# The radix dial

- 2026-09-13 [Proved] The fill law survives the radix dial: a radix design accepts every word over `F`, so it has `(card F)^level` words of length `level` at every ring, every base, every digit set and every twist, the twists appearing nowhere in the accept slot. The row restates the accept slot rather than proving anything beyond it. (witness: `mrlynum::radix::Radix::fill` against `words` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Proved] Every place map `phi_d(x) = (u_d x + d) / base` of a base of norm `q >= 2` is a similarity of ratio `q^(-1/2)` because a unit has modulus one, so all `card F` maps contract equally and the similarity dimension is `s = 2 log card F / log q` whatever the twists; this is `Hutchinson 1981` 5.1(2) and 5.1(3) at a ring base. (witness: `mrlynum::radix::Radix::dimension` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Proved] Today's plane designs are the untwisted real-base row of the dial: at `R = Z[i]`, `base = m`, digits the box `{a + c i : 0 <= a, c < m}` and every `u_d = 1`, the place map is `x -> (x + d)/m` on each coordinate, so the word `d_1 ... d_level` lands on the level-`level` cell of the plane design of the same code at base `m`, the code read in box row-major order `bit r q + c` and not in the canonical residue order. (witness: `mrlynum::radix::tile` against `mrlymath::bang::factory::create` in `lab/rs/radix-designs` verb `today`)
- 2026-09-13 [Proved] A twist keeps every count and moves only the place: the accept slot mentions no `u_d` so the word count stays `(card F)^level`, every ratio stays `q^(-1/2)` since a twist has modulus one so the dimension is untouched, and the twists enter the definition only through where an image sits. (witness: `mrlynum::radix::Radix::words` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Proved] An untwisted design on pairwise incongruent digits has no glue: reducing `sum_i d_i base^(level-i)` mod `base` recovers `d_level` because the digits are distinct residues, and induction on `level` gives the rest, so the distinct-point count equals `(card F)^level` at every level. The hypothesis is a hypothesis of the statement and not of the generator: `Radix::new` accepts any digit list, and `from_code` and `tile` are the two constructors that enforce it. (witness: `mrlynum::radix::Radix::distinct` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Proved] The canonical least-norm residue system of a real base `m` on `Z[i]` is the box `{a + c i : 0 <= a, c < m}` at `m = 2` and at no larger `m`: the box holds `m-1` of norm `(m-1)^2 >= 4` while its own class holds `-1` of norm `1`. (witness: `mrlynum::radix::Base::residues` in `lab/rs/radix-designs` verb `today`)
- 2026-09-13 [Proved] The four maps `z/3`, `e^(i pi/3) z/3 + 1/3`, `e^(-i pi/3) z/3 + 1/2 + i sqrt(3)/6`, `z/3 + 2/3` are `phi_d` coefficient for coefficient at base `3` on `Z[omega]` with digits `0, 1, 2+w, 2` and twists `1, 1+w, -w, 1`: `e^(i pi/3) = 1 + w`, `e^(-i pi/3) = -w` and `(2 + w)/3 = 1/2 + i sqrt(3)/6`, so `(u_d x + d)/3` at `(d, u) = (0, 1), (1, 1+w), (2+w, -w), (2, 1)` is that list in order. The `1.241e-16` printed at level `5` is a float evaluation of the same four maps and is a self-check of the crate's ring arithmetic, not an independent identification. (witness: `lab/rs/radix-designs` verbs `koch` and `compare`)
- 2026-09-13 [Verified] The Sierpinski gasket is the untwisted code `7` at base `2` on `Z[omega]`: against the three similarities of ratio `1/2` fixing the vertices of an equilateral triangle, written independently in `f64` and placed at `(1, 1)`, `(3, 1)`, `(2, 1 + sqrt 3)`, the `3^9 = 19683` words agree to `4.441e-16` after the translation and positive scaling that the statement leaves free, pinned by the two corresponding words `0^9` and `2^9` and then measured at every word, with turn residual `0`. (witness: `lab/rs/radix-designs` verb `compare`)
