The main and secondary terms of collision capacity leave a finer arithmetic fluctuation. Interpreting it requires separating that fluctuation from the bias introduced by the growing observation cutoff. We prove this separation for the gcd capacity J(n)=\frac13\sum_{r,s\le n}(r,s)^2/(rs). We determine the logarithmic and constant terms beneath the leading \pi^{-2}\log^2 n deficit. After this centering, the residual’s empirical distribution and second moment on dyadic integer intervals converge to those of the stationary profinite potential, with quantitative rates. The transfer is substantive because the canonical potential cutoffs diverge at every fixed integer, while the moving diagonal has a negative logarithmic drift. At resolutions one below a prime, the same centering gives a different law of mean -1/3, symmetric about that mean and with positive finite variance. For frozen cutoffs M\to\infty with M=o(X), the stationary second moment transfers precisely when M\log M=o(X); at the critical scale an independent Bernoulli-polynomial boundary term changes the limiting law. Exceptional residuals of both signs also persist when the endpoint-divisor contributions are bounded, so average control does not settle the pointwise maximum. The results give an explicit rule for recovering the stationary fluctuation from finite capacity measurements and identify the separate effects of prime sampling and the observation boundary.
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.
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.
Proposition 1 (Stationary input). The following facts hold for the weights in (1). \begin{aligned} &\sum_{k\ge2}w_k=1,\quad w_k\ll\frac{\log(2k)}{k^2},\quad \eta_q\ll\frac{\log(2q)}{q^2}, \\ &D_n=\frac23\sum_{k\ge2}w_kr_k(n),\quad D_*(M)=\frac13\sum_{k\le M}w_k(k-1), \\ &\left\|Y-Y_M\right\|_2\ll\frac{\log(2M)}{\sqrt M},\quad \sigma^2=\frac1{27}\sum_{q\ge2}J_2(q)\eta_q^2>0. \end{aligned} The Fourier coefficient of Y_M at a reduced nonzero frequency a/q is -\frac{2\eta_{q,M}}{3(1-\mathrm{e}(-a/q))}. For L\ge2, put \begin{aligned} m_L(a)&=\mathbb E(Y\mid r_L=a) =\frac23\sum_{k\ge2}w_kE_{(k,L)}(a),\\ C_L&=\frac13\sum_{\substack{q\mid L\\q\ge2}}\varphi(q)\eta_q,\qquad L_m=\operatorname{lcm}(1,\ldots,m),\qquad \Gamma=\frac{2e^\gamma}{\pi^2}. \end{aligned} Then m_L(0)=-C_L, m_L(-1)=C_L, C_{L_m}\sim\Gamma\log^2m, and m_{L_m}(1)=-C_{L_m}+2/3+O(\log(2m)/m). The law of Y is symmetric and has, for every c_0<\pi, \mathbb P(|Y|>H)\ll_{c_0}e^{-c_0\sqrt H}; the same upper bound holds uniformly for Y_M. Finally, define F_Q=\frac23\sum_{2\le q\le Q}\eta_qP_q. Then \sup_a|F_Q(a)|=\frac13\sum_{2\le q\le Q}\varphi(q)\eta_q =\frac{\log^2Q}{\pi^2}+O(\log(2Q)).
Proof. The mass and exact readout are [4]; the L^2 limit and Fourier normalization are [4]. The cylinder identities, sharp alignment, and translated mean are [4]. The symmetric law and uniform tails are [4], and the finite extrema are [4]. Their normalizations can also be checked directly. Reduced ratios in the gcd sum give 3J(n)=n+2\sum_{k\le n}a_k\lfloor n/k\rfloor. The primitive-pair normalization gives unit mass, and Euclidean division gives (8). The finite covariance \mathbb EE_kE_l=((k,l)^2-1)/12 gives (9). Finite Fourier summation gives (10). Conditioning on a residue modulo L gives \mathbb E(E_k\mid r_L=a)=E_{(k,L)}(a). The sharp alignment constants, uniform tails, and positive-regrouping extremum in (15) use those separate companion proofs; no arithmetic transfer result below is used there. ◻
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.
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.
Lemma 2 (Precise cumulative weight). Uniformly for real t\ge2, S_a(t):=\sum_{k\le t}a_k =\frac\alpha2\log^2t+\beta\log t+C_a +O\left(\frac{\log^2(2t)}t\right).
