An average digit-match score does not record which fractions agree. For a positional base b\ge2 and denominator n\ge2, we define the synchronized cross-alignment matrix \mathbf A_b(n) on the fractions \{k/n:1\le k<n\}. Its entries compare every pair at one common long-division depth and across one common periodic window. An explicit one-hot realization makes \mathbf A_b(n) a positive semidefinite Gram matrix.
For n\ge3, the pairwise alignment is its mean off-diagonal entry. The reference alignment is recovered from the first row after adding the terminating-fraction correction. For a prime p\nmid b, \mathbf A_b(p) is the identity exactly when p is digit-partitioning. For such a prime and n=ps with s b-supported, the matrix is an exact direct sum of all-ones blocks. Its rank is p-1 when s=1 and p when s\ge2.
For an arbitrary prime p\nmid b, multiplicative-coset ordering turns \mathbf A_b(p) into a matrix of circulant blocks. Fourier transformation along the orbit coordinate reduces the spectrum to small Hermitian Gram matrices. The primitive-root case is one circulant whose eigenvalues are the discrete Fourier transform of the cyclic digit-equality autocorrelation. Exact decimal examples give the two levels 2/3,4/3 at p=13 and nine levels from 1/4 to 7/4 at p=17.
At p=13, distinct fractions in the same multiplicative orbit never match at a synchronized decimal position. Partial matches occur between the two orbits. In multiplicative-orbit order, their comparison matrix is \mathbf A_{10}(13) =\begin{pmatrix}I_6&\frac13P^3\\[2pt] \frac13P^3&I_6\end{pmatrix}, where P is the unit cyclic shift on six positions. Two six-element orbits meet at one relative shift, and the spectrum has only the two levels 2/3 and 4/3. A scalar records how much agreement survives. The matrix records where it lives.
The fractional field \mathcal{F}(n) = \{k/n : 1 \le k \le n{-}1\} carries three scalar summaries. Reference alignment \alpha_b(n) compares every member with 1/n. Pairwise alignment \sigma_b(n) averages over distinct unordered pairs, and F_b(n)=\alpha_b(n)-\sigma_b(n) is their signed difference. None by itself records which fractions agree, whether agreement gathers into blocks, or which independent modes carry it.
Coordinatewise equality is Hamming similarity. Cyclic Hamming correlation is classical in sequence design [1], and Kak and Chatterjee apply it to full-period reciprocal digit sequences [2]. Repetend multiplication and its multiplicative-orbit organization are likewise established arithmetic [3]. The Gram realization below is standard linear algebra [5], and Fourier diagonalization of circulant blocks is classical [4]. The result here is their exact assembly over the complete denominator ensemble. It recovers the three scalar statistics, characterizes the identity and supported block cases, and reduces every prime matrix to explicit coset Fourier fibers without a primitive-root hypothesis.
The cross-alignment matrix keeps every synchronized pairwise comparison. Pairwise alignment is its mean off-diagonal entry. Reference alignment uses one marked row together with the terminating-fraction convention, and focused alignment is the residual between those two statistics. The matrix therefore restores the geometry compressed by the averages, while a termination marker completes their recovery.
All comparisons begin after one common clearing depth and use one common periodic window. No fraction is restarted at an independently chosen repetend phase. Here “fractional field” is a name for the displayed finite family; no algebraic-field structure is asserted.
Fix b\ge2. For n\ge2, write n=qs, where s is the largest divisor of n supported on the primes of b, and \gcd(q,b)=1. Thus s divides a power of b. If q>1, choose one clearing depth D with s\mid b^D and put L=\mathop{\mathrm{ord}}_q(b).
Definition 1. When q>1 and 1\le k<n, let d_t(k) be the base-b digit of k/n at the t-th position after the common clearing depth D. The synchronized cross-alignment matrix is the (n-1)\times(n-1) matrix \mathbf A_b(n)=(A_{ij}) defined by A_{ij}= \begin{cases} 1, & \text{if both $i/n$ and $j/n$ terminate},\\ 0, & \text{if exactly one of $i/n$ and $j/n$ terminates},\\ \displaystyle\frac1L\sum_{t=0}^{L-1} \mathbf 1_{\{d_t(i)=d_t(j)\}}, & \text{if neither fraction terminates}. \end{cases} If q=1, every fraction terminates and \mathbf A_b(n)=J_{n-1}, where J_m denotes the m\times m all-ones matrix. In every case A_{ii}=1.
