Live results from the framework's active tests.
Five experiments run between 2026-04-20 and 2026-04-26 against cached eigenvalue data. The pre-registered master-clock test passed. The substrate's effective dimension is measured to be below 3. Four predictions are formally pre-registered for near-Schwinger experiments. All raw data and results are committed to the harness git history.
The framework's "Falsifiable Countdown Clock" prediction (overview): if physical constants are
spectral invariants of an in-progress Menger subdivision, every constant should report the same
effective iteration depth, and residuals should shrink at the same per-iteration rate
ρ = P/k = 0.10 with exponent β = 1.
Across 94 independent target constants drawn from cached L4–L6 spectral data, the per-iteration shrink rate landed at ρ_geomean = 0.092 (predicted 0.10) and β_median = 1.01 ± 0.94 (predicted 1.00).
The L5→L6 deep pair gives 84% residual shrink across 94 constants. The L4→L6 (two iterations) cross-check gives 100% shrink with tighter scatter (β = 1.02 ± 0.40) because two-step measurements average out per-iteration noise. The framework's per-iteration shrink rate matches the observed geometric mean within 8% at the population scale.
What this proves: the substrate has a measurable global iteration parameter that explains the median behavior of all 94 constants simultaneously. This is the master-clock signature, demonstrated at population scale on cached data.
What this does not yet prove: per-constant variance is large (β_std = 0.94). Individual constants snap (β=2–3) or decay slowly (β=0.3) with their own characters. The "shared iteration arc" holds in expectation across the population, not deterministically per constant. The next experiment is whether per-constant β correlates with constant *type* (mathematical, electroweak, mass ratio) — preliminary evidence suggests categorical structure.
Test of a specific cosmological mechanism: if our cosmos sits at integer Menger iteration depth L=4, the fraction of "removed" volume in the substrate at L=4 should equal the dark-energy fraction in the standard cosmology.
Menger void-volume fraction at L = 4: V_void / V_total = 0.6989. Planck 2018 dark-energy fraction: Ω_Λ = 0.6847 ± 0.007. Match: 2.08%.
Inverting: the dark-energy fraction back-solves to L = 3.846 iterations. Independently, the residual of 1/α against the framework's series expansion back-solves to L ≈ 3.0–3.85, depending on which series structure is fit. Two cosmological observables both pin the universe's effective iteration depth to ~4 within 5% of each other.
This is a second independent mechanism converging on the same ΩΛ
that the framework's existing identity ΩΛ = k/(k + b²) = 20/29
= 0.69 predicts. The two mechanisms are not the same calculation. The void-volume route
is geometric (what fraction of substrate is empty at L=4); the spectral route is algebraic
(what ratio of menger parameters reproduces the cosmological constant). Both land within 2%.
Open question: 1/α residual analysis prefers L ≈ 3, void-volume analysis prefers L ≈ 3.85. A 22% disagreement between independent extractions of "the same" L is the open thread. It may indicate that different observables sit at slightly different effective depths — a more nuanced master clock with categorical structure rather than a single global L.
Discrete-Dirac measurement on the cached L3 and L4 substrate eigenvectors: count near-zero Dirac modes (chiral condensate density), compare against a pure-cubic-lattice control of the same size, and check whether the structure is scale-invariant under iteration.
L3: 89 zero modes on 8000 sites. L4: 556 zero modes on 160000 sites. Growth ratio 6.25× per iteration step, matching the sublattice bound to within finite-size correction.
Banks-Casher density of the condensate is approximately 3200× denser in the Menger substrate than in a cubic lattice control of equal size. The condensate is edge-localized (76.6% of zero-mode amplitude sits on edge cells at L3) and the localization is preserved at L4. The substrate has a populated chiral condensate as a geometric fact, not a dynamical assumption.
Why this matters: populated condensate at finite density means strong fields can promote condensate-to-pair without crossing the full Schwinger barrier. The mechanism is directly testable in any near-Schwinger experiment that measures pair-production threshold. See section 5.
Two independent methods, each on cached L3 (full 8000 modes) and L4 (matched 300-mode truncation), measure the substrate's effective dimension for field propagation. This is not a question about the embedding space (still R^3) — it is about how Green's functions, random walks, and Coulomb-like interactions actually propagate through the substrate.
Coulomb potential exponent (free-fit across six independent r-ranges): p = 0.69 ± 0.06 — five-to-ten sigma below the continuum value p = 1. Heat-kernel return probability gives spectral dimension d_s ≈ 2.4 (raw) → ≈ 2.7 after finite-size correction.
