Still preparing the reader…Long documents can take a little longer to load and typeset.
Still preparing the reader…Long documents can take a little longer to load and typeset.
Still preparing the reader…Long documents can take a little longer to load and typeset.
The finite collision boundary, with its prescribed square energy, uniquely selects a positive arithmetic mass of total one. Its capacity deficit at resolution N is two-thirds of that mass’s mean remainder. The compatible remainders form the profinite odometer. After subtraction of the truncated stationary mean, the age observables converge in Haar L^2 to a potential with an explicit Jordan-totient variance and a quantitative approximation rate. The deficit increment is its coboundary, so every stationary block sum has bounded variance, uniformly in the block length. A primitive additive-conductor expansion separates the odd sawtooth component from the even divisibility correction. The selected amplitudes satisfy an exact Jordan-totient refinement law and a mixed Dirichlet L(1)L(2) identity. Their conductor expansion gives the sharp leading constant in the Haar spectral tail. Conditional means have unique aligned extrema and sharp maximal order 2e^\gamma\pi^{-2}(\log\log L)^2. Positive Möbius regrouping also solves the finite conductor optimization, including the gap after its two extreme classes are removed. The stationary law is symmetric, atomless, and unbounded, with quantitative concentration and tail bounds. The canonical centered cutoffs nevertheless diverge at every ordinary integer. This identifies the distinct arithmetic transfer problem that remains after the stationary dynamics has been determined.
The capacity deficit admits an exact description by weighted remainders. Primitive-pair reduction of the affine boundary energy, followed by divisor inversion, determines the weight of every period. At every resolution, the same positive mass occupies the remainder classes of a countable collection of periods. Increasing resolution advances every remainder by one. Periods dividing the new integer reset, and the deficit records the change in their mean position.
The continuous quantity is J(N)=\frac13\sum_{r,s\le N}\frac{(r,s)^2}{rs}, \qquad D_N=N-J(N),\qquad D_0=0. It arises from the squared sawtooth response obtained by straightening the centered collision table in The Cubic Law for Digit-Collision Energy and The Secondary Term in Digit-Collision Energy [7, 8]. Define H_k^*=\sum_{\substack{1\le a<k\\(a,k)=1}}\frac1a,\qquad w_k=\frac{H_k^*}{k^2},\qquad a_k=kw_k\quad(k\ge2), and set w_1=a_1=0. The exact identities developed below are \sum_{k\ge2}w_k=1,\qquad D_N=\frac23\sum_{k\ge2}w_k(N\bmod k). The boundary response and its square-energy rule uniquely determine this probability mass. Equivalently, the weights are selected by the reciprocal-address budget \sum_{k\mid n}k^3w_k=nH_{n-1}\qquad(n\ge2).
Proposition 1 derives the mass and exact readout directly from the finite capacity identity. The potential is then constructed from these weights. No inverse theorem for a larger class of boundary kernels is required for this construction.
For a compatible residue system x\in\widehat{\mathbb Z}, let r_k(x) be its least nonnegative residue modulo k, and put E_k(x)=r_k(x)-\frac{k-1}{2},\qquad Y_M(x)=\frac23\sum_{2\le k\le M}w_kE_k(x). Haar expectation on \widehat{\mathbb Z} is denoted by \mathbb E. The central analytic conclusion is Y_M\longrightarrow Y\text{ in }L^2,\qquad g(x)=Y(x)-Y(x-1), where g(x)=\frac23\left(1-\sum_{k\ge2}a_k\mathbf 1_{k\mid x}\right). At positive integers, g(N)=D_N-D_{N-1}. If \eta_q=\sum_{q\mid k}w_k and J_2(q)=q^2\prod_{p\mid q}(1-p^{-2}), the potential has variance \sigma^2=\mathbb EY^2=\frac1{27}\sum_{q\ge2}J_2(q)\eta_q^2>0, \qquad \left\|Y-Y_M\right\|_2\ll\frac{\log(2M)}{\sqrt M}. Sections 3 and 4 prove the potential and coboundary identities. Theorem 12 defines the potential directly from boundary increments through cylinder averages, without using the weight formula in its defining condition.
Section 6 determines every \eta_q through a mixed odd-character L(1)L(2) moment and evaluates the leading Haar energy beyond a conductor cutoff. These exact moment identities recover the weights. Leading asymptotics alone do not: a nonzero perturbation at periods two, three and four preserves the truncated stationary profile and every sufficiently large conductor amplitude.
The underlying odometer and its rational spectrum are classical [3]. The arithmetic function 1+\sum_{k\mid n}a_k is the triangular gcd sum studied in [4]. In particular, their Theorem 5.1 gives 3J(N)=3N+O(\log^2(2N)), so the continuous leading law and this error bound are already known. The mass formula also occurs in the reduced-ratio capacity calculation of [8]. The conserved-state construction below recovers these facts directly. Its further results concern the square-integrable potential, the exact amplitude formulas and sharp spectral tail, maximal alignment and finite optimization, and quantitative concentration of the selected law. The cotangent and harmonic expansions used in the amplitude calculation are classical [2]. Related work on sawtooth distributions and arithmetic error terms includes [5]. The concentration argument uses Bernoulli activation and Sperner’s antichain estimate, in the classical framework discussed in [1].
All stationary assertions refer to Haar measure or to limits explicitly proved to equal Haar averages. The resolution variable N is an integer cutoff. The separate prime-square sampling defect is not estimated by replacing this cutoff with an independent random phase.
Use the midpoint convention \mathop{\mathrm{saw}}(t)= \begin{cases}\{t\}-\tfrac12,&t\notin\mathbb Z,\\0,&t\in\mathbb Z.\end{cases} At an odd prime p, the centered floor response has the form \mathcal F_p(a)= \sum_{d=0}^{p-1}\left( \left\lfloor\frac{(d(p+1)+1)a}{p^2}\right\rfloor -\left\lfloor\frac{d(p+1)a}{p^2}\right\rfloor\right)-\frac ap. Writing [u]_{p^2} for the least nonnegative residue, straightening gives \mathcal F_p([(1-p)A]_{p^2}) =2\sum_{r=1}^{p-1}\mathop{\mathrm{saw}}(rA/p^2). Indeed, multiplication by 1-p sends the two endpoint lists modulo p^2 to d and d+1-p. Converting the floors to fractional parts and pairing opposite indices gives the formula, including the midpoint values. This is the response underlying the continuous capacity in [8].
At integer resolution N, its boundary measure is \mathcal M_N=\sum_{r=1}^N\sum_{j=0}^{r-1}\delta_{j/r} =\sum_{k=1}^N\left\lfloor\frac Nk\right\rfloor \sum_{a\in U_k}\delta_{a/k}, where U_k=(\mathbb Z/k\mathbb Z)^\times for k\ge2 and U_1=\{0\}. Reducing each rational address proves the multiplicity on the right. The mean-zero periodic affine response is specified by \mathcal B_N'=N(N+1)\,dt-2\mathcal M_N,\qquad J(N)=\int_0^1\mathcal B_N(t)^2\,dt. The derivative and mean determine the response almost everywhere; midpoint jump values fix its sampled values. In particular \mathcal B_N(t)=2\sum_{r\le N}\mathop{\mathrm{saw}}(rt). The prescribed response and quadratic pairing are part of the selection.
The ramp overlap has a finite affine-cell proof. Write r=ga, s=gk, with (a,k)=1. The common dilation leaves the integral unchanged. On t=(j+u)/(ak), 0\le j<ak, 0<u<1, the residues j\bmod a and j\bmod k are independent and uniform. At fixed u, the two ramp means are (u-1/2)/k and (u-1/2)/a. Hence \int_0^1\mathop{\mathrm{saw}}(rt)\mathop{\mathrm{saw}}(st)\,dt=\frac{(r,s)^2}{12rs}. This proves (1) directly from (11). The resulting gcd/lcm kernel is classical [4].