Increasing D shifts every nonterminating tail by the same number of positions. Because each reduced period divides L, the length-L equality pattern is only rotated. The matrix is therefore independent of the chosen clearing depth.
Proposition 2 (Gram realization). For every n\ge2, the matrix \mathbf A_b(n) is real symmetric, positive semidefinite, and has unit diagonal.
Proof. When q=1, the assertion follows from \mathbf A_b(n)=J_{n-1}. Suppose q>1. In the orthogonal direct sum \mathbb R e_{\mathrm{term}}\oplus \mathbb R^{\{0,\ldots,L-1\}\times\{0,\ldots,b-1\}}, assign the vector e_{\mathrm{term}} to every terminating fraction. For a nonterminating k/n, assign v_k=\frac1{\sqrt L}\sum_{t=0}^{L-1}e_{t,d_t(k)}. The inner product of two nonterminating vectors is their synchronized digit-match proportion. A terminating vector is orthogonal to every nonterminating vector, while two terminating vectors have inner product one. Thus A_{ij}=\langle v_i,v_j\rangle, with v_i=e_{\mathrm{term}} in the terminating case. Hence \mathbf A_b(n) is a Gram matrix. ◻
Let N=n-1, and let \mathbf 1 denote the N-component all-ones vector. For n\ge3, pairwise alignment is the mean off-diagonal entry given by \sigma_b(n) =\binom{N}{2}^{-1}\sum_{1\le i<j\le N}A_{ij} =\frac{\mathbf 1^{*}\mathbf A_b(n)\mathbf 1-\operatorname{tr}\mathbf A_b(n)} {N(N-1)}. The statistic \sigma_b(2) is undefined because the field contains no pair of distinct elements.
If q>1, a fraction k/n terminates exactly when q\mid k, so exactly s-1 fractions terminate. The reference fraction 1/n is nonterminating, so its matrix row assigns zero to those mixed pairs, whereas the reference-alignment convention assigns them score one. Consequently, \alpha_b(n)=\frac{\displaystyle\sum_{j=1}^{N}A_{1j}+s-1}{N}. If q=1, then \mathbf A_b(n)=J_N and \alpha_b(n)=\sigma_b(n)=1 for n\ge3. In either case where \sigma_b is defined, focused alignment is recovered as F_b(n)=\alpha_b(n)-\sigma_b(n). Thus the matrix together with the termination marker recovers all three scalar statistics; the unmarked matrix alone does not determine the terminating correction.
For a prime p\nmid b, define the digit function \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor, \qquad 1\le r<p.
Theorem 3 (Identity characterization). Let p\nmid b be prime. The following conditions are equivalent.
\mathbf A_b(p)=I_{p-1};
\delta_{p,b} is injective on \{1,\ldots,p-1\};
p\le b+1.
Proof. If the digit function is injective, two distinct initial residues i,j remain distinct after multiplication by every power of b. Their synchronized digits therefore differ at every position, so all off-diagonal matrix entries vanish. Conversely, if \delta_{p,b}(r)=\delta_{p,b}(s) for distinct residues r,s, then the fractions r/p and s/p match at the first position and A_{rs}>0. Thus the matrix is the identity exactly when the digit function is injective. If p>b+1, then p-1 residues enter only b digit bins, so injectivity fails. Suppose p\le b+1. When b=p-1, \delta_{p,b}(r)=r-1. Otherwise b\ge p, and for 1\le r<s<p, \frac{bs}{p}-\frac{br}{p}\ge1, so \lfloor bs/p\rfloor>\lfloor br/p\rfloor. The digit function is injective in both cases. ◻
The next theorem gives the exact meaning of multiplicative-coset structure in the matrix.
