Integer resolution and stationary averages

An asymptotic profile gives the size of an arithmetic quantity, but the variation left around that profile can contain further structure. For collision capacity, the leading linear term and the quadratic-logarithmic deficit are already known. They leave two concrete questions. Does the remainder have a stable distribution, and does the stationary model describe the finite capacities actually being observed? Answering them would turn a bound on the error into a description of the frequency and size of the residual values.

The observation rule is the difficulty. The stationary model averages a fixed collection of remainder clocks over compatible phases before letting the collection grow. The finite capacity calculation instead reads every clock through period $n$ at the address $n$. Each increase in resolution changes both the readings and the collection being read. A mean-zero stationary law therefore needs a transfer theorem before it can predict the integer data.

A centered remainder clock has mean zero over a complete cycle. In the capacity calculation, a clock of period $k$ first enters at resolution $k$, at its least residue. The observation then advances while further periods enter. The unfinished cycles create a bias even though every completed cycle balances. Identifying this bias requires an estimate uniform in the growing period cutoff.

The result separates three effects that would otherwise be combined in the remainder. Explicit centering removes the moving-diagonal drift. Restriction to prime predecessors changes the phase law and its mean. A cutoff held fixed across an interval can contribute a Bernoulli boundary fluctuation of its own. This makes the stationary potential a usable description of finite observations, with the sampling rule and its corrections specified.

Set $$J(n)=\frac13\sum_{r,s\le n}\frac{(r,s)^2}{rs},\qquad D_n=n-J(n),\qquad w_k=\frac1{k^2}\sum_{\substack{1\le a<k\\(a,k)=1}}\frac1a.$$ All weight sums start at $k=2$; set $w_1=0$ and $a_k=kw_k$. For $x\in\widehat{\mathbb Z}$ let $r_k(x)$ be its least nonnegative residue modulo $k$. Define $$Y_M(x)=\frac23\sum_{k\le M}w_k \left(r_k(x)-\frac{k-1}{2}\right),\qquad y_n=Y_n(n).$$ The cutoffs $Y_M$ converge in Haar $L^2$ to a potential $Y$ by Conservation and Profinite Dynamics in Digit-Collision Energy [4]. They diverge at every fixed ordinary integer. That fixed-address statement does not determine the diagonal $y_n$.

For a positive integer $X$, write $$\mathbb E_X F=\frac1X\sum_{X<n\le2X}F(n),\qquad \left\|F\right\|_{X,2}=(\mathbb E_X|F|^2)^{1/2}.$$ Integer empirical laws use this uniform choice of $n$. Section 7 explicitly changes the ensemble to $n=p-1$. Haar expectation and variance are denoted by $\mathbb E$ and $\sigma^2=\mathbb EY^2$. There are explicit constants $b<0,c$ such that $$\begin{gathered} Z_n=y_n-b\log n-c,\qquad Z_n\xrightarrow{\rm law}Y,\\ \mathbb E_XZ_n^2=\sigma^2+O(X^{-1/5}\log^2(2X)). \end{gathered}$$ Here and throughout, an empirical limit means $X\to\infty$. The coefficient is $$b=\frac2{\pi^2}\bigl(\log(2\pi)-2\bigr).$$ The raw diagonal therefore has a negative logarithmic drift and a quadratic-logarithmic second moment. Its centered fluctuation retains the stationary law. This is an empirical statement on the specified integer intervals, not pointwise convergence of the residual.

A different limit arises if the period cutoff is frozen throughout the interval. The controlling parameter is $\lambda_X=M\log M/X$. For $M\to\infty$, $M=o(X)$, the stationary second moment transfers exactly when $\lambda_X\to0$. At $\lambda_X\to\lambda>0$ the limiting law becomes $$Y-\frac{\lambda}{3\zeta(2)T}B_2(U),$$ with independent $T$ uniform on $[1,2]$ and $U$ uniform on $[0,1]$. The polynomial $B_2(u)=u^2-u+1/6$ is the first surviving boundary correction in the tail integral.