Proposition 1 (Boundary selection and exact readout). The weights in (2) are positive, sum to one, and obey w_k\ll\log(2k)/k^2. They are the unique coefficients, with w_1=0, representing the boundary energy by (13) at every integer resolution. For every N\ge0, \begin{aligned} J(N)&=\frac N3+\frac23\sum_{k\ge2}w_k k \left\lfloor\frac Nk\right\rfloor,\\ D_N&=\frac23\sum_{k\ge2}w_kr_k(N),\\ D_N-D_{N-1} &=\frac23\left(1-\sum_{k\mid N}a_k\right)\quad(N\ge1). \end{aligned}
Proof. In (1), the diagonal contributes N/3. For an off-diagonal pair write (r,s)=(da,dk) with a<k and (a,k)=1. Its contribution before the factor 1/3 is 1/(ak), with \lfloor N/k\rfloor choices of d and two orientations. This proves (13). Its consecutive differences determine \sum_{k\mid n}kw_k, so finite Möbius inversion proves uniqueness. The elementary bound follows from H_k^*\le H_{k-1}\le1+\log k.
Normalization follows from the same primitive-pair geometry. For positive coprime a,b, put Q(a,b)=(a+b)/(a^2b^2). Then Q(a,b)-Q(a,a+b)-Q(a+b,b)=\frac1{ab(a+b)}. Every primitive pair except (1,1) has a unique parent obtained by subtracting its smaller coordinate from the larger. The square a,b\le M contains every such parent. Summation therefore gives \sum_{\substack{a,b\le M\\(a,b)=1}}\frac1{ab(a+b)} =2-\mathcal E_M,\qquad 0\le\mathcal E_M\le\frac{2(H_M+1)}M. For the last bound, an outgoing child has a+b>M. For fixed a there are a possible b\le M, and Q(a,a+b)\le1/(a^2M)+1/(aM^2). The other children give the reflected bound. Thus \mathcal E_M\to0. Putting k=a+b and symmetrizing nonnegative sums now gives \sum_{k\ge2}w_k =\sum_{\substack{a,b\ge1\\(a,b)=1}}\frac1{a(a+b)^2} =\frac12\sum_{\substack{a,b\ge1\\(a,b)=1}}\frac1{ab(a+b)}=1. The identity k\lfloor N/k\rfloor+r_k(N)=N, averaged against unit mass, proves (14). Its infinite sum converges because r_k(N)\le N. Finally r_k(N)-r_k(N-1)=1-k\mathbf 1_{k\mid N} proves (15). ◻
Corollary 2 (Reciprocal-address characterization). Among sequences with v_1=0, the identities \sum_{k\mid n}k^3v_k=nH_{n-1}\qquad(n\ge2) select exactly v_k=w_k. Among unit masses on k\ge2, the exact deficit D_N at every resolution, or its exact reset law, also suffices.
Proof. Reducing a/n to b/k partitions its reciprocal sum into \sum_{k\mid n}kH_k^*. Inversion gives v_n=\frac1{n^3}\sum_{d\mid n}\mu(n/d)dH_{d-1}. For any unit mass the age D_v(N)=\frac23\sum v_kr_k(N) is finite and satisfies D_v(N)-D_v(N-1)=\frac23(1-\sum_{k\mid N}kv_k). Finite inversion recovers each coefficient. ◻
On the disjoint union of cycles \Omega=\{(k,r):k\ge2,\ 0\le r<k\}, define \pi_N(k,r)=w_k\mathbf 1_{r=r_k(N)},\qquad T(k,r)=(k,r+1\bmod k). Then \pi_{N+1}=T_*\pi_N, and the mass on each cycle remains w_k. The age A(k,r)=r has expectation 3D_N/2 under \pi_N. This construction retains compatibility between periods; it does not assign independent phases to them.
The first downward reset already occurs at resolution six. The divisor sum and its effect on the deficit are \begin{array}{c|ccc} N&\sum_{k\mid N}a_k&g(N)&D_N\\ 4&5/6&1/9&13/9\\ 5&5/12&7/18&11/6\\ 6&6/5&-2/15&17/10 \end{array} At six, the periods two, three, and six reset together. Their weighted reset exceeds the unit advance. The total mass is unchanged, while its mean remainder decreases.
The next corollary recovers the continuous leading law already contained in [4], using the conserved mass.
Corollary 3. J(N)/N\to1 and D_N\ll\log^2(2N).
Proof. For fixed k, r_k(N)/N\to0, and this ratio is at most one. Dominated convergence against the unit weights proves D_N/N\to0. Splitting (14) at k=N gives D_N\le\frac23\left(\sum_{k\le N}kw_k+ N\sum_{k>N}w_k\right)\ll\log^2(2N). ◻
The finite energy in [8] satisfies E_p=p^2(J(p-1)-\mathcal S_p). The continuous leading law transfers to E_p\sim p^3 using the separate estimate \mathcal S_p=o(p). The stronger sampling limit \mathcal S_p\to1 is not a consequence of Corollary 3.
Proposition 4 (Truncated stationary age). Let D_*(M)=\frac13\sum_{k\le M}w_k(k-1). Then D_*(M)=\frac{\log^2M}{\pi^2}+O(\log(2M)). The cycle-stationary measure \pi_*(k,r)=w_k/k has infinite mean age. The Cesàro averages of \pi_N converge to \pi_* in total variation.
Proof. Writing H_u^{(2)}=\sum_{j\le u}j^{-2}, the identity \sum_{j\le t}\frac{H_{j-1}}j =\frac{H_{\lfloor t\rfloor}^2-H_{\lfloor t\rfloor}^{(2)}}2 and Möbius inversion give \sum_{k\le M}a_k =\frac12\sum_{d\le M}\frac{\mu(d)}{d^2} \bigl(H_{\lfloor M/d\rfloor}^2-H_{\lfloor M/d\rfloor}^{(2)}\bigr) =\frac{\log^2M}{2\zeta(2)}+O(\log(2M)). Subtracting \sum_{k\le M}w_k=O(1) and dividing by three proves (16). The mean age is infinite by monotone convergence. For the total-variation assertion, first retain finitely many cycles. Their time averages become uniform. The discarded mass is bounded by \sum_{k>K}w_k, independently of the averaging length. ◻
The truncated stationary mean has the same leading coefficient as the secondary law. Comparing it with the synchronized remainder positions requires further arithmetic estimates. In particular, D_N-D_*(N)=Y_N(N)+\frac23N\sum_{k>N}w_k, and the last term is O(\log(2N)).
The compact group \widehat{\mathbb Z}=\varprojlim_L\mathbb Z/L\mathbb Z consists of compatible residue systems. Its Haar probability measure is uniform on each finite quotient. Translation \tau x=x+1 visits every class modulo L once in each block of length L. Approximation of continuous functions by finite-quotient functions proves unique ergodicity. The functions \mathrm{e}(ax/q)=\exp(2\pi i a r_q(x)/q), with a/q reduced modulo one, form a complete orthonormal system. Translation acts on each by multiplication by \mathrm{e}(a/q). Its spectrum is simple and pure point.
Lemma 5 (Remainder covariance). For positive integers k,l, \mathbb EE_kE_l=\frac{(k,l)^2-1}{12}.
Proof. Put d=(k,l). Conditional on r_d=t, the quotients in the residues t+dj modulo k and t+dh modulo l are independent and uniform, by the Chinese remainder theorem. Both centered conditional means are t-(d-1)/2. Their product averages to (d^2-1)/12. ◻
Theorem 6 (Square-integrable potential). The functions Y_M converge in Haar L^2 to a mean-zero, nonconstant function Y. Equations (8) hold. Its law is symmetric, with Y(-1-x)=-Y(x) almost everywhere.
Proof. The divisor identity d^2=\sum_{q\mid d}J_2(q) and Lemma 5 give, for any finite nonnegative sequence v_k, \left\|\frac 23\sum_k v_kE_k\right\|_2^2 =\frac1{27}\sum_{q\ge2}J_2(q) \left(\sum_{q\mid k}v_k\right)^2. Since w_k\ll\log(2k)/k^2, \eta_q\ll\frac{\log(2q)}{q^2}. For the omitted mass t_q(M)=\sum_{k>M,\ q\mid k}w_k, summation over multiples gives t_q(M)\ll \begin{cases}\log(2M)/(qM),&q\le M,\\ \log(2q)/q^2,&q>M.\end{cases} The series \sum_{q\ge2}J_2(q)\eta_q^2 converges. Formula (19) proves the Cauchy property, the exact variance, and the squared tail bound O(\log^2(2M)/M). The term q=2 is positive. Reflection sends each E_k(x) to -E_k(-1-x), and passes to the L^2 limit. ◻
The full uncentered stationary age is infinite almost everywhere. Indeed its partial sums are Y_M+D_*(M), increase with M, and have uniformly bounded centered variances. Chebyshev’s inequality and D_*(M)\to\infty show that their limit cannot be finite on a set of positive measure.