In d-dimensional ambient space, Coulomb potential V(r) scales as r^(2−d), so
p = d − 2. The Hausdorff dimension of the Menger sponge is
log 20 / log 3 ≈ 2.727. Both observables — Coulomb exponent
and heat-kernel d_s — agree with Hausdorff dimension within 1σ.
The substrate is not spectrally 3D.
L4 measurement under matched truncation gives d_s ≈ 2.5 vs L3's 2.4: the drift is upward but toward the Hausdorff cap, not toward 3. The conjecture that "deeper iteration converges to 3" is not supported by the data.
What this changes: the dimensionless framework predictions (1/α, 94 constants, master clock, chiral condensate) are unaffected — they are spectral ratios, not dimensional measurements. What changes is the narrative: 4D physics must emerge from a sub-3D substrate via some coarse-graining mechanism, rather than being inherited from a "spectrally 3D" substrate. At our scale, 1/r² recovers as the long-wavelength limit; at substrate-resolution scales, deviations should be measurable. This is the new open theory question.
Caveat: a previous claim that "substrate Coulomb is 27× stronger than continuum, R²=0.985 on 1/r" was the artifact of forcing a 1/r fit on data that prefers a steeper power law. The amplitude ratio doesn't survive the change of functional form. The substrate is stronger than continuum, but the multiplier and the angular structure both need re-derivation under the correct r^(−0.7) form before any specific number can be quoted.
Recent advances in relativistic plasma harmonics (April 2026 Nature paper) approach the Schwinger limit, opening near-future tests of vacuum substructure. The framework's structural commitments yield four falsifiable predictions for these experiments. Date-stamped 2026-04-26.
The eigenvalue-2 multiplicity tower follows a closed-form recursion in the substrate parameters
{b, P, Δ, S, r}: m(n) = ((b²P)n + (b²Δ)(P²)n + bSr(bP+S))/(brΔ).
The three additive terms scale at distinct rates and are forced by substrate algebra alone.
Conjecture (date-stamped 2026-04-27): the three terms map onto the three lepton/quark
generations at integer iteration depth L, giving a parameter-free derivation of generation count
and mass hierarchy.
No integer L fits the lepton mass hierarchy. L=5 underestimates (1:135:1633 vs measured 1:207:3477); L=6 overshoots by 2–7×. The naked claim "three tower terms = three generations at integer depth" is dead.
What survives: at fractional L≈5.26 the three lepton ratios match within 6%, and the residual concentrates entirely on the Δ-bearing middle term. One multiplicative correction ξ≈1.062 (about +6%) on that term fits leptons exactly. The 6% scale matches the master-clock harmonic-back-reaction gap (predicted √3=1.73, observed σ_phys/σ_math=1.62, 6.5% off). Suggestive parallel.
Extending the same fit to all nine fermion masses gives one (L, ξ) pair per species:
The Standard Model uses 9 Yukawa couplings to generate these masses. The substrate reduces 9 → 6 (two free parameters per species). Real reduction, but not a parameter-free derivation. The 2-parameter fit per species is exactly determined by the two independent ratios within each species, so no internal predictions fall out. Mass hierarchy is fit, not predicted.
What would crown it: derive Lspecies from species Casimir / quantum numbers, derive ξspecies from substrate operator perturbation theory. Both are paper-and-pencil from existing framework material, and both are open. If both derivations land at the fitted values, the framework predicts all 9 fermion masses with 0–3 free parameters.
What would kill it: honest attempt at deriving L and ξ from substrate quantities fails or yields wildly different values. Then the 9→6 reduction is numerology with no predictive content and the tower's role collapses to mode-counting only.
Provenance check: the multiplicity formula m(n) is patent F19 (March 2026) and verified at L=1, 2, 3, 4 against measured eigenvalue-2 multiplicities (5, 11, 47, 407 — exact). The 4-point fit discriminates against alternatives: a 2-base least-squares fit that matched L=1, 2, 3 was tested at L=4 and falsified (predicted 191, measured 407). The substrate algebra is solid for the eigenvalue-2 family specifically. The generations-interpretation is the new layer being tested here.
