Carry Boundaries and Bernoulli Spectra in Long Division
Abstract
Long division emits a digit when multiplication crosses an integer wall of the floor. Each crossing leaves a centered sawtooth. On primitive odd characters, its transform is a generalized Bernoulli number. The floor supplies one universal response. A digit rule supplies the signed boundary on which that response acts.
Let F_w be the centered carry count attached to a weight w, and let S_w(\chi) be its signed boundary flux. Every primitive odd Dirichlet character satisfies the exact factorization \widehat F_w(\chi)=-B_{1,\bar\chi}\,S_w(\chi). Every finite-prefix digit relation for a denominator coprime to the base reduces to this form after its bulk and terminal endpoint are removed. Ordinary boundary fluxes recover exactly the boundary derivative on the unit group. Primitive transforms use that flux. Induced transforms use a separate conductor-depth pairing.
For the terminal collision diagonal modulo an odd prime power at depth at least two, the signed boundary has an exact unit-boundary norm, cancels at the relevant lower conductor, and reverses under reflection. Those facts force the exact primitive odd second moment \frac1{n_{\mathrm{po}}} \sum_{\chi\ \mathrm{primitive\ odd}}|S_G(\chi)|^2 =2|\partial_{\mathrm{form}}G|. Base 15 shows why the conductor condition matters. Lower-conductor boundary mass survives, and the mean falls from 60 to 28. A conductor ledger records that obstruction exactly. The same boundary law therefore explains both prime-power closure and composite failure. Squaring the primitive and conductor-depth factorizations gives an energy built from finite sawtooth correlations. Its unit-index terms are finite Möbius combinations of classical Dedekind sums.
Introduction
Digit collisions, agreements, transitions, and blocks look like different combinatorial questions. In long division they are built from the same event. Multiplication moves a remainder, and the floor records the integer wall crossed by that motion.
Fix an integer base b\ge2 and a modulus q coprime to b. For r\in\mathbb{Z}/q\mathbb{Z}, let \langle r\rangle_q denote its representative in \{0,\dots,q-1\}. Long division is governed by \mathop{\mathrm{T}}_b(r)=br\bmod q, \qquad \delta_q(r)=\left\lfloor\frac{b\langle r\rangle_q}{q}\right\rfloor . The first operation moves the remainder. The second records the wall crossed by that motion. The floor is not passive notation for a digit. It is the mechanism that emits the digit.
Repeated wall crossings leave a centered sawtooth. Its primitive odd transform is the generalized Bernoulli value B_{1,\bar\chi}, independent of the digit rule. The rule marks a finite region. Interior differences cancel, leaving its signed boundary. Each primitive odd transform is \text{universal floor potential}\times\text{selected boundary flux}. Every rule on the first N digits enters this form after its bulk and terminal endpoint are removed. Ordinary fluxes see only the unit boundary. Primitive transforms use that flux. Induced transforms retain the depth weights developed below.
The same boundary mechanism explains both a clean law and its failure. At terminal collision diagonals over odd prime powers of depth at least two, lower-conductor cancellation and antipodal reflection force an exact second moment. Base 15 retains lower-conductor boundary mass, and the mean changes. The conductor ledger records the obstruction exactly.
Sawtooth transforms, generalized Bernoulli numbers, Gauss sums, character orthogonality, and Dedekind sums are classical [5, 6, 7]. The new arithmetic lies in the finite boundary selected by a digit rule.
Finite determination for the two-point collision diagonal is established in [1]. Its finite-group character transform is developed in [2]. The prime-base depth-two Bernoulli factorization and its Parseval consequence occur in [3]. For the prime-base depth-two indicator, the two formulas agree after multiplying the normalized coefficient in [3] by \varphi(b^2) and matching the conjugate flux.
The carry-boundary reduction applies to arbitrary weighted finite cylinders. It treats nonunit endpoints through every prime-power conductor depth, identifies the unit-boundary image and kernel, and proves the terminal prime-power mean law and the composite conductor ledger.
Character sums over integers selected by fixed-base digit restrictions have also been studied in [9]. Those sums range over digit-restricted integers. The object here is instead the signed carry boundary produced by a denominator slice. The base-three calculation in Section 7 displays the carry table, boundary flux, and floor potential directly.
The reduction covers weighted digit rules, collisions, agreements, transitions, and block patterns whenever they depend on a finite prefix. Whole-orbit statistics require a finite boundary of their own.
Cylinder-to-Carry Reduction
A finite digit rule depends on a fixed number of digits. At level N, each digit word is represented by one cell index. The count therefore descends to a function of one multiplier residue.
Lemma 1 (Digit locality). For \frac rq=\sum_{i\ge1}d_i b^{-i}, \qquad 1\le r<q, let the digits be the greedy long-division digits d_j=\left\lfloor\frac{b^j r}{q}\right\rfloor -b\left\lfloor\frac{b^{j-1}r}{q}\right\rfloor. The readout after \ell remainder steps is the (\ell+1)-st digit, \delta_q\!\left(\mathop{\mathrm{T}}_b^\ell(r)\right) =\left\lfloor\frac{b^{\ell+1}r}{q}\right\rfloor\bmod b =d_{\ell+1}. Hence an observable built from readouts at lags \ell_1<\cdots<\ell_t depends on r/q only through d_1,\dots,d_{\ell_t+1}.
Proof. Write b^\ell r=qQ+\rho, \qquad 0\le\rho<q. Then \mathop{\mathrm{T}}_b^\ell(r)=\rho and \delta_q\!\left(\mathop{\mathrm{T}}_b^\ell(r)\right) =\left\lfloor\frac{b\rho}{q}\right\rfloor =\left\lfloor\frac{b^{\ell+1}r}{q}\right\rfloor-bQ. The last two integers are congruent modulo b, while the first lies in \{0,\dots,b-1\}. It is therefore the residue modulo b of \lfloor b^{\ell+1}r/q\rfloor. The base-b expansion of that integer shows that this residue is d_{\ell+1}. ◻
At level N, the first N digits form the single cell index \lfloor b^N r/q\rfloor. Counting digit words is therefore the same as counting how often that index enters each marked cell.
Proposition 2 (Finite-prefix slice identity). Let m=b^N, let w:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}, and let q=hm+a with h\ge 0, 1\le a<m, and (a,m)=1. Define the finite-prefix weighted count and the carry sum \begin{aligned} C_w(q) &=\sum_{1\le r<q} w\!\left(\left\lfloor \frac{mr}{q}\right\rfloor\right), \\ A_w(a) &=\sum_{n\bmod m}w(n)\left( \left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor\right). \end{aligned} Then C_w(q)=h\sum_{n\bmod m}w(n)+A_w(a)-w(m-1), and therefore C_w(q)-\frac{q}{m}\sum_{n\bmod m}w(n)+w(m-1) = A_w(a)-\frac{a}{m}\sum_{n\bmod m}w(n). The right side is the centered weighted carry count attached to the residue a.