Proposition 7 (General weight criterion). For any nonnegative weights v_k with \sum_{k\ge2}v_k=1, put \xi_q=\sum_{q\mid k}v_k. The centered ages \frac23\sum_{k\le M}v_kE_k converge in L^2 if and only if \sum_{q\ge2}J_2(q)\xi_q^2<\infty. This is also the condition that \frac23(1-\sum k v_k\mathbf 1_{k\mid x}), initially in L^1, have a mean-zero L^2 potential for translation.
Proof. Formula (19) and monotone convergence prove the first equivalence. The Fourier calculation in Section 5 gives the unique possible nonconstant coefficients of a potential. Their square sum is exactly \frac1{27}\sum J_2(q)\xi_q^2. Conversely the convergent centered ages give a potential by taking differences as in Theorem 9. Translation-invariant L^2 functions are constant, which proves uniqueness after centering. ◻
For weights proportional to k^{-s}, s>1, the potential criterion is s>3/2. A finite uncentered stationary mean requires the stronger condition s>2. These two integrability thresholds concern different observables.
Conservation and the completion identity also hold for every unit mass: N-D_v(N)=\frac N3+\frac23\sum_{k\ge2}kv_k\lfloor N/k\rfloor, \qquad D_v(N)/N\longrightarrow0. The limit follows by dominated convergence, since r_k(N)\le N. Identifying this energy with (11) is the selecting condition in Proposition 1.
Proposition 8 (A mass perturbation). For 0<|\varepsilon|<1/24, set v_k=w_k+\varepsilon (\mathbf 1_{k=2}-2\mathbf 1_{k=3}+\mathbf 1_{k=4}),\qquad Z=\frac23(E_2-2E_3+E_4). These are positive unit masses with Y_v=Y+\varepsilon Z,\qquad D_v(n)=D_n+\varepsilon Z(n), \qquad \mathbb EZ=0,\quad \left\|Z\right\|_2^2=\frac{56}{27}. Their truncated stationary means agree with D_*(M) for every M\ge4, although their potentials differ.
Proof. The weights at two, three and four are 1/4,1/6,1/12. The changes (1,-2,1) have zero total and first moments, giving positivity, normalization and equality of the stationary profiles. The same cancellation removes the centered constants in Z, proving the age identity. The function Z has period twelve. Formula (19), expanded bilinearly for signed coefficients, gives its norm and shows that it is nonzero. ◻
Theorem 9 (Exact compensation). Let f(x)=\sum_{k\ge2}a_k\mathbf 1_{k\mid x} and g=\frac23(1-f). Then f\in L^r for every finite r\ge1, \mathbb Ef=1, and g=Y-Y\circ\tau^{-1}\quad\hbox{almost everywhere}. For every integer H\ge1, \sum_{j=1}^Hg(x+j)=Y(x+H)-Y(x),\qquad \mathbb E\left|\sum_{j=1}^Hg(x+j)\right|^2\le4\sigma^2.
Proof. Minkowski’s inequality applies because \sum a_k k^{-1/r}<\infty. Also \mathbb Ef=\sum a_k/k=1. The exact finite difference of Y_M is Y_M(x)-Y_M(x-1) =\frac23\sum_{k\le M}w_k(1-k\mathbf 1_{k\mid x}). Its left side converges in L^2 and its right side in L^1 to the asserted expressions. Telescoping and the triangle inequality prove (22). ◻
Proposition 10 (Covariance and recurrence). With K(h)=\mathbb Eg(x)g(x+h), \begin{aligned} K(h)&=\frac49\sum_{k,l\ge2}a_ka_l \left(\frac{\mathbf 1_{(k,l)\mid h}}{[k,l]}-\frac1{kl}\right) \\ &=\frac49\sum_{q\ge2}\eta_q^2c_q(h), \end{aligned} where c_q(h)=\sum_{a\in(\mathbb Z/q\mathbb Z)^\times}\mathrm{e}(ah/q). The series converge absolutely, K(0)>0, and K(1)<0. For L_M=\operatorname{lcm}(1,\ldots,M), K(L_M)\longrightarrow K(0),\qquad \left\|Y(\,\cdot+L_M)-Y\right\|_2\ll\frac{\log(2M)}{\sqrt M}.
Proof. The two divisibility events have joint probability 1/[k,l] precisely when (k,l)\mid h. Their product sum is finite by f\in L^2. This proves (23). Formula (24) also follows from the Fourier coefficients computed below. Its absolute convergence follows from |c_q(h)|\le\varphi(q) and the bound on \eta_q. At h=1, terms with (k,l)=1 vanish and all others are negative, including k=l=2. Nonconstancy of g gives K(0)>0. The truncations Y_M and g_M have period L_M. Their L^2 approximations prove both recurrence assertions. ◻
Prime resolutions have g(p)=\frac23(1-H_{p-1}/p)\to2/3. Section 7 gives arbitrarily large negative increments. Thus no profile P(N) with P(N)-P(N-1)\to0 can leave a bounded error D_N-P(N) at every integer. A bounded error would force bounded increments, contrary to those resets.
Theorem 11 (Integer laws for fixed increments). For every fixed list of nonnegative shifts h_1,\ldots,h_r, the empirical joint law of g(n+h_i) as n\to\infty equals its Haar joint law. Every fixed joint moment converges. In particular, \mathbb E\prod_{i=1}^r f(x+h_i) =\sum_{k_1,\ldots,k_r\ge2} \frac{\prod_i a_{k_i}}{[k_1,\ldots,k_r]} \mathbf 1_{(k_i,k_j)\mid h_i-h_j\ \text{for all }i,j}. For each fixed H, the empirical second moment of D_{n+H}-D_n tends to the left side of (22).
Proof. Finite divisor truncations are periodic and have exactly the limiting uniform residue law. For unshifted positive integers, the empirical L^r norm of the tail is bounded, uniformly in the averaging length, by \sum_{k>M}a_k k^{-1/r}, using \lfloor X/k\rfloor/X\le1/k and Minkowski. Fixed shifts change only finitely many endpoints; the elementary bound f(n)\ll\log^2(2n) makes their normalized contribution tend to zero. Thus periodic approximation proves joint distribution and, using a higher fixed moment, joint moment convergence. Tonelli’s theorem and the generalized Chinese remainder theorem prove (25); its sum is finite by Hölder. Finally D_{n+H}-D_n=\sum_{j=1}^Hg(n+j). ◻
The block length is fixed when the integer average is taken. Equation (22) is uniform in H in the stationary space; it does not assert a simultaneous uniform integer-average limit in H and X.
Theorem 12 (Intrinsic boundary potential). Define J by (10) and (11), put J(0)=0, and set d_\partial(n)=1-J(n)+J(n-1). The potential Y is the unique mean-zero L^2(\widehat{\mathbb Z}) function satisfying, for every locally constant \phi, \mathbb E\bigl[(Y(x)-Y(x-1))\overline{\phi(x)}\bigr] =\lim_{X\to\infty}\frac1X\sum_{n\le X} d_\partial(n)\overline{\phi(n)}.
Proof. Proposition 1 identifies d_\partial(n)=g(n). Truncate its divisor sum at M, retaining the corresponding constant \frac23\sum_{k\le M}w_k. The resulting function is periodic, and its Cesàro and Haar pairings with \phi agree. The omitted terms have both Haar and upper Cesàro L^1 norm at most \frac43\sum_{k>M}w_k. Thus the limit exists and equals \mathbb E(g\overline\phi). Theorems 6 and 9 give existence. Density of locally constant functions makes the difference of any two solutions translation invariant in L^2. Ergodicity and the prescribed mean give uniqueness. ◻
This definition uses only the boundary and its square energy. It prescribes Haar pairings rather than values of an arbitrary representative on the null integer orbit.