Proof. Möbius inversion of the coprime harmonic sum and the elementary harmonic product identity give exactly S_a(t)=\frac12\sum_{d\le t}\frac{\mu(d)}{d^2} \bigl(H_{\lfloor t/d\rfloor}^2- H_{\lfloor t/d\rfloor}^{(2)}\bigr). For u\ge1, the inner expression including its factor 1/2 is \tfrac12\log^2u+\gamma\log u+ \tfrac12(\gamma^2-\zeta(2))+O(\log(2u)/u). The summed error is at most C t^{-1}\sum_{d\le t}\log(2t/d)/d\ll\log^2(2t)/t. Extending the main polynomial in \log(t/d) to all d\ge1 costs O(1/t) by the integral test. Expansion of that polynomial gives exactly (16). ◻
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.
Proposition 3 (Exact diagonal identities). With W_n=\sum_{k\le n}w_k, \begin{aligned} y_n&=\frac23\sum_{k\le n}a_k\psi(n/k)+\frac13W_n, \\ y_n&=\frac23\sum_{d\le n}\frac{\mu(d)}{d^2} \sum_{j\le n/d}\frac{H_{j-1}}j\psi(n/(dj)) +\frac13W_n, \\ \sum_{n\le N}y_n &=-\frac13\sum_{k\le N}w_k (r_k(N)+1)(k-1-r_k(N))\le0. \end{aligned}
Proof. The equality r_k(n)-(k-1)/2=k\psi(n/k)+1/2 proves (20), and Möbius inversion proves (21). For the last identity interchange the finite sums. Complete cycles of a centered remainder have sum zero, while \sum_{n=k}^N E_k(n) =-\tfrac12(r_k(N)+1)(k-1-r_k(N)). ◻
The incomplete cycles in this formula give the moving diagonal a nonpositive cumulative sum.
Lemma 4 (Uniform tail comparison). For integers 2\le M\le X<n\le2X, \begin{split} y_n-Y_M(n) ={}&\frac23\int_M^n\frac{\alpha\log t+\beta}{t}\psi(n/t)\,dt\\ &+O\left(\frac{X\log^2(2X)}{M^2}+\frac{\log(2M)}M\right). \end{split} Consequently, with Z_n=y_n-b\log n-c, |Z_n-Y_M(n)| \ll\frac{X\log^2(2X)}{M^2} +\frac{M\log(2X)}X+\frac{\log(2M)}M.
Proof. Apply summation by parts in (20), using (17). Its remainder E(t) is O(\log^2(2t)/t) on either side of an integer. Between its jumps, \psi(n/t) has derivative -n/t^2. Its jumps are at t=n/j. Besides the endpoint terms, the error is bounded by n\int_M^n\frac{\log^2(2t)}{t^3}\,dt +\sum_{j\le n/M}\frac jn\log^2(2n/j) \ll\frac{n\log^2(2n)}{M^2}. At coincident jumps the convention \psi(j)=-1/2 is retained; the atom and either one-sided remainder obey the same bound. The term (W_n-W_M)/3 costs O(\log(2M)/M). This proves (23).
The change of variable u=n/t transforms its integral into \frac23\int_1^{n/M} \frac{\alpha\log n+\beta-\alpha\log u}{u}\psi(u)\,du. The omitted tail to infinity is O(M\log(2n)/n), by the bounded periodic primitive B_2/2. The infinite integral equals b\log n+c by (18) and (19). This proves (25). ◻
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.
Lemma 5 (Growing-cutoff comparison). For 2\le L\le M, \left\|Y_M-Y_L\right\|_{X,2} \ll\frac{\log(2L)}{\sqrt L} +\sqrt{\frac MX}\log^{3/2}(2M).
Proof. At reduced frequency a/q, the coefficient is -2(\eta_{q,M}-\eta_{q,L})/[3(1-\mathrm{e}(-a/q))]. For q\le L, its mass difference is O(\log(2L)/(qL)); for q>L, it is O(\log(2q)/q^2). The primitive cosecant-square identity \sum_{(a,q)=1}\csc^2(\pi a/q)=J_2(q)/3 therefore bounds the Haar squared norm on a dyadic block Q<q\le2Q by C\begin{cases} Q\log^2(2L)/L^2,&Q\le L,\\ \log^2(2Q)/Q,&Q>L. \end{cases} A block intersecting L may be split.