Proof. The condition \lfloor mr/q\rfloor=n is equivalent to nq/m\le r<(n+1)q/m. For 0\le n<m-1 the number of integers r in this half-open interval is \left\lfloor\frac{(n+1)q}{m}\right\rfloor-\left\lfloor\frac{nq}{m}\right\rfloor =h+\left\lfloor\frac{(n+1)a}{m}\right\rfloor-\left\lfloor\frac{na}{m}\right\rfloor . For n=0, the formula also removes the excluded endpoint r=0. Every other lower endpoint is nonintegral because (q,m)=1. At n=m-1, the interval ends at q, but the count uses r<q. Removing that endpoint subtracts w(m-1). Multiplying by w(n) and summing gives the first identity. Substituting q=hm+a gives the second. ◻
Example 3 (A nine-state slice). Take b=3, N=2, and mark the two-digit words whose digits agree. Then m=9 and G=\{0,4,8\}. For q=11=9+2, the first two digits agree exactly for the four numerators r\in\{1,5,6,10\}, since \lfloor9r/11\rfloor lies in G exactly at those values. At the residue a=2, the marked cells contribute two carries, so the slice identity reads 4=1\cdot|G|+A_G(2)-\mathbf1_G(8)=3+2-1. The intrinsic digit count has separated into its bulk, its carry count, and its terminal endpoint.
Remark 4. After removing the bulk h\sum w and the terminal endpoint, the count depends only on a=q\bmod m. A new finite-prefix observable therefore begins by identifying its digit support or weight.
The indicator form makes the reduction explicit. The weighted form follows.
Theorem 5 (Cylinder-to-Carry Reduction). Let b\ge2, N\ge1, m=b^N, and let \mathcal D\subseteq\{0,\dots,b-1\}^N be any set of length-N digit words. Put G_{\mathcal D} =\{0\le n<m:\ (\mathrm{dig}_0(n),\dots,\mathrm{dig}_{N-1}(n))\in\mathcal D\}, where \mathrm{dig}_0(n),\dots,\mathrm{dig}_{N-1}(n) are the N padded base-b digits of n from most significant to least significant. For q=hm+a with h\ge0, 1\le a<m, and (a,m)=1, let \mathcal C_{\mathcal D}(q) =\#\{1\le r<q:\ (d_1(r/q),\dots,d_N(r/q))\in\mathcal D\}. Then \mathcal C_{\mathcal D}(q)=C_{\mathbf 1_{G_{\mathcal D}}}(q), and hence \mathcal C_{\mathcal D}(q)-\frac{q|G_{\mathcal D}|}{m} +\mathbf 1_{G_{\mathcal D}}(m-1) =A_{G_{\mathcal D}}(a)-\frac{a|G_{\mathcal D}|}{m}. Every finite-prefix digit relation with (q,b)=1 is therefore a carry-boundary observable, up to the standard endpoint normalization.
Proof. For 1\le r<q set n=\lfloor mr/q\rfloor. Since m=b^N, the base-b digits of n are exactly the first N long-division digits d_1(r/q),\dots,d_N(r/q). Hence the digit word of r/q lies in \mathcal D exactly when n\in G_{\mathcal D}. This gives \mathcal C_{\mathcal D}(q)=C_{\mathbf 1_{G_{\mathcal D}}}(q). Since m=b^N, the conditions (a,m)=1 and (q,b)=1 are equivalent. Proposition 2 with w=\mathbf 1_{G_{\mathcal D}} gives the centered identity. ◻
Corollary 6 (Weighted cylinder reduction). Let \Phi:\{0,\dots,b-1\}^N\to\mathbb{C} be a finite-prefix digit observable and set w_\Phi(n)=\Phi(\mathrm{dig}_0(n),\dots,\mathrm{dig}_{N-1}(n)). Under the hypotheses of Proposition 2, with q=hm+a, 1\le a<m, and (a,m)=1, the intrinsic weighted count \mathcal W_\Phi(q)=\sum_{1\le r<q}\Phi(d_1,\dots,d_N) equals C_{w_\Phi}(q). Define its centered table by \mathcal R_\Phi(a) =A_{w_\Phi}(a)-\frac am\sum_{n\bmod m}w_\Phi(n).
Proof. The digit word of r/q at level N is the base-b word of n=\lfloor mr/q\rfloor, so \Phi(d_1,\dots,d_N)=w_\Phi(n). Sum over r and apply Proposition 2. ◻
Specializing the digit-word set \mathcal D recovers the standard observables.
Corollary 7 (Collision diagonal). For the two-point collision at lag \ell, with \mathcal D=\{d_1=d_{\ell+1}\} and m=b^{\ell+1}, and for q coprime to b, the intrinsic collision count \mathcal C_{\ell,b}(q)=\#\{1\le r<q:\ d_1(r/q)=d_{\ell+1}(r/q)\} reduces to the carry-boundary observable attached to G_{\ell,b}=\{0\le n<m:\ \lfloor n/b^\ell\rfloor\equiv n\ (\mathrm{mod}\ b)\}.
Corollary 8 (Three-point agreement). For 0<i<j, m=b^{j+1}, and q coprime to b, the intrinsic three-point count \#\{1\le r<q:\ d_1=d_{i+1}=d_{j+1}\} reduces to the carry-boundary observable attached to G^{(3)}_{i,j}=\{n:\ \mathrm{dig}_0(n)=\mathrm{dig}_i(n)=\mathrm{dig}_j(n)\}.
Corollary 9 (Adjacent transitions and block patterns). For q coprime to b and for a set \mathcal A\subseteq\{0,\dots,b-1\}^2 of allowed digit pairs at level two, the adjacent-transition count reduces to the carry-boundary observable attached to G_{\mathcal A}=\{n<b^2:\ (\lfloor n/b\rfloor,n\bmod b)\in\mathcal A\}. For a set \mathcal P\subseteq\{0,\dots,b-1\}^L of allowed words, again with q coprime to b, the block-pattern count reduces to the carry-boundary observable attached to the level-L support G_{\mathcal P}.
Proof of Corollaries 7–9. Each is Theorem 5 for the stated digit-word set. The four supports encode, respectively, the collision relation d_1=d_{\ell+1}, the agreement d_1=d_{i+1}=d_{j+1}, the pair condition on (d_1,d_2), and the word condition on (d_1,\dots,d_L). In each case n=\lfloor mr/q\rfloor carries the relevant digits, so the intrinsic count equals the count over the named support. ◻
Carry-Boundary Observables
The reduction leaves one weight on each cell. Carries occur at the edges where that weight changes.
Definition 10 (Carry-boundary observable). Let m\ge 2 and w:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}. The associated carry-boundary observable is F_w(a) \;=\; -\sum_{n\bmod m}w(n)\bigl(\mathop{\mathrm{saw}}\!\bigl((n+1)a\bigr)-\mathop{\mathrm{saw}}(na)\bigr), \qquad a\bmod m, where \mathop{\mathrm{saw}} is the centered sawtooth of §4. Its boundary flux against a character \chi is S_w(\chi)=\sum_{n\bmod m}w(n)\bigl(\chi(n+1)-\chi(n)\bigr). For G\subseteq\{0,\dots,m-1\} and w=\mathbf 1_G we write F_G and S_G. Characters are extended by zero off the unit group, and we use the transform \widehat F_w(\chi)=\sum_{a\mod m}F_w(a)\overline{\chi(a)}.
The summand of F_w is a carry. For 0\le a<m the increment c_a(n):=\left\lfloor \frac{(n+1)a}{m}\right\rfloor -\left\lfloor \frac{na}{m}\right\rfloor\in\{0,1\} equals 1 exactly when \{na/m\}+a/m\ge 1. This is precisely the step from na to (n+1)a crossing a multiple of m. Thus c_a(n) records one carry. The basic object is the carry across a marked boundary. A collision diagonal chooses which steps to mark.