For q\ge2, define R_q(x)=\sum_{d\mid q}\mu(q/d)d\,\mathop{\mathrm{saw}}(r_d(x)/d),\qquad P_q(x)=\sum_{d\mid q}\mu(q/d)E_d(x). These are additive-frequency rows. Their conductors refer to reduced rational frequencies and are distinct from the conductors of multiplicative Dirichlet characters.
Proposition 13 (Spectrum and parity). The rows P_q have precisely the primitive additive frequencies of denominator q, and \begin{aligned} P_q&=R_q-\tfrac12c_q, \\ Y&=\frac23\sum_{q\ge2}\eta_qP_q=O+\tfrac12g, &O&=\frac23\sum_{q\ge2}\eta_qR_q. \end{aligned} All series converge in L^2. Distinct denominators are orthogonal. The function O is odd under x\mapsto-x, and g is even. Moreover, \begin{aligned} \left\|R_q\right\|_2^2&=\frac{J_2(q)-3\varphi(q)}{12}, &\left\|P_q\right\|_2^2&=\frac{J_2(q)}{12}, \\ \widehat g(a/q)&=-\frac23\eta_q, &\widehat Y(a/q)&=-\frac{2\eta_q}{3(1-\mathrm{e}(-a/q))} \quad((a,q)=1). \end{aligned} The row R_2 vanishes, but P_2=-c_2/2 does not.
Proof. Direct finite Fourier summation gives coefficient -1/(1-\mathrm{e}(-a/q)) for E_k at a nonzero reduced frequency a/q when q\mid k, and zero otherwise. Möbius inversion in (27) retains only denominator q. The pointwise endpoint identity is E_d=d\,\mathop{\mathrm{saw}}(r_d/d)+\frac12-\frac d2\mathbf 1_{d\mid x}. Summing with \mu(q/d) and using c_q(x)=\sum_{d\mid q}d\mu(q/d)\mathbf 1_{d\mid x} proves (28). The finite identity Y_M=\frac23\sum_{q\le M}\eta_{q,M}P_q, where \eta_{q,M}=\sum_{k\le M,\ q\mid k}w_k, now proves the limiting expansion. The centered divisibility coefficients give those of g.
The trigonometric identity \sum_{a=1}^{q-1}\csc^2(\pi a/q)=(q^2-1)/3, obtained by differentiating the cotangent multiplication formula, gives by divisor inversion \sum_{(a,q)=1}\csc^2(\pi a/q)=J_2(q)/3. Since |1-\mathrm{e}(-a/q)|^2=4\sin^2(\pi a/q), Parseval yields \left\|P_q\right\|_2^2=J_2(q)/12. Odd and even parity are orthogonal; \left\|c_q\right\|_2^2=\varphi(q) then gives the other norm. The same calculations prove (31) and all convergence claims. ◻
Lemma 14 (Recovery of the period masses). For every admissible mass in Proposition 7, v_k=\sum_{j\ge1}\mu(j)\xi_{kj}\qquad(k\ge2), with absolute convergence. Its potential, or its full coboundary, therefore determines every mass.
Proof. Since J_2(q)\ge q^2/\zeta(2), the energy condition and Cauchy–Schwarz imply \sum_{q\ge2}\xi_q<\infty. For fixed k, positivity gives \sum_{j,m\ge1}v_{kjm}=\sum_j\xi_{kj}<\infty. Interchanging the absolutely convergent sums leaves \sum_{j\mid m}\mu(j)=\mathbf 1_{m=1} and proves the inverse. The Fourier calculation (31) applies with \eta_q replaced by \xi_q, proving the final assertion. ◻
In particular, the spectral measures for translation are \rho_g=\frac49\sum_{q\ge2}\eta_q^2\sum_{(a,q)=1}\delta_{a/q}, \qquad \rho_Y=\frac19\sum_{q\ge2}\eta_q^2 \sum_{(a,q)=1}\csc^2(\pi a/q)\delta_{a/q}. Every rational nonzero frequency occurs because \eta_q>0. Consequently the closed span of the translates of g is the mean-zero subspace of L^2(\widehat{\mathbb Z}). To isolate one frequency, average \mathrm{e}(-ja/q)g(x+j) over j and use the simple pure point spectrum.
The Dedekind pairing of the odd rows can also be read directly on a common finite grid. For units a\bmod d, b\bmod e, put B_{d,a}(x)=\mathop{\mathrm{saw}}(a r_d(x)/d) and define s(h,g)=\sum_{t=0}^{g-1}\mathop{\mathrm{saw}}(t/g)\mathop{\mathrm{saw}}(ht/g). Conditioning on the residue modulo g=(d,e) and using the sawtooth multiplication formula gives \mathbb EB_{d,a}B_{e,b}=\frac{g}{de}s(b\bar a,g). The conditional mean of the first row is (g/d)\mathop{\mathrm{saw}}(at/g). For the second it is (g/e)\mathop{\mathrm{saw}}(bt/g). Averaging their product proves the formula. Refining the common grid preserves this norm for a fixed coefficient array. The carrier is therefore intrinsic to the remainder observable.
The rows in (29) occur for every admissible age mass. The boundary selects their amplitudes. Define \Delta(q)=\prod_{p\mid q}(1-p^{-3}),\qquad J_3(q)=q^3\Delta(q), \qquad F(q)=\sum_{m\ge1}\frac{H_{qm-1}}{(qm)^2}. Set \eta_1=1. The series defining F is absolutely convergent.
Proposition 15 (Exact amplitude refinement). For every q\ge1, \begin{aligned} \eta_q&=\frac1{\zeta(3)\Delta(q)} \sum_{d\mid q}\frac{\mu(d)}{d^3}F(q/d), \\ \sum_{d\mid q}J_3(d)\eta_d&=\frac{q^3F(q)}{\zeta(3)}. \end{aligned} In particular F(1)=\zeta(3).
Proof. The primitive-pair representation of the mass gives \eta_q=\frac12 \sum_{\substack{a,b\ge1\\(a,b)=1\\q\mid a+b}}\frac1{ab(a+b)}. Removing coprimality by Möbius inversion and rescaling both coordinates therefore gives \eta_q=\sum_{d\ge1}\frac{\mu(d)}{d^3} F\!\left(\frac q{(q,d)}\right). Indeed \frac12\sum_{a=1}^{n-1}1/[a(n-a)n]=H_{n-1}/n^2. Absolute summation is justified by the same positive sum without the divisibility restriction and \sum d^{-3}<\infty. For squarefree d, split its primes into those dividing q and those coprime to q. The latter contribute 1/(\zeta(3)\Delta(q)), proving (34). At q=1, the already proved unit normalization gives F(1)=\zeta(3). Multiplication by q^3\Delta(q) and finite divisor inversion prove (35). ◻
Put \begin{aligned} A_0&=\frac{\zeta(2)}{\zeta(3)},& \kappa_0&=\gamma-\frac{\zeta'(2)}{\zeta(2)},\\ \rho(q)&=\frac{\varphi(q)/q}{\Delta(q)} =\prod_{p\mid q}\frac{p^2}{p^2+p+1},& \nu(q)&=\sum_{p\mid q}\frac{\log p}{p-1}. \end{aligned}
Proposition 16 (Uniform conductor expansion). For every q\ge2, \eta_q=\frac{A_0\rho(q)}{q^2} \bigl(\log q+\kappa_0+\nu(q)\bigr)+\epsilon_q, where |\epsilon_q|\le \frac{\zeta(4)}{12\zeta(3)q^4\Delta(q)} \prod_{p\mid q}(p+1).
