The Collatz carry

• 2026-09-14 [Proved] In base 2 read least significant digit first, 3n + 1 = n + 2n + 1 has digit i equal to n_i + n_(i-1) + c_i over GF(2) with n_(-1) = 0, c_0 = 1 and c_(i+1) = MAJ(n_i, n_(i-1), c_i), and the substitution M = 2n + 1 clears the constant from the skeleton: M xor 2M = 2 (n xor 2n xor 1) + 1 for every n, so the carry-free step in the M coordinate is M -> M xor 2M, elementary rule 60 with no boundary term, mirroring to rule 102 in the most significant digit first render (witness: lab/rs/carry-skeleton README THE METHOD, 0 mismatches on all three identities over every n < 2^18).
• 2026-09-14 [Proved] The Collatz carry is a four-state Mealy transducer on the state (n_(i-1), c_i) reading least significant digit first, and its dependence radius is unbounded: with u_0 = 1 and u_j = 1 - (j mod 2) for 1 <= j < level, every majority from c_1 on propagates, so c_i = u_0 for 1 <= i <= level and flipping digit 0 alone changes every digit of 3n + 1 from 2 to level. A MrlyMath automaton is a triple (dim, mask, kind) with a finite offset mask (automata.md section 7), so the step is no rung of that ladder at any level; the dial grades acceptors and not transducers, so what it grades here is the step's zero-carry set and not the step (witness: lab/rs/carry-skeleton README THE CONTROLS, the radius witness at every level = 2..40).
• 2026-09-14 [Proved] For odd m the zero-carry set of m n + 1 in base 2 is exactly the even integers accepted by the width-(deg m + 1) rule W of beneath.md's memory dial forbidding two 1 digits at a distance in the difference set {abs(j - j') : j, j' in supp m}: digit 0 of the sum reads n_0 + 1 and digit i >= 1 reads sum_(j in supp m) n_(i-j), so the set depends on that difference set alone and m = 11 and m = 15 share one rule (witness: lab/rs/carry-skeleton README READS, set equality with 0 mismatches below 2^16 at m = 3, 5, 7, 9, 11, 15).
• 2026-09-14 [Proved] The Collatz carry fires with density exactly 1/2: on uniform independent digits the state (n_(i-1), c_i) is an irreducible aperiodic Markov chain on four states with stationary vector (1/3, 1/6, 1/6, 1/3) on (0,0), (0,1), (1,0), (1,1), whose carry-on mass is 1/6 + 1/3 = 1/2 (witness: lab/rs/carry-skeleton README READS, exact integer balance residuals all 0, and the mean of d_loc over every n < 2^level reading level/2 + 1/3 - (-1)^level/(3 * 2^level) as an exact integer identity at every level = 8..22).
• 2026-09-14 [Proved] The carry-free map T_free(n) = n/2 for even n and (n xor 2n xor 1)/2 for odd n never raises the base-2 digit count, and has exactly one cycle on the positive integers, {1}, which every orbit reaches: an odd n read as f in GF(2)[x] with f(0) = 1 has T_free equal to A(f) = ((1 + x) f + 1)/x, affine with a = (1 + x)/x and a + 1 = 1/x, so A^k(f) + 1 = a^k (f + 1) and A^k(f) = f forces a^k = 1 or f = 1, and (1 + x)^k = x^k fails at every k >= 1 by the constant term. The carry-free drift is therefore exactly zero and all of the growth of a Collatz step is carry (witness: lab/rs/carry-skeleton README READS, 0 digit-count increases and 0 values failing to reach 1 over every n < 2^20).
• 2026-09-14 [Proved] The carry-on block of the Collatz carry's transfer matrix is [[0, 1], [1, 1]], characteristic polynomial x^2 - x - 1, so a carry-on run survives one further digit at rate phi/2, and the worst-case carry run over level digits is level + 1, attained at n = 2^level - 1 (witness: lab/rs/carry-skeleton README READS, worst case pinned at level = 1, 2, 4, 8, 16, 32, 40).
• 2026-09-14 [Verified] The zero-carry rules of m n + 1 carry codes 7, 95, 23, 22015, 279 at m = 3, 5, 7, 9, 11 and widths 2, 3, 3, 4, 4 under mrlynum::memory::Rule::new(1, k, code) with Rule::allowed, with card W of 3, 6, 4, 12, 5, rho of 1.618034, 1.618034, 1.465571, 1.618034, 1.380278 and kappa of 0.098239, 0.167412, 0.115204, 0.201999, 0.115524; m = 15 repeats the m = 11 row (witness: lab/rs/carry-skeleton README READS, every code rebuilt from the difference set and rho, kappa taken by mrlynum::memory::perron and mrlynum::memory::kappa).
• 2026-09-14 [Verified] The golden rule, code 7 at width 2, and the supergolden rule, code 23 at width 3, are the zero-carry rules of 3n + 1 and 7n + 1, and mrlynum::memory::kappa reads 0.098239 and 0.115204 on them, reproducing beneath.md's two published couplings from the Collatz side with no shared code (witness: lab/rs/carry-skeleton README THE CONTROLS, the two values asserted to 5e-7).
• 2026-09-14 [Conjecture] The longest carry run of a Collatz step over level uniform digits grows like log_(2/phi) level, base 2/phi = sqrt 5 - 1 = 1.236068. The mechanism is Proved, the carry-on block being [[0, 1], [1, 1]] with Perron root phi; the constant is not established, sampled mean depths 6.338, 12.116, 18.462, 24.967, 31.481, 38.029 at level = 16, 64, 256, 1024, 4096, 16384 over 200000 uniform strings each, standard error at most 0.014 per mean, giving increments per quadrupling 5.778, 6.346, 6.504, 6.514, 6.548 against the predicted 6.541119, rising towards it from below with the offset settling near -7.76 (witness: lab/rs/carry-skeleton README READS, the depth table, no exponent fitted).
• 2026-09-14 [Refuted] The local carry count d_loc(n) = popcount((3n + 1) xor (n xor 2n xor 1)) is no function of popcount, of the longest run of 1 digits, of v_2, of the digit count, nor of all four read at once: the least clashing pairs are 1, 2 at d_loc 2, 0 for popcount and for the longest run, 1, 3 at 2, 3 for v_2, 2, 3 at 0, 3 for the digit count, 3, 5 at 3, 4 for (popcount, v_2) and 19, 25 at 3, 4 for all four, with 256217518, 417177932, 659301399, 662864636, 88157572, 9331881 unordered pairs below 2^16 sharing the statistic and disagreeing on d_loc; the last row subsumes every pair, so no pair is a summary either (witness: lab/rs/carry-skeleton README READS, the refutation table, six witnesses pinned as integer pairs).