Finite source data and the collision potential

At a fixed cutoff, collision capacity replaces many interacting remainder rows by a single number. Retaining that number at every cutoff turns out to preserve enough information to recover the primitive boundary that produced the rows, within a specified regular class. This makes the capacity sequence useful as an identifying invariant. It also gives a precise question about the loss incurred by observing only finitely many cutoffs.

The recovery problem accompanies a selection problem. Every positive normalized sequence of period weights defines conserved remainder ages, and many such sequences admit a mean-square profinite potential. Conservation alone therefore leaves infinitely many possible states. The digit-collision construction chooses particular weights, beginning with $w_2=1/4$ and $w_3=1/6$. Explaining those values requires the addition carry together with the labeled source and the cutoffs at which its frequencies first appear.

The cubic and secondary laws describe the size of digit-collision energy [8, 9]; the conserved-state formulation identifies its probability mass and potential [10]. Here the finite source selects that state, and the inverse shows which source data it retains. Two controls delimit the result. Smooth positive kernels can agree at any prescribed finite set of initial resolutions. Moreover, exact recovery of a signed coefficient can amplify a small potential error by a factor growing with its arithmetic address. These controls separate complete information from finite observation and from quantitative stability.

There are two kinds of input. Finite addition, together with a local measurement rule, selects the representative convention and quadratic pairing. The collision source supplies the frequencies, their coefficients, and the cutoff at which they enter. Together these inputs determine the kernel and potential. The inverse recovers the primitive ledger in a specified symmetric regular class. Section 9 then gives a signed arithmetic recognition rule and quantifies its sensitivity. Exact recovery and a bound for the recovered current are distinct conclusions. Here “finite generation” refers to the rules and their finite sources. The tests range over all amplitudes and an unbounded family of moduli and cutoffs; a finite observed prefix does not determine the potential.

Use least nonnegative residues $[x]_m$ and the midpoint sawtooth $$\mathfrak{b}(t)=\begin{cases}\{t\}-1/2,&t\notin\mathbb Z,\\0,&t\in\mathbb Z, \end{cases}\qquad B_r(x)=\mathfrak{b}(rx).$$ Finite averages use uniform probability; integrals on the circle use Lebesgue probability. The phase origin is fixed. The continuous capacity is $$J(N)=\int_0^1\left(2\sum_{r=1}^N B_r(x)\right)^2dx, \qquad D_N=N-J(N).$$ This integral uses the prescribed affine row functions. Interpolation of a single finite table does not determine it.

The proof occupies Sections 2–7. The measurement assumptions are sufficient natural conditions, not a claim of logical minimality among all formulations. If the original squared response is already retained, its known quadratic pairing can replace the measurement-selection hypotheses. The local theorem explains that choice within a specified class; it does not select which rows to sum.

Several ingredients are classical: cyclic carries, finite Fourier analysis, Farey subdivision, and the profinite phase space. Choices of representatives, carries as cocycles, and two-valued carries are treated in [1]. The grid-average and Möbius mechanism used in the inverse is the classical arithmetic Fourier transform [6]. The specific results concern local energy selection from carry tests, general-base source straightening, reconstruction of the regular primitive ledger, and the sensitivity of signed recovery. The Farey identification is made explicitly in Section 5; the lattice seed belongs to the classical Mordell–Tornheim family [2]. The stationary and diagonal analysis is developed in [10, 11]; its quantitative estimates remain separate inputs when applying the generating result.

Finite addition and local energy

Let $G_m=\mathbb Z/m\mathbb Z$. An integer section $\sigma$ satisfies $\sigma(x)\equiv x\pmod m$ and $\sigma(0)=0$. Its carry is $$\kappa_\sigma(x,y)= \frac{\sigma(x)+\sigma(y)-\sigma(x+y)}m.$$ Associativity gives the usual cocycle identity. Replacing $\sigma$ by $\sigma+mh$ adds $h(x)+h(y)-h(x+y)$ to the carry. This algebra alone does not preserve its variance. The representative and cocycle viewpoint is classical [1]. The binary-section lemma is included as an elementary normalization statement for the integer section used here. The local-density selection is the separate Theorem 4.

