dimensions.md

--- title: Complex dimensions lead: Complex dimensions: every design in the lattice class, the carpet proved not Minkowski measurable with its explicit profile, and the arithmetic pole at s_0 + 2 pi i/log 3 certified genuine. figure: research-dimensions slug: dimensions ---

The core page gives every design one real number, the dimension log(fill)/log(base). Fractal-string theory promotes that number to the real part of an infinite family - the complex dimensions, the poles of a zeta function attached to the set's gaps. This page computes them for the 1D designs, watches the imaginary parts surface as an oscillation in the box count, and follows the theory to its structural consequence, which needs careful qualifying.

Every claim carries a tag. Proved means a proof is given or restated here; Verified means recomputed from scratch by a lab study; Conjecture means neither. lab/py/complex-dimensions regenerates every number below except those of the arithmetic-pole section, which lab/py/burnol-residue prints; both print only and keep no log.

The string of a design

A 1D design is a subset F of its base's digits, the filled ones; it draws the set of x in [0,1] whose digits all lie in F. That set is the attractor of fill = |F| maps x -> (x + f)/base, every one with the same contraction ratio 1/base, and its dimension is log(fill)/log(base). The core page's parity rule at base 3 fills digits {0,2} - the middle-thirds Cantor set, log(2)/log(3) = 0.630930.

The complement of the set is a multiset of gaps, and the gap multiset of a one-base design satisfies G = G_1 + fill copies of G/base. So its geometric zeta function - the sum of g^s over all gaps - has the closed form

zeta(s) = Z(s) / (1 - fill*base^(-s)), Z(s) = sum of g^s over level-1 gaps,

and its poles sit where the complex Moran equation fill*base^(-s) = 1 holds:

s = dimension + 2*pi*i*m/ln(base), m in Z.

One vertical line of poles, equally spaced at omega = 2*pi/ln(base). Sets whose complex dimensions line up on such an arithmetic progression are called lattice; sets with incommensurable ratios, whose poles spread out, are nonlattice. Every one-base design is lattice for the trivial reason that all its ratios are equal.

Verified (lab/py/complex-dimensions). The progression is derived in closed form above and then evaluated: for the Cantor design and three others (table below), all 81 predicted poles at m = -40..40 kill the denominator to 5e-14, and the numerator Z(s) stays bounded away from zero at every one of them - minimum |Z(s)| = 0.500000 for the Cantor design - so no zero is cancelled and every one is a genuine pole. The identity Z(1) = 1 - fill/base, the statement that the gaps of a measure-zero set fill the whole interval, holds for all four.

One caveat: the pole set is the zero set of the denominator only where the numerator does not vanish, and a design with no gaps (fill = base), with one filled digit (fill = 1, a point), or with none has an empty gap multiset, an identically zero zeta function, and no complex dimensions at all.

The oscillation in the box count

The imaginary parts are not bookkeeping; they are visible. Let N(eps) be the number of eps-cells the set meets and detrend it: g(u) = ln N(exp(-u)) - dimension*u. A lattice set's g oscillates at angular frequency omega = 2*pi/ln(base) forever.

Verified (lab/py/complex-dimensions). For all four objects the periodogram peak of g(u) lands within 1% of the predicted 2*pi/ln(base); for the Cantor design the Blackman periodogram on the direct DFT grid reads 5.7024 against the predicted 5.719202. Four-decimal peak values sit inside one frequency bin of the estimator's resolution and are properties of the estimator, not of the object; what is stable, and what ships, is the sub-1% agreement.

A second reading folds u modulo each candidate period and asks how much of the variance of g the folded profile explains. At 40 bins on a fixed window:

Verified (lab/py/complex-dimensions). Each lattice object folds best at its own ln(base), and the two controls fold well at none. But the two lattice signatures are not parallel in strength: the base-15 design's folding explains 66.7% of the variance at its period, the Cantor design's only 19.2% at its own. The Cantor figure is genuinely modest - the box count is a step function and grid alignment injects a large aperiodic component - and it moves between roughly 0.16 and 0.23 as the bin count and window vary (Conjecture; the study prints the 40-bin value only), so it is also partly a property of the estimator. The ordering is stable; the percentages are not constants of the objects.