Proof. The positive-real harmonic expansion, with its first neglected term, is [2] H_{n-1}=\log n+\gamma-\frac1{2n}+e_n,\qquad -\frac1{12n^2}<e_n<0. Consequently F(q)=\frac{\zeta(2)(\log q+\kappa_0)}{q^2} -\frac{\zeta(3)}{2q^3}+r(q),\qquad -\frac{\zeta(4)}{12q^4}<r(q)<0. Apply (34). The isolated cubic-power term vanishes for q>1, since \sum_{d\mid q}\mu(d)=0. The other main terms use \sum_{d\mid q}\frac{\mu(d)}d=\frac{\varphi(q)}q,\qquad \sum_{d\mid q}\frac{\mu(d)\log d}d =-\frac{\varphi(q)}q\nu(q). The remainder is bounded by \zeta(4)/(12\zeta(3)q^4\Delta(q)) times \sum_{d\mid q}|\mu(d)|d=\prod_{p\mid q}(p+1). ◻
The error is O_\varepsilon(q^{-3+\varepsilon}) uniformly. For a prime p the exact formula retains the lower-conductor value: \eta_p=\frac{F(p)-p^{-3}F(1)}{\zeta(3)(1-p^{-3})}. For fixed p and a\to\infty, (36) instead gives \eta_{p^a}=\frac{A_0\rho(p)}{p^{2a}} \left(a\log p+\kappa_0+\frac{\log p}{p-1}\right) +O_p(p^{-4a}). The local main factor \rho is multiplicative; the amplitudes are not. In fact, (36) implies \eta_{2\cdot3^a}/\eta_{3^a}\to\rho(2)/4=1/7, whereas Theorem 18 below gives \eta_2=1/2.
Theorem 17 (Sharp Haar spectral tail). Let \Pi_{\le Q} be the orthogonal projection onto additive conductors at most Q, and define \mathcal C_{\rm sp}=\prod_p\left( 1+\frac{(1-p^{-2})\rho(p)^2-1}{p}\right)>0. Then \left\|Y-\Pi_{\le Q}Y\right\|_2^2 \sim\frac{A_0^2\mathcal C_{\rm sp}}{27}\frac{\log^2Q}{Q}.
Proof. Orthogonality gives the exact tail \frac1{27}\sum_{q>Q}J_2(q)\eta_q^2. Set h(q)=J_2(q)\rho(q)^2/q^2. This multiplicative function satisfies 0<h(q)\le1 and is constant on positive prime powers, with h(p)=1+O(p^{-1}). Writing h=1*c, the function c is supported on squarefree integers and has local value h(p)-1. Thus \sum_d|c(d)|/d<\infty, and dominated convergence in \frac1x\sum_{q\le x}h(q) =\sum_{d\le x}c(d)\frac{\lfloor x/d\rfloor}{x} gives the limit \sum_dc(d)/d=\mathcal C_{\rm sp}. The product is positive and converges absolutely.
Counting prime divisibility in the first and second moments of \nu gives \sum_{q\le x}\nu(q)=O(x),\qquad \sum_{q\le x}\nu(q)^2=O(x). The required convergent prime sums are \sum_p\frac{\log p}{p(p-1)},\qquad \sum_p\frac{(\log p)^2}{p(p-1)^2}. Since h\le1, partial summation shows that terms involving \kappa_0+\nu(q) in the square of (36) contribute O(\log Q/Q) to the tail. The error (37), with any fixed \varepsilon<1/2, makes its squared and cross terms o(\log^2Q/Q). Finally partial summation of \sum_{q\le x}h(q)\sim\mathcal C_{\rm sp}x yields \sum_{q>Q}\frac{h(q)\log^2q}{q^2} \sim\mathcal C_{\rm sp}\frac{\log^2Q}{Q}, proving the assertion. ◻
This projection uses full conductor amplitudes. It differs from the period cutoff Y_M, and its Haar norm gives no pointwise evaluation bound on the distinguished integer orbit.
Theorem 18 (Mixed-moment characterization). For every q\ge2, \eta_q=\frac12- \frac{\displaystyle\sum_{\substack{\chi\bmod q\\\chi(-1)=-1}} L(1,\overline\chi)L(2,\chi)} {\zeta(3)\Delta(q)\varphi(q)}. The sum is over all odd characters, including induced characters, and is real. Moreover \eta_2=1/2 and 0<\eta_q<1/2 for q>2. Among the admissible masses of Proposition 7, requiring (40) at every conductor selects v=w.
Proof. Define the absolutely convergent series \mathcal L(z)=\sum_{n\ge1}\frac{H_{n-1}}{n^2}z^n,\qquad C_3(\theta)=\sum_{n\ge1}\frac{\cos(n\theta)}{n^3},\qquad \operatorname{Cl}_2(\theta)=\sum_{n\ge1}\frac{\sin(n\theta)}{n^2}. Inside the unit disk, \mathcal L'(z)=\log^2(1-z)/(2z). For 0<\theta<2\pi, use \log(1-e^{i\theta})=\log(2\sin(\theta/2))+i(\theta-\pi)/2. The log-series identity gives \operatorname{Cl}_2'(\theta)=-\log(2\sin(\theta/2)). Differentiating both sides and matching their values at zero proves \Re\mathcal L(e^{i\theta}) =\frac{\zeta(3)+C_3(\theta)}2 +\frac{\theta-\pi}{2}\operatorname{Cl}_2(\theta). Differentiation can first be justified by Abel limits away from the endpoints; absolute convergence gives continuity at the endpoints.
Average (42) over the q-th roots of unity. The finite sine transform \sum_{a=1}^{q-1}\left(\frac aq-\frac12\right) \sin(2\pi an/q)=-\frac12\cot(\pi n/q) \quad(q\nmid n) vanishes when q\mid n. It follows that F(q)=\frac{\zeta(3)}2(1+q^{-3})-\frac{\pi}{2q}C(q),\qquad C(q)=\sum_{\substack{n\ge1\\q\nmid n}}\frac{\cot(\pi n/q)}{n^2}. Let C^{\rm u}(q) denote the same sum restricted to (n,q)=1, and set C^{\rm u}(1)=0. Descent by (n,q) gives C(q)=\sum_{d\mid q}(d/q)^2C^{\rm u}(d),\qquad \sum_{d\mid q}\frac{\mu(d)}{d^2}C(q/d)=C^{\rm u}(q). Applying (34) therefore yields \eta_q=\frac12-\frac{\pi}{2q\zeta(3)\Delta(q)}C^{\rm u}(q). The constant transforms to one half, and the pure cubic power vanishes for q>1.
The cotangent partial-fraction expansion [2], with positive and negative terms paired, gives for every odd character \sum_{a\in U_q}\overline{\chi(a)}\cot(\pi a/q) =\frac{2q}{\pi}L(1,\overline\chi). This uses oddness, not primitivity. The even coefficients vanish. Character orthogonality and absolute convergence at two give C^{\rm u}(q)=\frac{2q}{\pi\varphi(q)} \sum_{\chi(-1)=-1}L(1,\overline\chi)L(2,\chi). Substitution proves (40). At q=2 there are no odd characters. For q>2, pair residues a,q-a, with 0<a<q/2. The cotangent and the difference \sum_{m\ge0}[(a+mq)^{-2}-(q-a+mq)^{-2}] are both positive. Thus C^{\rm u}(q)>0; the lower bound follows from the positive mass. Finally the moment identities fix all \eta_q, and Lemma 14 recovers every w_k. ◻
For example, with G=L(2,\chi_{-4}) denoting Catalan’s constant, \eta_3=\frac12-\frac{3\sqrt3\pi}{52\zeta(3)}L(2,\chi_{-3}),\qquad \eta_4=\frac12-\frac{\pi G}{7\zeta(3)},\qquad \eta_6=\frac{15\eta_3-4}{7}. The last identity retains the Euler factors of the induced odd character modulo six.
The finite perturbation in Proposition 8 leaves every \eta_q with q>4 unchanged. It therefore preserves the large-conductor asymptotic and the sharp tail constant, while breaking the mixed-moment characterization. The exact amplitudes contain the selection information that these asymptotics omit.
Proposition 19 (Cylinder means). For L\ge2 and a\bmod L, \begin{aligned} m_L(a):=\mathbb E(Y\mid r_L=a) &=\frac23\sum_{k\ge2}w_kE_{(k,L)}(a) =\frac23\sum_{\substack{q\mid L\\q\ge2}}\eta_qP_q(a),\\ C_L&:=\frac13\sum_{k\ge2}w_k((k,L)-1) =\frac13\sum_{\substack{q\mid L\\q\ge2}}\varphi(q)\eta_q. \end{aligned} The minimum is -C_L, attained only at zero. The maximum is C_L, attained only at minus one. For h\ge0, -C_L\le m_L(h)\le-C_L+2h/3.