Distinct reduced frequencies in this block are separated on the circle by at least 1/(4Q^2). The unnormalized finite-interval Gram kernel has magnitude at most \min(X,C/\|\theta-\theta'\|). In each row, frequency separation bounds the sum of off-diagonal absolute values by CQ^2\log(2Q). Schur’s matrix estimate gives \left\|A_Q\right\|_{X,2}^2 \le C\left(1+\frac{Q^2\log(2Q)}X\right)\left\|A_Q\right\|_2^2 for every polynomial in that block. Apply this to (27), take square roots, and sum over dyadic Q. The Haar contributions sum to O(\log(2L)/\sqrt L), and the remaining geometric sums are O(\sqrt{M/X}\log^{3/2}(2M)). ◻
Lemma 6 (Periodic first and second moments). For L\ge2, \begin{aligned} |\mathbb E_XY_L|&\ll L\log(2L)/X, \\ |\mathbb E_XY_L^2-\left\|Y_L\right\|_2^2| &\ll L^2\log^2(2L)/X. \end{aligned} The first bound holds with a factor h(n/X) for any fixed bounded-variation function h, with a constant depending on its supremum and total variation.
Proof. Each centered residue has period k, mean zero, and incomplete sum O(k^2). Summing with weights uses \sum_{k\le L}w_kk^2\ll L\log(2L). Each product E_kE_l has period [k,l]\le kl and magnitude O(kl), so its incomplete-average error is O(k^2l^2/X). Summing proves (30). Partial summation proves the weighted first bound. ◻
Theorem 7 (Centered diagonal transfer). For Z_n in Lemma 4, \begin{aligned} \mathbb E_X Z_n&=O(X^{-1/3}\log^2(2X)),\\ \mathbb E_X Z_n^2&=\sigma^2+O(X^{-1/5}\log^2(2X)). \end{aligned} The empirical joint law of (Z_n,n/X) tends to (Y,T), with T uniform on [1,2] and independent of Y. For every fixed bounded-variation h, \mathbb E_X h(n/X)Z_n=O_h(X^{-1/3}\log^2(2X)).
Proof. Take M=\lfloor X^{3/5}\rfloor in (25) and L=\lfloor X^{2/5}\rfloor in (26). Then \left\|Z-Y_L\right\|_{X,2}\ll X^{-1/5}\log^2(2X). Lemma 6 and (9) give the asserted second moment by Cauchy–Schwarz. For the first moment take M=\lfloor X^{2/3}\rfloor in (25) and apply (29), including its bounded-variation form. This proves (31) and (33).
For the law, keep L fixed first. The pair (Y_L(n),n/X) has limiting law equal to an independent uniform residue and macroscopic coordinate. Lemmas 4 and 5, with M=\lfloor X^{3/5}\rfloor, give \limsup_{X\to\infty}\left\|Z-Y_L\right\|_{X,2} \ll\log(2L)/\sqrt L. Let L\to\infty using the Haar L^2 limit. This proves the joint law with an explicit uniform approximation, rather than an application of unique ergodicity to a changing function. ◻
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.
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.
Corollary 8 (Centered capacity law). The residual \mathcal R_n=D_n-\frac{\log^2n}{\pi^2}-B\log n-C satisfies \mathcal R_n=Z_n+O(\log^2(2n)/n). Its empirical law is that of Y, and its first and second moments satisfy (31) and (32). In particular, for every H>0, \frac1X\#\{X<n\le2X:|\mathcal R_n|>H\} \le\frac{\sigma^2+o(1)}{H^2}.
Proof. Insert (19), (36), and (37) into the conserved-state equation. The logarithmic and constant terms combine to (38). Theorem 7 and Chebyshev’s inequality finish the proof. ◻
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.
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.
Proposition 9 (The unit-phase limit). There is a limit Y^\times\in L^2(\mathcal U,\nu) with \left\|F_M-Y^\times\right\|_{L^2(\nu)} \ll\frac{\log^{3/2}(2M)}{\sqrt M}. Its mean is -1/3. The law of O^\times=Y^\times+1/3 is symmetric, and its variance satisfies \sigma_\times^2:=\mathbb E_\nu (O^\times)^2\ge\frac1{324}.
