlled by the number of visible multiples, so larger primes are soft. The probe’s condition number grew from about \(2.2\times10^2\) at \((X,\tau)=(30,0.002)\) to \(7.4\times10^2\) at \((60,0.002)\). At \(X=30,\tau=0.05\), however, it rose to about \(2.5\times10^4\): stronger smoothing enlarged the observed basin while collapsing the weakest Hessian eigenvalue. This is a resolution–stability tradeoff. Choosing \(\tau\) is part of the missing natural structure, not a neutral numerical convenience. [v0.5, rerun] The \(\tau=0.05\) output is reproduced. Its softest eigenvector is dominated by an antisymmetric \(17/19\) exchange (weights \(-0.60,+0.65\)), with smaller components on \(11,13,23\). Since \(\log19-\log17\approx0.11<\sqrt\tau\approx0.22\), the interpretation is persuasive: the kernel has difficulty resolving their separation. But \(\sqrt\tau\) is a resolution heuristic, not a proved universal exchangeability threshold.
3. [v0.5] Part of the sensitivity belongs to the coordinate change. The forward and inverse derivatives are convolution by the integer and Möbius measures, whose truncated total variations are of exponential size at the classical point. This explains why severe amplification is unsurprising. The observed Hessian spectrum also depends on the target, smoothing, spatial weight, generator parameterization and discretization, so it should not be identified wholesale with an invariant condition number of the Beurling map.

4.4 What the experiment should test, revised

1. Fix \(L\), \(m\), the chamber, \(\tau\), \(w_L\) and the enumeration margin. Report all of them; the result is a function of all of them.
2. Compute the Hessian at the classical configuration. Report its spectrum, the prime-weights of its extreme eigenvectors, and how the condition number scales with \(L\). (Done at toy scale; see Appendix C.)
3. Map the basin radius as a function of \(\tau\) and \(L\). If it is always of order \(\sqrt\tau\) the flow is telling you about the kernel, not the primes.
4. Replace the exact target by progressively weaker observables and record the point at which selectivity is lost (Project B). Here the Dirichlet-series form helps: weakening the target is restricting the Mellin window or replacing \(\zeta\) by a coarser function such as \(1/(s-1)\).
5. [v0.2] Run the same flow from Dedekind or Dirichlet-twisted systems and from the Diamond–Montgomery–Vorhauer examples. A flow that cannot tell \(\zeta_{\mathbb Q(i)}\) from \(\zeta\) at a given window is not measuring arithmetic.

4.5 The circularity ladder, revised [v0.2]

Until a selector in the middle rows is either proved or refuted, the flow of §4.2 is a research instrument and the programme has no theorem of the kind the notebook hoped for. That is a clear statement of where the programme stands.

5. The Julia–Gibbs ensemble: what is exact, what is standard

For \(\beta>1\), define a random positive integer \(N\) by

\[ \Pr_\beta(N=n)=\frac{n^{-\beta}}{\zeta(\beta)}. \]

This is the canonical Gibbs distribution for energy \(E(n)=\log n\), inverse temperature \(\beta\), and partition function \(\zeta(\beta)\). In probability it is the zeta (or Zipf) distribution; the old notes plotted these probabilities and followed the maxima as \(\beta\) changed. [v0.2] v0.1 asked which of the resulting statements are standard. The answer, item by item:

5.1 Independent prime occupations — standard

Unique factorization and the Euler product imply that the valuations \(v_p(N)\) are independent geometric random variables:

\[ \Pr_\beta\bigl(v_p(N)=k\bigr)=(1-p^{-\beta})p^{-k\beta}, \qquad k=0,1,2,\ldots. \]

This observation is Golomb (1970); the distribution and its moments are treated systematically by Lin and Hu (2001), who also show it is infinitely divisible. The mean energy and Shannon entropy are

\[ \mathbb E_\beta[\log N]=-\frac{\zeta'(\beta)}{\zeta(\beta)},\qquad H(\beta)=\log\zeta(\beta)-\beta\frac{\zeta'(\beta)}{\zeta(\beta)},\qquad H'(\beta)=-\beta\,\operatorname{Var}_\beta(\log N)<0 . \]

The last identity is the universal thermodynamic relation \(dS/d\beta=-\beta\,\mathrm{Var}(E)\) for any Gibbs family; nothing arithmetic enters. It is still worth writing down, because it is the honest content behind the notebook’s “has anyone applied entropy?” (2004 revision, §10.1): yes, and it is this.