Proof. Conditional on r_L=a, the residue modulo k is uniform among those congruent to a modulo d=(k,L). Its centered mean is E_d(a). The resulting series converges absolutely since d\le L. Conditional expectation passes through the L^2 limit and retains exactly the frequencies whose denominator divides L. The identity d-1=\sum_{q\mid d,\ q\ge2}\varphi(q) proves (47). All weights are positive, and the term k=L forces uniqueness at both endpoints. Finally 0\le r_d(h)\le h. ◻
The conditional expectations along L_{2^j}=\operatorname{lcm}(1,\ldots,2^j) form a martingale. Their differences are the sums of \eta_qP_q over newly admitted divisors, and \left\|Y-m_{L_{2^j}}\right\|_2\ll j\,2^{-j/2}. This follows because conditional expectation is the best L^2 approximation on its finite quotient and Y_{2^j} belongs to that quotient.
Theorem 20 (Maximal alignment and reset orders). Put \kappa=e^\gamma/\zeta(2) and \Gamma=2e^\gamma/\pi^2=\kappa/3. Then, as m\to\infty, f(L_m)\sim\kappa\log^2m,\qquad C_{L_m}\sim\Gamma\log^2m. Furthermore \limsup_{n\to\infty}\frac{f(n)}{(\log\log n)^2}=\kappa,\qquad \limsup_{L\to\infty}\frac{C_L}{(\log\log L)^2}=\Gamma. Thus \limsup -g(n)/(\log\log n)^2=2\Gamma.
Proof. Harmonic expansion in the Möbius formula gives uniformly for k\ge2 a_k=\frac{\varphi(k)}{k^2} \left(\log k+\gamma+\nu(k)\right) +O\left(\frac{\sigma_1(k)}{k^3}\right), \qquad \nu(k)=\sum_{p\mid k}\frac{\log p}{p-1}. Here H_{u-1}=\log u+\gamma-(2u)^{-1}+O(u^{-2}). The linear correction cancels since \sum_{d\mid k}\mu(d)=0. The displayed errors have bounded sum, including over any divisor set.
For v\ge1 define A_p(v)=1+p^{-1}-p^{-v-1},\qquad t_p(v)= \frac{\sum_{j=1}^v(1-p^{-1})p^{-j}(j+(p-1)^{-1})}{A_p(v)}. Euler-factor multiplication in the main term of (48) gives f(n)=\prod_{p^v\parallel n}A_p(v) \left(\gamma+\sum_{p^v\parallel n}t_p(v)\log p\right) -\gamma+O(1). The local numerator equals [1+p^{-1}-(v+1)p^{-v}+(v-1)p^{-v-1}]/(p-1). Hence A_p(v)\le1+p^{-1} and 0\le t_p(v)\le(p-1)^{-1}, with equality in the limits as v\to\infty.
The classical Mertens estimates are \prod_{p\le y}(1-p^{-1})^{-1}\sim e^\gamma\log y,\qquad \sum_{p\le y}\frac{\log p}{p}=\log y+O(1). See [6]. They imply \prod_{p\le y}(1+p^{-1})\sim\kappa\log y. For arbitrary n, split its prime factors at y=\log n. The larger ones satisfy \sum_{\substack{p\mid n\\p>y}}p^{-1} \le\frac{\log n}{y\log y},\qquad \sum_{\substack{p\mid n\\p>y}}\frac{\log p}{p-1} \ll\frac{\log n}{y}. These inequalities and (49) give the upper bound for f. For L_m, the exponents are v_p=\lfloor\log m/\log p\rfloor. Splitting at \sqrt m shows that replacing the finite local factors by their limiting factors costs O(m^{-1/2}) relatively in the product and O(\log m/\sqrt m) in the logarithm-weighted sum. Thus f(L_m)\sim\kappa\log^2m.
For C_L the corresponding factors, now for v\ge0, are \begin{aligned} K_p(v)&=1+p^{-1}-\frac{p^{-v}}{p+1},\\ u_p(v)&=\frac{\sum_{j\ge1}(1-p^{-1})p^{-2j+\min(j,v)} (j+(p-1)^{-1})}{K_p(v)}. \end{aligned} Because 3C_L+1=\sum_{k\ge2}w_k(k,L), formula (48) yields 3C_L+1=\prod_pK_p(v_p(L)) \left(\gamma+\sum_pu_p(v_p(L))\log p\right)-\gamma+O(1). The error is uniform since (k,L)\le k. The factor at p is the sum of \varphi(p^j)p^{\min(j,v)-3j} over j\ge0, so the product is absolutely convergent for each fixed L. Its bounds are K_p(v)\le1+p^{-1},\quad u_p(v)\le(p-1)^{-1},\quad K_p(0)=1+\frac1{p(p+1)},\quad u_p(0)\ll p^{-2}. For the second bound, the finite-v weights are the infinite-v weights multiplied by the nonincreasing factor p^{-(j-v)_+}. This decreases the average of the increasing function j+(p-1)^{-1}\mathbf 1_{j\ge1}. Primes not dividing L contribute a convergent product and a convergent logarithm-weighted sum; they cannot be omitted from (51). More explicitly, with y=\log L, their factors beyond y satisfy \prod_{\substack{p>y\\p\nmid L}}K_p(0)=1+O(y^{-1}),\qquad \sum_{\substack{p>y\\p\nmid L}}u_p(0)\log p \ll\frac{\log(2y)}y. The primes at most y are bounded by the limiting factors 1+p^{-1} and (p-1)^{-1}, while the larger prime divisors of L obey the two estimates used above for f. This makes the upper bound uniform in L. For L=L_m, K_p(\infty)=1+p^{-1} and u_p(\infty)=(p-1)^{-1}. With v_p=\lfloor\log m/\log p\rfloor, splitting at \sqrt m gives \sum_{p\le m}\frac{K_p(\infty)-K_p(v_p)}{K_p(\infty)} \ll m^{-1/2},\qquad \sum_{p\le m}\bigl(u_p(\infty)-u_p(v_p)\bigr)\log p \ll\frac{\log m}{\sqrt m}. The absent primes p>m have the convergent tails just estimated. These bounds and (51) give C_L\le(\Gamma+o(1))(\log\log L)^2 and C_{L_m}\sim\Gamma\log^2m. Finally \log L_m\sim m by the prime number theorem [9]. This proves the lower maximal orders and the assertion about g. ◻
The reset sum is a classical gcd observable: 1+f(n)=\sum_{j=1}^n\frac{(j,n)^2}{jn}. Grouping by the reduced denominator proves this identity. The new alignment quantity C_L averages the same harmonic weights against \gcd(k,L)-1. A single prime-power tower is uniformly bounded in the relevant quantities: f(p^v)+C_{p^v}\ll\frac{1+\log p}{p}\qquad(v\ge1). For f sum a_{p^j}\ll(1+j\log p)/p^j. For C, write k=p^jm, (p,m)=1, bound (k,p^v)-1\le p^j, and sum the convergent series in m^{-2} and p^{-j}.
Define the fixed-coefficient truncation F_Q(a)=\frac23\sum_{2\le q\le Q}\eta_qP_q(a),\qquad V_Q=\frac13\sum_{2\le q\le Q}\varphi(q)\eta_q. This differs from the period cutoff Y_Q, whose conductor coefficients are \eta_{q,Q}.
Lemma 21 (Positive Möbius regrouping). For 2\le d\le Q, let \beta_d(Q)=\sum_{j\le Q/d}\mu(j)\eta_{dj}. Then F_Q(a)=\frac23\sum_{d=2}^Q\beta_d(Q)E_d(a),\qquad \beta_d(Q)>\frac12w_d.