Definition 11 (Signed boundary chain). For a weight w, the signed boundary chain is the cyclic difference \mu_w(x)=w(x-1)-w(x),\qquad x\bmod m, so that the boundary flux is the pairing S_w(\chi)=\sum_{x\bmod m}\mu_w(x)\,\chi(x) =\langle\chi,\mu_w\rangle.
Neighboring cells with the same weight contribute nothing at their common edge. Only a change of weight leaves signed boundary mass.
Definition 12 (Formal boundary). Write a support G\subseteq\{0,\dots,m-1\} as a disjoint union G=\bigsqcup_i[\alpha_i,\beta_i] of maximal integer intervals. The formal boundary \partial_{\mathrm{form}}G is the multiset of interval endpoints, one left endpoint \alpha_i and one post-right endpoint \beta_i+1 per interval. Thus |\partial_{\mathrm{form}}G|=2\cdot\#\{\text{maximal intervals of }G\}. The cyclic signed boundary chain \mu_G is recovered from the formal boundary by summing signs at each residue modulo m. Endpoints that coincide modulo m may cancel, so \mu_G can carry less total mass than |\partial_{\mathrm{form}}G|.
The formal boundary counts walls before residues are identified. The chain \mu_G records the cancellations after identification. Ordinary fluxes and primitive transforms see only \mu_G|_{(\mathbb{Z}/m\mathbb{Z})^{\times}}. Induced transforms use the conductor-depth pairing of Section 9.
The formal count is the carry-wall normalization used in the boundary mean law. For the terminal collision diagonal G_N modulo p^N, with N\ge2, there are p^{N-1} singleton cells. Its formal boundary has size 2p^{N-1}. The two endpoints at residue 0 cancel in \mu_{G_N}.
For G=\{0,4,8\}, the six formal endpoints reduce modulo 9. Two meet and cancel at 0. The four unit endpoints that remain have signed mass +1 at 1, -1 at 4, +1 at 5, and -1 at 8. Ordinary fluxes and primitive transforms see exactly these four endpoints.
The chain is now visible. A weight gives a signed boundary, the boundary gives a flux, and a primitive character supplies the Bernoulli response w\longmapsto\mu_w\longmapsto S_w(\chi) \longmapsto-B_{1,\bar\chi}S_w(\chi).
Lemma 13 (Endpoint normalization). For a\in(\mathbb{Z}/m\mathbb{Z})^{\times}, define the centered carry table A_w^\circ(a)=A_w(a)-\frac am\sum_{n}w(n). Then F_w(a)=A_w^\circ(a)+\tfrac12\bigl(w(0)-w(m-1)\bigr). Thus the sawtooth-normalized observable and the centered carry table differ by a constant independent of a. The observable F_w gives the cleaner boundary formula, while F_w and A_w^\circ have identical primitive odd transforms.
Proof. For a\in(\mathbb{Z}/m\mathbb{Z})^{\times} and 1\le n\le m-2, neither na nor (n+1)a is congruent to 0 modulo m. Hence \mathop{\mathrm{saw}}((n+1)a)-\mathop{\mathrm{saw}}(na)=\frac{a}{m}-c_a(n), where c_a(n)=\lfloor (n+1)a/m\rfloor-\lfloor na/m\rfloor. At n=0 the value \mathop{\mathrm{saw}}(0)=0 gives the extra -\tfrac12 in the difference, and after the outer minus sign this contributes +\tfrac12 w(0). At n=m-1, (n+1)a\equiv0, giving an extra +\tfrac12 in the difference and hence -\tfrac12 w(m-1) after the outer minus sign. Summing the remaining terms gives A_w(a)-\frac am\sum_n w(n)=A_w^\circ(a). ◻
Corollary 14 (Endpoint constants are spectrally invisible). Extend A_w^\circ off (\mathbb{Z}/m\mathbb{Z})^{\times} by the same formula, or arbitrarily; only unit values contribute to Dirichlet transforms. For every nonprincipal Dirichlet character \chi\bmod m, \sum_{a\bmod m}F_w(a)\bar\chi(a) =\sum_{a\bmod m}A_w^\circ(a)\bar\chi(a).
Proof. The two differ by the constant \tfrac12(w(0)-w(m-1)) of Lemma 13, and \sum_a\bar\chi(a)=0 for nonprincipal \chi. ◻
This corollary justifies passing between the intrinsic centered table of §2 and the sawtooth-normalized F_w in every nonprincipal Dirichlet transform.
Remark 15 (Fixed level versus complexity). At a fixed level N, every support is a union of cells. Its boundary flux is therefore a finite character sum. The minimal level records how many digits the rule actually needs.
The Floor Potential
The digit rule is now gone. What remains is the response supplied by the floor.
The analytic engine is the centered sawtooth. Let m\ge2 and define on \mathbb{Z}/m\mathbb{Z} \mathop{\mathrm{saw}}(n) = \frac{n}{m}-\frac12 \quad (1\le n\le m-1), \qquad \mathop{\mathrm{saw}}(0)=0 . This is the centered fractional-part function on residue classes. Through \lfloor x\rfloor=x-\{x\} it is the oscillatory part of the floor. The single property that drives the theory is that \mathop{\mathrm{saw}} is odd, \mathop{\mathrm{saw}}(-n)=\mathop{\mathrm{saw}}(m-n)=\frac{m-n}{m}-\frac12 = -\mathop{\mathrm{saw}}(n).
The normalization below is the standard first generalized Bernoulli convention for Dirichlet characters; see, for example, [7].
Lemma 16 (Boundary-to-Bernoulli). For \widehat{\operatorname{saw}}(\chi) =\sum_{n\bmod m}\mathop{\mathrm{saw}}(n)\,\bar\chi(n), one has \widehat{\operatorname{saw}}(\chi)=0 whenever \chi is even. For every primitive character \chi\bmod m, \widehat{\operatorname{saw}}(\chi)=B_{1,\bar\chi}, \qquad B_{1,\psi}=\tfrac1m\sum_{a=1}^{m}a\,\psi(a). In particular, B_{1,\bar\chi}=0 when \chi is primitive and even.
Proof. The substitution n\mapsto-n and the oddness of \mathop{\mathrm{saw}} give \widehat{\operatorname{saw}}(\chi) =-\bar\chi(-1)\widehat{\operatorname{saw}}(\chi). The transform therefore vanishes when \chi is even. A primitive character of modulus m\ge2 is nonprincipal, so \sum_a\bar\chi(a)=0. It follows that \widehat{\operatorname{saw}}(\chi) =\sum_{a=1}^{m-1}\Bigl(\frac{a}{m}-\frac12\Bigr)\bar\chi(a) =\frac1m\sum_{a=1}^{m-1}a\,\bar\chi(a) =B_{1,\bar\chi}. For an even primitive character, the first part makes both sides zero. ◻
Even characters vanish because the sawtooth is odd. On a primitive odd character, the value is exactly a Bernoulli number with no extra Gauss factor. We call this universal factor B_{1,\bar\chi} the floor potential.
The Carry-Boundary Factorization
The boundary and the floor response are now separate. The factorization puts them back together and accounts for nonunit endpoints.
One uniform evaluation drives the proof. Primitivity enters through standard Gauss-sum separation [5].
Lemma 17 (Primitive Gauss-sum separation). Let \psi\bmod m be primitive and \tau_c(\psi)=\sum_{a}\psi(a)e(ca/m) with e(x)=e^{2\pi i x}. Then for every integer c, \tau_c(\psi)=\overline{\psi}(c)\,\tau_1(\psi), with the convention \psi(c)=0 when (c,m)>1. In particular \tau_c(\psi)=0 whenever (c,m)>1.
