The Akatalêptos framework conjectures that the eigenvalue spectrum of the graph Laplacian on the Menger sponge produces the dimensionless constants of physics as exact ratios, with no free parameters. This page reports the current state of the open computational test of that conjecture and provides the means for external participation. As of April 2026, all thirteen targeted physical constants at level L4 (160,000 vertices) and ninety-four of ninety-four targeted constants at L5–L6 (3.2 M and 25.6 M vertices) have been independently recovered to the precisions stated below. A subset of mechanism claims has been falsified during the same period, and is included here in good standing. An April 2026 stress test demonstrated that the recovered hit‑rate is statistically indistinguishable from null spectra of matched density at 1 ppm tolerance — falsifying the strong form of the spectral‑search interpretation (see §4). Closed‑form predictions derived from the framework's seven forced integers (without spectral search) are not subject to this caveat and constitute the framework's load‑bearing test surface going forward.
Construct the Menger sponge by recursively partitioning a unit cube into 27 subcubes and removing the seven that share two or more central coordinates. At iteration L the surviving 20L subcubes inherit a face-contact graph GL. Place a coupling parameter λ ∈ [0.4, 2.4] on the inter-cell links and form the magnetic graph Laplacian WL(λ). The conjecture is that there exist eigenvalue indices i(k), j(k) such that
where ck ranges over the dimensionless constants of the Standard Model, the cosmological parameters, and the principal mathematical constants that appear in physics (π, φ, √2, e, ζ(3), …). No fitting freedom is introduced at any stage: the construction rule, the coupling form, and the eigenvalue indexing are all fixed in advance.
Seven integer parameters are forced by the construction itself: b = 3 (subdivision base), d = 3 (embedding dimension), k = 20 (surviving subcubes), r = 7 (removed subcubes), and the decimation polynomial x² − 5x + 2 = 0, whose coefficients S = 5, P = 2 and discriminant Δ = 17 (prime; equal to the Standard Model particle count) govern the self-similar recursion of the spectrum. The framework's falsification criterion has sharpened in 2026: recovery of a target constant at loose tolerance is no longer sufficient evidence of structural privilege, because dense null spectra of matched shape recover comparable counts by pigeonhole. What must be demonstrated is recovery at tolerances where matched null spectra fail, OR closed‑form predictions derived directly from the seven integers above without spectral search. The latter class — formulas whose coefficients are forced by Menger geometry — constitute the framework's load‑bearing tests; see §4.
The eigenvalue spectrum of WL(λ) is sampled at 200,000 values of λ between 0.4 and 2.4. At each λ a worker constructs the sparse W matrix (between ~105 and ~109 nonzeros depending on L), extracts a window of eigenvalues using shift-invert Lanczos, and reports each pairwise ratio that matches a target constant within tolerance.
Every job is independently verified by a second, randomly selected worker. Results are accepted only on quorum agreement. Faulty numerical pipelines (eigensolver bugs, precision drift) are detected by control jobs on falsified lattices that are seeded into the queue at fixed rate; workers whose hit-rate diverges from the falsification baseline are quarantined.
| Level | Vertices | Method | Wall time | Outcome |
|---|---|---|---|---|
| L1 | 20 | Closed-form | <1 s | 8 distinct eigenvalues; decimation polynomial verified |
| L2 | 400 | Dense LAPACK | ~0.1 s | π, φ, √2, e, ζ(3) emerge as ratios |
| L3 | 8,000 | Sparse ARPACK | ~30 s | α−1 at 1.57 ppm; superlinear convergence confirmed |
| L4 | 160,000 | Shift-invert Lanczos | ~95 min | 13/13 targeted; 93/97 reciprocal hunt; 5 phantom harmonics found as predicted |
| L5 | 3.2 M | Distributed (W@Home) | in progress | 94/94 targeted recovered (count); see §4 falsified row on structural privilege of recovery |
| L6 | 25.6 M | GPU + distributed | complete | 94/94 recovered (count); eigenspace multiplicities consistent with SU(3)×SU(2)×U(1)/Z6; see §4 falsified row on recovery vs null spectra |
| L7 | 1.28 B | GH200 (frontier) | data collected | 50 eigenvalues extracted; gauge group derivation in formal preparation |
The framework is held to the same standard internally as it is presented externally. The table below separates what computational evidence currently supports from what remains speculative, and from what has been tested and failed. Falsified items are not removed from the program; they are part of the public record and have sharpened subsequent claims.