Proof. The identity follows by interchanging finite divisor sums. Dropping the extra coprimality restriction in H_{dm}^* gives H_{dm}^*\le H_d^*+\varphi(d)H_{m-1}/d. Pairing r with d-r gives H_d^*\ge2\varphi(d)/d, including d=2. Thus w_{dm}\le \frac{w_d}{m^2}(1+\tfrac12H_{m-1}). For J=\lfloor Q/d\rfloor, \beta_d(Q)=\sum_{m\ge1}w_{dm}\sum_{j\mid m,\ j\le J}\mu(j). For m\ge2 the inner sum is at least -N(m), where N(m)=2^{\omega(m)-1}-1\ge0. Its Dirichlet series is \mathcal A(s)=\sum_{m\ge2}\frac{N(m)}{m^s} =\frac{\zeta(s)^2}{2\zeta(2s)}-\zeta(s)+\frac12. Using H_{m-1}\le1+\log m and \frac12\log m\le e^{-1}\sqrt m<(37/100)\sqrt m yields \frac{\beta_d(Q)}{w_d} \ge1-\tfrac32\mathcal A(2)-\tfrac{37}{100}\mathcal A(3/2) >1-\frac{3979}{8000}>\frac12. Here \mathcal A(2)=7/4-\zeta(2)<1/8 and \mathcal A(3/2)<67/80. These estimates follow from \zeta(2)>13/8, \zeta(3)>6/5, and \zeta(3/2)<27/10. The first two are certified by sums through 50 and 16; for the last, the integral test gives \zeta(3/2)<2+5/14+1/5+1/8=751/280. Substitution in the displayed Dirichlet series proves the stated rational bounds. ◻
Theorem 22 (Complete and restricted finite extrema). On residues modulo L_Q the unique minimum and maximum of F_Q are -V_Q at zero and V_Q at minus one. Moreover, V_Q=\frac{\log^2Q}{\pi^2}+O(\log(2Q)). For Q\ge3, removing these two classes changes the extrema to -V_Q+\delta_Q and V_Q-\delta_Q, where \delta_Q\asymp\frac{\log Q}{Q^2}. For all sufficiently large Q, \delta_Q=\frac23\min_{Q/2<p\le Q}\eta_p \sim\frac{2\zeta(2)}{3\zeta(3)}\frac{\log Q}{Q^2}. For all sufficiently large Q, the restricted minimizers have one nonzero prime-power coordinate, a\equiv1\pmod p, with all other coordinates zero, for a prime attaining that minimum.
Proof. Positive regrouping makes each remainder simultaneously minimal at zero. Equality forces every d\le Q to divide a. Reflection a\mapsto-1-a gives the maximum. The endpoint value is P_q(0)=-\varphi(q)/2. Absolute divisor inversion gives \sum_{j\ge1}\mu(j)\eta_{dj}=w_d, and hence |\beta_d(Q)-w_d|\ll\frac{\log(2Q)}{dQ}. At zero this compares V_Q with D_*(Q) to within O(\log(2Q)), proving the leading value.
If Q/2<p\le Q, the only row d\le Q involving p is d=p, and \beta_p(Q)=\eta_p. The CRT address a\equiv0\pmod{L_Q/p}, a\equiv1\pmod p has gap 2\eta_p/3. The conductor expansion (36) gives \eta_p=\frac{\zeta(2)}{\zeta(3)}\frac{\log p}{p^2}+O(p^{-2}). Here \rho(p)=1+O(p^{-1}) and \nu(p)=O(\log p/p).
Before the factor 2/3, a nonzero residue at a prime p\le Q/2 costs at least \frac12w_p\ge\log p/(2p^2). This is at least 2\log(Q/2)/Q^2 for large Q, exceeding the candidate coefficient \zeta(2)/\zeta(3)<2. If p^j\le Q, j\ge2, is the first missing power of a prime, its remainder is at least p^{j-1}. Since H_{p^j}^*\ge\frac12\log(p^j), its cost is at least \log(p^j)/(4p^{j+1})\ge\log Q/(4Q^{3/2}). It also exceeds the candidate. Only primes in (Q/2,Q] can therefore occur in a restricted minimizer. Their single-row costs add positively, proving the exact classification. The prime number theorem supplies primes with p/Q\to1, and (55) proves (54). Finitely many smaller Q are absorbed into the two-sided order. ◻
The perturbation in Proposition 8 preserves these complete finite extremal values at every Q\ge4. Its conductor amplitudes change only at 2,3,4, by 2\varepsilon,-2\varepsilon, \varepsilon. Hence F_Q^v=F_Q+\varepsilon Z and V_Q^v=V_Q. The regrouped weights change by \varepsilon,-2\varepsilon,\varepsilon at d=2,3,4 and remain positive by Lemma 21 and |\varepsilon|<1/24. The unique complete extrema are still zero and minus one. Exact boundary readout contains more information than these extremal values.
Individual conductor rows need not have endpoint extrema. For example, (P_6(0),\ldots,P_6(5))=(-1,-2,-1,1,2,1). More generally, if s=\operatorname{rad}(q) and v=q/s, then P_q(a)=vP_s(\lfloor a/v\rfloor). To prove this, write a=vb+t and apply E_{ve}(a)=vE_e(b)+t-(v-1)/2 in the Möbius sum; its constant part vanishes. Thus the addresses (6^J-1)/5 align the interior minima of all rows 6^j, 1\le j\le J. Their weighted total remains bounded by a constant times \sum j/6^j. The positivity theorem concerns the entire weighted aggregate.
Proposition 23 (Position stability and its divisibility limit). Relative near-minimality controls mean relative position. Define \nu_L(k)=\frac{w_k((k,L)-1)}{3C_L}\qquad((k,L)>1). Then \frac{m_L(a)+C_L}{2C_L} =\sum_{(k,L)>1}\nu_L(k)\frac{r_{(k,L)}(a)}{(k,L)-1}. A gap at most \varepsilon C_L gives \nu_L-mass at most \varepsilon/(2\delta) to relative positions exceeding \delta. Nevertheless \begin{aligned} m_{L_m}(1)&=-C_{L_m}+\frac23+O(\log(2m)/m),\\ F_Q(1)&=-V_Q+\frac23+O(\log(2Q)/Q). \end{aligned} Thus relative near-extremality does not require divisibility by small primes.
Proof. The identity is a rearrangement of (45); Markov’s inequality proves the stability assertion. At address one, the exact conditional gap is \frac23\sum_{(k,L_m)>1}w_k. This sum contains every 2\le k\le m. The omitted mass is O(\log(2m)/m). For (57), use P_q(1)-P_q(0)=-\mu(q) and \sum_{q\ge2}\mu(q)\eta_q=-1. ◻
The analogous finite probability weights are \beta_d(Q)(d-1)/(3V_Q). There is also an absolute, much smaller divisibility threshold. If Z\le Q and F_Q(a)+V_Q<\log Z/(6Z^2), then L_Z\mid a. Indeed a missing prime power t\le Z has w_t\ge\log t/(2t^2) and contributes at least w_t/3 to the gap. This is an absolute-gap assertion, consistent with the failure of relative divisibility stability.
Theorem 24 (Tails and finite moments). For every 0<c<\pi, \mathbb P(|Y|>H)\ll_c \exp(-c\sqrt H)\qquad(H\ge1). The same bound is uniform for Y_M. Consequently Y_M\to Y in every finite L^r, \left\|Y\right\|_r\ll r^2, and \mathbb E\exp(c\sqrt{|Y|})<\infty for c<\pi. The support is unbounded in both directions. For every d>1/\sqrt\Gamma, eventually \mathbb P(Y>H)=\mathbb P(Y<-H)\ge\exp\{-\exp(d\sqrt H)\}.
Proof. The exact supremum of |Y_L| is D_*(L). Choose c<d_0<\pi and L=\lfloor\exp(d_0\sqrt H)\rfloor. Equation (16) gives D_*(L)\le(1-\varepsilon)H for some fixed \varepsilon>0 when H is large. Chebyshev applied to Y-Y_L gives O(\log^2L/(LH^2))\ll_c e^{-c\sqrt H}. For Y_M, if M\le L the event is empty; otherwise the same tail-variance argument is uniform in M. These uniform tails imply all the moment and integrability statements.