The arithmetic pole, certified

The string above lives in [0,1]; its arithmetic twin is the set S of positive integers whose base-3 digits all lie in {0,1} (1, 3, 4, 9, 10, 12, 13, ..., the mobius page's S_F at base 3), with counting function A(x) and Dirichlet series K(s) = sum n^(-s), abscissa s_0 = log_3 2 = 0.630930. Burnol 2026 continues K meromorphically to C with simple poles among the same lattice s_(m,k) = s_0 - m + 2 pi i k/log 3 (Proposition 4.1), proves the real pole s_0 genuine with positive residue, gives every off-real residue as a limit of level sums, lambda_(0,k) = (log 3)^(-1) lim_j sum_(3^j <= n < 3^(j+1), n in S) n^(-s_(0,k)) (Proposition 5.1), and shows that a vanishing lambda_(0,k) empties the whole column s_(m,k), m >= 1 (Proposition 7.1); the paper states that it does not study the off-real residues further and prints none. Whether a single off-real pole is genuine is therefore open at source, and the mobius page accordingly says among. This section closes it for k = 1..10 by a certificate.

Proved. Write G(u) = A(3^u)/2^u. Splitting an element n = 3n' + a, a in {0,1}, gives A(3x) = 2A(x) + 1 - delta(x) for x >= 1, where delta(x) = A(x) - A(x - 1/3) is 0 or 1; so G(u+1) - G(u) = (1 - delta(3^u))/2^(u+1) lies in [0, 2^(-u-1)], G(u+j) increases in j, and Phi(u) = lim_j G(u+j) exists, is 1-periodic, satisfies 0 <= Phi(u) - G(u) <= 2^(-u), and is continuous: at any point the oscillation of Phi is at most the jump of G(u+j) there plus twice the uniform distance, 3 * 2^(-u-j) for every j. Hence A(x) = x^(s_0) Phi(log_3 x) + E(x) with |E(x)| <= 1. The profile is explicit on a third of its period: no element of S lies strictly between 11...1 = (3^(level+1) - 1)/2 and 100...0 = 3^(level+1), so A = 2^(level+1) - 1 on [(3^(level+1) - 1)/2, 3^(level+1)) and Phi(u) = 2^(1-u) for u in [1 - s_0, 1], from Phi(1 - s_0) = 2^(s_0) = 1.548562 down to Phi(1) = Phi(0) = 1. The profile is not constant, and that much is elementary.

Proved. For Re s > s_0, K(s) = s int_1^inf A(x) x^(-s-1) dx. Substituting the profile and x = 3^u, the Phi part is s log 3 F(s)/(1 - 3^(s_0 - s)) with F(s) = int_0^1 3^(u(s_0 - s)) Phi(u) du entire, and the E part is holomorphic on Re s > 0. So on Re s > 0 the poles of K are among the zeros s_(0,k) of 1 - 3^(s_0 - s), all simple with derivative log 3 there, and Res_(s_(0,k)) K = s_(0,k) c_k with c_k = int_0^1 Phi(u) e^(-2 pi i k u) du. The pole at s_(0,k) is genuine exactly when the k-th Fourier coefficient of the profile is nonzero; non-constancy says only that some c_k is nonzero and nothing about k = 1.

