# 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).