The millennium map

Seven problems, one small fractal project, and the honest distances between them. The research notes have kept a running score of how the project's threads sit next to the Clay Millennium Prize Problems - first as a single number per problem, later as two. This essay is that map, redrawn with everything that has since been re-run in public.

One ground rule before any table: a score is a judgement, not a computation. What can be checked is the evidence a score leans on, and every number quoted below has been recomputed in the open lab - research/lab/millennium/, research/lab/porous/, research/lab/yangmills/, research/lab/levels/, research/lab/complex_dimensions/.

The first survey

The first pass scored each problem once, 0 to 10, on "how real is the link".

The scores are self-reported and nothing in them can be rerun, but the arithmetic under the reasons can. research/lab/millennium/survey.py rebuilds the two that are computations: lcm(1..n)^(1/n) walks to e - at n = 3000 the lcm holds 4330 bits against 4329 for e^3000 - and the exact Mobius count agrees with 1/zeta(d) to a few parts in ten thousand at d = 2, 3, 4. The reasons are sound. The scores stayed opinions.

Two axes

A later pass split the single number in two: link - is the mathematical connection real - and handle - could anything computable move the problem. The split matters because the two questions turn out to have opposite answers in the most interesting row.

Poincare is solved and left unscored. Only one row moved from the first survey: Riemann, from an undifferentiated ~3 to link 6, handle 0. It is also the only row with something checkable underneath; the other six remain judgements and should be read as judgements.

The Riemann amendment

Lay the same grid on the unit interval at every scale n = 1..Q and add the layers up. The bright points - the nodes many scales agree on - are exactly the Farey fractions F_Q, the reduced fractions with denominator at most Q. That identification is proved, not an analogy: the stack's nodes are not like the Farey sequence, they are the Farey sequence.

Which matters because of two theorems from 1924. Franel proved that sum delta_j^2 = O(Q^(-1+eps)) for every eps > 0 - with delta_j the distance of the j-th Farey node from perfect equidistribution - is equivalent to the Riemann hypothesis; Landau, in the note published immediately after, proved the same for sum |delta_j| = O(Q^(1/2+eps)). So the question "how evenly are the stack's bright points spread?" is not related to RH. At this level of precision it is RH.

The meter reads what RH predicts: S2*Q flattens near 0.656 as Q runs from 125 to 8000 and its local exponent walks to -1.00, the Franel rate, while S1/sqrt(Q) falls from 0.20 to 0.11, well under Landau's threshold of 0.5. And here the second axis earns its keep: RH is already verified numerically far beyond any range this meter can reach, so the table can only ever illustrate what is known. The connection is exact, and it is untouchable

• link 6, handle 0, and both halves deserve to be said together. The full

result, tagged claim by claim, is public at primes; the meter is research/lab/millennium/franel.py.

Navier-Stokes and the ordering effect

The sharpest physics-neighbourhood result started life on shaky paper: the research notes reported it citing a script that nobody has since been able to find. That is no pedigree to publish on, so the effect was rebuilt from scratch - and it is real.

The object: three carpet tiles at bases 3, 5 and 7, pore fractions 8/9, 21/25 and 40/49, composed by Kronecker product into one 105 x 105 sponge, with a choice of which base sits outermost, middle and innermost. Kronecker products commute in count, so all six orderings fill exactly the same 6720 pore cells - porosity cannot tell them apart. Simulated Darcy/Stokes flow can: conductance runs from 0.3399751735 (ordering 3-5-7) to 0.3439279767 (7-5-3), a spread of 1.16 percent, strongly anti-correlated with the outermost tile's pore fraction, Pearson -0.831251. The sign, stated so it cannot be read backwards: flow is highest when carpet(7), the tile with the lowest pore fraction, sits outermost. Coarse blockage costs more than the same blockage subdivided - carpet(3) outermost puts one solid 35 x 35 square in the middle of the sample, while carpet(7) outermost breaks the same blocked fraction into nine 15 x 15 squares that flow can route between.

Two honest fences. First, the original -0.83 was a measurement recorded in the research notes with its script since lost; it ships here on the strength of the fresh runs, not the old note. Four independent routes now agree, three of them public - research/lab/porous/flow.py, research/lab/porous/check.py and research/lab/porous/recheck.py, different geometry constructions and different solvers, all six conductances identical to ten decimal places - with a fourth adversarial re-derivation during verification. Second, two clauses of the original claim did not survive the rebuild: conductance is not monotonic in the outermost pore fraction (the orderings interleave, which is exactly why the correlation is -0.83 and not -1), and the isotropy and contact-count checks the note offered as evidence are automatic consequences of the tiles' symmetry and prove nothing. What carries the weight is independent solvers agreeing to 1e-10 while the orderings differ by 4e-3.