Proof. For (c,m)=1, substitute u=ca to get \tau_c(\psi)=\psi(c^{-1})\tau_1(\psi)=\overline\psi(c)\tau_1(\psi). For (c,m)>1, the additive character a\mapsto e(ca/m) has conductor m/(c,m)<m; summing the primitive multiplicative character over the corresponding cosets gives zero, which is the standard primitive Gauss-sum separation. ◻
Lemma 18 (Twisted sawtooth evaluation). Let \chi\bmod m be primitive. For every integer k, \sum_{a\bmod m}\mathop{\mathrm{saw}}(ka)\,\bar\chi(a)=\chi(k)\,B_{1,\bar\chi}. In particular the sum vanishes whenever (k,m)>1.
Proof. Use the finite Fourier expansion on \mathbb{Z}/m\mathbb{Z}, \mathop{\mathrm{saw}}(n)=\sum_{j\bmod m}\hat g(j)e(jn/m), \qquad \hat g(j)=\frac1m\sum_{u\bmod m}\mathop{\mathrm{saw}}(u)e(-ju/m). Since \mathop{\mathrm{saw}} has mean zero, \hat g(0)=0. Therefore \sum_{a}\mathop{\mathrm{saw}}(ka)\bar\chi(a) =\sum_{j\bmod m}\hat g(j)\tau_{jk}(\bar\chi) =\chi(k)\Bigl(\tau_1(\bar\chi)\sum_{j\bmod m}\hat g(j)\chi(j)\Bigr), by Lemma 17. The parenthesized factor is independent of k; setting k=1 identifies it as B_{1,\bar\chi} via Lemma 16. ◻
The uniformity in k is the point. Separability sends every nonunit index to the exact value \chi(k)B_{1,\bar\chi}=0. The factorization therefore holds for every weight without a coprimality hypothesis on the boundary.
Theorem 19 (Carry-Boundary Factorization). Let m\ge 2, let w:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}, and let \chi\bmod m be primitive. Then \widehat F_w(\chi)=0 for even \chi, and for primitive odd \chi, \boxed{\;\widehat F_w(\chi) \;=\; -\,B_{1,\bar\chi}\cdot S_w(\chi)\;}, \qquad S_w(\chi)=\sum_{n\bmod m}w(n)\bigl(\chi(n+1)-\chi(n)\bigr).
Proof. Expand and exchange sums, \widehat F_w(\chi) = -\sum_{n}w(n)\sum_{a}\bigl(\mathop{\mathrm{saw}}((n+1)a)-\mathop{\mathrm{saw}}(na)\bigr)\bar\chi(a). By Lemma 18, the two inner sawtooth transforms are \chi(n+1)B_{1,\bar\chi} and \chi(n)B_{1,\bar\chi}. Hence the preceding sum is -B_{1,\bar\chi}S_w(\chi). Even-character vanishing follows from F_w(-a)=-F_w(a). ◻
Remark 20 (Where the non-unit indices went). Lemma 18 absorbs every boundary index not coprime to m. By primitivity, such an endpoint contributes \chi(k)B_{1,\bar\chi}=0. A nonunit cell can still create a neighboring unit endpoint. The primitive transform sees that unit boundary difference, as Theorem 24 makes precise.
Corollary 21 (Constant weights have no spectrum). If w\equiv c is constant, then \mu_w\equiv0, so S_w(\chi)=0 and \widehat F_w(\chi)=0 for every \chi. A uniform digit weight has no boundary and no transform.
Proof. Directly from Definition 10, F_w(a)=-c\sum_{n\bmod m} \bigl(\mathop{\mathrm{saw}}((n+1)a)-\mathop{\mathrm{saw}}(na)\bigr)=0 by cyclic telescoping. Hence every transform of F_w vanishes. ◻
Corollary 22 (Spectral factorization of finite-prefix observables). Let b\ge2, N\ge1, and m=b^N. Let \Phi:\{0,\dots,b-1\}^N\to\mathbb{C} be a finite-prefix digit observable with cylinder weight w_\Phi. Its centered table \mathcal R_\Phi from Corollary 6 satisfies, for every primitive odd \chi\bmod m, \sum_{a\bmod m}\mathcal R_\Phi(a)\bar\chi(a) =-B_{1,\bar\chi}\,S_{w_\Phi}(\chi).
The Unit-Boundary Flux
For an interval support, interior differences cancel and only the signed endpoints remain.
Lemma 23 (Interval Stokes formula). If G=\bigsqcup_i[\alpha_i,\beta_i] is a disjoint union of integer intervals, then interior differences cancel and S_G(\chi)=\sum_i\bigl(\chi(\beta_i+1)-\chi(\alpha_i)\bigr) =\sum_{p\in\partial G}\varepsilon(p)\,\chi(p), where \partial G is the signed boundary, with a left endpoint \alpha_i carrying sign - and a post-right endpoint \beta_i+1 carrying sign +.
Proof. In \sum_{n\in G}(\chi(n+1)-\chi(n)) the value \chi(n) at an interior point n appears once with each sign and cancels, leaving the endpoints. ◻
Thus S_G(\chi)=\chi(\partial G) is the character integrated over the boundary. For primitive odd \chi, Theorem 19 reads \widehat{F}_G(\chi) = -\underbrace{B_{1,\bar\chi}}_{\text{floor potential}} \cdot \underbrace{\chi(\partial G)}_{\text{boundary flux}} . An interval contributes only its endpoints. The same fact holds for every weight. Ordinary flux depends only on the boundary derivative at unit residues.
Theorem 24 (Unit-boundary image and kernel). Let m\ge2 and w:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}, and recall \mu_w(x)=w(x-1)-w(x). For every Dirichlet character \chi\bmod m, extended by zero off the unit group, S_w(\chi)=\sum_{x\bmod m}\mu_w(x)\chi(x). Thus the full ordinary-flux family recovers \mu_w exactly on the unit group and recovers nothing on nonunits. Explicitly, \mu_w(x)=\frac1{\varphi(m)} \sum_{\chi\bmod m}S_w(\chi)\overline{\chi(x)}, \qquad x\in(\mathbb{Z}/m\mathbb{Z})^{\times}. Two weights have the same ordinary fluxes exactly when their unit boundary derivatives agree S_{w_1}(\chi)=S_{w_2}(\chi)\ \text{for all }\chi \iff \mu_{w_1}(x)=\mu_{w_2}(x)\ \text{for all }x\in(\mathbb{Z}/m\mathbb{Z})^{\times} . For a subfamily \mathcal X, equality means \sum_{x\in(\mathbb{Z}/m\mathbb{Z})^{\times}} \bigl(\mu_{w_1}(x)-\mu_{w_2}(x)\bigr)\chi(x)=0 \qquad(\chi\in\mathcal X). Under the usual Hermitian inner product, the unit-boundary difference is orthogonal to \mathrm{span}\{\bar\chi:\chi\in\mathcal X\}.