On the cylinder A=\{x=0\bmod L\}, the conditional mean is -C_L. For H<C_L, \mathbb E((-Y-H)\mathbf 1_{A\cap\{Y<-H\}})\ge(C_L-H)/L. Cauchy–Schwarz and \mathbb EY^2=\sigma^2 therefore give \mathbb P(Y<-H)\ge(C_L-H)^2/(L^2\sigma^2). Choose L=L_m with m=\lfloor\exp(d_0\sqrt H)\rfloor and 1/\sqrt\Gamma<d_0<d. Theorem 20 and \log L_m\sim m yield the lower bound. Reflection gives the other sign. ◻
Lemma 25 (Bernoulli activation). Suppose independent Bernoulli variables have probabilities b_i\le1/2. If changing any coordinate from zero to one increases a function by at least a>0, then the probability that its value lies in any interval of length less than a is O((1+\sum_i b_i)^{-1/2}).
Proof. Such an inverse image is an antichain in the coordinate order. Write each Bernoulli variable as C_i\varepsilon_i, with C_i of probability 2b_i and \varepsilon_i fair, independently. Conditional on the C_i, Sperner’s estimate bounds the probability by O((1+\sum C_i)^{-1/2}). The variance of \sum C_i is at most its mean. Splitting at half that mean and applying Chebyshev gives the assertion. This is the probabilistic antichain method of [1]. ◻
Theorem 26 (Concentration and absence of atoms). For 0<\varepsilon\le1, \sup_u\mathbb P(Y\in[u,u+\varepsilon])\ll\varepsilon^{1/4}. The law of Y therefore has no atoms. The laws of f, g, and of every fixed nonempty increment block have no atoms either.
Proof. Let F_h(x)=\sum_{j=1}^h f(x+j) and select primes P/2<p\le P, where h\le P/4. The event that p divides one of x+1,\ldots,x+h has probability h/p\le1/2; there is at most one hit. These events are independent across primes under Haar measure.
For each selected prime, sample a Bernoulli activation indicator with parameter h/p. Independently sample a hit location uniformly from \{1,\ldots,h\} and a positive valuation e with probabilities \mathbb P(e=j)=(p-1)/p^j, j\ge1, together with the conditional p-adic tail. Also sample an independent inactive p-adic coordinate, conditioned to avoid the h hit classes. These auxiliary choices are independent of the activation indicators and independent across primes. They reproduce Haar measure: conditional on a hit, its location is uniform and its valuation has the displayed law. Fix the auxiliary choices and all other prime coordinates. Activating p leaves all divisors prime to p unchanged and adds only positive divisor terms, including the term a_p=H_{p-1}/p at the hit. Terms involving several selected primes are also nondecreasing. This argument applies first to finite divisor sums. Their increasing limit is almost surely finite by Theorem 9. For the fixed finite set of selected primes, every activation pattern has positive probability. Removing the union of its finitely many exceptional auxiliary sets makes the full sums finite simultaneously for all patterns. The same monotonicity and increment lower bound therefore hold for the full sums on that common set. Thus Lemma 25 and the prime number theorem give \sup_u\mathbb P(F_h\in[u,u+\delta]) \ll\min\left(1,\sqrt{\frac{\log P}{h}}\right), \quad \delta<\frac{\log(P/2)}P.
Let A=\{Y\in I\} with |I|=\varepsilon and \rho=\mathbb P(A). By (22), A\cap\tau^{-h}A places F_h in an interval of length 3\varepsilon. Take P=\lfloor\log(1/\varepsilon)/(10\varepsilon)\rfloor and N=\lfloor P/4\rfloor, for sufficiently small \varepsilon. Stationarity and the second moment of \sum_{j=0}^{N-1}\mathbf 1_A(x+j) give \rho^2\le\frac{\rho}{N} +\frac2{N^2}\sum_{h=1}^{N-1}(N-h)\mathbb P(A\cap\tau^{-h}A) \ll\sqrt{\frac{\log P}{N}}. Taking square roots proves (58). For a fixed h, select instead all primes 2h<p\le P. Their activation probabilities have divergent sum as P\to\infty. The level set of a single value is an antichain for every finite set of selected primes, since all increments are positive. Lemma 25 then forces every atom of F_h to have mass zero. This includes f=F_1\circ\tau^{-1} and hence g. ◻
The characteristic function has the explicit finite approximation \Phi(t)=\mathbb Ee^{itY} =\lim_{M\to\infty}\frac1{L_M}\sum_{a\bmod L_M}e^{itY_M(a)}, \qquad |\Phi(t)-\Phi_M(t)|\ll |t|\frac{\log(2M)}{\sqrt M}. It is real, even, and smooth, by symmetry and finite moments. Neither analyticity nor a density follows from these assertions. Combining (58) with the L^2 tail estimate gives the uniform distribution-function approximation \sup_u|\mathbb P(Y_M\le u)-\mathbb P(Y\le u)| \ll\left(\frac{\log^2(2M)}M\right)^{1/9}. Indeed compare at distance \varepsilon, with errors O(\varepsilon^{1/4}) and O(\log^2(2M)/(M\varepsilon^2)), and balance them.
There are no gaps in the support longer than 2/3. If a gap (a,b) with b-a>2/3 separated two parts of positive probability, the bound g\le2/3 would imply that \{Y\le a\} is forward invariant modulo null sets. Measure preservation would make it invariant, contradicting ergodicity. Unbounded support ensures positive mass on both sides of any alleged gap. This does not identify the full support or prove absolute continuity.
Theorem 27 (Canonical cutoff singularity). For every fixed nonnegative integer n and every M\ge n, Y_M(n)=D_n-D_*(M)-\frac23n\sum_{k>M}w_k. In particular Y_M(n)\to-\infty. At every fixed negative integer it tends to +\infty. Nevertheless D_n=\lim_{M\to\infty}\bigl(Y_M(n)-Y_M(0)\bigr).
Proof. For k>M\ge n, the residue r_k(n) equals n. Separate these terms in (14) and subtract D_*(M). The tail tends to zero at fixed n, whereas D_*(M)\to\infty. Reflection around -1/2 gives the negative integers. Subtracting the formula at zero proves the final identity. ◻
The integer orbit is dense in \widehat{\mathbb Z} and has Haar measure zero. Theorem 27 concerns the canonical cutoffs on that orbit; it does not assign point values to an L^2 equivalence class. At a growing resolution the exact equation is instead D_n-D_*(n)=Y_n(n)+\frac23n\sum_{k>n}w_k. The changing observable Y_n must be evaluated at the changing integer address n. Its distribution cannot be deduced from unique ergodicity for a fixed continuous function.
The moving-diagonal identity isolates a separate arithmetic transfer problem. Determining the distribution of this changing observation, either on integer intervals or at resolutions p-1, requires additional averaging estimates. The stationary theory alone does not provide that transfer. The mixed-moment identity in Theorem 18 recovers every spectral amplitude and hence the original boundary mass.
[1]M. Aizenman, F. Germinet, A. Klein, and S. Warzel, On Bernoulli decompositions for random variables, concentration bounds, and spectral localization, Probab. Theory Related Fields 143 (2009), 219–238, https://doi.org/10.1007/s00440-007-0125-7.
[2]F. W. J. Olver et al., eds., NIST Digital Library of Mathematical Functions, Sections 4.22 and 5.11, https://dlmf.nist.gov/.
[3]M. Foreman and B. Weiss, Odometer based systems, https://arxiv.org/abs/2009.10162.
[4]T. Hilberdink, F. Luca, and L. Tóth, On certain sums concerning the gcd's and lcm's of k positive integers, Int. J. Number Theory 16 (2020), no. 1, 77–90, https://doi.org/10.1142/S1793042120500049.
[5]R. J. Lemke Oliver and K. Soundararajan, The distribution of consecutive prime biases and sums of sawtooth random variables, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 1, 149–169, https://doi.org/10.1017/S0305004118000592.
[6]J. D. Lichtman, Mertens' prime product formula, dissected, https://arxiv.org/abs/2002.03361.
[7]A. S. Petty, The Cubic Law for Digit-Collision Energy, https://doi.org/10.5281/zenodo.20547910.
[8]A. S. Petty, The Secondary Term in Digit-Collision Energy, https://doi.org/10.5281/zenodo.21875096.
[9]F. K. Richter, A new elementary proof of the prime number theorem, https://arxiv.org/abs/2002.03255.
Discussion
Sign in to join the discussion.