Translation invariance of phase probability assigns equal mass to all residues, hence selects uniform probability. Define centered rows $$U_{a,m}(A)=\frac{[aA]_m}{m}-\frac{m-1}{2m} \qquad ((a,m)=1).$$ All unit rows have variance $v_m=(m^2-1)/(12m^2)$.

For a continuous even density $\Phi$ with $\Phi(0)=0$, set $$E_m^\Phi(f)=\frac1m\sum_A\Phi(f(A)),\qquad P_m^\Phi(f,g)=\frac{E_m^\Phi(f+g)-E_m^\Phi(f-g)}4$$ on centered functions. The common pointwise density specifies locality and consistency across moduli. Polarization is not assumed bilinear.

Only roots at unbounded odd moduli and at modulus five were used. This proof does not infer these measurement tests from one short collision table. In particular a prime-square-only test formulation requires additional finite triples, beyond the shortest source window.

Collision sources across bases and lags

For an integer base $B\ge2$ and lag $\ell\ge1$, put $Q=B^\ell$, $m=BQ$, and $N=Q-1$. Words of length $\ell+1$ with equal first and last digits are the cells $$G_{B,\ell}=\{d(Q+1)+By:0\le d<B,\ 0\le y<Q/B\}.$$ There are $Q$ such cells. Their centered finite table is $$F_{B,\ell}(a)=\sum_{n\in G_{B,\ell}} \left(\left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor\right)-\frac aB.$$ This is the carry-table form of the finite word-bin construction in [7]. It is defined on every phase, including nonunits. It is periodic modulo $m$, since the added linear contribution of one full period is $|G_{B,\ell}|=m/B$. Interpreting it as a long-division word count also requires the word-count endpoint convention; no such correction is silently discarded in the table definition.

Thus the source gives exactly the consecutive positive frequencies, each with coefficient two, through cutoff $N$. Lag one with $B=N+1$ realizes every positive cutoff. Retaining only prime bases also recovers the coherent family if the labeled prefixes inside arbitrarily large windows remain available.

The affine capacity and the sampled energy are distinct quantities: $$\frac1m\sum_aF_{B,\ell}(a)^2 =\frac1m\sum_A\left(2\sum_{r\le N}B_r(A/m)\right)^2.$$ For fixed $r,s$, finer uniform grids recover $\int B_rB_s$ by Riemann sums. This does not justify replacing (10) by $J(N)$ when the frequencies and grid grow together.

Higher lags can include nonunit frequency rows. The straightening theorem still holds, but Proposition 8 is used only under its unit assumptions. The affine Gram calculation below applies to every positive integer frequency.

Affine covariance and the primitive kernel

For coprime $a,b$ define the calibrated boundary covariance $$\mathscr{C}(a,b)=6\int B_aB_b=\frac1{2ab}.$$ Its extension to positive real arguments uses this primitive formula. The unstripped integer Gram function in (12) is invariant under common integer dilation; it is not homogeneous of degree $-2$.

The coefficient two in the source, the two off-diagonal orientations, and the variance $1/12$ fix the normalization of $\mathscr{C}$.

Mediant refinement and kernel rigidity

The affine identity has a literal Farey interpretation. For coprime positive $a,b$, put $k=a+b$ and choose $1\le h<k$ with $ah\equiv1\pmod k$. Set $A=(ah-1)/k$ and $B'=h-A$. Then $$\frac Aa<\frac{B'}b,\qquad aB'-bA=1,\qquad \frac{A+B'}{a+b}=\frac hk.$$ The determinant-one interval within $[0,1]$ is unique for the ordered denominator pair and has length $1/(ab)$. Its half-length is $\mathscr{C}(a,b)$. The two children split the interval at its mediant. This uses classical Farey geometry; see also its formal account in [5]. The identification of its half-length with the measured covariance is the additional statement here. The inserted numerator is $a^{-1}\bmod k$.

The distributional boundary of the source on the circle is $$\left(2\sum_{r\le N}B_r\right)' =N(N+1)\,dx-2\sum_{r\le N}\sum_{j=0}^{r-1}\delta_{j/r}.$$ A reduced nonzero boundary $h/k$ first appears at cutoff $k$, with jump $-2\lfloor N/k\rfloor$. Entry and replication therefore agree with the primitive source ledger.