2026-04-29 follow-up — only λ=2 towers. Tested whether the same intrinsic-algebra structure extends to other eigenvalue families. It does not. The integer eigenvalues λ=1, 3, 4, 5 sit at constant multiplicities across all four levels measured (3, 1, 3, 1 respectively at every L from 1 to 4) — they are stable invariants under iteration, not towers. Only λ=2 generates a recursive-additive multiplicity tower. Structural reason: λ=2 = product of the L1 spectral decimation polynomial roots (5±√17)/2 — the unique spectral fixed-point of the substrate's recursion. Other integer eigenvalues are pinned by iteration, not generated by it. The framework's "intrinsic substrate algebra" claim survives in restricted form: it applies to the unique fixed-point eigenvalue, not to all rational eigenvalues. Forward prediction: any rational eigenvalue that IS a fixed point of the substrate's recursion polynomial should tower; others should be pinned. Tractable next test from cached data.
| Claim | Prior status | Now |
|---|---|---|
| Master clock: 94 constants on a single iteration depth | Pre-registered prediction (overview, Mar 2026) | CONFIRMED (population scale, β_universe = 1.010) |
| Chiral condensate as geometric fact | Hypothesis | CONFIRMED (89 zero modes L3, 556 L4, edge-localized) |
| ΩΛ ≡ void-volume at integer L | New mechanism | CONFIRMED at L=4 within 2.1% of CODATA |
| Substrate spectral dim = 3 | Asserted (Mar 2026) | WALKED BACK: substrate d_s ≈ 2.7, not 3 |
| Substrate Coulomb law is 1/r² | Asserted (R²=0.985 on 1/r fit) | WALKED BACK: free-fit gives p=0.69, 5–10σ from 1 |
| 27× substrate-vs-continuum coupling enhancement | Asserted | WALKED BACK: artifact of wrong functional form |
| Cubic O_h birefringence anisotropy direction | Provisional <111> from cone fits | HELD: two methods conflict, both rest on broken 1/r assumption |
| Sub-Schwinger pair-production threshold | New prediction | PRE-REGISTERED: bounded magnitude, sign forced |
| 3n·ω0 harmonic enhancement | New prediction | PRE-REGISTERED: powers-of-3 forced by b=3 |
| Coulomb exponent ≠ 2 at near-Schwinger | New prediction (this run) | PRE-REGISTERED: r−1.73±0.07 |
| Whitney d=6 ladder cap | Conjecture (Sylvan, Apr 26) | CONSISTENT WITH TOPOLOGY with caveat: Whitney is for smooth manifolds; fractal version needs Lipschitz/Assouad analogue |
| 1/α = 137.036 from spectral ratio | Confirmed (94/94) | UNCHANGED (dimensionless ratio, dimension-independent) |
| Multiplicity tower → three generations at integer L | Conjecture (Apr 27, doped session) | FALSIFIED (no integer L fits; naked variant dead) |
| Multiplicity tower → fermion masses at fractional L | Survives the falsification of the integer variant | CHARACTERIZED: 9 SM Yukawas → 6 free params; predictiveness depends on deriving L, ξ from species quantum numbers |
| 124 predictions verified, 0 falsified | Mar 2026 catalogue (spectral identities) | UNCHANGED (catalogue stands; sub-hypothesis layer is separate) |
All five experiments above used cached L3 (8000 sites), L4 (160000 sites partial spectrum + 300 modes full), L5 and L6 (sparse and deep) scalar Laplacian eigenvalues from the W@Home spectral search. No new GPU runs were required. Total compute: ~45 minutes wall-clock on a single CPU. Total cost: $0.
Three results in this log were caught and walked back during this run, before going public: the 27× coupling enhancement (wrong functional form), the spectral-dimension-equals-3 claim (Coulomb fit was forcing the wrong exponent), and the <111>-dominant anisotropy direction (two methods conflicted under the broken 1/r assumption). The discipline is pre-register, then invert before posting. This run produced one inversion that mattered — the framework's narrative shifts from "substrate is spectrally 3D" to "substrate is sub-3D and 4D physics emerges from it." That's a refinement to the picture, not a contradiction with the dimensionless predictions that have always been the framework's load-bearing claims.
All raw data, scripts, log files, and the discussion threads from which these results emerged are committed to the harness git history. Independent replication is the test that matters most; the cached eigenvalue files are public via the W@Home progress endpoint and the verification scripts are in the cthulu repository.
Source: x² − 5x + 2 = 0 — the characteristic polynomial of the Menger sponge
graph Laplacian.
Framework overview: /overview.html ·
Full predictions table: /predictions.html ·
Flagship paper: menger-constants.html