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 1s 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)