The evidence

A skeptic-facing summary of what in this project is actually checked. It is written for someone deciding whether the rest of these pages are worth reading, so the caveats sit in the body rather than in the footnotes, and the results that came out weaker under recomputation than the research notes claimed are on the page rather than off it.

Every claim carries a tag. Proved means proved here or classical; Verified means recomputed from scratch for this page and reported as measured; Conjecture means supported and open. Nothing is quoted that was not re-run. The four scripts behind the page are in lab/evidence/, and the protocols, parameters and failure modes are written out in that directory's RUN.txt.

The object, in one paragraph

A design fills some corners of the parity cube {0,1}^d; a cell of an n^d grid is filled when its vector of coordinate parities is one of them; the fractal is that tile substituted into itself by the Kronecker product. The Sierpinski carpet, the Menger sponge and their siblings are single choices in a finite universe — 16 designs in 2D, 256 in 3D, 2^(2^d) in dimension d (Proved, counting subsets of the parity cube) — so the shapes are enumerated rather than invented. The core sets this up and the bijection proves the count of distinct ones is the count of Boolean functions up to cube symmetry. Everything below is a consequence of that one setup.

The fill-ratio law

Fix a design filling k of the 2^d corners. At side n the filled count is a sum over those k corners of a product over coordinates, each factor being the number of even or of odd residues in 1..n; dividing by n^d turns every factor into a fraction that tends to one half, so

filled fraction  ->  k / 2^d

The limit sees nothing about the shape except how many corners it fills. (Proved, one line; Verified — brute force over the whole n^d grid agrees with the product formula for all four families at d <= 3 and n = 3, 5, 7, 9, and the generalised rules reproduce the library's own designs: codes mrly_07, mrly_03, mrly_06 in 2D and mrly_023, mrly_003, mrly_024 in 3D, filling 8 of 9 in 2D and 20, 12 and 6 of 27 in 3D; the codes and the first three of those fills are what the core page publishes. The generalised net comes out as mrly_14 in 2D and mrly_232 in 3D — the parity flip of carpet, hence the same symmetry class, which is exactly the collapse the core page records.)

Carpet and net fill 1 + d corners, tree and void fill 2, so the four named families collapse to two limits.

(Proved: (1+d)/2^d and 2^(1-d); Verified to d = 12 by exact rational arithmetic.) It is a limit and not an identity — at d = 3 the carpet's filled fraction runs 0.7407, 0.5830, 0.5147, 0.5015 at n = 3, 9, 51, 501 (Verified).

Three consequences: the research notes get the first right, leave the second open, and get the third wrong.

• Carpet and net have identical limits in every dimension, because

C(d,0) + C(d,1) = C(d,d) + C(d,d-1). That is not a coincidence between two shapes: net is the parity flip of carpet, so on the infinite tiling they are one symmetry class, as the core page records, and the binomial identity is that flip counted at the level of corners. What differs is the truncation to a finite n, which is why their finite fills differ all the way down the tables on this page. (Proved, binomial symmetry.)

• An earlier write-up asked whether the 2D ratio equals the 4D ratio. It does

not: 3/4 against 5/16. Both closed forms are strictly decreasing in d, so no family ever repeats a limit of its own in another dimension. (Proved; Verified by scan to d = 32.)

• The 1/2 coincidence — the 3D carpet, the 2D tree and the one-corner 1D design

all filling half the space — was called a one-off. The value very nearly is: (1+d)/2^d = 1/2 only at d = 3, and 2^(1-d) = 1/2 only at d = 2. The crossing is not. (1+d)/2^d = 2^(1-e) rearranges to 1 + d = 2^(d-e+1), so the two closed forms meet exactly when 1 + d is a power of two — at d = 1, 3, 7, 15, 31, ... for ever, each time at a smaller value. (Proved; Verified, an exact scan to d = 32 returns exactly (1,1), (3,2), (7,5), (15,12), (31,27) and nothing else.)