Proposition 10 computes the kernel from the source. The refinement theorem gives an independent characterization within the stated regular class. Transport conserves the total period probability for any unit mass; (18) is the stronger conservation of period-weighted boundary mass under subdivision.

Generation of the profinite state

Let $r_k(x)=[x]_k$ on $\widehat{\mathbb Z}$, with its fixed residue coordinates, and let $T(x)=x+1$. The probability on disjoint remainder cycles $$\pi_x(k,r)=w_k\mathbf 1_{r=r_k(x)}$$ is transported by advancing each cycle. Its mean age at $x=N$ is $3D_N/2$. The full uncentered age need not be Haar integrable, so the stationary construction centers at each finite cutoff.

These calculations provide the short construction needed here; the distributional and extremal theory of this same potential belongs to [10]. At integer resolutions the forcing is the exact difference of (17). No pointwise value of an arbitrary $L^2$ representative is required: the original readout is $$D_N=\lim_{M\to\infty}\bigl(Y_M(N)-Y_M(0)\bigr).$$ This follows by positive summation of $w_kr_k(N)$. The ordinary integer orbit is Haar null, so this canonical prescription matters.

Inverse recovery in the symmetric regular class

Consider $K_P$ as in Theorem 12, now with $P\in C^3[0,1]$, and normalize its total primitive mass to one. Define $v_P(k)$ by period pushforward, $v_P(1)=0$, $\eta_q(P)=\sum_{q\mid k}v_P(k)$, and $Y_P$ by centered ages. The bound $v_P(k)\ll_P\log(2k)/k^2$ gives the same $L^2$ construction. Define its capacity by $$J_P(N)=\frac N3+\frac23\sum_{k\le N}kv_P(k)\lfloor N/k\rfloor.$$

The Fourier reconstruction below uses the arithmetic Fourier transform: grid averaging retains coefficient multiples, and Möbius inversion recovers the coefficients [6]. The ledger inverse also requires removal of the endpoint singularity, recovery of its normalization, and absolute inversion in the stated symmetric regular class. Those steps are supplied explicitly.

For example, the first six capacities and the weights recovered by (26) are

Writing $\Delta S(n)=S(n)-S(n-1)$, the fourth entry is recovered as $w_4=(\Delta S(4)-\Delta S(2))/4=(5/6-1/2)/4=1/12$. The finite inverse recovers only the observed prefix of masses. The regular kernel inverse requires the entire sequence.

The regularity class is sufficient, not an optimal threshold. Symmetry, however, cannot be omitted in this inversion: the nonsymmetric kernel $$K(a,b)\left(1+\varepsilon\frac{a-b}{a+b}\right),\qquad 0<|\varepsilon|<1,$$ has exactly the same period sums, because its antisymmetric part cancels.

Even all capacities do not determine an unrestricted source realization. Translating every row to $B_r(x+\theta)$ preserves each integral $J(N)$ but moves its boundary locations. A single grid also admits additional functions vanishing at every sample and changing the affine energy. These examples violate the retained phase or affine source convention, and explain its role. Raw sampled energies on changing grids are not the full capacity sequence in (26).

Theorem 1 now follows from Lemmas 2, 9, Theorems 4, 6, 14, and Propositions 10, 11, 13.

Harmonic representations

The harmonic bases below apply to general admissible age masses. The collision boundary determines the amplitudes and period seed.