Scope, plainly: level 1 in each scale, one 105 x 105 grid per ordering, two dimensions. Nothing is claimed about deeper levels, about 3D, or about the Navier-Stokes equations themselves. This is the physics neighbourhood, which is what link 4, handle 2 says.

Yang-Mills and the band gap

The code-23 design - the same rule that draws the Menger sponge at base 3 - built at base 2 has a tile graph that is the star K_{1,3}, and its Laplacian opens an exact band gap. At every level tested the value 4 is an eigenvalue exactly and exactly once, nothing lies anywhere in [2, 4), and the top of the lower band climbs to 2: 1.000000 at level 1, 1.975680 at level 3, 1.999605 at level 5. Exact arithmetic certifies the empty interval through level 4 and dense spectra confirm it through level 6, so the width converges to 2 = k - 2 with k = 4 the popcount of the tile. The defect 2 - lo shrinks by a factor climbing toward 8 per level - ratios 5.80, 7.09, 7.75, 7.94, reading 8.0000 by level 11 in the lab's deepest run - an 8^(-L) rate that is a numerical fit, not a theorem.

The score stays at link 3, handle 1, because a spectral gap in a graph Laplacian shares a word with the Yang-Mills mass gap, not the gauge theory. The public spectral work is on complexity; the gap scripts are in research/lab/yangmills/.

Two supporting threads

Both belong to the Riemann row's clean-negative half and its structural counterpart. The spacings story is written up on complexity; the lattice story on dimensions.

Level spacings are Poisson-side, not GUE. Had the fractal spectra shown the level repulsion the zeta zeros show, that would have been a headline. They show the opposite: every fractal tested - carpet cell graphs in 2D, Menger cell and slice graphs in 3D - reads clustered, at roughly three times the GOE prediction for small spacings, with a random-graph control reading GOE and a square-lattice control reading clustered, as they should. The mechanism is self-similarity forcing degeneracy: the largest eigenvalue multiplicity in the slice graph grows from 12 at level 2 to 48 at level 3, against multiplicity 1 for the random control. Established on spectra up to 4096 nodes; research/lab/levels/recheck.py.

Every design is lattice-class. The box-counting function of every proper design - anything strictly between a single point and the full cube - oscillates log-periodically forever, at the period the theory predicts - the measured peak sits within a tenth of a percent of 2*pi/ln 3 for the base-3 Cantor design - and composing designs only multiplies the bases: base 3 with base 5 lands at the base-15 period, still lattice. No composition of the existing operators can leave the class; escaping it needs unequal cell sizes inside one subdivision level, which no parity rule on a grid can express. One honest trim from the re-derivation: "lattice implies not Minkowski measurable" is a theorem on the line and a conjecture in higher dimension, so for the 2D and 3D designs that clause is conjectured, not proved. research/lab/complex_dimensions/spectrum.py and research/lab/lattice/.

P vs NP, and the remainder

The P vs NP thread comes from an earlier and narrower survey in the research notes - three problems, no scores - and re-running it mostly shrank it. Reading designs as Boolean functions, certificate complexity equals block sensitivity for every design at D <= 4 - all 2^(2^D) of them, by exhaustion - and block sensitivity exceeds sensitivity by at most 1. True, verified, and empty as evidence: at dimension D the designs are all Boolean functions on D variables, and a collapse at so few variables is exactly what the classical bounds predict, since the known separating constructions need far more. A fact about small functions, not about P versus NP. research/lab/complexity/collapse.py.

Hodge, Birch-Swinnerton-Dyer and Poincare keep their near-zero rows for the reasons in the first table: real Betti numbers with no Hodge content, no elliptic curve anywhere, and a solved problem admired from a distance.

What the map is for

The standing note the research notes have always carried belongs at the end of the public version too: nothing here claims a Millennium problem, approaches one, or expects to. Five links score 4 or less, the one that scores 6 has a handle of 0, and the seventh problem is solved. The value is the map itself - knowing which threads touch something real (a genuinely RH-equivalent observable, a reproducible transport effect), which are vocabulary (a band gap that shares a name with a mass gap), and which are nothing (an elliptic curve the project has never met). A small project that knows its distances can walk anywhere without falling into a famous hole.