# 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.