Claims · 11 dated claims on Adjacent digits, each with its tag and its witness; newest 2026-10-05.
Proved For every base b >= 3 the integers with no two equal adjacent base-b digits are the accepted words of a (b+1)-state automaton read from the units digit whose state is the last digit read; they hold exactly (b-1)^L integers of L digits, form no digit design since 1 and b lie in the set and b + 1 does not, have no digit allowed in every state and no reset word, and their automaton completed with a sink synchronizes on the word 00. Witness: notes/adjacent.md, section The automaton.
Verified The automaton of the integers with no two equal adjacent base-b digits agrees with direct membership below b^6 at b = 3, 4, 5, 10, with 597870 members at b = 10; the split into at most 2(b-1) blocks per scale cut by the top digit matches direct membership at 420 values of x; the units-digit identity holds at all 450 cells b = 3..12, j = 1..6. Witness: lab/py/adjacent-digits, verb count.
Proved Let S be read by a digit automaton at base b whose start state keeps at most b digits and every live state at most fill, split below every x with L digits into at most N_0 blocks per scale, each of at most fill^j integers with transform a product of the automaton matrices against an end vector, plus at most N_0 points, with A_S(x) >= fill^(L-1)/N_0; with a shifted-grid certificate alpha_1 < 1/5 from every state against every end vector and unit vector; with a contraction c_E < 1 of the live rows and of the start row over m_0 steps whose first frequency lies at distance at least 3/(8b) from Z; and with (b/phi(b)) times the block counts coprime to b equal to kappa times the block size within E_0. Then abs(M_S(x)) and abs(psi_S(x) - kappa A_S(x)) are at most C A_S(x) exp(-c sqrt(log x)); the Kummer set (3, 1) is the two-state case. Witness: notes/adjacent.md, section The theorem on a digit automaton.
Proved In every block P b^j + W of the integers with no two equal adjacent base-b digits, W the words of length j with top digit not f, (b/phi(b)) times the number of elements coprime to b equals (b-1)^j + (-1)^(j-1)(1 - (b/phi(b))[f a unit mod b]), so the prime count constant on the set is kappa_b = 1 at every base. Witness: notes/adjacent.md, section The blocks and the units digit.
Proved For the automaton of no two equal adjacent base-b digits every one-step digit-state row of the transform matrix has modulus sum b - 1 and the start row b, while every two-step row sums to at most b abs(S_b(v)) + 2b - 1 with S_b the full digit sum at the first frequency v, a contraction c_E = (b sigma_b + 2b - 1)/(b-1)^2 with sigma_b = sin(3 pi/8)/sin(3 pi/(8b)), below 0.7874 at every b >= 1153. Witness: notes/adjacent.md, section The two-step contraction.
Proved The transform T_i of the words of length i with no two equal adjacent base-b digits satisfies T_i(u) = sum over r < i of (-1)^(r-1) S_b(rep_r u) T_(i-r)(b^r u) plus (-1)^(i-1) S_b(rep_i u), rep_r = (b^r - 1)/(b - 1), and its shifted grid sums are at most (1 + Phi*(y)) (b/y)^i whenever an explicit closed form Q_b(y) is at most 1, a certificate with b^alpha_1 = b/((b-1) y) and rate at most (2/pi + o(1)) (log b)^(2/3). Witness: notes/adjacent.md, sections The run expansion and The certificate.
Proved The chain certificate of the integers with no two equal adjacent base-b digits has alpha_1 < 1/5 at every base b >= 1153, certified at 120 bits at every base 1153..2400 with a decreasing tail bound beyond and failing at 1152, 1/5 - alpha_1 >= 1.1311 * 10^-5 at 1153, and alpha_1 < 1/4 at every base b >= 185, failing at 184. Witness: notes/adjacent.md, section The wall; lab/py/adjacent-digits, verb wall.
Proved At every base b >= 1153, with A(x), M(x) and psi_R(x) the count and the sums of mu(n) and Lambda(n) over the n <= x with no two equal adjacent base-b digits, abs(M(x)) <= C A(x) exp(-c sqrt(log x)) and abs(psi_R(x) - A(x)) <= C A(x) exp(-c sqrt(log x)) at every x >= 2 with C and c computable from b; so there are infinitely many primes with no two equal adjacent base-b digits, (A(x)/log x)(1 + O(1/log x)) of them up to x. Witness: notes/adjacent.md, section The theorem on R_b.
Verified The chain certificate of the integers with no two equal adjacent base-b digits reads alpha_1 at most 0.5155374 at b = 10, 0.2752730 at 100, 0.2029957 at 1000, 0.1646604 at 10^4 and 0.0912137 at 10^9; it sits below 0.22 at every base 489..1000 and below 0.18 at every base 3452..5000; the exact grid sums by FFT read 2.4144 per digit at b = 10, level 6, against the certificate's 3.27746. Witness: lab/py/adjacent-digits, verbs wall and rate.
Verified Over the integers with no two equal adjacent digits, psi_R(x)/A(x) with prime powers reads 0.9995 at base 10 and 0.9991 at base 7 at x = 10^8, A counted exactly through the blocks, and abs(M(10^8))/A(10^8)^(1/2) reads 0.655 at base 10, sanity prints far below the wall. Witness: lab/py/adjacent-digits, verb primes.
Conjecture At every base b >= 3 the sum of Lambda over the integers up to x with no two equal adjacent base-b digits is asymptotic to their count; the readings at b = 3, 4, 5, 7, 10 at x = 10^8 sit within 0.0066 of 1. Witness: lab/py/adjacent-digits, verb primes.