The honest size of this: it is an exact derivation of a known-style result. The value is that a question left open in the research notes is now closed, in both directions.

Pi, counted rather than assumed

Stack the grid at every scale and keep the points visible from the corner. Their density is the classical coprimality density 6/pi^2 (Proved, Dirichlet and Mertens), so pi can be read off a count: at N = 100000 the stack holds 6079301507 lit points, a density of 0.6079301507 against 6/pi^2 = 0.6079271019, and pi = sqrt(6/density) = 3.14158478 against pi = 3.14159265 — five figures right, the sixth low by one, an absolute error of 7.9e-06. (Verified, recomputed by totient sieve and cross-checked against a brute gcd scan at N = 10, 50, 100, 200.)

Pi out of the stack carries this properly, including why a single carpet cannot hold pi, why the estimator is slow and non-monotone, and how the dimension picks which zeta constant falls out. The constant is classical; what is new here is the geometric realisation, which makes this a strong expository result and not a theorem. The fractal-restricted versions of the density — the Sierpinski triangle, the carpet, the Vicsek fractal — are Conjecture in the ledger, with numerical support and no proof of the interchange of limits, and nothing on this page upgrades them.

Farey structure, and where the primes sit

The bright nodes of the same stack are the Farey fractions (Proved on the primes page); a node a/b has brightness floor(N/b) over scales 1..N (Verified by direct counting at N = 30); and the nodes a scale n contributes for the first time number exactly phi(n), whose running maxima fall exactly on the primes (Verified for n = 2..30 by set difference over reduced fractions). The stack is an RH-observable carries the full account, including the sharpest link in this project to a named open problem: the discrepancy of those nodes is the object of a pair of 1924 theorems equivalent to the Riemann hypothesis. That page carries its own tags and its own literature check, and this page does not restate them.

One sub-claim the research notes asserted without showing a check is upgraded here. For Farey neighbours a/b < c/d one has b*c - a*d = 1, and any p/q strictly between them satisfies p/q - a/b >= 1/(q*b) and c/d - p/q >= 1/(q*d); adding and using c/d - a/b = 1/(b*d) gives q >= b + d, with equality only for the mediant (a+c)/(b+d). Brightness floor(N/q) is non-increasing in the denominator, so no node in the gap is brighter than the mediant — though a larger denominator can tie with it, so the sharp statement is that the mediant is the unique node of least denominator. (Proved; Verified — the smallest denominator in every gap of F_Q for Q = 10, 30, 60, 120, 5798 neighbour pairs, is the mediant every time.)

Sound through a patterned slab

A 2D scalar-wave simulation drives a broadband pulse through a slab of fixed thickness patterned by each family at levels 1 to 3, filled cells carrying half the free wave speed. Transmission is the recorded spectrum over a no-slab reference, averaged across the band the pulse actually excites.

Finer subdivision at fixed thickness suppresses transmission from level 1 to level 2 in all four families, and keeps suppressing it to level 3 only for carpet and net; tree and void turn back up at level 3, where their filled fractions have fallen to 0.296 and 0.088. The same pattern holds in the low and high halves of the band taken separately, so it is not an artefact of the averaging window. (Verified, recomputed here. The research notes report the trend as monotone in all four families; under an independent implementation two of the four are not.)

The control is the part that matters. Against each family and level, a random mask on the same grid with the same number of filled cells:

At levels 2 and 3 the random mask blocks more than the fractal in every single comparison. (Verified, four filled-cell counts per level, two seeds each.) So the suppression is carried by fineness and sparsity, not by self-similarity, and this experiment supplies no support for the claim that a self-similar pattern makes a better acoustic barrier than an equally sparse random one. The scope is narrow — one scalar model, one impedance contrast, one thickness — but within that scope the structured-versus-random control fails, and a suppression table without that control does not mean what it appears to mean.