Proof. Shifting the index in the first sum, \begin{aligned} S_w(\chi) &=\sum_x w(x-1)\chi(x)-\sum_x w(x)\chi(x) \\ &=\sum_x\bigl(w(x-1)-w(x)\bigr)\chi(x) =\sum_x\mu_w(x)\chi(x). \end{aligned} Characters vanish off (\mathbb{Z}/m\mathbb{Z})^{\times}, so the sum sees only \mu_w|_{(\mathbb{Z}/m\mathbb{Z})^{\times}}. The characters modulo m form an orthogonal basis of functions on (\mathbb{Z}/m\mathbb{Z})^{\times}, giving the stated determinacy and the subfamily orthogonality criterion. ◻
Corollary 25 (No unit boundary flux). If \mu_w is supported on nonunit residues, then S_w(\chi)=0 for every Dirichlet character modulo m. Consequently \widehat F_w(\chi)=0 for every primitive character \chi\bmod m. Imprimitive odd transforms are governed instead by the conductor-depth pairing and need not vanish.
Proof. Immediate from Theorem 24. The pairing sees only the unit part of \mu_w, which is zero. ◻
The limit of the corollary is already visible modulo 6. Let G=\{0,1\}. Then \mu_G=-\delta_0+\delta_2, so every ordinary Dirichlet flux vanishes. Let \chi\bmod6 be induced from the nontrivial character modulo 3, with \chi(1)=1 and \chi(5)=-1. Directly, F_G(a)=-\mathop{\mathrm{saw}}(2a), \qquad F_G(1)=\frac16, \qquad F_G(5)=-\frac16, \qquad \widehat F_G(\chi)=\frac13. The ordinary flux is zero, but the imprimitive odd transform is not.
The modulus-six example marks the limit exactly. Ordinary flux ignores nonunit endpoints, and primitive transforms inherit that loss through Theorem 19. An induced transform uses the conductor-depth pairing instead. It may retain an endpoint that survives at the inducing conductor.
The Base-Three Test Case
The nine-state example displays the whole mechanism before conductor depth enters.
Let b=3, N=2, m=9. The lag-one collision diagonal is G=\{0\le n<9: \lfloor n/3\rfloor\equiv n\ (\mathrm{mod}\ 3)\}=\{0,4,8\}. This is the support already used in Example 3. Its visible signed boundary is \mu_G=\delta_1-\delta_4+\delta_5-\delta_8. Let \chi be the primitive odd character modulo 9 with \chi(2)=\zeta_6=e^{2\pi i/6}, so \begin{array}{c|rrrrrr} a&1&2&4&5&7&8\\ \chi(a)&1&\zeta_6&\zeta_6^2&\zeta_6^5&\zeta_6^4&-1 \end{array} \qquad \begin{array}{c|rrrrrr} a&1&2&4&5&7&8\\ \textstyle\sum_{n\in G}c_a(n)&1&2&2&1&1&2\\[2pt] F_G(a)&\tfrac23&\tfrac43&\tfrac23&-\tfrac23&-\tfrac43&-\tfrac23. \end{array} A direct computation gives S_G(\chi)=3-i\sqrt3, \qquad B_{1,\bar\chi}=-1+\tfrac{i\sqrt3}{3}, \qquad -B_{1,\bar\chi}S_G(\chi)=2-2i\sqrt3, and independently \widehat F_G(\chi)=\sum_{a\in(\mathbb{Z}/9\mathbb{Z})^{\times}}F_G(a)\bar\chi(a)=2-2i\sqrt3. The carry table, the boundary flux, and the floor potential agree exactly on the nine-state system.
Collision Supports and the Three-Point Hierarchy
The leading-equals-trailing diagonal at lag \ell, level N=\ell+1, is G_{\ell,b} = \bigl\{\,0\le n<b^{\ell+1}\;:\;\lfloor n/b^{\ell}\rfloor \equiv n \!\!\pmod b\,\bigr\}. The collision invariant, transform, and spectrum study this diagonal [1, 2, 3]. Under the carry-boundary reduction, its distinguishing data are the signed endpoints of G_{\ell,b}.
Corollary 26 (Collision boundary factorization). For primitive odd \chi\bmod b^{\ell+1}, \widehat{F}_{G_{\ell,b}}(\chi)=-B_{1,\bar\chi}S_{G_{\ell,b}}(\chi). By Corollary 7 the intrinsic finite-prefix collision table has this same factorization, up to the endpoint normalization.
The three-point agreement support at level N=j+1, G^{(3)}_{i,j}=\{n:\mathrm{dig}_0(n)=\mathrm{dig}_i(n)=\mathrm{dig}_j(n)\}, is the intersection of two diagonals, hence again digit-defined.
Corollary 27 (Three-point factorization). For primitive odd \chi\bmod b^{j+1}, \widehat{F}_{G^{(3)}_{i,j}}(\chi)=-B_{1,\bar\chi}S_{G^{(3)}_{i,j}}(\chi). The floor potential is identical to the two-point case; only the boundary \partial G^{(3)}_{i,j} changes.
Remark 28 (A hierarchy by boundary, not by mechanism). Two-point and higher agreement supports share the same floor potential and differ only in boundary flux. The number of matching positions changes. The mechanism does not.
Conductor Stratification
A primitive transform keeps full residue resolution. It pairs the Bernoulli response with ordinary flux on the units. An induced transform first descends the sawtooth to its inducing conductor. Its depth weight records the endpoints that survive.
Definition 29 (Conductor depth weight). Let p be prime, N\ge1, and let \chi\bmod p^N be induced by a primitive \chi^\ast\bmod p^t, with 1\le t\le N. Only positive conductor levels are needed. The principal character at t=0 is even and does not enter the odd decompositions. For k\not\equiv0\pmod{p^N} write k=p^v u with v=v_p(k) and p\nmid u. The conductor depth weight is \eta_{\chi}(k)= \begin{cases} p^v\chi^\ast(u), & k=p^v u,\ p\nmid u,\ 0\le v\le N-t,\\[2pt] 0, & k\equiv0\ (\mathrm{mod}\ p^N)\text{ or }v>N-t. \end{cases}
The power of p in an endpoint records how far that endpoint survives under this loss of resolution.
Theorem 30 (Conductor-stratified twisted sawtooth). With p,N,\chi,\chi^\ast,t as above and \mathop{\mathrm{saw}}_N the sawtooth modulo p^N, \sum_{a\bmod p^N}\mathop{\mathrm{saw}}_N(ka)\,\overline{\chi(a)} =B_{1,\overline{\chi^\ast}}\,\eta_\chi(k) \qquad(k\in\mathbb{Z}/p^N\mathbb{Z}). For primitive \chi this is Lemma 18.
Proof. If k\equiv0\pmod{p^N}, both sides vanish. Now write k=p^v u with p\nmid u. For v\le N-t, write units modulo p^N as a=a_0+p^t\ell with a_0\in(\mathbb{Z}/p^t\mathbb{Z})^{\times}, and put M=N-v. Since \mathop{\mathrm{saw}}_N(p^v x)=\mathop{\mathrm{saw}}_M(x), the p^{N-t} lifts of a_0 reduce to p^{M-t} lower-level fibers, each repeated p^v times. The factor p^v comes from that multiplicity. Applying \sum_{j=0}^{p^{M-t}-1}\mathop{\mathrm{saw}}_M(A+p^t j)=\mathop{\mathrm{saw}}_t(A) for p\nmid A gives \sum_{a}\mathop{\mathrm{saw}}_N(ka)\overline{\chi(a)} =p^v\sum_{a_0\in(\mathbb{Z}/p^t\mathbb{Z})^{\times}}\mathop{\mathrm{saw}}_t(ua_0)\overline{\chi^\ast(a_0)} =p^v\chi^\ast(u)B_{1,\overline{\chi^\ast}}, by Lemma 16 at conductor p^t. For v>N-t, the inner sawtooth depends only on a_0\bmod p^{N-v} with N-v<t; since \chi^\ast is primitive modulo p^t, its sum over each fiber vanishes. ◻
Theorem 31 (Conductor-stratified factorization). With notation as above, for every weight w:\mathbb{Z}/p^N\mathbb{Z}\to\mathbb{C}, \begin{aligned} \widehat F_w(\chi) &=-\,B_{1,\overline{\chi^\ast}} \sum_{n\bmod p^N}w(n) \bigl(\eta_\chi(n+1)-\eta_\chi(n)\bigr),\\ &=-\,B_{1,\overline{\chi^\ast}} \sum_{x\bmod p^N}\mu_w(x)\eta_\chi(x). \end{aligned} Lower-conductor odd transforms pair the same boundary derivative with endpoints weighted by conductor depth.
