flavio.np_prediction directly. We do not claim a detection of new physics, nor that the kernel is the unique q² shape consistent with the anomaly. We do report what the data shows: the same fixed kernel describes all five datasets without retuning, the sign is uniform across all five, the cross-channel ratio matches a predicted basis-correction factor, and the kernel variant was selected by a pure-geometry criterion decided before the LHCb data was looked at.
The B→K*μμ Anomaly
The angular distribution of the rare flavour-changing neutral-current decay B⁰→K*⁰μ⁺μ⁻ has shown a persistent tension with Standard-Model predictions for over a decade. The conventional interpretation parametrises the gap as a constant negative shift in the semileptonic vector Wilson coefficient C₉:
The most recent LHCb comprehensive analysis at 8.4 fb⁻¹ (LHCb 2025) confirms the pattern with full angular and branching-fraction observables on the same dataset. The free-shift interpretation makes no commitment to the underlying q² shape: by construction it averages over all bins. If the underlying physics has a non-trivial q²-dependence, that information is invisible to it.
A Deliberately Constrained Question
Suppose, instead, that the q²-dependent response shape is fixed a priori by an external geometric construction — with no shape parameters tunable to the data — and one fits only its overall amplitude. Does the same fixed shape describe multiple datasets and channels with no per-dataset retuning?
ΔC₉ᵛ𝕱𝔡(q²) = −A · κ(q²)
The kernel κ(q²) is identical across every dataset and decay channel. The kernel shape itself never moves. The only data-tunable parameter is the dimensionless amplitude A — one per dataset, no shape retuning, no per-bin freedom.
The Kernel: 600-Cell + Golden Ratio
The kernel κ(q²) is the discrete Green's response of a finite graph carrying binary-icosahedral symmetry — the 600-cell V₆₀₀, with 120 vertices and 720 edges — regularised by the golden ratio φ = (1+√5)/2 as a discrete mass scale:
The kernel κ(q²) is the shell-mean projection of ψ onto a dimensionless coordinate anchored at the J/ψ – ψ(2S) midpoint. The continuum limit is the simple exponential κ(x) = e−|x|/φ. The choice of φ⁻¹ as the mass scale is the unique scale that ties the operator to the binary-icosahedral graph V₆₀₀ — the vertex coordinates of V₆₀₀ are built from φ as the only irrational unit.
Continuum φ-tuned
One-dimensional Green's function of Lφ = −∂² + φ⁻². Yields the exponential kernel e−|x|/φ.
Bounded Dirichlet mode
Lowest eigenmode of Lφ on the kinematic interval with Dirichlet conditions. Single-mode shape, no continuous tunables.
Discrete 600-cell lift
2I-equivariant graph realisation on V₆₀₀. The Layer-3 kernel is the headline shape; Layers 1–2 are theoretical controls.
Five Fits, One Shape
We test the kernel against five public flavour-physics datasets — spanning two collaborations, two isospin partners (B⁰ and B⁺), and three decay channels (B⁰→K*⁰, B⁺→K*⁺, Bₛ→φ) — using the non-linear flavio.np_prediction backend. One amplitude per dataset, no shape retuning across datasets.
| Dataset | Channel | n | A (fit) | ΔC₉eff | ΔAIC |
|---|---|---|---|---|---|
| LHCb 2015 | B⁰→K*⁰ | 32 | +1.24 | −0.96 | −0.24 |
| LHCb 2021 | B⁺→K*⁺ | 32 | +2.06 | −1.59 | +0.17 |
| CMS 2025 | B⁰→K*⁰ | 18 | +1.05 | −0.81 | +0.47 |
| LHCb 2025 | B⁰→K*⁰ | 32 | +1.14 | −0.86 | +1.09 |
| LHCb 2015 | Bₛ→φ (S-basis) | 24 | +4.98 | −3.85 | −0.24 |
Each row is one dataset, fitted with one amplitude. The shape is the same shape in every row.
What the Data Shows
Four independent statements survive the non-linear refit:
The same fixed kernel shape describes all five datasets — two collaborations, two isospin partners, three decay channels — with one amplitude per dataset. No shape retuning across datasets, channels, or isospin partners. The kernel itself does not move from fit to fit.
A > 0 in 5/5 fits. ΔC₉eff < 0 in 5/5 fits. The fitted amplitude reproduces the established direction of the anomaly across all five independent measurements. Under a random-sign null hypothesis, the probability of 5/5 same-sign is 2⁻⁵ ≈ 0.03.
The B→K* fits live in the Krüger–Matias P-basis; the Bₛ→φ fit lives in the CP-averaged S-basis. A predicted basis-correction factor ⟨1/√(FL(1−FL))⟩ ≈ 2.2 explains most of the B→K* ↔ Bₛ→φ amplitude ratio. The predicted basis-corrected amplitude AP × 2.2 ≈ 2.5 sits a factor of ∼2 below the fitted Bₛ→φ value of +4.98 — a residual overshoot the published 2015 statistics cannot currently resolve. The sign and the order of magnitude come out right by construction.