Proved (computer-assisted, lab/py/burnol-residue). Let Lev_j(w) = sum n^(-w) over the 2^j elements of S in [3^j, 3^(j+1)); Burnol's Proposition 5.1 reads lambda_(0,k) log 3 = lim_j Lev_j(s_(0,k)). Three elementary lemmas make the limit computable with a bound. (1) Splitting n = 3n' + a and expanding (1 + a/(3n'))^(-w) = sum_l (-1)^l ((w)_l/l!) (a/(3n'))^l, absolutely convergent since a/(3n') <= 1/3, then summing over the finitely many n' of level j: Lev_(j+1)(w) = 3^(-w) sum_(l >= 0) (-1)^l ((w)_l/l!) 3^(-l) gamma_l Lev_j(w + l) with gamma_0 = 2 and gamma_l = 1 for l >= 1. (2) |Lev_j(w)| <= 2^j 3^(-j Re w) for Re w >= 0: 2^j terms, each of modulus n^(-Re w) <= 3^(-j Re w). (3) At w = s_(0,k) + i, cutting the l-sum at T costs at most 2 * 3^(-ji) sum_(l > T) ((|w|)_l/l!) 3^(-l(j+1)): termwise |(w)_l| <= (|w|)_l, gamma_l <= 2 (slack by a factor 2, kept for the general engine), and (2) at w + l with 2^j 3^(-j s_0) = 1; the full sum is the binomial series (1 - 3^(-j-1))^(-|w|), increasing in |w| so an upper bound of |w| may stand in, and the cut is that closed form minus the interval partial sum; the same bound at i = 0, cut by (1 - x)^(-a) - 1 <= a x (1 - x)^(-a-1) for 0 <= x < 1 and a >= 0 and summed over j >= level, gives the level tail E_level = (3/2) |s| (1 - 3^(-level-1))^(-|s|-1) 3^(-level-1). The generator runs (1) on the vector of shifts i = 0..I from Lev_0 = 1 to level level = 40 in mpmath.iv at 128 bits, 3^(-s_(0,k)) = 1/2 being exact so that s_(0,k) and its Pochhammer symbols are the only transcendental inputs, adds the box of (3) at every step and E_level at the end, and prints endpoints floored and ceiled from exact fractions. At k = 1, I = 66: lambda_(0,1) lies in [0.231891517689918, 0.231891517689919] + i [-0.501067414481069, -0.501067414481068], width 3.4e-18, level tail 2.4e-19, at distance at least 0.552125193 from zero. So the pole of K at s_(0,1) = 0.630930 + 5.719202 i is genuine, c_1 is nonzero, and the counting function of S carries the frequency 2 pi/log 3 at the fundamental with a certified nonzero amplitude. The claim rests on Burnol's Proposition 5.1, the three lemmas above, and the outward rounding of mpmath's interval arithmetic; every bound used is proved.

Proved (computer-assisted, lab/py/burnol-residue --band). The same certificate at k = 0..10, level = 40, I from 52 to 164, widths from 2e-18 to 4e-13, the table printed by the generator:

The k = 0 row is Burnol's positive real residue recovered, its imaginary interval the enclosure's own outward rounding and not a reading, since s_(0,0) is real and the residue's imaginary part is exactly 0. By Proposition 7.3 of the same paper, under its hypothesis 1 < N < base (here N = 2 at base 3, so the Pochhammer prefactors relating his mu_(m,k) to lambda_(m,k) never vanish), a nonzero lambda_(0,k) makes s_(m,k) a pole exactly when s_(m,0) is one, so each certified k settles its whole column against the real axis.

Verified (lab/py/burnol-residue). Two independent floating-point computations land on the certificate. Direct enumeration of Lev_20(s_(0,1)) over its 2^20 terms gives 0.231891518 - 0.501067414 i after division by log 3, at distance 1.9e-10 from the enclosure against its own bound 7.5e-10, the successive level differences 1.13e-08, 3.77e-09, 1.26e-09, 4.19e-10 contracting by 1/3. The counting side, from exact counts A(x) on 2^17 midpoints of one period at J = 30, gives c_1 = -0.082138842 - 0.049607499 i and s_(0,1) c_1 = 0.231891457 - 0.501067454 i, at distance 7.2e-08 (1.0e-06 on 2^15 midpoints: the quadrature error of a rough profile, not a certified bound); the sampled profile runs over [1.000003, 1.548557] and matches 2^(1-u) on [1 - s_0, 1] to 7.2e-10. A second certified enclosure by the functional-equation route, R_1 = 1 + sum_(m >= 1) (-1)^m ((s)_m/m!) 2^(-1) 3^(-m) K(s + m), each K(s+m) summed in intervals to level 13 plus its tail 3^(-13m)/(1 - 3^(-m)) and the m-series cut at 36 plus its tail, gives [0.23189098, 0.23189280] + i [-0.50106814, -0.50106632], width 1.8e-06, meeting the first. The engine is tested on two designs with known answers. On {0,2}, whose elements are twice those of {0,1}, the enclosure [0.135292875345483, 0.135292875345484] + i [0.329874091244466, 0.329874091244467] agrees with 2^(-s_(0,1)) times the {0,1} enclosure to every printed digit. On the full digit set {0,1,2}, where K is the Riemann zeta function, k = 0 returns [0.999999999999999, 1.000000000000001] around the residue 1 and k = 1 returns a box of width 1.7e-18 around 0, the regular point it must be.

