One number at every odd side

• 2026-10-02 [Proved] At odd side N >= 3 a real x in [0, 1] lies in the even-digit design E_N iff N^j x mod 2 lies in the closed arc [0, 1] for every j >= 0, and p/q in lowest terms lies in it iff N^j p mod 2q lies in {0, 1, ..., q} for every j >= 0; the endpoint q is needed, 1/3 lying in E_3. Witness: cobham Object E, bullet The membership law.
• 2026-10-02 [Proved] For 0 < p < q in lowest terms the odd sides holding p/q are periodic mod 2q from side 1 with no transient, their least period is exactly 2q, and p/q and (q-p)/q are held by the same sides. Witness: cobham Object E, bullet The period.
• 2026-10-02 [Proved] For 0 < p < q in lowest terms the share of the odd sides holding p/q is 1/2 at q = 2 and from q = 3 on lies in [2/q, (q+1)/(2q)] at odd q and in [2/q, 1/2] at even q, so at every q the largest share of a number of (0, 1) is 2/3, attained at 1/3 and 2/3 alone. Witness: cobham Object E, bullet The share.
• 2026-10-02 [Proved] At odd q side N holds p/q iff every least residue N^j p mod q is 0 or has the parity of p; at odd prime q the share of p/q is 1/q times 1 plus the sum of phi(d) over the divisors d of m for which every least residue of p G_d has the parity of p, with m the odd part of q - 1 and G_d the subgroup of order d of (Z/q)^*. Witness: cobham Object E, bullet The exact rule.
• 2026-10-02 [Verified] The orbit rule agrees with a walk on the digits at every p/q in [0, 1] in lowest terms with q <= 200, over the odd sides up to 6q + 1 at q <= 40 and one period above, 1661996 checks; at all 12231 fractions in (0, 1) with q <= 200 the least period 2q, symmetry, share bounds, rule mod q, coset count and prime formula hold; shares above 1/2 occur only at 1/3, 2/3 and 1/2 only at q = 2, 4, 10, 12. Witness: lab/py/sides-holding-a-number verb period 200.
• 2026-10-02 [Proved] At an odd prime q, 1 <= p <= q - 1 and a the odd one of p, q - p, the Legendre symbol (a/q) is -1 to the number of odd sides 3 <= N < q at which the first base-N digit of p/q is odd. Witness: cobham Object E, bullet The Eisenstein link.
• 2026-10-02 [Proved] At an odd prime q and 1 <= p <= q - 1 the sum of floor(p N/q) over the odd N < 2q is (2p - 1)(q - 1)/2 + p, the even-multiplier sum of Eisenstein's lemma cancelling, so the full period of odd sides carries no Legendre symbol. Witness: cobham Object E, bullet The full period carries no symbol.
• 2026-10-02 [Verified] The half-period parity law holds at all 4180 pairs with q an odd prime at most 200, the full-period identity holds at all 4180, and its parity agrees with the symbol at exactly 2090. Witness: lab/py/sides-holding-a-number verb eisenstein 200.
• 2026-10-02 [Proved] For irrational x in (0, 1) and every level L >= 1 the odd sides at which the first L base-N digits of x are even have density exactly 2^(-L), so a number of [0, 1] is rational iff the odd sides holding it have positive density. Witness: cobham Object E, bullet An irrational, level by level.
• 2026-10-02 [Proved] c(k) = card{odd N : 3 <= N <= 2k, every base-N digit of 2k even} satisfies abs(c(k) - (1 - log 2) k) <= sqrt(2k) + 1 for every k >= 1. Witness: cobham Object E, bullet The integer count.
• 2026-10-02 [Verified] At every k <= 10^6, abs(c(k) - (1 - log 2) k)/sqrt(k) <= 0.547191, attained at k = 74, the error is at most 0.357533 of sqrt(2k) + 1, and c(10^6) = 306665. Witness: lab/py/sides-holding-a-number verb count 1000000.
• 2026-10-02 [Conjecture] c(k) = (1 - log 2) k - kappa sqrt(k) + o(sqrt(k)) with kappa = (2 - sqrt 2) abs(zeta(1/2))/4 = 0.213864; over 60 random k per decade the mean of the error over sqrt(k) reads -0.220703 at 10^6 rising to -0.215000 at 10^11. Witness: lab/py/sides-holding-a-number verb second 11.
• 2026-10-02 [Proved] The integers held by every odd side are 0 and 2, the reals held by every odd side are 0 and 1, and E_N lies in E_(N^e) for every e >= 1, strictly at every e >= 2, N + 1 and (N + 1)/N^e lying at side N^e and not at side N. Witness: cobham Object E, bullets What every side holds and Powers of a side.
• 2026-10-02 [Proved] The union of the E_N over the odd sides N >= 3 is Lebesgue null and meagre, has Hausdorff dimension 1 attained by no E_N, holds every rational of [0, 1], and has Fourier dimension 0: every Borel probability measure with abs(hat mu(xi)) <= C abs(xi)^(-eps) gives it measure 0. Witness: cobham Object E, bullets The union of all sides and Fourier dimension 0.
• 2026-10-02 [Proved] K_(p^a) is the set of k for which no carry of k + k in base p leaves a position = a - 1 mod a, so 20 lies in K_9 although 9 divides binomial(40, 20); and K_15 and K_3 cap K_5 are incomparable, 10 lying in the second only and 2 in the first only. Witness: cobham Object E, bullet Composite sides.
• 2026-10-02 [Verified] Below 10^12 the odd sides {3, 5} hold 10072 integers, {3, 5, 15} hold 50, {3, 5, 7} hold 17, exactly twice the terms of A030979 below 5 10^11, {3, 5, 7, 11} hold 0, 2, 6320, {3, 5, 7, 13} hold 0, 2, 1512, 1514, 6500, {3, 5, 7, 15} hold 0, 2, 1512, 1514, 15302, and {3, 5, 7, 11, 13} and {3, 5, 7, 11, 15} hold 0, 2. Witness: lab/py/sides-holding-a-number verb inter 1e12.
• 2026-10-02 [Verified] The odd sides {3, 5, 7, 11} hold 0, 2, 6320 and no other integer below 10^1000. Witness: lab/py/sides-holding-a-number verb deep 1000.