The capacity originates in the finite collision energy [5, 6]. Its transfer here is an all-integer averaging problem for the continuous quantity $J(n)$, followed by a separate transfer to the prime-indexed resolutions $p-1$. In that ensemble the mean-zero Haar law is replaced by a unit-phase law of mean $-1/3$. The change is forced by the admissible residues of primes at every fixed period. The prime-square sampling limit is a separate problem. The underlying gcd sum is classical. [2] already gives $J(n)=n+O(\log^2(2n))$. Sawtooth limiting laws, moments, and large values for different arithmetic weights and ensembles appear in [3]. For prime averages, [1] gives a general theorem for limit-periodic functions under quantitative second-moment, spectral-tail, and progression-discrepancy hypotheses. The present prime law is proved through a separate unit-Haar construction and explicit cutoff estimates; it is not asserted to follow from that theorem for every nonlinear test function. The specific contributions are the diagonal centering constants and finite-interval rates, the distinct prime-indexed law, the critical Bernoulli boundary transition, and the translated exceptional witnesses. Their proofs use a precise cumulative weight, a finite-frequency Gram bound, and explicit treatment of the observation boundary.

Stationary inputs and exact normalizations

Let $$\eta_q=\sum_{q\mid k}w_k,\qquad \eta_{q,M}=\sum_{\substack{k\le M\\q\mid k}}w_k,\qquad J_2(q)=q^2\prod_{p\mid q}(1-p^{-2}).$$ For $E_d(x)=r_d(x)-(d-1)/2$, define the primitive additive row $$P_q(x)=\sum_{d\mid q}\mu(q/d)E_d(x)\qquad(q\ge2).$$ The additive conductor is the denominator of a reduced rational Fourier frequency.

In particular the core diagonal transfer uses only the mass, coefficient bounds, covariance, and Fourier formulas. The progression constructions later require the additional alignment theorem. The exceptional-set estimate uses empirical mean-square approximation and a deterministic cutoff bound. For the trivial modulus, set $m_1=0$ and $C_1=0$.

The exact boundary-energy rule selects these weights uniquely [4]. Section 11 shows that their smooth centering constants alone do not. The sharper Haar spectral-tail asymptotic in [4] concerns a stationary conductor projection; the integer estimates below require their own finite-interval comparison.

Cumulative weights and centering constants

Write $\alpha=1/\zeta(2)$, and let $\gamma$ be Euler’s constant. The absolutely convergent constants $$s_j=\sum_{d\ge1}\frac{\mu(d)(\log d)^j}{d^2}\quad(j=1,2)$$ satisfy $s_1=\zeta'(2)/\zeta(2)^2$ and $s_2=(1/\zeta)''(2)$. Define $$\beta=\gamma\alpha-s_1,\qquad C_a=\frac{s_2}{2}-\gamma s_1+ \frac{\gamma^2-\zeta(2)}2\alpha.$$

Use the fractional-part convention $\psi(u)=\{u\}-1/2$ at every real $u$, so $\psi(j)=-1/2$ at integers. This differs from the midpoint sawtooth at its jumps. The conditionally convergent integrals $$I_0=\int_1^\infty\frac{\psi(u)}u\,du =\frac12\log(2\pi)-1,\qquad I_1=\int_1^\infty\frac{\psi(u)\log u}u\,du =1+\frac12\zeta''(0)$$ define the constants $$b=\frac23\alpha I_0 =\frac2{\pi^2}(\log(2\pi)-2),\qquad c=\frac23(\beta I_0-\alpha I_1).$$ The bounded periodic primitive $B_2(\{u\})/2$ proves convergence. Summing the integrals over unit intervals and applying Stirling’s formula evaluates $I_0$. For $I_1$, differentiate at zero the identity $$\zeta(s)=\frac1{s-1}+\frac12 -s\int_1^\infty\psi(u)u^{-s-1}\,du.$$ Integration by parts continues the integral to a neighborhood of zero, and its derivative there is $-I_1$. This gives (18). In particular $b\approx-0.03285297505$ and $c\approx-0.03640007885$.

The incomplete cycles in this formula give the moving diagonal a nonpositive cumulative sum.

Large-period tail

The estimate tends to zero for $M=X^{3/5}$ and for $M=X^{2/3}$. This overlap is essential. It leaves a growing stationary approximation while evaluating the remaining periods with an arithmetic tail integral.

Finite-interval averaging and the diagonal law

For the raw moments, put $\ell_1=2\log2-1$ and $\ell_2=2(\log2)^2-4\log2+2$. Expansion of $y_n=b\log n+c+Z_n$ gives $$\begin{aligned} \mathbb E_Xy_n&=b\log X+c+b\ell_1+O(X^{-1/3}\log^2(2X)),\\ \mathbb E_Xy_n^2 &=b^2\log^2X+2b(c+b\ell_1)\log X +c^2+2bc\ell_1+b^2\ell_2+\sigma^2+o(1). \end{aligned}$$ The cross terms use (33) with $h=1,\log t$. Subtracting only the dyadic mean leaves limiting variance $\sigma^2+b^2(\ell_2-\ell_1^2)$. Subtracting $b\log n+c$ at each individual integer leaves exactly $\sigma^2$.