- 2026-09-13 [Verified] That level-system reading is the code `7` at base `2+w` on `Z[omega]` with twists `1, w, 1`: its `3^8` words are the segment starts word for word at level `8` to `7.511e-16`. The reading is the terdragon's own level-system and is carried by no source read here, so the name stays [Conjecture] until one is. (witness: `lab/rs/radix-designs` verb `compare`)
- 2026-09-13 [Verified] The five codes as printed: code `7` at base `2` on `Z[omega]` of dimension `1.584963`, code `3` at `1+i` on `Z[i]`, code `7` at `2+w` and code `127` at `3+w` on `Z[omega]` each of dimension `2`, all four untwisted with `card F = 3, 2, 3, 7`, and code `147` at base `3` on `Z[omega]` twisted, `card F = 4`, dimension `1.261860`. (witness: `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Verified] Every plane code at `q = 2` and `q = 3` is the untwisted real-base radix design of the same code at level `2`: `528` codes checked, `0` mismatches, through the pixel map bit `r q + c` of the code is the cell at row `r` and column `c`, the column the real part and the row the imaginary part, which is box row-major order and not canonical residue order. (witness: `lab/rs/radix-designs` verb `today`)
- 2026-09-13 [Verified] The distinct-point count equals the fill for code `7` at base `2` to level `11`, code `3` at `1+i` to `17`, code `7` at `2+w` to `11`, code `127` at `3+w` to `6` and the twisted code `147` at `3` to `8`, so the Koch twist glues nothing inside that reach. (witness: `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Verified] The classes of digit CODES under the residue action are `12, 4, 12, 8, 6, 84, 28` over `16, 4, 32, 16, 8, 512, 128` codes at `Z[i]` bases `2, 1+i, 2+i` and `Z[omega]` bases `2, 2+w, 3, 3+w`; a Burnside count over the group and a direct orbit walk over all `2^q` codes agree at every base. These are classes of codes and never designs up to similarity. (witness: `lab/rs/radix-designs` verb `census`)
- 2026-09-13 [Verified] The group of a base is the units acting on residues by multiplication, joined by conjugation exactly when `conj(base)` is an associate of `base`, the mirror failing at `2+i` and at `3+w`: the abstract group `R^* semidirect <conj>` has order `8, 8, 4` on `Z[i]` at `2, 1+i, 2+i` and `12, 12, 12, 6` on `Z[omega]` at `2, 2+w, 3, 3+w`, and it acts on the residues through an image of order `2, 1, 4, 6, 2, 12, 6`. The action is not faithful: at `1+i` every element is the identity permutation. Burnside over the abstract list stays correct, because the list is the image of one abstract group with each element once. (witness: `mrlynum::radix::Base::group` in `lab/rs/radix-designs` verb `census`)
- 2026-09-13 [Verified] The canonical residue systems printed: `0, 1, i, 1+i` at `Z[i]` base `2`; `0, 1` at `1+i`; `0, 1, i, -1, -i` at `2+i`; `0, 1, 1+w, w` at `Z[omega]` base `2`; `0, 1, 1+w` at `2+w`; `0, 1, 1+w, w, -1, -1-w, -w` at `3+w`; and `0, 1, 1+w, w, -1, -1-w, -w, 2+w, 1+2w` at `3`. (witness: `mrlynum::radix::Base::residues` in `lab/rs/radix-designs` verb `census`)
- 2026-09-13 [Verified] The twist vectors at base `3` on `Z[omega]` number `7^9 = 40353607` summed over the `512` codes, not over the `84` classes: `sum_k binom(9, k) 6^k = 7^9` counts one `6^(card F)` for each code, and the two quotients are different quotients. No class count is claimed for twists, because no action of the group on twist vectors is defined here, and `7^9` counts only designs whose representative vector is canonical. (witness: `lab/rs/radix-designs` verb `census`)
