# The question mark

- 2026-09-21 [Proved] Stern-Brocot is a base-2 design with a rank-2 carry: `v(n) = (s(n), s(n+1))` on the Stern diatomic sequence A002487 obeys `v(2n + b) = v(n) M_b`, `M_0 = [[1,1],[0,1]]`, `M_1 = [[1,0],[1,1]]`, so `v(n) = (0,1) M_(d_1) ... M_(d_L)` over the digits of `n`, a linear representation and hence 2-regular. Witness: lab/py/question-mark, verb carry, 0 mismatches below 2^16
- 2026-09-21 [Proved] `M_0 M_1 = [[2,1],[1,1]]` and `M_1 M_0 = [[1,1],[1,2]]`, so `s(5) = 3` and `s(6) = 2` on the words `101` and `110` (least pair without leading zeros; `011`, `101` with them, `s = 2, 3`), and no level `L >= 2` of the Stern sequence is a Kronecker power of a one-digit table, level 2 reading `0, 1, 1, 2`. Witness: lab/py/question-mark, verb carry
- 2026-09-21 [Verified] The Stern-Brocot row at depth `d` is the Calkin-Wilf row `s(n)/s(n+1)`, `2^d <= n < 2^(d+1)`, under bit reversal of the position, depths 0..12. Witness: lab/py/question-mark, verb carry, 0 mismatches
- 2026-09-21 [Verified] On the 8193 nodes of the Farey subtree of [0,1] to depth 12, computed exactly by the mediant recursion: Denjoy's formula, the tent conjugacy `?(F(x)) = T(?(x))`, the branch law `?(1/(a + x)) = 2^(-a)(2 - ?(x))` at `a = 1..5`, and the address law `?(node i of row d) = (2i + 1)/2^(d + 1)`, each with 0 mismatches. Witness: lab/py/question-mark, verb question
- 2026-09-21 [Proved] For finite or cofinite `A`, `?(E_A)` is the set of binary words whose runs after the first lie in `A`, a width-`(max A + 1)` or width-`(max F + 2)` rule of the memory dial; the accepted words outside `?(E_A)` are exactly those whose leading run refuses `A`, and at `A = {1..m}` that surplus is the cylinder `[0^m] = 2^(-m-1)(2 - ?(E_A))`. Witness: lab/py/question-mark, verb question, 0 missing on all 18 codes, 233 surplus words of 1220 at code 126 all opening on 00
- 2026-09-21 [Proved] The Perron root of a run-length rule is the largest real root of `P_A`, `x^m - sum_(a in A) x^(m-a)` at finite `A` and `x^(f+1) - 2x^f + (x-1) sum_(j in F) x^(f-j)` at `A = N \ F`, from the counting series `2(B + B^2/(1 - C))`. Witness: lab/py/question-mark, verb table, `P_A` divides the exact characteristic polynomial on all 18 codes
- 2026-09-21 [Proved] The Hausdorff dimension of the accepted set of a run-length rule is `log_2 rho`: the cover gives the upper bound for every rule, Hutchinson 1981 5.3(1) on the similitudes `y -> 2^(-a)(2 - y)` under the open set condition gives the lower bound at finite `A`, and truncation carries it to cofinite `A`. Holds for every run-length rule at every width; the 18 codes are the census rows at widths 2 to 4. Witness: lab/py/question-mark, verb table, 18 codes on 11 alphabets
- 2026-09-21 [Verified] The pressure zero of `L_(A,s)` at `A = {1,2}` by 40-mode Chebyshev collocation and bisection reads 0.5312805062772050 against Jenkinson and Pollicott's 0.5312805062772051416..., gap 1.1e-16, moving at most 8.9e-16 between 24 and 56 modes; `A = N` reads 1.0000000000000000. Witness: lab/py/question-mark, verb table, controls
- 2026-09-21 [Verified] The table `(k, code, A, rho, log_2 rho, dim_CF)` on the 18 run-length codes at dim 1, widths 2 to 4: `{1,2}` at code 126 reads `0.694241913630` against `0.531280506277`; `N \ {1}` at 219 the same `rho` against `0.840884586414`; `{1,3}` at 32190 `0.551463089745` against `0.454489077661`; `{2,3}` at 21450 `0.405685231375` against `0.337436780806`; `{1,2,3}` at 32766 `0.879146421606` against `0.705660908028`; `N \ {2}` at 64959 `0.811370462751` against `0.929965925781`; `N \ {1,2}` at 53643 `0.551463089745` against `0.785953471982`. Witness: lab/py/question-mark, verb table
- 2026-09-21 [Conjecture] At cofinite `A` the printed pressure zero is `dim_H E_A`; the identification for an infinite alphabet is Mauldin and Urbanski's theorem, not read here. Witness: lab/py/question-mark, verb table, four cofinite rows
- 2026-09-21 [Proved] `dim_H E_A >= alpha log_2 rho` with `alpha = log 2/(2 log phi)` the Holder order of `?`, from the cover argument for Holder maps; at `A = {1,2}` the floor is exactly 1/2. Witness: lab/py/question-mark, verb table, asserted on all 18 rows