5.2 The critical scaling law and Karamata’s theorem [v0.3, qualified]

Write \(\beta=1+\varepsilon\). v0.1 showed by Laplace transforms that \(\varepsilon\log N\to\operatorname{Exp}(1)\) as \(\varepsilon\downarrow0\), and for a Beurling system with \(\zeta_\Pcal(s)\sim C(s-\alpha)^{-m}\) that \((\beta-\alpha)\log N\to\operatorname{Gamma}(m,1)\). Both implications are correct and follow from classical regular-variation theory. [v0.3] The unqualified converse asserted in v0.2 was stronger than the limit law warrants.

For the classical system \(m=1\), \(\alpha=1\), \(C=1\), and (ii) reads \(\sum_{n\le e^x}1/n\sim x\). In physical language, Gibbs energy fluctuations at this critical point are exponentially distributed; Julia’s 1994 paper is already in this territory. Conversely, convergence to \(\operatorname{Gamma}(m,1)\) yields the Laplace-transform ratio \(\zeta_\Pcal(\alpha+\lambda t)/\zeta_\Pcal(\alpha+t)\to\lambda^{-m}\), hence regular variation of index \(-m\). It still permits a slowly varying factor \(L(1/t)\), and therefore does not by itself recover the pure pole constant \(C\). The leading limit law is standard; normalization, remainders and converse hypotheses remain legitimate questions.

[v0.5, qualified] Esseen-type smoothing inequalities give a plausible route from characteristic-function control to a probability metric. Here \(\phi_\varepsilon(y)=\zeta_\Pcal(\beta-iy\varepsilon)/\zeta_\Pcal(\beta)\), so quantitative convergence requires uniform information about \(\zeta_\Pcal\) in a shrinking complex neighbourhood of \(\alpha\). v0.4’s direct conclusion “local error \(O(|s-\alpha|^\delta)\) gives Kolmogorov distance \(O(\varepsilon^\delta)\)” omitted the growth of the integration range and decay of the limiting characteristic function; the resulting exponent depends on those details. The sound research question is to state and prove a transfer theorem with explicit uniformity assumptions. Analytic singularities or zeros matter only insofar as they affect that neighbourhood.

5.3 The moving maxima — verified

For each fixed \(n>1\), the value of \(\beta\) at which \(\Pr_\beta(N=n)\) is maximal is the unique solution \(m_n>1\) of \(-\zeta'(m_n)/\zeta(m_n)=\log n\). Near the pole, since \(-\zeta'/\zeta(s)=(s-1)^{-1}-\gamma+O(s-1)\),

\[ m_n=1+\frac1{\log n+\gamma}+O\!\left((\log n)^{-3}\right). \]

[v0.2] Fable recomputed these independently (mpmath, 20 digits): \(m_2=1.8791006723\), \(m_3=1.6351665161\), \(m_4=1.5329591778\), \(m_5=1.4743970400\), agreeing with the 2004 notebook; and \(m_{10}=1.3547124102\), \(m_{20}=1.2838026937\), \(m_{30}=1.2542036802\), \(m_{40}=1.2367192641\), agreeing with v0.1 and confirming that the notebook’s \(m_{10},\dots,m_{40}\) were wrong. The asymptotic \(1+1/(\log n+\gamma)\) gives \(1.3472\) at \(n=10\) and \(1.2344\) at \(n=40\), consistent with an \(O((\log n)^{-3})\) remainder.

5.4 Where new work could begin [v0.2]

• Fluctuations beyond the pole. Karamata sees only regular variation. Analytic continuation, nearby singularities and zero-free information can constrain remainder terms, but v0.2’s statement that the next correction is simply “governed by the zeros near \(\Re s=\alpha\)” was too strong. A precise problem is to identify hypotheses under which an explicit analytic remainder transfers to a quantitative distance between the rescaled Gibbs law and its Gamma limit.
• Prime-factor structure at criticality. For the classical zeta distribution, one may ask for the limiting ranked log-sizes of prime factors as \(\beta\downarrow1\), in relation to Poisson–Dirichlet. A pole of order \(m\) alone contains too little local intensity information. [v0.5, refined] The infinitely divisible representation of the Gibbs law gives a Poisson process of logarithmic prime-power jumps with intensity \(e^{-(\beta-\alpha)u}d\Pi_\alpha(u)\). A natural sufficient hypothesis is the Mertens-type estimate

\[ \Pi_\alpha([0,u])=m\log u+c+o(1), \]