- 2026-09-13 [Verified] The twisted glue witness holds as stated: `Z[i]`, base `2`, `F = {0, 1}`, twists `1, -1`, level `2` has fill `4` and `3` distinct points, the words `01` and `11` both landing on `1/4`, that is on `1` after scaling by `base^2`. (witness: `mrlynum::radix::Radix::distinct` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Proved] The distinct-point count of that twisted design is `2^(level-1) + 1` at every level: the scaled point of the word whose `1`s sit at positions `j_1 < ... < j_t` is `sum_(k=1..t) (-1)^(k-1) 2^(level - j_k)`, an alternating sum of strictly decreasing powers of two with top exponent at most `level-1`; such a sum is `0` or lies in `[1, 2^(level-1)]`, since the alternating tail is smaller than the leading term; and every integer `n` of `[1, 2^(level-1)]` is reached by exactly one choice, the greedy one, taking `2^a` for the least `a` with `2^a >= n` and recursing on `n - 2^a`, whose modulus is below `2^(a-1)`. Printed and checked at every level to `16`. (witness: `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Verified] The code census is not the design census: at base `3` on `Z[omega]` the `84` three-digit codes fall in `13` orbits of the residue action and in `9` similarity classes of the untwisted canonical digit sets, computed in exact arithmetic over `Q(w)`, and the two partitions cross. Codes `131`, digits `0, 1, 2+w`, and `137`, digits `0, w, 2+w`, share an orbit and are not similar, though they are affinely conjugate; codes `7`, digits `0, 1, 1+w`, and `42`, digits `1, w, -1-w`, are similar and sit in different orbits. Among the `36` two-digit codes the census gives `7` orbits where similarity gives `1` class. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-13 [Proved] The Hausdorff dimension of a radix design is at most its similarity dimension `2 log(card F) / log q`, with no hypothesis at all, by `Hutchinson 1981` 5.1(4)(i), which gives `H^s(K) < infinity` and `dim K <= s` for arbitrary contractions. (witness: `mrlynum::radix::Radix::dimension` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Verified] the Koch quintuple places `256` words on `256` distinct points at level `4`, similarity dimension `1.261860`, no digit canonical - `mrlydemo::radix::radix_read`, `site/check.ts` row `radix koch design`.
- 2026-09-13 [Verified] the twindragon quintuple places `1024` words on `1024` distinct points at level `10`, similarity dimension `2.000000` - `mrlydemo::radix::radix_read`, `site/check.ts` row `radix twindragon tiles`.
- 2026-09-13 [Verified] the carpet at base `3` on `Z[i]` with the BOX digits fills `64` words at level `2` and has similarity dimension `1.892789` - `mrlydemo::radix::radix_read`, `site/check.ts` row `radix carpet fill`.
- 2026-09-13 [Verified] that same carpet codes `479` over the canonical classes where it codes `495` in box row-major order, so the two readings of one design differ - `mrlydemo::radix::radix_read`, `site/check.ts` row `radix carpet fill`.
- 2026-09-13 [Verified] the digits `0, 1` at base `2` on `Z[i]` twisted by `1, -1` glue `4` words onto `3` points at level `2`, and untwisted they glue nothing - `mrlydemo::radix::radix_read`, `site/check.ts` row `radix twisted glue`, `crates/mrlydemo/tests/radix.rs`.
- 2026-09-13 [Verified] four digits reach level `8` and eight digits level `5` at the budget of `2^16` points - `mrlydemo::radix::radix_cap`, `site/check.ts` row `radix koch dimension`.