Proof. For a dyadic block M<k\le2M, set Q=\operatorname{lcm}(1,\ldots,\lfloor2M\rfloor). For every nonnegative function of the residue modulo Q, its average over units is at most Q/\varphi(Q) times its average over all residues. The elementary Mertens product bound gives Q/\varphi(Q)=\prod_{\ell\le2M}(1-1/\ell)^{-1}\ll\log(2M). Translation preserves the additive Haar norm. Thus (9) gives \left\|F_{2M}-F_M\right\|_{L^2(\nu)} \ll\frac{\log^{3/2}(2M)}{\sqrt M}. The same argument applies to an incomplete block. Summing over dyadic blocks proves convergence and (41) for all M. Equation (40) and W_M\to1 prove the mean and symmetry.
For nondegeneracy, let \chi_3 be the nonprincipal character modulo three. If 3\nmid k, the unit coordinates modulo 3 and k are independent, so their character correlation vanishes. If 3\mid k, inclusion–exclusion gives \sum_{a\in U_k}a\chi_3(a) =-\frac k3\prod_{\substack{\ell\mid k\\\ell\ne3}} (1-\chi_3(\ell))\le0. Indeed, \sum_{j<h}j\chi_3(j)=-h/3 for 3\mid h; terms in inclusion–exclusion with divisor divisible by three vanish. Every remaining local factor is zero or two. Consequently every nonzero summand of \mathbb E_\nu O_M\chi_3 is nonpositive. At k=3, w_3=1/6 and the summand is -1/18. Taking the L^2 limit gives \mathbb E_\nu O^\times\chi_3\le-1/18. Since \left\|\chi_3\right\|_{L^2(\nu)}=1, Cauchy–Schwarz proves (42). ◻
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.
Lemma 10 (Small periods on primes). Fix A,D>0. Set L=\lfloor(\log X)^A\rfloor. For j=1,2 the moment comparison is \mathbb E_X^{\rm pr}Y_L^j=\mathbb E_\nu F_L^j+O_{A,D}((\log X)^{-D}). The second-moment comparison also holds for Y_L-Y_R and F_L-F_R, uniformly for 2\le R\le L. The constants may be ineffective.
Proof. Siegel–Walfisz, in the form stated in [9], followed by removal of prime powers and partial summation, gives for each fixed A,B>0 \frac{\#\{p\in\mathcal P_X:p\equiv a\pmod q\}}{P_X} =\frac1{\varphi(q)}+O_{A,B}((\log X)^{-B}) uniformly for (a,q)=1 and q\le(\log X)^{2A}. Each term in a first or second moment has period q=[k,l]\le L^2 (with one period for the first moment) and magnitude O(kl). Summing the residue-class errors and the weights costs at most L^2(\log X)^{-B} \left(1+\sum_{k\le L}kw_k\right)^2 \ll L^2(\log X)^{-B}\log^4(2L). Choose B sufficiently large in terms of A,D. The same calculation applies to any subinterval of period indices, which proves the assertion for differences. In particular no modulus as large as \operatorname{lcm}(1,\ldots,L) is required for these moments. ◻
Theorem 11 (Prime-indexed diagonal transfer). With p uniform on \mathcal P_X, Z_{p-1}\xrightarrow{\rm law}Y^\times,\qquad \mathcal R_{p-1}\xrightarrow{\rm law}Y^\times. For every fixed D>0, \begin{aligned} \mathbb E_X^{\rm pr}Z&=-\frac13+O_D((\log X)^{-D}),\\ \mathbb E_X^{\rm pr}(Z+1/3)^2 &=\sigma_\times^2+O_D((\log X)^{-D}). \end{aligned} Both estimates also hold with Z replaced by \mathcal R. The constants in these bounds may be ineffective.
Proof. For any sequence H on the integer interval, \left\|H\right\|_{{\rm pr},X,2} \le\sqrt{X/P_X}\left\|H\right\|_{X,2} \ll\sqrt{\log X}\left\|H\right\|_{X,2}. Take M=\lfloor X^{3/5}\rfloor and L=\lfloor(\log X)^A\rfloor, where A>1 is fixed. Apply this domination to Lemma 5, and use the uniform bound (25) directly for Z-Y_M. It follows that \left\|Z-Y_L\right\|_{{\rm pr},X,2} \ll X^{-1/5}\log^2(2X) +\frac{\sqrt{\log X}\log(2L)}{\sqrt L}. The loss from the sparse prime ensemble is explicit. A fixed L would not suffice for this bound; a sufficiently large logarithmic power does.