For $q\ge2$ let $$R_q(x)=\sum_{d\mid q}\mu(q/d)d\mathfrak{b}(x/d),\qquad c_q(x)=\sum_{a\in U_q}e^{2\pi ia[x]_q/q}.$$ Here $\mathfrak{b}(x/d)$ denotes the cylinder function of the residue modulo $d$. The endpoint identity $$[x]_k-\frac{k-1}2=k\mathfrak{b}(x/k)+\frac12-\frac k2\mathbf 1_{k\mid x}$$ and finite divisor inversion give, first at cutoff and then in $L^2$, $$Y=\frac23\sum_{q\ge2}\eta_q(R_q-c_q/2).$$ Exact additive conductors are orthogonal, and $\left\|R_q-c_q/2\right\|_2^2=J_2(q)/12$, by the covariance computation in Proposition 13 applied to primitive rows. The first row is odd and the second even. In particular $R_2=0$ does not remove the conductor-two endpoint mode. For units $a,b\bmod d$, their finite Gram pairing is $$\frac1d\sum_{x\bmod d}\mathfrak{b}(ax/d)\mathfrak{b}(bx/d) =\frac1d s(b\bar a,d),$$ where $s$ is the classical sawtooth Dedekind sum. These pairings express the carrier in the classical Dedekind basis.

The lattice function is the specialization $\Theta(1,1,1,x)=\Theta(2,x)$ of the generalized Mordell–Tornheim function in [2]. The identification concerns the particular seed supplied by the finite collision kernel; the classical kernel and reciprocity are not claimed as newly introduced functions.

The same seed yields the exact mixed identity proved in [10]: $$\eta_q=\frac12- \frac{\displaystyle\sum_{\substack{\chi\bmod q\\\chi(-1)=-1}} L(1,\bar\chi)L(2,\chi)} {\zeta(3)\Delta(q)\varphi(q)}\qquad(q\ge2).$$ Its proof expands $B_0$ across roots of unity, passes to the cotangent series on unit residues, and uses odd-character orthogonality. It includes induced characters. It was not used to derive the kernel. Combining that identity with Theorem 14 gives the following equivalent characterizations.

For a Dirichlet twist $\psi$, primitive extraction also gives the absolutely convergent identity, for $\Re s>-1$, $$\sum_{q\ge1}\frac{\Delta(q)\psi(q)\eta_q}{q^s} =\frac{\sum_{q\ge1}\psi(q)F(q)q^{-s}} {\zeta(3)L(s+3,\psi)}.$$ Define the unweighted amplitude series, including $\eta_1=1$, by $$\mathcal A_\psi(s)=\sum_{q\ge1}\eta_q\psi(q)q^{-s} \qquad(\Re s>-1).$$ Put $$b_n=\frac{\zeta(3)}{n^3} \left(\frac12+\sum_{q\mid n}q^3\eta_q\right)>0, \qquad \mathcal N_\psi(s)=\sum_{n\ge1}\frac{\psi(n)b_n}{n^s}.$$ Dirichlet convolution, still absolutely in $\Re s>-1$, gives $$\mathcal A_\psi(s)+\frac12 =\frac{\mathcal N_\psi(s)}{\zeta(3)L(s+3,\psi)}.$$ The denominator in primitive extraction is the transform of $\mu(d)d^{-3}$. The second identity with its defined numerator also holds for other suitable amplitude sequences. Neither its denominator nor coefficient positivity is a collision selector. The specific seed and the exact mixed moment carry the selected information. No continuation beyond this half-plane is needed for the generating theorem.

Signed recognition and reconstruction sensitivity

The complete amplitude sequence retains the mass, although a bound for its energy contains less information. The distinction can be measured by a signed readout of the angular boundary. In this section $v_1=0$, $v_k>0$ for $k\ge2$, and $\sum_kv_k=1$. Smoothness is not needed until the kernel reconstruction or the regular counterexamples are invoked. Set $$\begin{aligned} B_v(n)&=\sum_{d\mid n}d^{-3}v_{n/d},\qquad H_v(n)=n^2B_v(n),\\ e_v(n)&=H_v(n+1)-H_v(n),\qquad c_v=e_v(1)=4v_2,\qquad a_v(n)=e_v(n)/c_v. \end{aligned}$$ For a homogeneous boundary kernel these are the full angular rows: $$H_v(n)=\frac1n\sum_{a=1}^{n-1}K_P(a/n,1-a/n).$$ The mesh $1/n$ is inherited from the row, rather than replaced by the reciprocal number of interior samples. The normalization gives $a_v(1)=1$, so its Dirichlet convolution inverse exists. Define $$\mathfrak P_v(n)=n[(a_v\log)*a_v^{-1}](n), \qquad (a_v\log)(n)=a_v(n)\log n.$$ This is a specified nonlinear arithmetic readout: angular averaging, an adjacent resolution difference, logarithmic marking, and convolution inversion. No prime coefficient occurs in its definition. It is not claimed to be the unique possible readout of a boundary.