Verified (lab/py/burnol-residue). Burnol's Proposition 7.1 recurrence carried down the k = 1 column from the certified lambda_(0,1): lambda_(1,1) in [0.6950303416, 0.6950303417] + i [0.3777908610, 0.3777908611], lambda_(2,1) in [-0.1945129694, -0.1945129693] + i [0.2161122817, 0.2161122818], lambda_(3,1) in [0.0645979040, 0.0645979041] + i [0.1353703406, 0.1353703407], all nonzero. The first two meet the closed forms Theorem 7.4 forces, lambda_(m,k) = (-1)^m (s - 1)...(s - m) lambda_(0,k) [t^m] (1/E(t)) at s = s_(0,k), E the moment generating function of the Cantor measure of {0,1} with moments 1/4 and 3/32, so [t^1] = -1/4 and [t^2] = 1/64: lambda_(1,1) = (s - 1) lambda_(0,1)/4 and lambda_(2,1) = (s - 1)(s - 2) lambda_(0,1)/64.

What is Burnol's and what is new. The continuation, the lattice, the positivity at s_0, the limit formula and the column proportionality are Burnol 2026, cited; that A(x) x^(-s_0) oscillates is elementary (A(3^level) = 2^level against A((3^(level+1) - 1)/2) = 2^(level+1) - 1, the profile lemma) and nothing here discovers it. New is the certificate at k = 1..10, the k = 0 row recovering Burnol's theorem: the pole at s_(0,1), and at every s_(0,k) for k = 1..10, is genuine, the residue enclosed to 4e-18 at k = 1 and 4e-13 at k = 10, which is what a complex-dimension claim on the arithmetic side needs. The engine takes any base and digit set F containing 0; only base 3 is certified here, and the zeta check and the {0,2} scaling are its tests. No statement is made about k > 10, about other bases, or about the real-axis residues lambda_(m,0) beyond m = 0.

Composition multiplies the base

Compose two rules by alternating bases across levels: base 3 with digits {0,2} at odd levels, base 5 with digits {0,4} at even ones.

Verified (lab/py/complex-dimensions). The alternation produces exactly the one-base design at base 15 with digits {5*d1 + d2} = {0,4,10,14} - checked as integer arithmetic and then as geometry, eight alternating levels and four base-15 levels producing the identical 256 intervals as exact fractions. So the composite is a different lattice period, omega = 2*pi/ln(15) = 2.320188, not a departure from the lattice class - and the table above shows its signature is the sharpest of the family, 66.7% of variance at its own period.

Two qualifications, both load-bearing. First, base 5 with digits {0,4} is not a mrly design: the core page's move one fills by parity, and the even digits of base 5 are {0,2,4}, three of them - nor is the base-15 composite a parity design. The parity-faithful versions - base 5 {0,2,4} and their composite base 15 {0,2,4,10,12,14} - run alongside and behave identically, which is the real point: the argument turns on the base, not on which digits the rule picks.

Second, the structural claim that one contraction ratio per level implies lattice is false without a periodicity hypothesis. (Refuted.) An aperiodic control that alternates bases 3 and 5 on a Thue-Morse schedule uses exactly one ratio per level and is not self-similar at all, so it is neither lattice nor nonlattice: its best folding is 0.115 against the composite's 0.667, and its periodogram peak matches no 2*pi/ln(base). What survives, and needs no computation, is the statement for mrly designs proper: move two is a Kronecker power of one tile, so the schedule is constant, every level subdivides by the same base, and periodic cross-base alternation multiplies into one product base. Neither move can express two ratios inside one level - move one only chooses which cells of a fixed base^dim grid survive, and every cell of that grid is the same size. (Verified for the alternation; the one-tile argument is read off the definition.)