Averaged capacity expansion

The exact conserved-state equation is $$D_n-D_*(n)=y_n+\frac23n\sum_{k>n}w_k.$$ Summation by parts in (17) gives $$\begin{aligned} \frac23n\sum_{k>n}w_k &=\frac23(\alpha\log n+\alpha+\beta) +O(\log^2(2n)/n),\\ D_*(n)&=\frac\alpha6\log^2n+\frac\beta3\log n+ \frac{C_a-1}{3}+O(\log^2(2n)/n). \end{aligned}$$ For the first identity use $n\sum_{k>n}a_k/k=-S_a(n)+n\int_n^\infty S_a(t)t^{-2}\,dt$; for the second use $D_*=(S_a-W_n)/3$.

Define the explicit constants $$B=\frac{\beta+\alpha\log(2\pi)}3,\qquad C=\frac{C_a-1}{3}+\frac\beta3\log(2\pi)-\frac\alpha3\zeta''(0).$$ Numerically $B\approx0.60489835319$ and $C\approx0.37754122174$.

Thus the refinement $D_n=\pi^{-2}\log^2n+B\log n+C+\mathcal R_n$ has a bounded limiting second moment after centering. The exceptional constructions in Section 10 show that the fluctuation cannot be replaced by a pointwise bounded error.

Prime-indexed resolutions

Put $\mathcal U=\widehat{\mathbb Z}^\times$, and let $\nu$ be its multiplicative Haar probability measure. Its projection modulo each integer $k$ is uniform on the units $U_k$. This measure is singular with respect to additive Haar measure on $\widehat{\mathbb Z}$. An additive-Haar $L^2$ function therefore cannot simply be restricted to $\mathcal U$. Define the unit-phase observable from its finite cutoffs by $$F_M(u)=Y_M(u-1),\qquad W_M=\sum_{k\le M}w_k.$$ For $u\in\mathcal U$, the least residue $r_k(u)$ is nonzero, so $$O_M(u):=F_M(u)+\frac{W_M}{3} =\frac23\sum_{k\le M}w_k\left(r_k(u)-\frac k2\right).$$ The right side is odd under $u\mapsto-u$. In particular $\mathbb E_\nu F_M=-W_M/3$ exactly.

Write $$\mathcal P_X=\{p\text{ prime}:X<p-1\le2X\},\quad P_X=\#\mathcal P_X,$$ $$\mathbb E_X^{\rm pr}H=\frac1{P_X}\sum_{p\in\mathcal P_X}H(p-1).$$ The prime number theorem gives $P_X\asymp X/\log X$. The next step needs uniformity for powers of $\log X$, rather than equidistribution at each fixed modulus alone.

In particular the smooth constant term for the prime ensemble is $C-1/3$. The residual $$D_{p-1}-\frac{\log^2(p-1)}{\pi^2} -B\log(p-1)-(C-1/3)$$ has a symmetric limiting law of positive finite variance. The unit-phase law is not a stationary law for addition on $\widehat{\mathbb Z}$. Its definition and proof do not inherit atomlessness or sharp tails from the additive-Haar law. Those questions for $Y^\times$ remain separate, as does the prime-square sampling defect. The theorem concerns the continuous capacity evaluated at $p-1$.

Period-cutoff transition

Every fixed power $M=X^\theta$, $0<\theta<1$, lies below this transition. The logarithmic threshold separates cutoffs that a power classification does not distinguish. The endpoint $M=X$ is another regime. Directly from (23) and $\int_1^t\psi(u)\,du/u=t-1-\frac32\log t$ for $1\le t\le2$, $$\left\|\frac{Y_X(n)}{\log X} -\frac\alpha3\left(\log(2\pi)-2\frac nX+ 3\log\frac nX\right)\right\|_{X,2} \longrightarrow0.$$ This frozen endpoint is distinct from the full moving diagonal $Y_n(n)$.