- 2026-09-13 [Proved] An untwisted radix design obeys the fill law: with every unit `u_d = 1`, the word `d_1 ... d_level` lands on `base^(-level) sum_i d_i base^(level-i)`, and reducing that integer modulo `base` recovers `d_level` because the digits are distinct residues, so induction gives distinct points for distinct words and `fill(level) = card F^level` at every ring, base and digit set. (witness: beneath.md, The fill law, and where it stops)
- 2026-09-13 [Proved] No plane radix design carries a rotation of order `5` or `8`: a unit of `Z[i]` solves `a^2 + c^2 = 1` with four solutions and a unit of `Z[omega]` solves `a^2 - ac + c^2 = 1` with six, so every available twist has order `1, 2, 3, 4` or `6`; and a rotation preserving a rank-2 lattice is an integer matrix in a lattice basis with trace `2 cos theta` in `{-2,-1,0,1,2}`, the classical crystallographic restriction. (witness: beneath.md, What a plane lattice will not carry)
- 2026-09-13 [Proved] A twist keeps every count the accept slot computes and every contraction ratio: the accept slot is the full shift on `F` and mentions no `u_d`, and every place map `phi_d(x) = (u_d x + d)/base` has ratio `q^(-1/2)` because a unit has modulus one, so the similarity dimension `2 log(card F) / log q` is untouched and the twists move only where an image sits and which words collide. (witness: beneath.md, The twist law)
- 2026-09-13 [Conjecture] That design is the Koch curve: `Hutchinson 1981` 3.3(2) gives the Koch curve as the attractor of four similitudes each carrying `a_1 a_5` to `a_i a_(i+1)` with positive determinant, and the four maps above are exactly those for the polyline `0`, `1/3`, `1/2 + i sqrt(3)/6`, `2/3`, `1`, but that polyline is read from its Figure 3.2 and not from its text, so the name rests on a figure and not on a sentence. (witness: `lab/rs/radix-designs` verb `compare`)
- 2026-09-13 [Conjecture] The twindragon is the untwisted code `3` at base `1+i` and the flowsnake is the untwisted code `127` at base `3+w`: each is compared only against the maps `(z + d)/base` over the residues of its own base, which is its definition as a radix set, so the comparison is a self-check at `0` and `2.259e-16` and no independent witness for either name exists here. (witness: `lab/rs/radix-designs` verb `compare`)
- 2026-09-13 [Conjecture] That bound is an equality: equality needs the open set condition, `Hutchinson 1981` 5.3(1), which is a hypothesis per base and per twist and is checked at no base here. (witness: `mrlynum::radix::Radix::dimension` in `lab/rs/radix-designs` verb `named`)
- 2026-09-13 [Refuted] A code over residue classes does not name a radix design: the place moves with the chosen representative, `d + base m` shifting the image of `phi_d` by `m`, and the Koch digits `0, 1, 2+w, 2` are not the canonical representatives of their classes, since `2` and `-1` share a class mod `3` on `Z[omega]` and the canonical system holds `-1`. A design is a quintuple, ring, base, code, representative vector, twist vector. (witness: `mrlynum::radix::Radix::canonical` in `lab/rs/radix-designs` verb `koch`)
- 2026-09-13 [Refuted] The terdragon is the untwisted code `7` at base `2+w`: read the terdragon's own level-system `F -> F + F - F` at `120` degrees as a turtle, three segments to a level, normalise by the endpoint, and the untwisted design misses the `3^8 = 6561` segment starts by `1.060` at level `8`. (witness: `lab/rs/radix-designs` verb `compare`)
- 2026-09-13 [Refuted] The fill law is not inherited by a twisted radix design: at `R = Z[i]`, `base = 2`, canonical residues `0, 1, i, 1+i`, digit set `F = {0,1}`, twists `u_0 = 1` and `u_1 = -1`, the words `01` and `11` both land on `1/4`, so two words of length two name one point and the cell count is `3` where `card F^level` is `4`. A twisted design owes its fill law a proof of its own. (witness: beneath.md, The fill law, and where it stops)