Usable surface, not just surface

Surface area is the figure of merit for a porous solid, but in a real catalyst the outer surface screens the interior. Wall faces — internal faces where a filled cell meets a void cell — are counted exactly, twice, by two different code paths that agree as sets: 24 for carpet, net and tree at level 1 and 26 for void, then 672, 192, 432 and 198 at level 2. (Verified. Carpet and net agree at level 1 because they are complementary solids there and a complement has the same interface — Proved.)

Then reactant is random-walked in from one open face, reacting on contact with probability P and marking the face it used. Two numbers come out and they must not be confused.

• Every wall face at levels 1 and 2, in all four families, is reachable from the

open face by a breadth-first search through the void. Nothing is geometrically sealed off. (Verified, exact, no sampling involved.)

• Utilisation is therefore kinetic, not geometric: it is a statement about a

fixed reactant budget. At 400 walkers per entry cell with P = 0.5, level-2 utilisation is 0.478 for carpet against 0.896 for net, 0.894 for void and 0.708 for tree. Raising the budget through 50, 100, 400, 1600, 6400 walkers moves carpet's level-2 figure through 0.237, 0.312, 0.493, 0.637, 0.775. (Verified; repeat draws move a figure by about 0.02, so only two digits mean anything.)

Holding the budget fixed and sweeping the reaction probability gives the Thiele behaviour: carpet at level 2 uses 0.564, 0.565, 0.558, 0.479, 0.417 of its wall at P = 0.05, 0.1, 0.25, 0.5, 0.9. Fast reactions are consumed at the mouth, slow ones penetrate. (Verified.)

The obvious next claim — that net's pore space is the better catalyst because it uses nearly twice the fraction of its wall that carpet does — does not survive its own control, and this is the second place on the page where that happens. A budget quoted per entry cell is not a matched budget: the families differ in how many void cells the open face has and in how many wall faces they own, so at 400 walkers per entry cell the reactant delivered per wall face is 10.1 for carpet, 41.7 for tree, 161.6 for void and 166.7 for net — a factor of sixteen, in the same order as the utilisations. Rerun with each family given 160 walkers per wall face and the spread nearly closes: 0.749, 0.859, 0.884, 0.891 in the same order, against 0.491, 0.718, 0.899, 0.901 at the per-entry-cell budget. (Verified.) A family difference survives, but it is small and it is not the factor of two. What the run does establish is the negative half: all of the surface is reachable, so nothing here is a statement about sealed pores.

What this is not

• It is not new research-level mathematics. The strongest results here are exact

derivations of known-style facts — the fill limits, the coprimality density, the Farey structure — and their value is exactness, synthesis and pedagogy.

• It is not an attack on a famous open problem. The Riemann link on

the primes page is exact and untouchable in the same breath: the picture renders a genuinely RH-equivalent object and supplies no route to a proof.

• The primality-display idea runs into an assembly wall, and the wall is the

honest reason it is a display and not an oracle: a device that must realise every period up to n at once is as large as lcm(1..n), which grows like e^n. (Proved, classical — the Chebyshev psi asymptotic; Verified, log(lcm(1..n))/n = 0.9974 at n = 4000.)

• The metamaterial claim does not survive its own control, as above, and the

catalysis claim loses most of its size to a matched-budget control.

• The fractal-restricted coprimality densities remain Conjecture in

the ledger. No page here upgrades them, and a claim of proof for any of them should be read against that entry first.

Where the numbers live

lab/evidence/dimlaw.py is the fill-ratio law and its consequences. lab/evidence/stack.py is the pi count, the totient count, the mediant check and the lcm wall. lab/evidence/pores.py is the wall-face census, the reachability search and the diffusion run. lab/evidence/slab.py is the wave simulation and its random-mask control. Each is self-contained and reproduces every number on this page; lab/evidence/RUN.txt states the protocols, the seeds and the one bug that had to be fixed along the way. Sequence-level results and their verification standard are in the ledger.