"Read off the definition" has a proof behind it, and it covers more than one tile. Block reduction, Proved: for any designs c_1, ..., c_p and any level >= 1, the periodic word (c_1, ..., c_p)^level equals (A_(c_1) (x) ... (x) A_(c_p))^((x) level), by associativity of the Kronecker product and nothing else. (The case p = 2 is the base-15 alternation Verified above; a check on six test cases at periods 2 and 3, lengths to 6, matching the flat word cell for cell against the self-Kronecker power of the composite, is Verified, lab/rs/magic-words.) So every periodic schedule is the ordinary self-similar theory of its one-period composite tile, of base prod_i base_i and fill prod_i fill_i; the base-15 composite above is the case p = 2, and its agreement was never in doubt. The corollary is the sharper half. The first genuinely non-stationary behaviour requires an aperiodic word, which is exactly why the Thue-Morse control above is neither lattice nor nonlattice rather than being a third kind of composition. For an aperiodic word at one common base with fill_i = fill(c_i), the scale dimension is lim_level (sum_(i <= level) log fill_i) / (level log base) when the limit exists; characterizing which words make it exist, and which leave the dimension fluctuating, is open. (Conjecture, untouched.)

The genuine way out is a two-ratio system: maps of ratio 1/3 and 1/5 mixed within one level, outside the mrly family, with dimension 0.518370 solving 3^(-s) + 5^(-s) = 1. Verified (lab/py/complex-dimensions). Its 21 complex dimensions in the box Re in [-3,3], Im in [-40,40] - a complete list, by the argument principle: the winding number over the box is 21 and 21 roots are found - have real parts spread from -0.699926 to 0.518370 and fit no arithmetic progression, the worst offset being 0.43, 0.17 and 0.38 of a step for the three candidate spacings. Nonlattice is a real, different behaviour, and no mrly design or composition exhibits it.

Measurability, with its hypotheses

A set is Minkowski measurable when M(eps) = eps^(dimension-1) * V(eps) - V the inner tube, the length of the set's eps-neighbourhood inside the gaps - has a limit as eps -> 0. The lattice/nonlattice split decides this, and what is known is narrower than it looks.

Proved. The Cantor design {0,2} at base 3 is not Minkowski measurable. Splitting the tube sum at the scale of eps gives the exact limit profile

M -> 2^(1-dimension) * (t^(dimension-1) + t^dimension), t in [1/3, 1),

one fixed profile traversed each time eps is divided by 3, with minimum 2.494975716 at t = (1-dimension)/dimension = 0.584963 and maximum 2.583040469 at the ends - a swing of 3.53%, so the profile is not constant and the limit does not exist. The measured tube matches the closed form to 4.9e-9 at the minimum (lab/py/complex-dimensions).

Verified (lab/py/complex-dimensions). The other three lattice objects behave the same way: the swing of M(eps) over successive windows is flat from u = 15 out to u = 60 (eps = 8.8e-27), and each object satisfies M(eps) = M(eps/base) at its own base to 1e-9 or better and at neither other candidate. The two-ratio control does the opposite: its swing decays monotonically 3.79% to 0.42% and is still falling - converging, as the nonlattice side predicts.

The literature, read rather than recalled, is not symmetric. Nonlattice self-similar sets under the open set condition are Minkowski measurable in every dimension (Gatzouras 2000). Lattice sets are not - but as a theorem only on the line, for a nontrivial set of non-integer dimension (Falconer 1995, completed by Kombrink and Winter 2020; for self-similar strings, Lapidus and van Frankenhuijsen 2006 - that attribution rests on secondary citations, the book itself being unopened for this page). In dimension 2 and above the lattice direction is an open conjecture of Lapidus, proved under a pluriphase hypothesis and for particular families, open in general.

So the claim that every mrly design is not Minkowski measurable ships only with two qualifications, and both bite on real designs.