- 2026-09-14 [Proved] The conjugacy group of the untwisted canonical base family is the centraliser of `1/base` extended by translations. Conjugating `phi_d(x) = (x + d)/base` by an invertible real affine `h(x) = H x + s` gives `(y + H d + s(base - 1))/base`, again an untwisted place map exactly when `H` commutes with multiplication by `1/base`, and `s(base - 1)` sweeps the plane since `N(base) >= 2` forces `base != 1`. At a non-real base that centraliser is `C`, so the group is the similarity group; at a real base `1/base` is the scalar `(1/base) I` and the group is all of `GL_2(R)` semidirect `R^2`. This bites at `2` on `Z[i]` and at `2` and `3` on `Z[omega]`. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Proved] The mirror `x -> v conj(x) + t` preserves the untwisted base family exactly when `conj(base) = base`, and being an associate is not enough: a direct conjugacy keeps the derivative `1/base` and a mirror one sends it to `1/conj(base)`, so the mirrored object is a place map at base `conj(base)`. The code census admits conjugation at `1+i` and `2+w`, where the conjugacy group does not. At a real base the mirror is one element of the full affine group and not the only new one, so it is load-bearing for the similarity quotient alone. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Verified] The similarity census of the untwisted canonical digit sets runs at every base of the code census and every digit count `card F` from `0` to `q`, in exact arithmetic over `Q(i)` and `Q(w)`: the classes total `5, 3, 8, 6, 4, 117, 22` at `Z[i]` bases `2`, `1+i`, `2+i` and `Z[omega]` bases `2`, `2+w`, `3`, `3+w`, against the code classes `12, 4, 12, 8, 6, 84, 28`, and by `card F = 0` to `9` at base `3` on `Z[omega]` they are `1, 1, 1, 9, 23, 30, 29, 16, 6, 1` against code classes `1, 3, 7, 13, 18, 18, 13, 7, 3, 1`. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Verified] The conjugacy census of the same sets, the similarity group at the four non-real bases and `GL_2(Q)` semidirect `Q^2` at the three real ones, totals `5, 3, 8, 5, 4, 88, 22` over the seven bases, `135` over the `41` cells against `165` similarity classes and `154` code classes. At base `3` on `Z[omega]` the affine classes by `card F = 0` to `9` are `1, 1, 1, 2, 11, 23, 26, 16, 6, 1`, so `512` codes give `84` code classes and `88` affine classes and the design count still exceeds the code count under either name. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Verified] No two of the three quotients are comparable. Over the `41` cells the similarity count is below the code count in `17`, equal in `19` and above it in `5`, the two partitions crossing in `6`; the affine count is below in `19`, equal in `18` and above in `4`, crossing in `5`. The lost crossing is `Z[omega]` base `2` at `card F = 3`, where two orbits meet two similarity classes with codes `7` and `11` split and codes `11` and `14` merged, while all four triples are non-degenerate and fall in one affine class. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Verified] At base `3` on `Z[omega]` and `card F = 3` the `84` codes fall in `13` orbits, `9` similarity classes and `2` affine classes, the collinear triples against the rest. Codes `131`, digits `0, 1, 2+w`, and `137`, digits `0, w, 2+w`, share an orbit and are not similar, squared side lengths `1, 1, 3` against `1, 3, 4`, yet are affinely conjugate by `H = [[0, 2], [1, -1]]` of determinant `-2`; codes `7`, digits `0, 1, 1+w`, and `131` are affinely conjugate by the unimodular `H = [[1, 1], [0, 1]]`; codes `7` and `42` are similar in different orbits. (witness: `lab/rs/radix-designs` verb `affine`)
- 2026-09-14 [Verified] The census is controlled by two explicit conjugacies asserted in the verb, `H = [[0, 2], [1, -1]]` carrying code `131` onto code `137` and `H = [[1, 1], [0, 1]]` carrying code `7` onto code `131`, and by the assertion that the similarity classes refine the affine classes pair by pair in every cell. The per-size orbit counts summing to `12, 4, 12, 8, 6, 84, 28` constrains the code column alone, and the `f64` rerun of the similarity normal form checks the exact arithmetic and not the group. (witness: `lab/rs/radix-designs` verb `affine`)