Conductor decomposition of the drift

Möbius inversion gives the exact finite expansion $$y_n=\frac23\sum_{2\le q\le n}\eta_{q,n}P_q(n).$$ The elementary estimates $|P_q(n)|\le\sigma_1(q)/2$ and $\eta_q-\eta_{q,n}\ll\log(2n)/(qn)$ imply, for $Q\le X<n\le2X$, $$\left|\frac23\sum_{q\le Q}(\eta_{q,n}-\eta_q)P_q(n)\right| \ll Q\log(2X)/X.$$ The sum of $\sigma_1(q)/q$ up to $Q$ is $O(Q)$ by divisor interchange.

The low and intermediate additive conductors approximate the persistent fluctuation. Their complement carries the deterministic diagonal drift in empirical mean square. This is not a uniform pointwise tail bound. The cutoff $Q$ here retains full coefficients $\eta_q$, whereas the period cutoff $M$ retains $\eta_{q,M}$. The threshold in Theorem 13 is proved for the latter.

Progression means and signed exceptional resolutions

The construction does not assert signed asymptotics at the single canonical integers $L_m$ and $L_m-1$. It gives explicitly bounded progressions and a definite finite selection rule. For fixed $0<\vartheta<1$, the negative progression also contains $\gg_\vartheta X C_L^2/L^2$ integers with $\mathcal R_n<-\vartheta C_L$, when its mean is $-C_L+o(C_L)$. Indeed the total negative excess beyond this threshold is $\gg(1-\vartheta)XC_L/L$. Cauchy–Schwarz and $\sum_{X<n\le2X}\mathcal R_n^2=O(X)$ give the count. The positive statement follows in the same way.

For each fixed modulus $L$, the empirical conditional law on $n\equiv a\pmod L$ also tends to the Haar conditional law of $Y$. The proof approximates by fixed periodic $Y_M$; restricting a global mean-square estimate to this progression costs at most a factor $O(L)$ in the squared norm. Let $X\to\infty$ and then $M\to\infty$. This argument does not establish a conditional law for growing $L$. The growing-modulus result used above is specifically the first-moment estimate (59).

Mass perturbations preserving the centering

The boundary selection and the arithmetic transfer have different information requirements. The exact boundary fixes every period mass, whereas the following perturbations preserve all four smooth constants.

For the explicit mass from [4], $$h_k=\varepsilon(\mathbf 1_{k=2}-2\mathbf 1_{k=3}+\mathbf 1_{k=4}), \qquad 0<|\varepsilon|<1/24,$$ the potential changes by the nonzero period-twelve function $Z_h=\frac{2\varepsilon}{3}(E_2-2E_3+E_4)$. For every $M\ge4$, $$\mathcal R_n^v-Y_M^v(n)=\mathcal R_n-Y_M(n).$$ For $n,M\ge4$, the same exact identity holds with the centered diagonals $y_n^v-b\log n-c$ and $Z_n$ in place of the residuals. Thus the Bernoulli boundary term in (50) and its cutoff scale remain unchanged. The stationary part of the limiting law is $Y_v$.

Only the conductor amplitudes at two, three and four change. Every larger amplitude and the sharp Haar spectral-tail constant remain the same, but the exact mixed-moment characterization in [4] fails. These masses share the smooth profiles without representing the same boundary energy. Bounded changes also preserve the signed lower constants in (61); they supply no sharper uniform pointwise estimate.

Exceptional sets and empirical moments

The input $O(\log^{5/3}n)$ is the continuous capacity estimate from [6]. Its proof passes through the weighted fractional-part and Möbius-reduction Lemmas 6–7 and the fractional-floor Proposition 8 there. The first step uses [8]. The exponent $2/3$ in that fractional-part estimate acquires an additional logarithm under partial summation with the harmonic weight. The mean-square transfer above does not improve this uniform exponent. In particular the finite-$X$ error in (64) must remain when considering extreme values. A small stationary tail probability alone does not determine the maximum along integer addresses.

Translated alignment with bounded endpoint contributions

The signed extremes need not coincide with large current divisibility. This can be stated directly in the conductor expansion. Define $$\mathcal U_n=\frac23 \sum_{\substack{2\le q\le n\\q\nmid n,\ q\nmid n+1}} \eta_{q,n}P_q(n)-b\log n-c.$$

Thus a residual at the known lower scale can persist after the divisibility producing a nearby reset has disappeared. The exact one-sided increment bound $D_{n+1}-D_n\le2/3$ is consistent with this persistence. A large negative residual cannot recover in a single forward step. The smooth centering changes by only $O(\log n/n)$.

Theorem 21 disproves bounds of the form $|\mathcal U_n|\ll1+C_n+C_{n+1}$. It matches the known lower scale and constant; it supplies neither a larger construction nor an upper bound of that order.