1. Nontrivial fill and non-integer dimension. The theorem's own hypotheses exclude integer dimension, and the exclusion is not exotic: bang dim 2, code 3, the core page's pin(y), fills 2 of 4 at base 2, has dimension log(2)/log(2) = 1 exactly, draws a segment - and a segment is Minkowski measurable. The solid, single-point and empty designs have no gaps and no oscillation to have. At base 3 in 1D the only design of non-integer dimension is {0,2} itself. (Proved: the counterexample and the census of which designs the statement covers are read off the definition.)
2. Dimension one, and the carpet. For 1D designs with 2 <= fill < base and non-integer dimension, non-measurability is a theorem, and for the Cantor design it is proved outright above. For the carpet it is a theorem too: proved below with its explicit profile, and a corollary of the pluriphase theorem of Kombrink, Pearse and Winter 2016. For the sponge, lattice membership holds and non-measurability stays a Conjecture, here as in the literature: the pluriphase theorem does not reach it with the open cube as its open set, because the level-1 hole is the plus of seven cubes whose six outer windows and whose arm walls, carpets with windows of their own, are not in the sponge - the window centre (1/2, 1/2, 1) lies on the hole's boundary at distance 1/6 from the sponge and the wall-window centre (2/3, 1/2, 5/6) at distance 1/18 - so the tube inside the hole is not the parallel volume of the hole, and Lapidus, Pearse and Winter 2011 name the Menger generator as neither convex nor pluriphase.

The carpet, proved, and the class it opens

Proved. The Sierpinski carpet F, base 3 with the eight digit pairs other than (1,1), of dimension log(8)/log(3) = 1.892789 at dim 2, is not Minkowski measurable, and its tube has an explicit limit profile. Three facts about the holes make the tube a sum. First, the carpet contains the boundary of the unit square: a coordinate equal to 0 or 1 reads 0.000... or 0.222... in base 3, whose digit pairs never read (1,1); so the carpet contains the image of that boundary under every word of maps, the boundary of every retained square at every level. Second, the complement of the carpet in the open unit square is the disjoint union over levels m >= 1 of 8^(m-1) open squares of side 3^(-m), the middle squares of the retained level-(m-1) squares: a point outside the carpet leaves the retained squares at a first level m; it lies in a retained level-(m-1) square Q but in none of the eight retained subsquares of Q, and since bd Q lies in those eight subsquares, it lies in the open middle of Q, whose boundary consists of edges of the eight surrounding retained squares, hence lies in the carpet by the first fact. Third, for a point x of a hole H, dist(x, F) = dist(x, boundary of H): the boundary lies in F, and every point of F lies outside H, so the segment to it crosses the boundary first. The inner parallel area of an open square of side s is h(s, eps) = 4 eps s - 4 eps^2 for 2 eps < s and s^2 otherwise, so the tube inside the unit square is exactly V(eps) = sum_(m >= 1) 8^(m-1) h(3^(-m), eps), and the part of the eps-neighbourhood outside the unit square is the collar 4 eps + pi eps^2, which the factor eps^(dimension-2) sends to 0 because the dimension exceeds 1. Write eps = 3^(-j) t with t in [1/3, 1). The level-m hole is fully covered when 3^(-m) <= 2 eps, which reads m >= j + 1 for t < 1/2 and m >= j for t >= 1/2; the two geometric sums give, up to terms of size (3/8)^j and 8^(-j),

M(eps) = eps^(dimension-2) V(eps) -> G(t) = t^(dimension-2) (1 + 4t/5 - 4t^2/7)        for t in [1/3, 1/2),
                                          = t^(dimension-2) (9/8 + 3t/10 - t^2/14)     for t in [1/2, 1).