Proposition 9 and Lemma 10 bound the first two prime moments of Y_L and give their limits. Use (49) and Cauchy–Schwarz for the difference of second moments. Choosing, for example, A>2D+3 gives (47)–(48) with the stated error.
For the full law, fix R. Equidistribution of primes modulo \operatorname{lcm}(1,\ldots,R) gives the law of Y_R(p-1) as that of F_R. The difference version of Lemma 10 and (41) imply \limsup_{X\to\infty} \left\|Z-Y_R\right\|_{{\rm pr},X,2} \ll\frac{\log^{3/2}(2R)}{\sqrt R}. First apply bounded Lipschitz test functions and let X\to\infty, then let R\to\infty. This proves the first law in (46). Finally Corollary 8 gives a uniform O(\log^2 X/X) difference between \mathcal R and Z throughout the interval, proving all assertions for \mathcal R. ◻
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.
Lemma 12 (Bernoulli boundary term). Uniformly for 2\le M\le X<n\le2X, \begin{split} Z_n={}&Y_M(n)+\frac{(\alpha\log M+\beta)M}{3n} B_2(\{n/M\})\\ &+O\left(\frac{X\log^2(2X)}{M^2} +\frac{\log(2M)}M+\frac{M^2\log(2M)}{X^2}\right). \end{split}
Proof. The omitted tail in the proof of Lemma 4 is \int_{n/M}^{\infty} \frac{\alpha(\log n-\log u)+\beta}{u}\psi(u)\,du. Its first integration-by-parts boundary term is -\frac{M}{2n}(\alpha\log M+\beta)B_2(\{n/M\}). The periodic polynomial B_2 has mean zero and bounded primitive B_3/3, where B_3(u)=u^3-\frac32u^2+\frac12u. A second integration by parts bounds the remainder by O((M/n)^2\log(2M)). The omitted tail enters Z_n-Y_M(n) with factor -2/3. This fixes both the sign and normalization in (50). Add the summation-by-parts error from (23). ◻
Theorem 13 (Sharp frozen-period threshold). Let M=M(X) be integer valued, M\to\infty, M=o(X), and set \lambda_X=M\log M/X.
If \lambda_X\to0, the empirical law of Y_M(n) tends to that of Y, its mean tends to zero, and its second moment tends to \sigma^2.
If \lambda_X\to\lambda\in(0,\infty), the limiting law is Y-\frac{\alpha\lambda}{3T}B_2(U), where Y,T,U are independent, T is uniform on [1,2], and U is uniform on [0,1]. The mean tends to zero and \mathbb E_XY_M(n)^2\longrightarrow \sigma^2+\frac{\alpha^2\lambda^2}{3240} =\sigma^2+\frac{\lambda^2}{90\pi^4}.
If \lambda_X\to\infty, the law of Y_M(n)/\lambda_X tends to -\alpha B_2(U)/(3T), and its second moment tends to \alpha^2/3240.
Within the stated regime, stationary second-moment transfer holds if and only if M\log M=o(X).
Proof. When M\le X^{3/5}, Lemma 5 approximates Y_M in empirical mean square by Y_L for fixed L, with error tending to O(\log(2L)/\sqrt L). Periodic averaging and then L\to\infty give the first conclusion in this range. For larger M with \lambda_X\to0, equation (50) shows that Y_M-Z tends uniformly to zero. The boundary is O(\lambda_X), and its final error is O(\lambda_X M/X).
For a positive finite limit of \lambda_X, the cutoff has order X/\log X. Every error in (50) tends to zero. The boundary coefficient approaches \alpha\lambda/(3(n/X)). Jointly with any fixed residue class, (n/X,\{n/M\}) tends to the independent uniform macroscopic and phase coordinates. To see this, on a progression of fixed period the normalized sum of each fixed nonzero phase mode is O(M/X); for large M there is no alias with that fixed period. Summation by parts allows continuous macroscopic weights. Approximate Z by fixed periodic Y_L as in Theorem 7, then let L\to\infty. The approximation is in mean square, so it also transfers the mixed second moments with the bounded boundary term.