• which should make the rescaled intensity converge vaguely to \(m e^{-v}dv/v\). Normalized ranked jumps of that limiting process have the Poisson–Dirichlet law \(\mathrm{PD}(0,m)\). This hypothesis should not be renamed a “tilted Beurling PNT” or asserted equivalent to whole-line analytic continuation without further work. Conjecture 5.2, revised. Under this Mertens-type hypothesis plus conditions controlling repeated prime-power aggregation, the normalized ranked logarithms of actual Beurling prime factors converge to \(\mathrm{PD}(0,m)\). The passage from Poissonian prime-power jumps to the factorization of \(N\) is part of the conjecture, not already a theorem.
• Uniformity over families. Rates of convergence uniform over classes of Beurling systems with controlled pole data; stability of the Gamma law under perturbation of \(\alpha,m,C\).

5.5 Julia’s chemical-potential flow, and a productive conflation [v0.3]

The 2004 notebook (§1.6) records Julia’s Sathe–Selberg/chemical-potential parameter as a “flow in the space of Beurling theories”. v0.2 identified it with \(\zeta(s)^z\). That is a related and useful family, but not the chemical-potential deformation itself.

Give every occupied prime factor a fugacity \(z\). The grand partition function is

\[ Z(s,z)=\prod_p(1-zp^{-s})^{-1} =\sum_{n\ge1}\frac{z^{\Omega(n)}}{n^s}, \]

where \(\Omega(n)\) counts prime factors with multiplicity. Its logarithm has coefficients \(z^k/k\) at the prime powers, not \(z/k\). At fixed inverse temperature \(s=\beta\), \(z=e^{\beta\mu}\) is the usual chemical-potential fugacity. Equivalently, treating \(\mu\) as the path parameter shifts every logarithmic prime coordinate:

\[ u_p(\mu)=\log p-\mu,\qquad \Pi_\mu=\sum_{p,k\ge1}\frac1k\,\delta_{k(\log p-\mu)}. \]

For \(\mu<\log2\) this is a genuine strict Beurling path with generalized primes \(p e^{-\mu}\). It is conceptually valuable: Julia had supplied an honest one-parameter motion of the generators by a common translation. It does not select the classical point without an additional criterion.

The v0.2 family remains as a comparison. Scaling the ambient prime-power measure to \(z\Pi_{\mathrm{cl}}\) gives \(\zeta(s)^z\) and generalized-divisor weights \(d_z(n)\), but generally leaves the strict Beurling locus. The two deformations are linked by Selberg–Delange factorization:

\[ \prod_p(1-zp^{-s})^{-1}=\zeta(s)^z F(s,z), \]

with an Euler product \(F\) regular near \(s=1\) in the usual parameter range. Thus the conflation was mathematically productive: it exposes two distinct nearby paths—one translating strict Beurling generators, the other scaling an ambient convolution measure—whose separation should now be maintained.

[v0.4] Fable has checked the separation and confirms it: \(\log Z(s,z)\) has coefficients \(z^k/k\) at \(k\log p\), so \(Z(\cdot,z)\) is not the zeta function of any Beurling system for \(z\ne1\), while the translation path \(\{pe^{-\mu}\}\) is a Beurling system whose zeta function is \(Z(s,e^{\mu s})\)—the fugacity depends on \(s\). The two coincide only at a single fixed temperature. That said, the substance of v0.2’s remark survives on Julia’s true path, and in a more interesting form:

6. An actual evolution of zeta zeros, as a control case

The original notebook imagined zeta zeros as moving particles and asked whether repulsion might regulate a deformation. There is a rigorous comparison that belongs on the main page: the de Bruijn–Newman heat flow.

One writes a Fourier representation of the Riemann xi function and deforms its kernel:

\[ H_t(z)=\int_0^\infty e^{t u^2}\Phi(u)\cos(zu)\,du, \qquad \frac{\partial H_t}{\partial t}=-\frac{\partial^2H_t}{\partial z^2}. \]

There is a constant \(\Lambda\) such that all zeros of \(H_t\) are real exactly when \(t\ge\Lambda\). The Riemann hypothesis is equivalent to \(\Lambda\le0\), while Rodgers and Tao proved \(\Lambda\ge0\); Polymath 15 (2019) proved \(\Lambda\le0.22\). Thus, if RH holds, it is poised at the transition value \(\Lambda=0\).