G is continuous at t = 1/2, where both branches give (44/35) 2^(2-dimension), is C^1 there, and takes the rational value 379/280 at both ends t = 1/3 and t = 1. Its stationary points are roots of quadratics: the maximum 1.355617083 at t = 0.429638, the minimum 1.350670209 at t = 0.692137, a swing of 0.3662%, a tenth of the Cantor design's 3.53%. A swing above zero is the whole proof: G is not constant, so M(eps) has no limit. The non-constancy needs no digits either: on [1/3, 1/2) the function t^(2-dimension) G(t) is a polynomial, and a polynomial equal to c t^(2-dimension) on an interval with 2 - dimension not an integer forces c = 0, while G > 0. lab/py/complex-dimensions/carpet_tube.py checks the hole decomposition cell by cell to level 5, the level sum against the closed form as exact rationals at 96 values of eps, the profile against the level sum at eps = e^(-u), u in [50, 60], and a level-6 distance transform, which uses no hole lemma, against the closed form at eps = 21/729 as equal rationals.

Verified against the source. The statement is a corollary of Kombrink, Pearse and Winter 2016, Theorem 1.1(ii), restated as Theorem 3.4: the attractor of a lattice self-similar system under the open set condition, with non-integer Minkowski dimension, is not Minkowski measurable when some strong feasible open set O satisfies the projection condition and the attractor is pluriphase with respect to Gamma(O), the set O minus the images S_i(O), meaning that the area of F_eps inside Gamma is piecewise polynomial in eps (Definition 2.9). For the carpet with O the open unit square: O is strong feasible because (1/2, 1/4), digits 0.111... and 0.0202..., lies in F and in O; the projection condition holds because every point of a closed retained subsquare has all its nearest carpet points inside that subsquare, the segment to any other point of F crossing the subsquare's boundary first, so S_i O lies in the closure of pi_F^(-1)(S_i F); Gamma is the closed middle square together with the four grid segments, the area of F_eps inside it is 4 eps/3 - 4 eps^2 on (0, 1/6] and 1/9 beyond, one polynomial piece; and the paper's Figure 1 caption says that with this O the carpet F is monophase with respect to Gamma and that bd O lies in F, without spelling the conclusion out; the monophase case under that compatibility is Lapidus, Pearse and Winter 2013. The theorem is the literature's; what is added here is the explicit profile, its extrema and the one-paragraph proof.

Proved. The same argument covers a class. Let a one-base design at base base >= 3 in dimension dim >= 2 remove at least one digit vector, every removed vector having all coordinates in {1, ..., base-2} and any two differing by at least 2 in some coordinate. Then every hole is an isolated open cube surrounded by retained cells, its boundary lies in the set, the retained count fill lies strictly between base^(dim-1) and base^dim, so log(fill)/log(base) is never an integer, and with h(eps) = 1 - (1 - 2 eps)^dim for 2 eps < 1 and 1 otherwise the profile is G(t) = t^(dimension-dim) sum_(j in Z) fill^(j-1) base^(-j dim) h(t base^j), t in [1/base, 1), both tails geometric because fill base^(1-dim) > 1 and fill base^(-dim) < 1. G > 0, every term being nonnegative and the covered levels contributing the constant 1/(base^dim - fill); on [1/base, 1/2) the function t^(dim-dimension) G(t) is a polynomial and dim - dimension is not an integer, so G is not constant and the design is not Minkowski measurable. The parity carpets, the all-odd class removed at every odd base and in every dimension, are in this class; the carpet is base 3, dim 2, where the sum is the two branches above. A design whose removed cells form a plus, the sponge's hole shape, leaves the class twice: with the removed cells touching the boundary of the unit cube the hole's boundary is not in the set, and even for an interior plus the inner parallel volume of the hole stops being polynomial at the first radius where the arcs about its reentrant corners meet, so the closed form stops there and what remains is the characterisation of Kombrink, Pearse and Winter 2016, Corollary 3.2, measurability if and only if their periodic function is constant. The tube demo fattens a design by a radius and reads the inner tube back, where the Minkowski reading never settles but circles one log-periodic profile, exact in closed form wherever the holes are isolated squares.