Proof. Expand \widehat F_w(\chi) and apply Theorem 30 to k=n+1 and k=n. Shifting the first finite sum gives \sum_n w(n)\bigl(\eta_\chi(n+1)-\eta_\chi(n)\bigr) =\sum_x\bigl(w(x-1)-w(x)\bigr)\eta_\chi(x), which is the boundary-chain form. ◻
This pairing is not the ordinary flux S_w(\chi)=\sum_x\mu_w(x)\chi(x). The character \chi vanishes on nonunits. The weight \eta_\chi may retain an endpoint that survives conductor descent. The modulus-six example is the smallest instance of the difference.
A support lifted to a higher prime-power level becomes invisible to primitive characters there because dilation pushes every endpoint onto a nonunit.
Theorem 32 (Primitive higher-conductor invisibility of a lifted support). Let p be an odd prime, 1\le K\le N, and s=N-K. Let H\subseteq\{0,\dots,p^K-1\} and let G=\{p^s u+v:\ u\in H,\ 0\le v<p^s\}. Write \iota_s:\mathbb{Z}/p^K\mathbb{Z}\longrightarrow\mathbb{Z}/p^N\mathbb{Z}, \qquad \iota_s(x)=p^s x. Then the signed boundary chain of G is the pushforward \mu_G=(\iota_s)_*\mu_H. If s>0, every point in the support of \mu_G is divisible by p. Consequently S_G(\chi)=0 for every primitive character \chi\bmod p^N.
Proof. An interval [\alpha,\beta] of H lifts to [p^s\alpha,\ p^s(\beta+1)-1]. Its left endpoint becomes p^s\alpha, and its post-right endpoint becomes p^s(\beta+1), with the same signs. Summing over the maximal intervals gives \mu_G=(\iota_s)_*\mu_H. When s>0, every lifted endpoint is a nonunit modulo p^N. Corollary 25 then gives S_G(\chi)=0. ◻
The theorem says that a support lifted from level p^K cannot acquire primitive visibility above p^K. Visibility at p^K itself requires a nonzero primitive component of \mu_H there. The terminal collision support has such a component by Theorem 36.
Lift the nine-state support H=\{0,4,8\} from modulus 9 to modulus 27. The lifted support is \{0,1,2\}\ \cup\ \{12,13,14\}\ \cup\ \{24,25,26\}. Every surviving endpoint is divisible by 3. Primitive characters modulo 27 therefore see no boundary, while the original boundary remains visible at its native modulus 9.
Corollary 33 (Leading-lag conductor stratification). Let p be an odd prime, N\ge2, 1\le\ell\le N-1, k=\ell+1, s=N-k. The leading-lag collision support G_{N,\ell}\subseteq\{0,\dots,p^N-1\} is defined by equality of the first digit and digit \ell+1. It is the prefix pullback of the terminal collision support H_k at level k. If s>0, then S_{G_{N,\ell}}(\chi)=0 for every primitive \chi\bmod p^N. At conductor p^k, the support H_k obeys the boundary mean law of §10. Thus a leading-lag collision is visible at the conductor of the last digit it uses and invisible to higher primitive conductors.
Proof. The defining relation uses only the first k digits, so G_{N,\ell} is the pullback of H_k and Theorem 32 applies with this s. The native-conductor statement is Theorem 36 at level k. ◻
For terminal collisions, the source conductor identifies where the primitive boundary is visible. The mean law measures the size of what survives there.
The Boundary Mean Law
Two conditions give the exact second moment. Conductor balance removes the lower-resolution terms. Antipodal oddness makes the top reflection add rather than cancel. The criterion applies to any boundary weight with both properties.
For a signed boundary weight \mu, write S_\mu(\chi)=\sum_{x\bmod m}\mu(x)\chi(x). When \mu=\mu_w is the boundary derivative of w, this agrees with the flux notation S_w(\chi) above.
Definition 34 (Mean-admissible weight). Let p be an odd prime, N\ge2, m=p^N. A weight \mu:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C} supported on unit residues is mean-admissible if
(conductor balance) for every unit r\bmod p^{N-1}, \sum_{x\equiv r\,(p^{N-1})}\mu(x)=0;
(antipodal oddness) \mu(-x)=-\mu(x) for every unit x.
Theorem 35 (Boundary Mean Criterion). Let p be an odd prime, N\ge2, m=p^N, and let \mu be mean-admissible with U=\sum_{x\in(\mathbb{Z}/m\mathbb{Z})^{\times}}|\mu(x)|^2. Then \sum_{\substack{\chi\bmod p^N\\ \mathrm{primitive\ odd}}} |S_\mu(\chi)|^2 =\varphi(p^N)\,U, \qquad \frac1{n_{\mathrm{po}}}\sum_{\substack{\chi\bmod p^N\\ \mathrm{primitive\ odd}}} |S_\mu(\chi)|^2=\frac{2p}{p-1}\,U, where n_{\mathrm{po}}=\tfrac12(\varphi(p^N)-\varphi(p^{N-1})).
Proof. For units x,y, the primitive-odd kernel is \begin{aligned} \sum_{\substack{\chi\bmod p^N\\ \mathrm{prim\ odd}}} \chi(x)\overline{\chi(y)} &=\tfrac{\varphi(p^N)}{2} \bigl(1_{x\equiv y\,(p^N)}-1_{x\equiv -y\,(p^N)}\bigr) \\ &\quad -\tfrac{\varphi(p^{N-1})}{2} \bigl(1_{x\equiv y\,(p^{N-1})} -1_{x\equiv -y\,(p^{N-1})}\bigr), \end{aligned} the odd part of the conductor-exact Dirichlet kernel. Expand the second moment against this kernel. Condition (i) kills both terms at conductor p^{N-1}. At conductor p^N, the diagonal contributes U. Condition (ii) makes the antipodal contribution -U. Hence \frac{\varphi(p^N)}2\bigl(U-(-U)\bigr)=\varphi(p^N)U. Finally, n_{\mathrm{po}}=p^{N-2}(p-1)^2/2, and division by n_{\mathrm{po}} gives \varphi(p^N)/n_{\mathrm{po}}=2p/(p-1). ◻
The terminal collision boundary satisfies both conditions for a concrete reason. Every coarse unit class receives one entrance and one exit, and negation exchanges entrances with exits.