As long as the zeros \(x_k(t)\) are real and simple, differentiating \(H_t(x_k(t))=0\) gives the inverse-gap interaction

\[ x_k'(t)=2\sum_{j\ne k}\frac1{x_k(t)-x_j(t)}, \]

with the usual qualifications for an infinite zero set (Csordas–Smith–Varga 1994 for the finite-degree version; Rodgers–Tao for the infinite one). This is not just particle imagery: it is an actual evolution equation with repulsion between zeros.

v0.1 asked whether this comparison illuminates or decorates. Our answer remains: it decorates the particle picture and illuminates Project D. It should stay, reframed as above, and not be used to suggest that the programme has an “evolution of zeros” of its own. [v0.5] Lemma 5.4 supplies two honest paths of Beurling generators, but only the dilation path has an immediate conventional zero motion: \(\zeta(cs)\) rescales zeros by \(\rho\mapsto\rho/c\). The translation path generally has noninteger branch behaviour at the singularities of \(\zeta\), so it may not admit the kind of completed entire function whose zeros can be followed as particles. Project D must first decide what analytic continuation and “zero” mean along that path; comparison with dBN comes later, if at all.

The correct 2026 question is therefore not “was the particle picture right?” but “can a deformation of prime measures be functorially transported to a controlled deformation of completed zeta functions, in a way compatible with a zero flow and a positivity principle?” That is precise enough to fail, and therefore worth asking.

7. Metaphysical implications, cautiously stated

The psychological force of the prime distribution is not exhausted by the equations. Why should an internally necessary object feel discovered, alien, almost physical? The following interpretations are offered as philosophical consequences of the mathematics, not additional mathematical claims. Each subsection carries a verdict.

7.1 Static objects can be equilibria without having histories — clarifies

A canonical mathematical object may be characterized as a fixed point, attractor, critical point or rigid orbit in a space of alternatives. “Evolution” then means logical or variational dependence on a parameter, not a time before arithmetic existed. A circle minimizes an isoperimetric functional without having been physically squeezed into shape. The classical primes might similarly be a stable form in a moduli space even if no literal process produced them. This is the sentence that turns the 1998 experience into a research question and it should be kept.

7.2 Arithmetic sits at an intersection of incompatible-looking orders — clarifies

The positive integers form a perfectly regular additive lattice. Their irreducibles under multiplication form the irregular prime sequence. Passing to logarithms makes multiplication additive but bends the lattice. Analytic continuation then turns the residual discrepancy into a spectrum of complex oscillations. The primes may feel alien because they are the interface at which several notions of order—additive, multiplicative, logarithmic and spectral—must agree globally while none determines the local pattern cheaply. [v0.2] The additive lattice and the multiplicative generators are one datum seen in two coordinate systems; the “alienness” may be the phenomenological cost of the coordinate change.

[v0.3] The bijection adds a further metaphysical lesson. Explanation is not always the acquisition of more information. Sometimes it is the discovery of a representation in which an already complete datum becomes simple, local or inevitable. The same arithmetic object can look lawlike in one coordinate system and process-like in another. What would make one representation explanatory rather than merely redescriptive is precisely an independently natural structure—symmetry, locality, stability or universality—that is inexpensive there and contorted elsewhere.

[v0.5, qualified] v0.4 proposed that the convolution algebra and affine group are the only canonical structures available and that the prime–integer correspondence has a metric-independent exponential condition number. Neither uniqueness claim has been proved. What has been earned is more modest and still philosophically substantial: convolution and affine motion are two target-free structures already present; density fixes the classical point only along the affine orbit; and Möbius inversion can amplify perturbations on exponentially large windows in natural norms. Thus the phenomenology of the 1998 experience—regular integers appearing to transmit a far less locally legible prime signal—has a genuine mathematical analogue. The two configurations contain the same information, yet converting between their natural descriptions may be globally sensitive. What remains unexplained is why any independently privileged geometry should select this datum rather than merely encode it.

[v0.2] One may add a perceptual conjecture, clearly labelled as such: via the explicit formula, the prime-counting fluctuation is literally a superposition of oscillations \(x^{\rho}\), and minds are tuned to hear superposed oscillations (sound, light) as signals with sources. The sense that the primes are “the trace of a process” may be what a Fourier duality feels like from the inside.