Theorem 4 (Coset–Fourier reduction). Let p\nmid b be prime, put L=\mathop{\mathrm{ord}}_p(b) and h=(p-1)/L, and choose representatives c_1,\ldots,c_h for the cosets of \langle b\rangle in (\mathbb Z/p\mathbb Z)^*. Order the nonzero residues as c_a b^r\pmod p, \qquad 1\le a\le h,\quad 0\le r<L. In this ordering, \mathbf A_b(p) is an h\times h matrix of L\times L circulant blocks.
More precisely, for 0\le d<b define x_{a,d}(t)= \mathbf 1_{\{\delta_{p,b}(c_a b^t\bmod p)=d\}}, \qquad X_{a,d}(j)=\sum_{t=0}^{L-1}x_{a,d}(t)e^{-2\pi ijt/L}. Fourier transformation in the exponent coordinate reduces \mathbf A_b(p) to the direct sum of the h\times h Hermitian matrices M_j(a,a')=\frac1L\sum_{d=0}^{b-1} \overline{X_{a,d}(j)}X_{a',d}(j), \qquad 0\le j<L. Consequently, the spectrum of \mathbf A_b(p) is the union, with multiplicity, of the spectra of M_0,\ldots,M_{L-1}.
Proof. At synchronized position t, the row indexed by c_ab^r has digit \delta_{p,b}(c_ab^{t+r}\bmod p). Consequently, A_{(a,r),(a',r')} =\frac1L\sum_{t=0}^{L-1}\sum_{d=0}^{b-1} x_{a,d}(t+r)x_{a',d}(t+r'). After replacing t+r by a new index, this expression depends on r'-r modulo L. Every coset-to-coset block is therefore circulant. The length-L Fourier transform diagonalizes all these blocks simultaneously [4]. Taking the Fourier transform of the displayed cross-correlation gives the stated matrix M_j. For each j, M_j is the Gram matrix of the h vectors L^{-1/2}\bigl(X_{a,0}(j),\ldots,X_{a,b-1}(j)\bigr), so it is Hermitian and positive semidefinite. The simultaneous Fourier decomposition proves the spectral union. ◻
Corollary 5 (Primitive-root spectrum). If b is a primitive root modulo p, then h=1 and \mathbf A_b(p) is permutation-similar to one circulant matrix. Write X_d=X_{1,d}, and let R(\ell)=\#\{t:\delta_{p,b}(b^t\bmod p) =\delta_{p,b}(b^{t+\ell}\bmod p)\}, \qquad \ell\in\mathbb Z/L\mathbb Z. Its eigenvalues are the discrete Fourier transform of the normalized cyclic digit-equality autocorrelation. \lambda_j=\frac1L\sum_{\ell=0}^{L-1}R(\ell)e^{-2\pi ij\ell/L} =\frac1L\sum_{d=0}^{b-1}|X_d(j)|^2.
Proof. This is the case h=1 of Theorem 4. Its first row is \frac1L\bigl(R(0),\ldots,R(L-1)\bigr). The second identity is the one-vector specialization of the formula for M_j. ◻
The digit channels x_{a,d} record equality, rather than numerical products of digit values. Accordingly, the displayed power sum is the sum of the channel powers, not the power spectrum of the digits read as one numerical signal. For example, the six distinct digits of 1/7=0.\overline{142857} give normalized equality autocorrelation (1,0,0,0,0,0) and six eigenvalues equal to 1. This is a flat equality spectrum of a deterministic periodic sequence.
When h=2, the theorem gives two eigenvalue branches at every Fourier frequency. It does not force two global eigenvalue levels; the collapse at p=13 below is special.
Proposition 6 (Two exact decimal spectra). In base 10, the matrix \mathbf A_{10}(13) has eigenvalues 4/3 and 2/3, each with multiplicity six. The matrix \mathbf A_{10}(17) has exactly nine distinct eigenvalues, with minimum 1/4 and maximum 7/4.