Finally \int_0^1B_2(u)^2\,du=1/180 and \int_1^2t^{-2}\,dt=1/2. These integrals give (53). In the third regime divide (50) by \lambda_X. Its errors tend to zero, since M=o(X) and M\log M/X\to\infty, while Z/\lambda_X\to0 in mean square. The same phase argument gives the normalized law and moment. If \lambda_X does not tend to zero, a subsequence has a positive finite or infinite limit, so its second moment cannot tend to \sigma^2. ◻
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).
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.
Theorem 14 (Conductor allocation of the bias). For each fixed 0<\theta<1, with Q=\lfloor X^\theta\rfloor, \begin{aligned} \left\|Z-F_Q\right\|_{X,2}&\longrightarrow0,\\ \left\|\frac23\sum_{Q<q\le n}\eta_{q,n}P_q(n) -b\log n-c\right\|_{X,2}&\longrightarrow0. \end{aligned}
Proof. For fixed L\le Q, apply the block calculation in Lemma 5 to F_Q-Y_L. For q\le L the coefficient difference is bounded by the omitted mass \log(2L)/(qL), and for L<q\le Q by \log(2q)/q^2. Thus \left\|F_Q-Y_L\right\|_{X,2} \ll\frac{\log(2L)}{\sqrt L} +\sqrt{\frac QX}\log^{3/2}(2Q). Combine with the fixed-L approximation to Z, take X\to\infty, and then L\to\infty. This proves (57). Equations (55) and (56) prove (58). ◻
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.
Theorem 15 (Uniform progression means). Uniformly in a\bmod L, for 1\le L\le\sqrt X, \operatorname{Avg}_{\substack{X<n\le2X\\n\equiv a\ (L)}}\mathcal R_n =m_L(a)+O\left((L^2/X)^{1/3}\log^2(2X)\right).
Proof. On a progression modulo L, the sequence E_k(n) has period k/(k,L) in progression steps, magnitude O(k), and mean E_{(k,L)}(a). Its incomplete-average error is O(Lk^2/(X(k,L))). Therefore \left|\operatorname{Avg}_{n\equiv a\ (L)}Y_M(n) -\mathbb E(Y_M\mid r_L=a)\right| \ll LM\log(2M)/X. The conditional tail has absolute value O(L\log(2M)/M) by (11) and the weight tail. Finally (25) and Corollary 8 compare \mathcal R_n with Y_M(n) uniformly. Take M=\lfloor(X^2/L)^{1/3}\rfloor. All errors are bounded by the one in (59). The progression has \asymp X/L elements in the stated range, so normalization by its actual count changes only the absolute constants. ◻
Corollary 16 (Exceptional residuals of both signs). The capacity residual satisfies \limsup_{n\to\infty}\frac{\mathcal R_n}{(\log\log n)^2}\ge\Gamma,\qquad \liminf_{n\to\infty}\frac{\mathcal R_n}{(\log\log n)^2}\le-\Gamma. Witnesses may be selected by finite minimization and maximization in specified progressions with L_m^3<n\le2L_m^3.
Proof. Take L=L_m and X=L^3 in (59). Its error is O(L^{-1/3}\log^2(2X))=o(1). Choose the smallest minimizer in the class zero, and the smallest maximizer in the class minus one, within the indicated interval. Their values are at most -C_L+o(1) and at least C_L+o(1), respectively. Since C_{L_m}\sim\Gamma\log^2m and \log\log n=\log m+O(1), this proves (61). ◻
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).
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.
Corollary 17 (Moment-preserving perturbations). Let h_k be real numbers such that \sum_{k\ge2}k|h_k|<\infty,\qquad \sum_{k\ge2}h_k=\sum_{k\ge2}kh_k=0,\qquad v_k=w_k+h_k\ge0. Define Z_h(x)=\frac23\sum_{k\ge2}h_kE_k(x),\qquad Y_v=Y+Z_h, \qquad D_n^v=\frac23\sum_{k\ge2}v_kr_k(n), and form Y_M^v,y_n^v,D_{*,v}(M) by replacing w with v in the corresponding age definitions. Then Z_h is bounded, continuous and mean zero, and \begin{aligned} D_n^v&=D_n+Z_h(n),& D_{*,v}(M)-D_*(M)&\longrightarrow0. \end{aligned} With the same constants b,c,B,C as in (19) and (38), both y_n^v-b\log n-c and \mathcal R_n^v=D_n^v-\pi^{-2}\log^2n-B\log n-C have limiting empirical law Y_v, limiting mean zero, and limiting second moment \left\|Y_v\right\|_2^2. For any scale A(n)\to\infty, \frac{\mathcal R_n^v-\mathcal R_n}{A(n)}\longrightarrow0.