7.3 Physics as shared invariance — overreaches; trimmed

v0.1 proposed that mathematics and physics “both classify what remains invariant when descriptions change”. As a slogan about why spectral languages recur it is fine; as a thesis about the relation between the subjects it is too large to be earned by anything on this page, and it tempts the reader back toward “the integers are a physical system” by a different door. We keep only the modest form: the four mechanisms of §2 are sufficient to explain the recurrence of spectral language without any claim about shared substance.

7.4 Information as neutral language — metaphor; demoted to a remark

“Energy”, “entropy” and “spectrum” in §5 are not evidence that integers are made of matter; they are the vocabulary of Gibbs measures, which apply to any countable set with a real-valued function on it. That much is true and worth saying once. The stronger picture of “traffic in lossless and lossy encodings” is a metaphor and is moved here from the main text so that it cannot be mistaken for a result.

This reframing preserves the original experience without demanding that a private revelation carry evidential weight. The experience generated a good kind of question: not “what supernatural agency placed the primes?” but “in what larger space would this apparently arbitrary configuration become natural?”

8. A research programme small enough to begin

8.1 Questions parked for specialists and autumn work [v0.5]

The model-to-model exchange pauses here. Answered questions and retractions remain visible in the version record; the active list is deliberately shorter.

8.2 Standards for a credible result

• State the space, topology, cutoff and multiplicity conventions.
• Separate identities, proved propositions, numerical evidence, conjectures and metaphors.
• Test robustness under regularization and initial conditions.
• Compare with deliberately constructed nonclassical Beurling systems, including Dedekind systems and the DMV/BDV examples.
• Do not use RH, the classical zero set or the ordinary prime locations inside a functional advertised as explaining them—and remember that by Prop. 3.1 “the ordinary integers” is the same datum.
• Publish counterexamples and failed selection principles; they map the problem.

Appendix A. Paths that looked promising in 1999

The following ideas are not deleted. They are moved out of the argumentative load-bearing structure and kept as intellectual archaeology. Each caught a real resemblance; the mistake was to ask it to identify a unique mechanism.

Appendix B. Mathematical repairs to the earlier versions

The archival pages should remain historically intact. The following corrections prevent their slips from propagating into the 2026 argument.

Appendix C. The finite probe and what it showed [v0.3, rerun]

A deliberately tiny implementation of §4.2 was supplied with v0.2 and independently rerun for v0.3 (script: beurling_probe.py, attached alongside the preceding pass). Window \([0,\log X]\) with \(X=30\) (ten primes) and \(X=60\) (seventeen primes); Gaussian smoothing with variance \(\tau\); flat weight with a cosine taper over the last \(3\%\) of the window; generalized integers enumerated to \(\log X+6\sqrt\tau\); Hessian by central differences; basins by L-BFGS-B from random perturbations of the classical point. The published numerical values were reproduced.

Caveats: toy scale; one random seed and four trials per perturbation size; finite-difference Hessian; fixed grid; no convergence study in the six-standard-deviation enumeration margin; no claim about the full Gaussian energy or an infinite-\(L\) limit. Because gints changes its finite list when an atom crosses that margin, the implemented energy has exponentially small jumps; the v0.4 script comment now says so. [v0.5] Sol reran \(\tau=0.05\): condition number \(2.4696\times10^4\) confirmed. The softest eigenvector is dominated by the antisymmetric \(17/19\) components but also contains visible \(11,13,23\) components, so it should not be described as a pure two-prime mode. What the experiment supports is modest: positive Hessians in the tested discretizations, softness concentrated among larger primes, and a pronounced resolution–stability tradeoff. It does not establish a universal \(\sqrt\tau\) basin law or show that the flow “finds” the primes for a reason other than having been given the classical target.