Theorem 36 (Boundary Mean Law for terminal collisions). Let p be an odd prime, N\ge2, and m=p^N. Write each residue with its padded N-digit base-p expansion, and let G_N=\{0\le n<m:\text{ first and last digits agree}\} be the terminal collision diagonal. Let \mu_N be the signed boundary chain \partial G_N restricted to (\mathbb{Z}/p^N\mathbb{Z})^{\times}. Then |\partial_{\mathrm{form}}G_N|=2p^{N-1}, the unit boundary norm is U=2p^{N-2}(p-1)=\tfrac{p-1}{p}|\partial_{\mathrm{form}}G_N|, and \frac1{n_{\mathrm{po}}} \sum_{\substack{\chi\bmod p^N\\ \mathrm{primitive\ odd}}}|S_{\mu_N}(\chi)|^2 =2\,|\partial_{\mathrm{form}}G_N|. Equivalently, the primitive-odd flux has root-mean-square value \sqrt{2|\partial_{\mathrm{form}}G_N|}.
Proof. A point of G_N is n=ap^{N-1}+py+a with 0\le a\le p-1 and 0\le y<p^{N-2}. Each point is a singleton interval. Its left endpoint is L_{a,y}=ap^{N-1}+py+a with sign -, and its right endpoint is R_{a,y}=L_{a,y}+1 with sign +. There are p^{N-1} singletons, so |\partial_{\mathrm{form}}G_N|=2p^{N-1} (Definition 12). Modulo p one has L_{a,y}\equiv a and R_{a,y}\equiv a+1, so the unit endpoints are the L_{a,y} with a\ne0 and the R_{a,y} with a\ne p-1, giving U=2p^{N-2}(p-1)=\frac{p-1}{p}|\partial_{\mathrm{form}}G_N|.
We verify the two conditions for \mu_N. Modulo p^{N-1}, every unit residue has the form c+py with 1\le c\le p-1. It receives one negative endpoint L_{c,y} and one positive endpoint R_{c-1,y}. Their signed mass is zero, which gives conductor balance. Negation sends -L_{a,y}\equiv R_{p-1-a,\ p^{N-2}-1-y}\pmod{p^N}, exchanging unit endpoints with opposite signs. This gives antipodal oddness. The Boundary Mean Criterion now gives the moment \varphi(p^N)U, and \frac1{n_{\mathrm{po}}}\varphi(p^N)U =\frac{2p}{p-1}\cdot\frac{p-1}{p}\,|\partial_{\mathrm{form}}G_N| =2\,|\partial_{\mathrm{form}}G_N|. ◻
For the nine-state example, the formal boundary has size 6 and the visible unit boundary has squared norm 4. There are two primitive odd characters modulo 9, and each has squared flux magnitude 12. Their mean is therefore 12, exactly twice the formal boundary size.
Prime powers close because the lower fibers balance. For a general modulus, each divisor can retain part of the boundary, so the second moment becomes a signed ledger over conductors.
Theorem 37 (Conductor ledger identity). Let m\ge3, \mu:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}, and S_\mu(\chi)=\sum_x\mu(x)\chi(x). For each d\mid m define the unit pushdown M_d(r)=\sum_{x\in(\mathbb{Z}/m\mathbb{Z})^{\times},\ x\equiv r\,(d)}\mu(x) for r\in(\mathbb{Z}/d\mathbb{Z})^{\times}. For d=1, (\mathbb{Z}/1\mathbb{Z})^{\times} is understood as the trivial group. Then \begin{aligned} \sum_{\substack{\chi\bmod m\\ \mathrm{primitive\ odd}}}|S_\mu(\chi)|^2 &=\frac12\sum_{d\mid m}\mu_{\mathrm{Mob}}(m/d)\varphi(d) \\ &\quad\cdot \left( \sum_{r\in(\mathbb{Z}/d\mathbb{Z})^{\times}}|M_d(r)|^2 -\sum_{r\in(\mathbb{Z}/d\mathbb{Z})^{\times}}M_d(r)\overline{M_d(-r)} \right). \end{aligned}
Proof. Only units contribute. Expand the second moment and apply the primitive-odd kernel \sum_{\substack{\chi\bmod m\\ \mathrm{prim\ odd}}}\chi(x)\overline{\chi(y)} =\tfrac12\sum_{d\mid m}\mu_{\mathrm{Mob}}(m/d)\varphi(d) \bigl(1_{x\equiv y\,(d)}-1_{x\equiv -y\,(d)}\bigr), obtained by Möbius inversion of orthogonality over conductors followed by the odd projection \chi\mapsto\tfrac12(\chi(x)-\chi(-x)). Grouping x,y by common residue modulo d gives the diagonal \sum_r|M_d(r)|^2 and the antipodal \sum_r M_d(r)\overline{M_d(-r)}. ◻
Corollary 38 (Proper-conductor leakage). If every proper pushdown M_d (d\mid m, d<m) vanishes and \mu is antipodally odd on (\mathbb{Z}/m\mathbb{Z})^{\times}, then \sum_{\mathrm{po}}|S_\mu(\chi)|^2 =\varphi(m)\sum_{x\in(\mathbb{Z}/m\mathbb{Z})^{\times}}|\mu(x)|^2. Only divisors with nonzero Möbius coefficient enter the ledger. Call them ledger-active. Vanishing of every proper pushdown is sufficient for the top-conductor mean. A nonzero pushdown at a proper ledger-active divisor is a leakage term.
Proof. Vanishing proper pushdowns leave only d=m in the ledger, where the diagonal is U and antipodal oddness gives -U. ◻
Base 15 shows the failure explicitly. The top-level symmetry survives, but two lower conductors still carry boundary mass.
Proposition 39 (Composite obstruction). In base 15 at level N=2, let m=225 and G=\bigl\{n<225:\text{ first and last base-}15\text{ digits agree}\bigr\}. The signed boundary of G has nonzero proper-conductor pushdowns modulo 45 and 75. The primitive-odd mean at conductor 225 is 28, while 2|\partial_{\mathrm{form}}G|=60.
Proof. The diagonal consists of the fifteen singletons \{16a\}, \qquad 0\le a\le14. Thus \mu_G has mass -1 at L_a=16a and mass +1 at R_a=16a+1, and |\partial_{\mathrm{form}}G|=30. Negation sends L_a to R_{14-a} modulo 225, so the unit-restricted chain is antipodally odd.
The ledger-active divisors are 15, 45, 75, and 225. Modulo 15 one has 16\equiv1, so each unit entrance L_c cancels the unit exit R_{c-1} in the same residue class and M_{15}=0. The proper pushdowns at 45 and 75 are nonzero already at the residue 1, since M_{45}(1)=M_{75}(1)=1. Writing D_d=\sum_{r\in(\mathbb{Z}/d\mathbb{Z})^{\times}}|M_d(r)|^2 and A_d=\sum_{r\in(\mathbb{Z}/d\mathbb{Z})^{\times}}M_d(r)\overline{M_d(-r)}, exact reduction of the thirty signed endpoints gives \begin{array}{c|cccc} d & 15 & 45 & 75 & 225\\ \mu_{\mathrm{Mob}}(225/d) & +1 & -1 & -1 & +1\\[2pt] \varphi(d) & 8 & 24 & 40 & 120\\[2pt] D_d-A_d & 0 & 32 & 32 & 32 \end{array} At each of d=45,75,225, one has 16^{-1}\equiv-14\pmod d. Same-sign endpoints are distinct. A cross-sign collision would require a-b=-14, which pairs only L_0 with R_{14}. Those are the excluded nonunit endpoints at residue 0. The sixteen unit endpoints therefore remain distinct, each with value 1 or -1. Antipodal oddness gives D_d=16 and A_d=-16, which explains the value 32 directly. Theorem 37 now gives \sum_{\substack{\chi\bmod225\ \mathrm{primitive\ odd}}}|S_{\mu_G}(\chi)|^2 =\frac12\bigl(120\cdot32-40\cdot32-24\cdot32+8\cdot0\bigr)=896. There are n_{\mathrm{po}}=32 primitive odd characters modulo 225, so the primitive-odd mean is 896/32=28. The top-conductor term alone would give 60=2|\partial_{\mathrm{form}}G|. The leakage terms at 45 and 75 lower that value by 32, so the prime-power identity fails. ◻
Remark 40 (Exact finite checks). Exact calculations in nfield [4] corroborate the terminal boundary identities for p\in\{3,5,7,11\} and N\in\{2,3\}. Every case returns the predicted primitive-odd mean 4p^{N-1}=2|\partial_{\mathrm{form}}G_N|. The base-15 ledger also returns mean 28 modulo 225, together with the stated conductor contributions, total moment, and primitive-odd character count. These calculations do not enter the proofs.