The door this shuts, and what would open it

Every design has an exact geometric zeta function: for fill pieces at base base, zeta(s) = 1/(1 - fill*base^(-s)), whose poles are the complex dimensions s = log_base(fill) + 2*pi*i*m/ln(base) this page already tabulates. What that bookkeeping meets is a thirty-year-old theorem. (ISP) at a dimension a asks: if a fractal string of dimension a has spectral counting function N(x) = W(x) - C*x^a + o(x^a) with C nonzero, must the string be Minkowski measurable? Lapidus and Maier 1995: (ISP) at a holds for all strings of that dimension if and only if zeta has no zeros on the line Re(s) = a. So (ISP) at every a in (0,1) except a = 1/2 is equivalent to RH, and (ISP) at 1/2 is false outright, the midfractal case being the obstruction. That is RH stated entirely in the language of fractal geometry, and it is the highest-adjacency RH equivalence this tree touches.

And it is vacuous here. Every one-base design is lattice - proved in the sections above, not restated - so the complex dimensions sit periodically on one vertical line and the Lapidus-Maier machinery has nothing to say about the degenerate case. Measurability is not out of reach here: it is trivially settled and therefore empty.

Scope guard, the same one this page already applies: the lattice/nonlattice dichotomy is exact for self-similar STRINGS and settles dimension one. A one-base carpet or sponge is certainly lattice, but lattice membership alone does not prove higher-dimensional non-measurability. State the dimension and the object class every time.

What would give it content: several incommensurable scaling ratios. Drop the single base and allow pieces scaled by r_1, ..., r_N with ln(r_i)/ln(r_j) irrational for some pair - a Moran construction, or a graph-directed self-similar set. Complex dimensions become quasiperiodic instead of periodic, measurability becomes a real question, and (ISP) acquires content. The two-ratio system verified above is the smallest instance of exactly this. "Several bases" is not automatically non-lattice: the contraction system and its separation hypotheses have to be specified before any of the above applies. This is independently the single most valuable generalization available to the tree, arrived at from two directions - the measurability question and the RH map both end on the same instruction.

Staircase schedules, the cheapest non-stationary object

• Instead of a constant word, stack carpet_3, then magic(3,5), then magic(3,5,7), and so on; Kronecker associativity flattens that to the staircase word 3 | 3,5 | 3,5,7 | 3,5,7,9 | ....
• Letter base_j occurs n - j + 1 times in the first n blocks, so the controls are immediate and non-negotiable: side = prod_j base_j^(n-j+1), fill = prod_j fill_j^(n-j+1), and dimension_n = Sum_j (n-j+1)*ln(fill_j) / Sum_j (n-j+1)*ln(base_j).
• The staircase word is aperiodic and not eventually periodic, so block reduction does not apply to it. It is the cheapest concrete non-stationary schedule available.
• The weights (n-j+1) are a Cesaro profile - the earliest letter carries weight n, the newest carries 1 - so if the letter dimensions converge the whole dimension converges to their limit; the interesting regime is letter dimensions that oscillate.
• The generalisation is what makes this a programme rather than an example: a staircase is one weight profile, any letter-multiplicity schedule is another, and the question "which dimension functions are realisable by a schedule and which are not" quantifies over all schedules and is native to the construction.
• Caveat that travels with every mixed number. If each factor is rendered at its native base, side_i = base_i, the filled points have the mixed-radix form x = a_1*(base_2...base_level) + ... + a_level with a_i in F_i, and that is the correct arithmetic object. A factor rendered at a side unrelated to its residue base is still a valid tile product, but it is not a mixed-radix digit construction and inherits no digit theorem for free.
• lab/py/slice-ladder-controls prints the five staircase dimensions - 1.892789261, 1.892315261, 1.893034267, 1.894190425, 1.895495742 at n = 1..5 - assuming the carpet at one base has fill base^2 - ((base-1)/2)^2; the run states that assumption before any number. (Verified under that assumption; confirm the definition against the core page before quoting any number from it.)

Where the numbers live

lab/py/complex-dimensions is the one pass behind every number on this page - the poles, the box-count periodogram and folding, the composition, the two-ratio control and the tube - and it prints only. lab/py/slice-ladder-controls prints the staircase dimensions, and lab/py/burnol-residue the arithmetic-pole section: the certified residues, the band, the controls and the column. The dimension formula this page extends, and the designs it names, are the core page; the spectral side of the same fractals is the complexity page.