References and routes onward

1. Historical source. M. R. Watkins, “Prime evolution” notes, third version (2004), with links there to the 1999 and 2000 versions.
2. Beurling systems, the monograph. H. G. Diamond and W.-B. Zhang, Beurling Generalized Numbers, AMS Mathematical Surveys and Monographs 213 (2016). [v0.2]
3. Beurling systems, fractal membranes. T. W. Hilberdink and M. L. Lapidus, “Beurling Zeta Functions, Generalised Primes, and Fractal Membranes” (2004; published 2006).
4. Sharpness of the classical error term. H. G. Diamond, H. L. Montgomery and U. M. A. Vorhauer, “Beurling primes with large oscillation”, Math. Ann. 334 (2006), 1–36; W.-B. Zhang, “Beurling primes with RH and Beurling primes with large oscillation”, Math. Ann. 337 (2007), 671–704. [v0.2]
5. Well-behaved systems. T. W. Hilberdink, “Well-behaved Beurling primes and integers”, J. Number Theory 112 (2005), 332–344. [v0.2]
6. Error-exponent constraints and examples. F. Broucke, G. Debruyne and Sz. Gy. Révész, “Some examples of well-behaved Beurling number systems” (2024 version; subsequently published), including the \(\max\{\alpha,\beta\}\ge1/2\) constraint used in §3.4(b). [v0.5]
7. Continuous-to-discrete approximation. F. Broucke and J. Vindas, “A new generalized prime random approximation procedure and some of its applications” (2021).
8. RH-quality primes, irregular integers. F. Broucke, G. Debruyne and J. Vindas, “Beurling integers with RH and large oscillation” (2020).
9. Small perturbations can change analytic behaviour. F. Broucke, A. Kouroupis and K.-M. Perfekt, “A note on Bohr’s theorem for Beurling integer systems” (2023).
10. Functional-equation rigidity over \(\mathbb N\). H. Hamburger, “Über die Riemannsche Funktionalgleichung der \(\zeta\)-Funktion”, Math. Z. 10 (1921), 240–254; J. Kaczorowski and A. Perelli, “On the structure of the Selberg class, I: \(0\le d\le1\)”, Acta Math. 182 (1999), 207–241. [v0.2]
11. Crystalline measures. N. Lev and A. Olevskii, “Quasicrystals and Poisson’s summation formula”, Invent. Math. 200 (2015), 585–606; Y. Meyer, “Measures with locally finite support and spectrum”, PNAS 113 (2016), 3152–3158; P. Kurasov and P. Sarnak, “Stable polynomials and crystalline measures”, J. Math. Phys. 61 (2020), 083501; A. Olevskii and A. Ulanovskii, “A simple crystalline measure” (2020). [v0.3: hypotheses separated]
12. Zeta distribution. S. W. Golomb, “A class of probability distributions on the integers”, J. Number Theory 2 (1970), 189–192; G. D. Lin and C.-Y. Hu, “The Riemann zeta distribution”, Bernoulli 7 (2001), 817–828. [v0.2]
13. Tauberian theory. W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, ch. XIII.5 (Karamata’s theorem). [v0.2]
14. Selberg–Delange. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, ch. II.5; A. Selberg, “Note on a paper by L. G. Sathe”, J. Indian Math. Soc. 18 (1954), 83–87. [v0.2]
15. Prime-factor structure. R. Arratia, A. D. Barbour and S. Tavaré, Logarithmic Combinatorial Structures, EMS (2003); P. Donnelly and G. Grimmett, “On the asymptotic distribution of large prime factors”, J. London Math. Soc. 47 (1993), 395–404. [v0.2]
16. Riemann–Beurling gases. B. Julia, “Thermodynamic limit in number theory: Riemann–Beurling gases”, Physica A 203 (1994), 425–436; B. Julia, “Statistical theory of numbers”, in Number Theory and Physics, Springer Proc. Phys. 47 (1990); Watkins’s Julia notes and partition-function archive preserve the chemical-potential description used in §5.5.
17. Arithmetic quantum statistical mechanics. J.-B. Bost and A. Connes, “Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory”, Selecta Mathematica 1 (1995), 411–457; see also Laca, Larsen and Neshveyev for the KMS-state phase structure cited in §2.
18. Arithmetic geometry and flow. A. Connes and C. Consani, “Geometry of the arithmetic site” (2015); C. Deninger, “Some analogies between number theory and dynamical systems on foliated spaces”, Doc. Math. ICM 1998, 163–186; A. Connes, “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function”, Selecta Math. 5 (1999), 29–106. [v0.2: Deninger, Connes added]
19. A genuine zero flow. B. Rodgers and T. Tao, “The de Bruijn–Newman constant is non-negative” (2018; published 2020); G. Csordas, W. Smith and R. S. Varga, “Lehmer pairs of zeros, the de Bruijn–Newman constant \(\Lambda\), and the Riemann hypothesis”, Constr. Approx. 10 (1994), 107–129; D. H. J. Polymath, “Effective approximation of heat flow evolution of the Riemann \(\xi\) function, and a new upper bound for the de Bruijn–Newman constant”, Res. Math. Sci. 6 (2019). [v0.2: CSV, Polymath added]
20. Wider physics context. See the archive sections on zeta functions as partition functions, random matrices, quantum field theory, noncommutative geometry and Beurling generalized primes.

Version record