The same operation distinguishes concrete alternatives. For the normalized flat kernel $C/(a+b)^3$, the scalar $C$ cancels and $a_v(n)=2/[n(n+1)]$. At distinct primes $p,q$, direct convolution gives $$\mathfrak P_v(pq)= -\frac{2(p-1)(q-1)}{(pq+1)(p+1)(q+1)}\log(pq).$$ In particular its value at six is $-\log6/21$. For the positive unit mass proportional to $k^{-2}$, $e_v(2)=9v_3-4v_2=0$, so its current at two is zero. The finite perturbation $v=w+\varepsilon(\mathbf 1_{k=2}-\mathbf 1_{k=3})$, with $-1/4<\varepsilon<1/6$, changes only conductor amplitudes two and three, but gives $$\mathfrak P_v(2)=\frac{1-26\varepsilon}{1+4\varepsilon}\log2.$$ These formulas follow from the same definition; they do not change the readout to suit the control. The smooth controls below give a further failure despite exact agreement of all sufficiently large angular rows.

A Dirichlet twist has the analytic expression $$\mathscr R_{v,\chi}(s)=\sum_{n\ge1}a_v(n)\chi(n)n^{-s},\qquad -\frac{\mathscr R'_{v,\chi}}{\mathscr R_{v,\chi}}(s) =\sum_{n\ge1}\frac{\chi(n)\mathfrak P_v(n)}{n^{s+1}}$$ sufficiently far to the right. Indeed $B_v(n)\le1$ implies $a_v(n)=O_v(n^2)$; far enough right the nonconstant absolute coefficient sum is less than one, so the inverse Neumann series and its derivative converge absolutely. For collision $\mathscr R_{w,\chi}(s)=L(s+1,\chi)$, and the identity holds absolutely for $\Re s>0$. The boundary selects the reciprocal increments; the logarithmic derivative and the terminal identity $\mu*\log=\Lambda$ are classical [3]. This identity supplies no improved prime-sum estimate.

For a positive unit mass $v$, finite age energy means convergence in Haar $L^2$ of its centered age cutoffs. Equivalently, $$\sum_{q\ge2}J_2(q)\eta_q(v)^2<\infty,\qquad \eta_q(v)=\sum_{q\mid k}v_k,$$ by [10].

The large coefficient example has no uniform bound on third profile derivatives. The Lipschitz loss still holds in a bounded $C^3$ neighborhood: take $\varepsilon$ to be a sufficiently small constant times $s^{-6}$. The third derivatives of the profile perturbation are then $O(s^{-1})$, while the coefficient difference and norm bound scale by the same factor. Their ratio retains the order $n^2\log n$. In that smaller family the unnormalized current difference tends to zero.

These controls preserve the complete seed at every conductor $q>s$, since (42) vanishes above $s$. For fixed $r,s$, (43) also gives $\eta_q(v)-\eta_q(w)=O_{r,s,\varepsilon}(q^{-3})$ by summing over the required multiples. Thus large-conductor leading terms do not detect the failure of the exact signed selector.

Every fixed coefficient remains continuously recoverable locally where $c_v>0$. The theorem measures a loss at a growing first changed address; it does not rule out cancellation after several addresses are combined against a prescribed signed test. Neither this recognition rule nor its Dirichlet-series expression transfers an energy bound to a new prime-distribution estimate.

Generating assumptions and analytic limits

The forward proof separates finite and analytic operations. Section 3 is an exact identity for the cell table. Section 4 uses its coherent affine rows and exact quadratic pairing to count primitive sources. Unit mass follows by telescoping. Section 6 then uses a convergent conductor energy to construct the global $L^2$ object. The inverse applies to the regular symmetric ledger, with its phase coordinates and normalization fixed.