Proof. For p=13, the two multiplicative-coset digit words are 076923,\qquad 153846. Each word has distinct digits. Their only shared digits are 3 and 6, and both align at relative shift three. If P is the unit cyclic shift on six positions, then \mathbf A_{10}(13) =\begin{pmatrix}I_6&\frac13P^3\\[2pt] \frac13P^3&I_6\end{pmatrix}. Write \mathbf A_{10}(13)=I_{12}+K/3, where K=\begin{pmatrix}0&P^3\\ P^3&0\end{pmatrix}. Since K^2=I_{12} and \operatorname{tr}K=0, the eigenvalues of K are +1 and -1, each six times. This gives 1\pm1/3.
For p=17, the primitive-root digit word is 0588235294117647. Its equality autocorrelation is 16 at shift zero, 2 at shifts \pm1,\pm3,\pm5, and zero elsewhere. Corollary 5 therefore gives \lambda_j=1+\frac14\left( \cos\frac{\pi j}{8}+ \cos\frac{3\pi j}{8}+ \cos\frac{5\pi j}{8}\right). For 0\le j\le8, the nine values are \frac14,\quad 1\pm\frac14\cos\frac{\pi}{8},\quad 1\pm\frac{\sqrt2}{8},\quad 1\pm\frac14\cos\frac{3\pi}{8},\quad 1,\quad \frac74. They are distinct because \cos(\pi/8)>\cos(\pi/4)>\cos(3\pi/8)>0; conjugate Fourier frequencies supply the remaining multiplicities. ◻
Proposition 7. Let p\nmid b be prime with p\le b+1, and let s\ge1 be b-supported. Then \mathbf A_b(ps) is permutation-similar to \mathbf A_b(ps) \cong J_{s-1} \oplus \underbrace{J_s \oplus \cdots \oplus J_s}_{p-1}, where J_0 denotes an empty block. If s=1, then \mathbf A_b(p)=I_{p-1},\qquad \operatorname{rank}\mathbf A_b(p)=p-1. If s\ge2, then \operatorname{rank}\mathbf A_b(ps)=p, and its spectrum is \begin{aligned} s &\quad\text{with multiplicity }p-1,\\ s-1 &\quad\text{with multiplicity }1,\\ 0 &\quad\text{with multiplicity }ps-p-1. \end{aligned}
Proof. Choose D with s\mid b^D and put u=b^D/s. Since p\nmid b, u is invertible modulo p. The s-1 nonzero multiples of p terminate and form the block J_{s-1}. Every other numerator begins after depth D at a nonzero remainder ku\pmod p. Each of the p-1 nonzero remainders has exactly s numerator representatives.
Members of one remainder class have identical synchronized digits. Members of different classes remain at distinct remainders at every position, and digit-function injectivity forces distinct digits. This proves the direct-sum formula. A nonempty J_m has one eigenvalue m and m-1 zero eigenvalues. Reading the ranks and multiplicities from the blocks gives the two cases above. ◻
The prime field s=1 has no terminating block. Once s\ge2, that block supplies the additional rank-one direction. Further growth of s enlarges only the null space inside the synchronized classes.
The matrix records the full synchronized pairwise geometry. \sigma_b is its global off-diagonal average. The marked reference row and the terminating correction recover \alpha_b, after which F_b is their signed difference. At prime level, the identity matrix characterizes digit partitioning exactly. For supported lifts ps, the block theorem shows precisely which new directions are created and which merely enlarge the kernel.
Theorem 4 reduces every prime matrix to finite Hermitian Fourier blocks. For composite rough parts, additional orbit and period interactions enter the matrix. How much of the prime spectrum can the digit-bin geometry reveal before the full matrix is constructed?
[1]A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inform. Theory 20 (1974), no. 1, 90–94. https://doi.org/10.1109/TIT.1974.1055169.
[2]S. C. Kak and A. Chatterjee, On decimal sequences, IEEE Trans. Inform. Theory 27 (1981), no. 5, 647–652. https://doi.org/10.1109/TIT.1981.1056394.
[3]N. J. Armstrong and R. J. Armstrong, Some properties of repetends, Math. Gaz. 87 (2003), no. 510, 437–443. https://doi.org/10.1017/S0025557200173619.
[4]P. J. Davis, Circulant Matrices, 2nd ed., Chelsea, 1994.
[5]R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013. https://doi.org/10.1017/CBO9781139020411.
Discussion
Sign in to join the discussion.