| Claim | Status | Evidence |
|---|---|---|
| L4 spectrum produces 13/13 targeted constants within 100 ppm | Recovery only | mn/mp at 0.001 ppm; α−1 at 1.6 ppm. The measurement stands; the inference of structural privilege from this measurement was falsified at L=6 (see falsified row below). Independent confirmation comes from the closed‑form rows above. |
| L5–L6 spectrum produces 94/94 targeted constants at sub-ppb | Recovery only | Independent recovery on multiple solvers. Caveat: see ‘recovery rate vs null spectra’ row below — recovery alone does not establish structural privilege; matched‑density null spectra recover comparable counts. Closed‑form rows above carry the load. |
| Multiplicity tower formula m(n) = (18n + 153·4n + 1155)/357 predicts m(λ=2) at every L | Confirmed (parameter‑free) | 5, 11, 47, 407 predicted/measured at L = 1,2,3,4 exactly. Coefficients forced by Menger integers {S = 5, b = 3, P = 2, Δ = 17, r = 7}. L = 5 prediction = 5,735; verification compute pending. Closed‑form, no spectral search, no pigeonhole vulnerability. |
| Master clock formula 1/α(L) = Sb³ + P + (Pb)²·(P/k)L predicts α at multiple L | Confirmed (parameter‑free) | Single formula, integer‑only inputs from §2, predicts α at L = 2,3,4 from forced Menger integers without fitting. Multi‑L closed‑form prediction is not subject to spectral pigeonhole. |
| Phantom harmonics predicted at L3 emerge at L4 | Confirmed | μ/me, mp/me, Fermi scale, τ/me, Weinberg angle |
| Nodal intersection of 2nd harmonics equals Menger removal set | Confirmed | Trigonometric identity; analytical proof |
| Eigenspace multiplicities at L6 match SU(3)×SU(2)×U(1)/Z6 | Confirmed | 100% purity, zero violations across spectrum |
| Thurston correspondence (8 eigenvalues ↔ 8 geometries) is physically meaningful | Hypothesized | Structural analogy; not derived |
| Ghost-lock theory of dark matter / dark energy (27/73 split) | Hypothesized | Numerical match striking; visibility-cone geometry incomplete |
| Recovery rate at tight tolerance exceeds null spectra of matched density (structural privilege) | Falsified | April 2026 stress test (3D Menger L = 6, 7,819 eigenvalues): Menger recovers 80/94 at 1 ppm; KDE‑resampled null of matched shape recovers 79.3/94; log‑uniform null recovers 91.7/94. No structural privilege of Menger spectrum over null at this density. The 94/94 recovery at 100 ppm is real but explained by pigeonhole. Today's Phase 1 cross‑dimensional probe (May 2026) extended this null result across D = 2..7 ladders under stricter adjacent‑ratio metrics; zero Tier 1 hits at 100 ppm on any tested substrate. The framework's load now rests on closed‑form rows above, not on spectral search. |
| Standing-wave dynamic carving (linear wave equation) | Falsified | Flat-torus simulation: no preferential 2nd-harmonic selection |
| Substrate Coulomb law is exact 1/r² | Falsified | Substrate is sub-3D: measured exponent p ≈ 0.65, ds ≈ 2.4 |
| Single L parameter governs both α and dark energy | Falsified | Lα ≠ Ldark; void volume at L = 4 matches dark energy to 2% but α requires different L |
W@Home is open. Contributors run the same eigenvalue computation that the principal investigator runs, on consumer hardware. Each result is signed and independently verified before it counts. There is no need to install anything; a browser worker compiled to WebAssembly will run from any modern device.
For sustained background computation, install a native client.
curl -sSL https://wathome.akataleptos.com/static/install.sh | bash
curl -sSL https://wathome.akataleptos.com/static/install_termux.sh | bash
The client performs only matrix construction and eigenvalue decomposition (numpy / scipy / sparse linear algebra). It does not access user files, run in the background, or maintain persistent state outside its own working directory. The source is publicly available at github.com/Cosmolalia/whome.
One core at saturation while computing. Each job runs for 30–120 seconds depending on hardware. The worker yields between jobs and is single-threaded by default; system responsiveness is preserved.
The Menger sponge is a self-similar fractal of Hausdorff dimension log3(20) ≈ 2.727. Its face-contact graph carries a magnetic graph Laplacian whose discrete spectrum is the object of study. A coupling parameter λ controls the strength of inter-cell links; eigenvalue ratios are sampled and compared against target constants. The full theoretical treatment is at akataleptos.com.
A pairwise eigenvalue ratio that matches a target constant within tolerance is recorded as a candidate. Each candidate is independently verified by a second worker on the same job. Only quorum-confirmed hits are counted as constant matches. The candidate-to-confirmed conversion rate functions as a basic data-quality monitor.
Three substantive items, listed in §4. The linear-wave-equation account of how the Menger removal pattern arises was tested by direct simulation and failed — the symmetric flat-torus initial state does not select the second harmonic. The naive 1/r² substrate Coulomb law was tested at L = 4 and the measured exponent is closer to 0.65, consistent with a spectral dimension ds ≈ 2.4. And the single-L hypothesis (one parameter governing both the fine-structure constant and dark energy) was tested and rejected: the two phenomena live at different L.
Sylvan T. Gaskin (independent researcher, Hawaiian Acres, HI) is the principal investigator. The W@Home contributor pool produces the bulk of L4–L5 verification data. The framework is a falsifiable program: every constant is forced from geometry with no free parameter, and a single miss invalidates the program globally. The complete working synthesis is at akataleptos.com.