# The radix census

- 2026-10-03 [Proved] A twisted radix design with digits `d_0, ..., d_(k-1)` has unit steps between consecutive words at every level if and only if `phi_a(z_e) = phi_(a+1)(z_0)` for every `a` from `0` to `k-2` and `J_a(l)` is a unit for `l` from 1 to the order of the unit group, 4 on `Z[i]` and 6 on `Z[w]`, where `z_0, z_e` are the fixed points of the first and last place maps and consecutive words first differing at `a, a+1` with `l` levels to go step by a unit times `J_a(l) = A_a b^(l-1) - u_(a+1) z_0 u_0^(l-1) + u_a z_e u_(k-1)^(l-1)`, `A_a = b (phi_(a+1)(z_0) - phi_a(z_e))`. Witness: toolpaths.md, The radix census, The step law; lab/rs/radix-census junction.
- 2026-10-03 [Proved] A twisted radix design with unit steps at every level whose attractor is not one point is conjugate, map for map, by `h(x) = (x - z_0)/(z_e - z_0)` to the walk of its twists, digits the partial sums `0, u_0, u_0 + u_1, ...`, its twists sum to the base, and its word order runs the vertices of the replacement curve with generator `u_0, ..., u_(k-1)` and every flag `F`, up to a translate at each level, exactly when `d_0 = 0` or its twists are all equal; with one point it fails, as for `Z[i]` base `2+i`, digits `i, 1+i, 1+2i`, twists `1, i, -1`, all maps fixing `(1+i)/2`. Witness: toolpaths.md, The radix census, The step law; lab/rs/radix-census census, junction.
- 2026-10-03 [Refuted] Unit steps at level 3 imply unit steps at every level for the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`; the pair at `1+i` with digits `0, 1` and twists `1, -1` has unit steps at levels 1 to 3 and not at level 4. Witness: lab/rs/radix-census junction.
- 2026-10-03 [Verified] Every one of the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, that has unit steps at level 4 has unit steps at every level. Witness: toolpaths.md, The radix census, The step law; lab/rs/radix-census census.
- 2026-10-03 [Proved] For a twisted radix design, two words of one level landing on one point, sharing a place map, or lying in copies that touch under unit adjacency is decided at every level by a finite automaton on the normalised difference `delta` and the relative turn `r`, with every state of `abs(delta) >= 2M/(abs(b) - 1)` and `abs(delta) > 1` pruned for `M` the largest digit modulus, and the level set is connected at every level exactly when the copy graph is connected at every level, which the eventually periodic touch sets decide. Witness: toolpaths.md, The radix census, Glue and area; lab/rs/radix-census junction.
- 2026-10-03 [Proved] A twisted radix design with `N(b)` digits has positive area if and only if no two words of one length share a place map. Witness: toolpaths.md, The radix census, Glue and area.
- 2026-10-03 [Proved] At the bases `2+i` of `Z[i]` and `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, every twist of the full canonical residue system leaves every level set of the untwisted design unchanged, so all `1024` and `279936` full twists are plane-filling with distinct points at every level. Witness: toolpaths.md, The radix census, Glue and area; lab/rs/radix-census census.
- 2026-10-03 [Refuted] A twisted radix design at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, that is a curve at level 4 is a curve at every level; the walk `1, w` at base `1+w` is a curve at levels 1 to 4 and lands two words on one point at level 5. Witness: lab/rs/radix-census junction.
- 2026-10-03 [Verified] At the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, the first revisit of a canonical pair or walk with unit steps comes at level 5 at the latest, reached by the walk `1, w` at `1+w`, the first split of a canonical twisted pair at level 13 at the latest, reached at base `3` by code `223` twisted by `1, w^4, w^3, w^5, w^2, 1, w^4`, and the first shared place map of a full residue twist at level 7 at the latest, reached at base `2` of `Z[w]` by the twists `w^2, w^4, w^5, w`. Witness: lab/rs/radix-census census, curves, junction.
- 2026-10-03 [Proved] The similarity `x -> v x` with `v` a unit carries the radix design with digit `d` and twist `u` to the digit `v d` with the same twist `u`, and `x -> v conj(x)` with `conj(b) = eps b` carries it to the digit `eps^(-1) v conj(d)` with the twist `eps^(-1) conj(u)`. Witness: toolpaths.md, The radix census, The census.