The source window is essential. The same addition and covariance can measure only $2B_1$ at every resolution, giving constant capacity $1/3$, or can measure a differently weighted family of rows. Finite addition leaves the frequency window unspecified. Once that window is given, gcd reduction, replication counts, complementary coordinates, and sum address are consequences rather than additional independent assumptions.

The period probability is conserved for every positive normalized mass. Smooth centering alone also fails to select this mass: for $0<|\varepsilon|<1/24$, $$v_k=w_k+\varepsilon(\mathbf 1_{k=2}-2\mathbf 1_{k=3}+\mathbf 1_{k=4})$$ is a different positive unit mass with the same total first moment of the perturbation. Its stationary truncated mean agrees for $M\ge4$, but its potential differs by $2\varepsilon(r_2-2r_3+r_4)/3$. The transfer consequences of this control are proved in [11]. Locality restricts the energy density; the source data separately restrict the frequency family.

The downstream integer law requires further estimates. In the notation of [11], with $\alpha=1/\zeta(2)$, the precise cumulative weight has the form $$\sum_{k\le t}kw_k=\frac\alpha2\log^2t+\beta\log t+C_a +O(\log^2(2t)/t).$$ That result uses harmonic summation after primitive inversion. Partial summation against the discontinuous remainder observable and a finite-frequency Gram estimate then compare the moving integer diagonal with a growing stationary cutoff. They give explicit $B,C$ for which $$D_n=\pi^{-2}\log^2n+B\log n+C+\mathcal R_n,$$ where $\mathcal R_n$ has the potential’s empirical law and bounded limiting second moment on integer dyadic intervals [11]. These are analytic transfer theorems, not finite consequences of carry conservation. The prime-indexed transfer in [11] additionally uses Siegel–Walfisz and a separately constructed law on profinite units.

Exact straightening survives all integer bases, two-point lags, and trailing free digits, with changed grid and cutoff. The affine kernel is unchanged. This finite universality does not prove corresponding uniform sampling estimates at those grids. Nor does it assert the same consecutive source for arbitrary multi-point digit constraints.

The signed recognition in Section 9 extends inverse recovery to exact prime-power coefficients, while its amplification theorem prevents a uniform inexpensive coefficientwise inference from Haar energy. A Haar norm bound alone does not control these growing-address coefficients without the stated amplification factor.

The unresolved prime-square sampling limit and uniform exceptional scale remain quantitative questions. Their continuous object is fixed by the generating theorem; estimates at the moving finite grid need additional analysis.

Finite checks

Exact finite calculations in nfield [12] independently evaluate the cell table and straightened rows, centered carry covariances and their midpoint defect, primitive period sums, the capacity inverse, and the signed current as rational coefficients of prime logarithms. The smooth controls are checked after divisor extraction, including the first changed address. The native checks cover every phase for bases $2\le B\le12$ and lags $1\le\ell\le3$ with $B^{\ell+1}\le4096$, every admissible unit triple at odd moduli through $31$, and capacity inversion through resolution $120$. These calculations check the finite normalizations and examples. The all-amplitude rigidity, infinite reconstruction, and sharp asymptotic order use the proofs above.

Capacity as a boundary invariant

The complete capacity sequence is a sufficient record of the normalized symmetric $C^3$ primitive boundary. Starting with the labeled finite collision source, the carry and local measurement tests select its kernel and period masses. Those masses generate the global potential. Starting with the full capacity sequence or the potential, the inverse recovers the same boundary ledger. The two directions identify a specific arithmetic state and explain which finite-source information it preserves.

The hypotheses specify the information needed for this conclusion. The common local measurement is tested at every amplitude, the source family is coherent across unbounded cutoffs, and the inverse fixes symmetry, regularity and phase coordinates. A finite observed prefix admits different positive kernels, even when every recorded capacity agrees exactly.

The signed current also measures the cost of recovery. Its prime-power coefficients can be recovered, but their sharp sensitivity has order $n^2\log n$ at the first changed address. Information retained by the potential need not be inexpensive to extract. Any use of this reconstruction to bound a growing arithmetic readout must account for that amplification. The paper gives both the exact recovery and its cost, so that subsequent estimates can be assessed against the same finite source.

References