Of three discrete Laplacian variants for the 600-cell, the unweighted choice wins on a pure-geometry criterion (correlation 0.997 with the continuum kernel e−|x|/φ) decided independently of any LHCb data. The same variant later wins on the LHCb data χ². The geometric and data criteria point at the same kernel. The ranking is consistent with selection on geometric grounds before the data is looked at.
These four findings are structural: they concern whether one fixed shape can describe heterogeneous data, not whether that shape is statistically preferred over a generic alternative. They are independent of the statistical-significance discussion in the next section.
An Independent Statement: The AIC Comparison
AIC compares per-parameter goodness-of-fit and does not reward predicting a specific functional form across datasets. It scores how well any single fit lands at low parameter cost; it does not score whether the same shape was used in every fit. The AIC tie and the universality result are therefore independent statements about the same data:
- Universality (Findings 1–4): one fixed shape describes all five datasets with one amplitude each. The kernel shape itself is geometry, not fit.
- AIC comparison (this section): on per-dataset goodness-of-fit, that fixed shape is statistically tied with a free constant Wilson-coefficient shift.
Both statements are facts about the data. Neither one falsifies the other. We are reporting both.
The Linearisation Diagnostic
An earlier project iteration evaluated all fits using a linearised expansion of the angular observables in ΔC₉ around the SM benchmark (the project's "Mode B"). The fitted ΔC₉ values are of order unity, well outside the linear regime. Under the non-linear refit using flavio.np_prediction directly, the LHCb 2025 ΔAIC drifted by +2.77 units — from −1.67 (linearised) to +1.09 (non-linear). The drift is larger than the linearised preference itself.
We treat the non-linear analysis as the headline. The linearised analysis is reported in the paper as a methodology diagnostic only. The +2.77 AIC drift between the two modes is the largest single methodological uncertainty in the project, and is flagged as such throughout. The structural-test claims (Findings 1–4) are made on the non-linear refit and are independent of the linearisation diagnostic.
How to Read This Result
Geometric coincidence
The kernel happens to look enough like a centre-peaked nuisance that it averages to a constant shift on AIC. The sign uniformity and cross-channel ratio are real but not specifically geometric.
Centre-peaked class
The data is consistent with a class of centre-peaked one-parameter shapes, of which the geometric kernel is one specific, geometrically motivated instance. A free-width charm-loop Gaussian fits comparably in χ² at the cost of one extra shape parameter.
Geometry is the right object
The four-way agreement (universality, sign, cross-channel ratio, pure-geometry variant pick) reflects real structure that AIC alone is not powerful enough to resolve. Belle II angular B→K* data, when released, will sharpen this.
Current data does not distinguish between these readings. We are not advocating for any of them. We name the readings, name the limitations, and let the next round of measurements settle it. What current data can say is what is in Findings 1–4: the structural test passes.
Pre-Registered Falsification Programme
Each of the following events would falsify the surviving structural claim — that one geometry-derived shape is direction-consistent with the b→sμμ angular anomaly across collaborations and channels:
- A future B→K*μμ measurement (any collaboration, any luminosity, any energy) gives A < 0 on the geometry-derived kernel under non-linear refit.
- Cross-dataset amplitude scatter on the P-basis fits exceeds a factor of ~5 once basis is harmonised.
- Belle II angular B→K* data, when released, gives an amplitude inconsistent with +1.5 ± 0.5 in the P-basis under non-linear refit.
- A cross-channel B⁺→K⁺μμ analysis (when public HEPData becomes available) gives A < 0 under non-linear refit.
A separate, pre-registered model-comparison trigger (linearised ΔAIC < 0 surviving the non-linear refit) has fired in the negative direction. We acknowledge this explicitly: on AIC, the kernel is not preferred over a constant Wilson-coefficient shift. The structural-test claims above are independent of, and unaffected by, this AIC tie.
Stated Limitations
- The kernel is not statistically preferred over a constant Wilson-coefficient shift on AIC (stacked wVFD = 0.35 vs wFREE_C9 = 0.65)
- A free-width Gaussian charm-loop proxy fits comparably in χ² at the cost of one extra shape parameter; the kernel is not the unique q² shape consistent with the anomaly
- The +2.77 AIC unit drift between linearised and non-linear modes is the largest single methodological uncertainty in the project
- Bₛ→φμμ amplitude has a residual ~50% overshoot above basis-corrected expectations, attributed to missing angular covariance and 3 fb⁻¹ statistics
- B→Kμμ has not been tested — public HEPData submissions for the LHCb and CMS analyses are not available; recorded as an open channel
- This paper does not claim that the underlying physics is geometric — only that the geometric kernel passes a structural multi-dataset test that AIC is not powerful enough to resolve on current data
What This Is, And Is Not
One fixed shape from geometry. Five datasets, three channels, two collaborations, two isospin partners. The same direction in 5/5 fits. The kernel never moves.
repro/run_all.sh regenerates every number in the paper from a clean checkout in ~5 minutes. MIT licence.