Proof. The bound |E_k|\le(k-1)/2 gives uniform convergence of Z_h, and \mathbb EE_k=0 gives its mean. The two vanishing moments remove the centered constants from the integer age difference and imply D_{*,v}(M)-D_*(M)=\frac13\sum_{k\le M}(k-1)h_k\to0. They also give y_n^v-y_n=Z_h(n)+o(1) uniformly as n\to\infty. The omitted part is bounded by \frac13\sum_{k>n}(k-1)|h_k|\longrightarrow0.
For a finite truncation Z_{h,K}, the joint empirical law of (\mathcal R_n,Z_{h,K}(n)) is the Haar law of (Y,Z_{h,K}). Indeed Z_{h,K} is constant on each residue modulo L_K, and the fixed-residue conditional transfer just proved applies to all these finitely many classes. Let K\to\infty using uniform convergence. This proves the law of \mathcal R_n+Z_h(n), and hence both asserted laws.
For the cross term in the second moment, apply Theorem 15 at fixed modulus L_K to obtain \mathbb E_X\mathcal R_nZ_{h,K}(n)\to\mathbb EYZ_{h,K}. The bounded empirical second moments of \mathcal R_n, together with uniform convergence of Z_{h,K}, permit K\to\infty. Periodic approximation also gives \mathbb E_XZ_h(n)^2\to\mathbb EZ_h^2 and \mathbb E_XZ_h(n)\to0. Expansion of the square proves the remaining moment assertions. The final statement follows from the bounded exact difference \mathcal R_n^v-\mathcal R_n=Z_h(n). ◻
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.
Theorem 18 (Quantitative exceptional-set bound). For every fixed 0<c_0<\pi, uniformly for 1\le H\le\log^2X/(4\pi^2), \frac1X\#\{X<n\le2X:|\mathcal R_n|>H\} \ll_{c_0}e^{-c_0\sqrt H} +\frac{X^{-2/5}\log^4(2X)}{H^2}. For every A>0 there is C_A such that |\mathcal R_n|\le C_A(\log\log X)^2 outside a set of O_A(X/\log^A X) integers in the dyadic interval. For every fixed positive integer r, \mathbb E_X\mathcal R_n^r\longrightarrow\mathbb EY^r.
Proof. The proofs of Theorem 7 give uniformly for 2\le L\le X^{3/5}, \left\|\mathcal R-Y_L\right\|_{X,2} \ll\frac{\log(2L)}{\sqrt L} +X^{-1/5}\log^2(2X). Choose c_0<d<\pi and L=\lfloor e^{d\sqrt H}\rfloor. The imposed range gives L\le X^{1/2}, up to a harmless constant for small H. The exact bound |Y_L|\le D_*(L) and D_*(L)=\pi^{-2}\log^2L+O(\log(2L)) leave a fixed fraction of H between D_*(L) and H once H is large. Chebyshev’s inequality therefore bounds the exceptional proportion by O\left(\frac{\log^2L}{LH^2} +\frac{X^{-2/5}\log^4(2X)}{H^2}\right), which gives (64); bounded H is absorbed into its constant. Taking H=C_A(\log\log X)^2 proves the exceptional-count assertion.
For the moment claim, the continuous secondary estimate in [6] gives \mathcal R_n=O(\log^{5/3}(2n)). This is below the upper range in (64) for large X. Integrating the tail estimate against rH^{r-1}, up to this pointwise maximum, gives uniform integrability of every fixed power. The exponentially decaying term has arbitrarily small tail after a fixed threshold; the second term contributes X^{-2/5} times a fixed power of \log X, hence tends to zero. Combine with the weak limit from Corollary 8. ◻
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.
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.
Lemma 19 (Endpoint separation). For n\ge2, \mathcal R_n=-C_n+C_{n+1}+\mathcal U_n +O(\log(2n)/\sqrt n).