- 2026-09-21 [Verified] The run-length codes number 2, 5, 11 at widths 2, 3, 4 among 4, 8, 64 codes fixed by both the digit flip and reversal; the orphans at widths 2 and 3 are codes 0, 9, 129, 165 with `rho <= 1`; at width 4, 19 of 53 orphans carry `rho > 1`, the least being code 11892, alphabet `{1,2}` with the pair `22` forbidden, at the supergolden root. Witness: lab/py/question-mark, verb obstruction
- 2026-09-21 [Proved] The `dim 1` case again by a second route: `K_W` is compact and `x -> 2x mod 1` forward invariant on the circle, the subshift `X_W` has entropy `log rho`, the box dimension is `log_2 rho` in house, and Furstenberg 1967, Proposition III.1, read in the restatements of Kenyon, Peres and Solomyak 2012 Section 1 and Austin 2021 Section 3 with Furstenberg's own text paywalled and unread, gives `dim_H = dim_B`; the statement read lives on the circle and does not reach `dim >= 2`. Witness: notes/beneath.md, `### Every rule is a graph-directed continued fraction set`
- 2026-09-21 [Proved] The Holder floor `dim_H ?^(-1)(K_W) >= alpha log_2 rho`, `alpha = log 2/(2 log phi)`, holds on every `dim 1` rule with `rho >= 1` and not only on the run-length rules, by the dimension identity above and the cover argument for Holder maps. Witness: notes/beneath.md, `### Every rule is a graph-directed continued fraction set`; lab/py/question-mark, verb graph, asserted on all 23 rows printed
- 2026-09-21 [Proved] Every `dim 1` width-`k` rule is a finite graph-directed continued fraction set: states the `2^(k-1)` words of `k - 1` digits, an edge `u -a-> v` when `a` copies of the digit opposite to the last digit of `u` close only allowed windows, the labels `a >= k` a single cofinite edge to `b^(k-1)`, the branches `x -> 1/(a + x)` with disjoint images; `E_u = union of 1/(a + E_v)` over the edges, the run-word preimage `R_W` of the accepted set is a countable union of bi-Lipschitz images of the `E_u`, and `dim_H R_W = max_u dim_H E_u`, the largest dimension of a strongly connected component; the parity of the position is the last digit of the state. Witness: notes/beneath.md, `### Every rule is a graph-directed continued fraction set`; lab/py/question-mark, verb graph
- 2026-09-21 [Verified] The graph form recovers the pressure zero of all 18 run-length codes of the table to a largest gap of `8.9e-16`, the states whose last exact run lies in `A` carrying a copy of `L_(A,s)`. Witness: lab/py/question-mark, verb graph, control
- 2026-09-21 [Verified] Code 11892 at level 14: 378 accepted words, 277 prefixes of `?(M)` for `M = {x : a_i in {1,2}, no pair 22}`, 0 missing, 101 extra opening on the leading runs `0` and `00`; by hand its accepted infinite words are exactly the run words with every run in `{1,2}` and no two consecutive `2`s, so `R_W` is `M` with two bi-Lipschitz pieces of it; its recurrent graph has 6 states, 10 finitely labelled edges and no cofinite edge. Witness: lab/py/question-mark, verb graph
- 2026-09-21 [Verified] Code 11892 reads `rho = 1.465571231876`, `log_2 rho = 0.551463089745`, `dim_CF = 0.416817764433` as the pressure zero of the matrix transfer operator on its graph, above its Holder floor `0.397169256795`. Witness: lab/py/question-mark, verb graph
- 2026-09-21 [Verified] The table `(code, forbids, states, tail, rho, log_2 rho, dim_CF, alpha log_2 rho)` on the 19 width-4 orphans with `rho > 1`, every row above its Holder floor: the 11 rows without a cofinite edge in a cycle lie below 0.58 and the 8 with one above 0.74; the golden `rho` prints six distinct `dim_CF` on its six orphans, `0.531280506277` to `0.887530492554`; `22506`, `48765`, `54699`, `55275` land on the table rows `{2,3}`, `{1,2}`, `N \ {1,2}`, `N \ {1}` to 12 digits, their infinite words being those alphabets plus the alternating or constant words; `60375` is two copies of code 23 and prints its `0.762395011393`; `11892` and `44661`, `14940` and `47709` differ by the two constant words and print one row. Witness: lab/py/question-mark, verb graph, 19 rows
- 2026-09-21 [Conjecture] The pressure zero `dim_CF` of an orphan's graph is the Hausdorff dimension of its continued fraction set `R_W`; the identification for a graph-directed Markov system is the theorem of Mauldin and Urbanski, not read here; on `22506` and `48765` it is the finite-alphabet identification of Jenkinson and Pollicott already read. Witness: lab/py/question-mark, verb graph
