Bin Derangements and the Gate Width Theorem
The ring
Every fraction of a prime is the same ring of digits, entered at a different point. 1/17 and 2/17 and 16/17 are the same repeating block: 0588235294117647. Connected end to end, rotating.
Spin the ring. At some rotation angles, digits land on copies of themselves: the ring recognizes itself. At other angles, every digit moves off its own position. Complete displacement. The ring is perfectly out of phase with itself.
In base 10, there are always exactly 9 angles of complete displacement. Always. For every prime. The count is b - 1, where b is the base.
The proof
The digit function delta(r) = floor(br/p) assigns each residue to a bin. The bins are contiguous intervals. But under the change of variables x = br mod p, those same bins become the residue classes modulo b. The geometry becomes algebra.
A collision (two residues in the same bin after multiplication by g) becomes a congruence: x congruent to gx mod b. The parameter c = b(1 - g)^{-1} mod p controls everything. When 1 <= c <= b - 1, a squeezing argument shows no collision can exist. When c >= b + 1, an explicit collision is constructed.
The deranging multipliers are exactly:
g = -u/(b - u) mod p, for u = 1, ..., b - 1.
These are b - 1 distinct values. The proof is complete and holds for all primes p > b, not just primitive-root primes.
What it means
The base provides b digit symbols. The gate blocks exactly b - 1 multipliers from producing any same-bin collision. The blocked multipliers form a specific algebraic family determined by b alone. The prime determines where they fall in the multiplicative group, but how many there are is fixed by the base.
The number b - 1 is not a coincidence, not an approximation, not a statistical artifact. It is forced by the interaction between the floor function (which creates contiguous bins) and modular arithmetic (which transforms those bins into congruence classes). The contiguity is essential: the proof uses the fact that bin membership is determined by a residue class, which holds only for the Beatty partition.
The mean formula
At the constructive lags (where R(l) > 0), the mean value satisfies:
$$\overline{R} = \frac{Q(b(Q-1) + 2R)}{b(Q-1) + R}$$
where Q = floor((p - 1)/b) and R = (p - 1) mod b. For primes congruent to 1 mod b, the mean is exactly Q. For the smallest non-DP primes, the mean is always 2, in any base.
The paper
Bin Derangements and the Gate Width Theorem (PDF)
./nfield spectral 17 # show R(ell) for p=17
./nfield spectral 47 # compare within same class
./nfield spectral 61 --base 6 # verify in base 6
.:.