Limits of uniform conductor truncation

For $2\le Q\le n$ put $$\mathcal T_Q(n)=\frac23\sum_{Q<q\le n}\eta_{q,n}P_q(n) -b\log n-c.$$ The exact decomposition and coefficient replacement give $$\mathcal R_n=F_Q(n)+\mathcal T_Q(n) +O\left(\frac{Q\log(2n)}n+\frac{\log^2(2n)}n\right).$$

This is a necessary constraint on uniform conductor approximation, not a sufficient threshold or a sharp pointwise maximum. The known all-integer upper estimate remains $O(\log^{5/3}n)$, while both signed lower constructions have scale $(\log\log n)^2$. The empirical laws and the period-cutoff theorem leave this gap open.

The difference between a small conductor cutoff and a full compatible cylinder is itself visible exactly. The cylinder $x=0\bmod L_m$ includes many composite conductors larger than $m$. Their total endpoint magnitude is $$C_{L_m}-\frac13\sum_{2\le q\le m}\varphi(q)\eta_q =\left(\Gamma-\frac1{\pi^2}+o(1)\right)\log^2m.$$ All these conductors use the same small prime-power coordinates. Their collective signed contribution survives a cutoff at $q=m$.

The translated residue classes in Theorem 21 retain fluctuations of both signs at the complete-alignment lower scale, even after both endpoint-divisor sectors have been removed. Controlling the remaining signed conductor sum uniformly is still necessary to determine the pointwise maximum.

Finite numerical results

Finite evaluations in nfield [7] use the coprime weights in (1). A divisor recurrence evaluates the capacity without enumerating the full square of source pairs, $$D_n-D_{n-1}=\frac23\left(1-\sum_{k\mid n}a_k\right).$$ The exact diagonal dictionary then gives $y_n$, $Z_n$ and $\mathcal R_n$. Independent rational checks compare these evaluations with direct coprime sums and the defining gcd capacity for every $1\le n\le240$. They also check the cumulative diagonal identity, finite conductor decomposition, CRT conditional means, unit-phase reflection, and the character correlation that supplies the variance lower bound.

Table 1 records long-double evaluations on complete integer windows. The prime column averages over exactly $X<p-1\le2X$. At $X=10^6$ it contains $70\,435$ primes. All displayed values are rounded to six decimal places.

For the boundary formula in Lemma 12, take $X=10^6$, $M=72\,382$ and the $401$ sample addresses $n_j=X+1+\lfloor j(X-1)/400\rfloor$, $0\le j\le400$. The root-mean-square difference $Z_{n_j}-Y_M(n_j)$ is approximately $0.00962$. Subtracting the stated Bernoulli boundary term reduces it to approximately $0.000117$. No constants are fitted. These finite evaluations illustrate the identities and scale separation. The asymptotic rates, independence and infinite limiting laws are established by the preceding proofs.

Arithmetic fluctuation and observation

The finer remainder in collision capacity can now be interpreted. For $D_n=n-J(n)$, subtracting only the quadratic-logarithmic profile leaves a deterministic term $B\log n+C$ as well as the fluctuation. A distribution formed from that partially centered remainder continues to drift. After the full explicit centering, the distribution and second moment over $X<n\le2X$ converge to those of the stationary potential. The transfer theorem supplies the missing connection between the profinite model and the sequence of finite capacities.

The sampling rule carries arithmetic information. At $n=p-1$, the admissible phases replace the mean-zero Haar law by a distinct law with mean $-1/3$. Holding the period cutoff fixed across each observation interval creates a different effect. Near $M\asymp X/\log X$, an independent Bernoulli boundary term changes the limiting distribution and contributes exactly $\lambda^2/(90\pi^4)$ to its second moment. The changed mean and variance therefore have identifiable causes in the observation rule.

These averaged laws leave a precise pointwise problem. The translated progressions produce large residuals even with bounded endpoint-divisor contributions, so an estimate of the maximum must also control the remaining signed conductor sum. The finer transfer from continuous capacity at $p-1$ to the original finite digit-grid energy requires a separate estimate for the sampling defect. Neither issue is resolved by the empirical law.

The main and secondary terms describe the smooth capacity profile. The present results explain how to read the variation around it. Balanced periodic components do yield a stable arithmetic law, once the bias from admitting unfinished cycles has been removed and the chosen sampling ensemble has been accounted for.

References