- 2026-09-21 [Verified] The four named codes read as parity-constrained continued fraction sets: code 7, even quotients 1, `dim_CF = 0.798858366966`; code 23, even quotients 1 and odd quotients at least 2 past the first, `0.762395011393`; code 54, even quotients 1 and odd quotients 1 or 2 past the first, `0.305702946078`; code 127, even quotients 1 or 2, `0.873619869023`; three continued fraction sets on the golden word count, `0.531280506277`, `0.798858366966`, `0.840884586414`. Witness: lab/py/question-mark, verb graph
- 2026-09-21 [Proved] At every `dim`, base 2, every width-`k` rule with `rho >= 1` has `dim_H K_W = log_2 rho` for its accepted set `K_W`, the intersection over `L` of the closed cells of the accepted words of length `L`, the total dimension in `[0,1]^dim`. Upper bound in house: the closed cells of level `L` meeting `K_W` lie between the extendable accepted words and `3^dim` times them, so `dim_B K_W = log_2 rho`. Lower bound by citation: `K_W = pi(X_W)` by Konig, the dead states pruned, one unit cube per live state spaced apart with the similarity `x -> c_u + (x - c_v + d)/2` of ratio `1/2` on each edge is a geometric graph directed construction, one edge per ordered pair at `k >= 2`, the images from one state among the `2^dim` nonoverlapping subcubes, loops split into two copies joined both ways, the theorem applied per undirected component; Mauldin and Williams 1988, Theorem 4, carried from the restatement of Abram and Lagarias 2012 Section 2.1 read at source, puts the dimension of the construction object at the largest `beta` with `2^(-beta) rho_H = 1` over the components with a cycle, `log_2 rho`, and `K_W` is a finite union of similar copies of the construction object. Retags the Conjecture of `### The transfer matrix` to Proved at every `dim`; at `rho = 0` the accepted set is empty. Witness: notes/beneath.md, `### Every rule is a graph-directed continued fraction set`
- 2026-09-21 [Proved] At cofinite `A` the printed pressure zero `dim_CF` is `dim_H E_A`: Mauldin and Urbanski 1996 Theorem 3.15, read in the restatement of Roy, Sumi and Urbanski 2008 Theorem 2.3, its hypotheses checked in house on the Gauss branches: open set condition on the disjoint intervals `(1/(a+1), 1/a)`, Mobius extension to `W = (-1/4, 3/2)` with the compact `S = [-1/8, 5/4]` forward invariant, bounded distortion `L = 9`, `alpha = 1` over every label, the cone condition trivial on intervals, contraction restored in the coordinate `y = log(1 + x)` where every branch contracts by at most `1/2` with distortion `L = 2`, `alpha = 1`, and the pressure equal to `log lambda(t)` by the sandwich `q_w^(-2)/4 <= abs(phi_w') <= q_w^(-2)`; supersedes the Conjecture row on the four cofinite rows. Witness: notes/beneath.md, `### The pressure zero is the dimension`; lab/py/question-mark, verb table
- 2026-09-21 [Proved] The pressure zero `dim_CF` of an orphan's graph, and of the four named codes, is the Hausdorff dimension of each strongly connected component's limit set and so of `R_W`: Mauldin and Urbanski 2003 Theorem 4.2.13, read in the restatement of Chousionis, Leykekhman, Urbanski and Wendt 2024 Theorem 2.7 with Definitions 2.1, 2.3 and 2.5, the hypotheses checked in house on the seeds `3u + [0, 1]` with the edge maps `x -> 3u + 1/(a + x - 3v)`, finitely irreducible on a strongly connected finite graph; supersedes the Conjecture row on the orphans. Witness: notes/beneath.md, `### The pressure zero is the dimension`; lab/py/question-mark, verb graph
- 2026-09-21 [Proved] `dim_H E_A > 1/2` for every `A` with `sum_(a in A) 1/a = infinity`, every cofinite `A` among them: `P(t)` is finite for `t > 1/2`, `P(t) >= log sum_(a in A) (a + 1)^(-2t)` rises to infinity as `t` falls to `1/2`, and `P(1) <= 0` since `Z_n(1) <= 4`, so the zero lies in `(1/2, 1]`; the divergence of `sum 1/a` is the hypothesis and not the infinitude of `A`, `A = {4^k}` having `P(1/4) <= 0` and `dim_H E_A <= 1/4` by the cylinder cover. Witness: notes/beneath.md, `### The pressure zero is the dimension`
- 2026-09-21 [Proved] The upper bound `dim_H <= inf{t : P(t) <= 0}` in house on every graph: the level-`n` cylinders, of diameter at most `q_w^(-2)`, cover the limit set with `sum diam^t <= Z_n(t)`, which tends to `0` when `P(t) < 0`, and `P(t + e) <= P(t) - 2 e log phi` by the Fibonacci growth of the continuants; the citation carries only the lower bound. Witness: notes/beneath.md, `### The pressure zero is the dimension`