- 2026-10-03 [Refuted] The code action of the radix dial, each twist carried to the image class, preserves whether a twisted radix design is one piece; the untwisted digits `1, w` at base `2` of `Z[w]`, `w = e^(i pi/3)`, are one piece at every level and conjugation sends them to the code with digits `1, -1+w`, apart at level 1. Witness: lab/rs/radix-census census.
- 2026-10-03 [Verified] There are `608, 16, 3104, 2376, 324, 40353552, 823500` of the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, in that order, in `324, 8, 816, 1217, 165, 20181403, 137775` orbits of the stabiliser of the canonical residues, whose orders are `2, 2, 4, 2, 2, 2, 6`. Witness: lab/rs/radix-census census.
- 2026-10-03 [Verified] Of the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, in that order, the ones with unit steps at every level number `4, 2, 0, 8, 6, 1, 0` and the curves, with unit steps and no two words on one point at any level, `4, 0, 0, 5, 0, 1, 0`, every curve an untwisted straight segment. Witness: toolpaths.md, The radix census, The census; lab/rs/radix-census census.
- 2026-10-03 [Verified] Of the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, in that order, the ones whose level set is connected under unit adjacency at every level number `245, 12, 1760, 1295, 179, 31610251, 702828`. Witness: lab/rs/radix-census census.
- 2026-10-03 [Verified] Of the canonical twisted radix pairs, a code over the canonical residues read in canonical order with a unit twist per digit and at least two digits, at the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, in that order, the ones with `N(b)` digits and no two words of one length sharing a place map number `128, 16, 1024, 324, 144, 82968, 279936`, and those with `N(b)` digits and no two words on one point at any level `9, 10, 1024, 10, 14, 12, 279936`. Witness: lab/rs/radix-census census.
- 2026-10-03 [Verified] No replacement curve with every flag `F` and order `N(c)` to a chord `c` of norm 2, 4 or 5 in `Z[i]` or 3, 4, 7 or 9 in `Z[w]` avoids points, its end vertex included: the walks of `N(b)` unit steps to the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, number `16, 2, 50, 34, 6, 116214, 4235`, and each lands two vertices on one point by level 4. Witness: toolpaths.md, The radix census, The walks; lab/rs/radix-census curves.
- 2026-10-03 [Verified] Below dimension 2 the arcs among the walks to the bases `2`, `1+i`, `2+i` of `Z[i]` and `2`, `1+w`, `3`, `2+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, are, up to reversal and mirror, the straight segments, the walk `1, w, w^5, 1` to `3`, the walk `1, i, 1` to `2+i` and the walk `1, w, 1` to `2+w`; every other walk lands two vertices on one point by level 5. Witness: lab/rs/radix-census curves.
- 2026-10-03 [Verified] With `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, the radix dial's Koch design is the walk `1, w, w^5, 1` to base `3` and no canonical pair, its terdragon is the canonical pair code `7` at `1+w` twisted by `1, w^2, 1`, which has unit steps at every level, positive area and two words on one point at level 2, and its twindragon and flowsnake are the untwisted full residue designs at `1+i` and `2+w`, with positive area, distinct points at every level and no unit steps. Witness: lab/rs/radix-census named.
- 2026-10-03 [Proved] Neither the Hilbert curve nor the Gosper curve is the word order of a radix design, since every copy at level 2 of a radix design is a turn of level 1 read forward. Witness: toolpaths.md, The radix census, The named curves; lab/rs/radix-census named.
- 2026-10-03 [Verified] The radix design at base `1+w` of `Z[w]`, `w = e^(i pi/3)`, so that `omega = w^2` and `1+w`, `2+w` are the radix dial's `2+omega`, `3+omega`, with digits `-1, 0, 1` and twists `1, w^2, 1` has unit steps at every level, shares no place map, never splits and lands no two words on one point at any level, so it is a plane-filling curve whose level path runs the midpoints of the terdragon's edges. Witness: toolpaths.md, The radix census, The walks; lab/rs/radix-census junction.