Proof. At q\mid n, P_q(n)=-\varphi(q)/2; at q\mid n+1 it equals \varphi(q)/2. These sets are disjoint for q\ge2. Replacing \eta_{q,n} by \eta_q on either divisor sector costs O\left(\frac{\log(2n)}n \sum_{q\mid n\ \text{or}\ n+1}\frac{\varphi(q)}q\right) \ll\frac{\log(2n)}{\sqrt n}, using the elementary divisor bound \tau(n)\le2\sqrt n. The absent row q=n+1 costs O(\log(2n)/n). Apply (55) and \mathcal R_n=y_n-b\log n-c+O(\log^2(2n)/n). ◻
Lemma 20 (A rough-factor bound). Suppose N=tv, with t in a fixed finite set, (t,v)=1, every prime factor of v exceeding m, and \log N=O(m). Then C_N=O(1), uniformly under these restrictions.
Proof. Since \varphi(q)\eta_q\ll\log(2q)/q, it is enough to bound \sum_{d\mid N}\log(2d)/d. For the rough factor, \sigma_{-1}(v)\le\prod_{p\mid v}(1-p^{-1})^{-1} \le\exp\left(O\left(\frac{\log v}{m\log m}\right)\right)=O(1). The logarithmically weighted local geometric sums give \sum_{d\mid v}\frac{\log d}{d} \le\sigma_{-1}(v)\sum_{p\mid v}\frac{\log p}{p-1} \ll\log v/m=O(1). The fixed factor t changes only the implied constant. ◻
Theorem 21 (Large partial alignments with bounded endpoints). There are sequences tending to infinity along which C_n+C_{n+1}=O(1) and, respectively, \limsup\frac{\mathcal U_n}{(\log\log n)^2}\ge\Gamma,\qquad \liminf\frac{\mathcal U_n}{(\log\log n)^2}\le-\Gamma. The same two bounds hold for \mathcal R_n along these sequences.
Proof. Take L=L_m, X=L^3. By Proposition 1, m_L(1)=-C_L+\frac23+O(\log(2m)/m),\qquad m_L(-2)=-m_L(1). Theorem 15 has error o(1) here. Select the smallest minimizer of \mathcal R_n in the class 1\bmod L and the smallest maximizer in the class -2\bmod L, both within X<n\le2X. Their residuals are at most -(\Gamma+o(1))\log^2m and at least (\Gamma+o(1))\log^2m.
For m\ge4, if n\equiv1\bmod L_m, neither n nor n+1 has a prime factor at most m, except for exactly one factor of two in n+1. For n\equiv-2\bmod L_m, the factor of two occurs instead in n. Since \log n=O(m) throughout the interval, Lemma 20 gives C_n+C_{n+1}=O(1) on both progressions. Apply (68) and \log\log n=\log m+O(1). ◻
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.
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).
Theorem 22 (Obstruction to short uniform truncation). Fix 0<A<\sqrt{2e^\gamma} and put Q(n)=\lfloor(\log n)^A\rfloor. Then \begin{aligned} \limsup_{n\to\infty} \frac{\mathcal T_{Q(n)}(n)}{(\log\log n)^2} &\ge\Gamma-\frac{A^2}{\pi^2}>0,\\ \liminf_{n\to\infty} \frac{\mathcal T_{Q(n)}(n)}{(\log\log n)^2} &\le-\left(\Gamma-\frac{A^2}{\pi^2}\right). \end{aligned} Consequently \mathcal T_{Q(n)}(n)=o((\log\log n)^2) cannot hold uniformly for any such exponent A.
Proof. The finite extremum (15) gives |F_{Q(n)}(n)| \le\left(\frac{A^2}{\pi^2}+o(1)\right)(\log\log n)^2. The error in (69) tends to zero. Apply the positive and negative sequences from Corollary 16, or from Theorem 21, and subtract the maximum allowed contribution of F_{Q(n)}. ◻
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 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.
| X | \mathbb E_X y_n | \mathbb E_X Z_n | \mathbb E_X Z_n^2 | \mathbb E_X^{\mathcal P} Z_{p-1} |
|---|---|---|---|---|
| 10^{2} | -0.201303 | -0.000805 | 0.095937 | -0.292598 |
| 10^{3} | -0.275399 | 0.000644 | 0.139772 | -0.305324 |
| 10^{4} | -0.351672 | 0.000007 | 0.162803 | -0.332542 |
| 10^{5} | -0.427253 | 0.000072 | 0.173603 | -0.330713 |
| 10^{6} | -0.502974 | -0.000002 | 0.178321 | -0.334007 |
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.
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.
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