Corollary 41 (Composite leakage). The second-moment boundary mean law holds for terminal collision diagonals at odd prime-power conductors of depth at least two (Theorem 36) and can fail at composite conductors (Proposition 39). The conductor ledger identifies every leakage term. Vanishing of all proper pushdowns is sufficient for the top-conductor mean. Base 15 shows how surviving leakage changes it.
Proof. Combine the prime-power mean law with the conductor ledger and the explicit base-15 computation of Proposition 39. ◻
The Quadratic Edge
Theorem 19 concerns linear observables. The natural quadratic invariant is the energy on the unit group, E(F_w)=\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^{\times}}|F_w(a)|^2 =\frac{1}{\varphi(m)}\sum_{\chi\bmod m}|\widehat F_w(\chi)|^2, which is Parseval on (\mathbb{Z}/m\mathbb{Z})^{\times}.
For a primitive character, the transform is its floor potential times its ordinary boundary flux. For an induced character, Theorem 31 replaces that flux with the depth-weighted boundary pairing. Energy sums their squared magnitudes by conductor.
Theorem 42 (Prime-power energy decomposition). Let p be prime, let N\ge1, let m=p^N, and let w:\mathbb{Z}/m\mathbb{Z}\to\mathbb{C}. Then E(F_w) =\frac{1}{\varphi(p^N)} \sum_{t=1}^{N} \sum_{\substack{\chi^\ast\bmod p^t\\ \mathrm{primitive\ odd}}} \bigl|B_{1,\overline{\chi^\ast}}\bigr|^2 \left|\sum_{n\bmod p^N}w(n) \bigl(\eta_\chi(n+1)-\eta_\chi(n)\bigr)\right|^2, where \chi is the character modulo p^N induced by \chi^\ast.
Proof. Parseval and the identity F_w(-a)=-F_w(a) show that only odd characters contribute. Every odd character modulo p^N is induced by a unique primitive odd character \chi^\ast\bmod p^t for some 1\le t\le N. Theorem 31 gives the corresponding coefficient. Substitution into Parseval gives the formula. ◻
For the base-nine table, the six centered carry values have total square mass 16/3. The two primitive odd Fourier coefficients each have square magnitude 16, while the induced odd channel vanishes. Parseval gives 32/6=16/3.
Remark 43 (The quadratic layer). The signed boundary representation gives F_w(a)=-\sum_{x\bmod m}\mu_w(x)\mathop{\mathrm{saw}}(xa). Consequently E(F_w) =\sum_{x,y\bmod m}\mu_w(x)\overline{\mu_w(y)} \sum_{a\in(\mathbb{Z}/m\mathbb{Z})^{\times}}\mathop{\mathrm{saw}}(xa)\mathop{\mathrm{saw}}(ya). The inner expressions are finite sawtooth correlations over the unit group. For q\ge2, write \mathop{\mathrm{saw}}_q(r)= \begin{cases} r/q-1/2,&1\le r<q,\\ 0,&r=0, \end{cases} \qquad \mathfrak s(h,q)=\sum_{r\bmod q}\mathop{\mathrm{saw}}_q(r)\mathop{\mathrm{saw}}_q(hr), and set \mathfrak s(h,1)=0. If x,y\in(\mathbb{Z}/m\mathbb{Z})^{\times} and h\equiv yx^{-1}\pmod m, Möbius inversion of the unit condition gives \sum_{a\in(\mathbb{Z}/m\mathbb{Z})^{\times}}\mathop{\mathrm{saw}}_m(xa)\mathop{\mathrm{saw}}_m(ya) =\sum_{d\mid m}\mu_{\mathrm{Mob}}(d)\, \mathfrak s(h,m/d). Multiplication by x first turns the unit variable into u=xa. Inserting 1_{(u,m)=1}=\sum_{d\mid(u,m)}\mu_{\mathrm{Mob}}(d), writing u=dr, and using \mathop{\mathrm{saw}}_m(dr)=\mathop{\mathrm{saw}}_{m/d}(r) gives the formula. Thus every unit-index correlation is a finite linear combination of classical Dedekind sums [10, 11, 12]. For prime-power m, Theorem 31 accounts for the nonunit terms on the character side.
For a primitive odd character \chi\bmod m, the standard functional equation [5] gives the exact magnitude relation |B_{1,\bar\chi}|=\frac{\sqrt m}{\pi}|L(1,\chi)|. The primitive part of the energy is therefore a boundary-weighted second moment of the values L(1,\chi) at one. At prime depth two with the collision indicator, this primitive summand is the Parseval moment of [3].
The Surviving Boundary
Start with the digit rule, not the character sum. The rule creates a signed boundary chain \mu_w, and S_w(\chi)=\langle\chi,\mu_w\rangle reads that chain on the unit group. The arithmetic object is the surviving unit boundary, not the mass inside the selected region. Its flux is a finite incomplete character sum over a structured support [8].
A nonunit cell can still create a neighboring unit endpoint. Ordinary fluxes and primitive transforms see the boundary on units. Induced transforms may retain nonunit endpoints through conductor depth. Two supports have the same ordinary fluxes exactly when their unit boundary derivatives agree.
The terminal prime-power boundary has the required unit norm, reverses under negation, and cancels at its only ledger-active lower conductor. Hence \frac1{n_{\mathrm{po}}}\sum |S_G(\chi)|^2 =2|\partial_{\mathrm{form}}G|. Base 15 preserves the reflection but not the lower-conductor cancellation. Its proper pushdowns survive and change the mean. The ledger identifies exactly which terms prevent the prime-power collapse.
Beyond Finite Windows
A finite-prefix rule already closes on a carry boundary. Whole-orbit statistics, recurrences, complete repetends, and products of reduced carry observables need a finite boundary of their own before the same factorization can apply.
For three-point and higher agreements, the support changes but the factorization is ready. The open question is whether the prime-power mean law persists.
For composite moduli, any mean law must retain the proper pushdowns exposed by the exact conductor ledger.
The collision invariant first exposed the structure, but digit equality is not its source. The carry is. The floor creates the odd sawtooth and its Bernoulli response. A digit rule marks a finite region. Interior differences cancel. Ordinary unit-boundary fluxes and conductor-depth pairings read what remains.
Long division does more than produce digits. It produces boundaries, and those boundaries are what survive the transform.
References
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