Drill down

The mathematics

Everything on this page is implemented in the repository and covered by tests. Where a statement is a conjecture or an assumption, it says so.

1. Action and field equation

The total action separates into bulk, boundary, and coupling pieces:

$$S = S_{\text{bulk}}[g,\Psi] + S_{\partial}[\phi] + S_{\text{int}}[g,\phi] + \dots$$

The boundary term is a scalar phase field on the surface Σ with induced metric h:

$$S_{\partial} = \frac{1}{2\kappa}\int_\Sigma d^{2}x\,\sqrt{|h|}\,h^{ab}\nabla_a\phi\nabla_b\phi - \int_\Sigma d^{2}x\,\sqrt{|h|}\,V(\phi)$$

and the metric coupling projects boundary gradients onto the normal direction:

$$S_{\text{int}} = \lambda\int_M d^{D}x\,\sqrt{|g|}\,P^{ab}_{\mu\nu}(\nabla_a\phi)(\nabla_b\phi)\,g^{\mu\nu},\qquad P^{ab}_{\mu\nu}=h^{ab}n_\mu n_\nu$$

Varying with respect to φ gives the boundary field equation, and varying with respect to the metric adds a boundary stress-energy T^φ to the Einstein equations. The free massless case on S² is the c = 1 compact boson, a standard two-dimensional conformal field theory. The dimensionless rigidity is identified as κ = z/2 = 3.

2. Spectrum on the sphere

For the round metric on S² of radius a, the scalar Laplacian has

$$-\nabla^2 Y_{\ell m} = \frac{\ell(\ell+1)}{a^2}\,Y_{\ell m},\qquad \ell=0,1,2,\dots,\quad \text{multiplicity } 2\ell+1.$$

The ℓ = 1 space is three-dimensional. It is a spatial spin-1 triplet under SO(3). It is not a spin-½ multiplet and does not by itself represent three copies of a fermion species. This distinction is the root of the September 2026 correction below.

The topological selection of S² is sound: a compact, orientable, simply connected closed surface is a sphere. Topology alone does not select the round metric or its degeneracies.

3. The fine-structure constant

$$\frac{1}{\alpha}=(\ln p)^2+\frac{z}{2}+\gamma-\frac{1}{2\pi}=133.62+3.00+0.577-0.159=137.039$$

Measured: 137.035999. The stated origins of the four terms are lattice phase-space screening (two propagators each scaling as ξ²/a² = ln p), the bare rigidity κ, the harmonic-number correction Σ1/k = ln(p−1) + γ + O(1/p), and a Kaluza–Klein zero-mode subtraction. Since p was selected by this formula, the agreement is anchored, not predicted. What is checkable is the sensitivity: d(1/α)/d(ln p) = 2 ln p ≈ 23, so a 1 percent change in p moves 1/α by about 0.2. Implementation: bpr/alpha_derivation.py.

4. Neutrino and CKM mixing

$$\sin^2\theta_{12} = \frac{1}{3} - \frac{1}{3.5\,\ln p} = 0.3086 \qquad \text{(JUNO 2025: } 0.3092\pm0.0087)$$

The tri-bimaximal 1/3 is derived; the coefficient 3.5 in the curvature correction is fitted, so this is labelled framework. The reactor angle uses a Gaussian boundary-localisation factor:

$$\sin\theta_{13} = \frac{e^{-1}}{\sqrt{2\,n_{\text{gen}}}} = 0.1502 \;\Rightarrow\; \theta_{13}=8.64^\circ \qquad \text{(PDG: } 8.54^\circ\pm0.15^\circ)$$

Note that this uses n_gen = 3 as an input. The CKM CP phase is

$$\delta_{CP} = \frac{\pi}{2} - \frac{1}{\sqrt{z+1}} = 68.3^\circ \qquad \text{(PDG: } 68.5^\circ\pm5.7^\circ)$$

and the three CKM angles follow from quark mass ratios in the flavor ansatz below, with the Jarlskog invariant J = 2.9 × 10⁻⁵ against a measured 3.1 × 10⁻⁵. Implementation: bpr/neutrino.py, bpr/qcd_flavor.py.

5. The flavor ansatz, honestly labelled

Fermion masses are modelled with effective labels ℓ_k and the rule m_k ∝ ℓ_k² (up-type quarks and charged leptons) or a shifted quadratic for down-type quarks. The labels are written as functions of z and n_gen:

Sector	Labels	z = 6, n_gen = 3	Status
Up quarks (u, c, t)	1, z(z−2), (z²−1)(z+n_gen+2−N_c)+n_gen	1, 24, 283	conjectural
Down quarks (d, s, b)	1, z−2, z(z−1)	1, 4, 30	conjectural
Charged leptons (e, μ, τ)	1, √(z(z²−1)), z(z+n_gen+1)−1	1, √210, 59	conjectural

These reproduce, for example, m_μ/m_e = 210 (measured 206.77) and m_u/m_t = 1/283² (giving m_u = 2.16 MeV against a reference 2.16 MeV). They are not eigenvalues of any fermionic Hamiltonian in the framework. ℓ² is not the exact Laplacian eigenvalue ℓ(ℓ+1), and √210 is not an integer angular momentum. Until September 2026 the labels were described as derived. They are retained as phenomenology only. Implementation: bpr.qcd_flavor.derive_l_modes, which now reports CONJECTURAL for every label and EMPIRICAL_INPUT for n_gen.

6. Two withdrawn derivations, with the corrections

6a. The color-bundle index behind ℓ_t = 283

The claim. The +n_gen = +3 offset in the top-quark label was attributed to an Atiyah–Singer index: index(D_color) = c₁(color bundle) = 3 on S².

Why it fails. For an ordinary SU(r) bundle E over S², the determinant line bundle is trivial and the connection is su(r)-valued, so

$$\operatorname{ind} D_E = \int_{S^2} c_1(E) = \frac{i}{2\pi}\int_{S^2}\operatorname{tr}F = 0.$$

The Â-genus has no degree-two term, and a second Chern number cannot contribute on a two-dimensional base. A nonzero index requires extra U(1) data: twisting by a line bundle O(q) gives c₁(E ⊗ O(q)) = rq, so index = rq. That q is not derived from p or z. And an index counts net zero modes; it does not add an integer to an angular-momentum label.

What survives, conditionally. On CP¹ with spin bundle K^1/2 = O(−1), a Dirac field twisted by O(q) has positive-chirality zero modes counted by H⁰(O(q−1)) and negative by H¹(O(q−1)), so

$$n_+=\max(q,0),\qquad n_-=\max(-q,0),\qquad \operatorname{ind}=q.$$

q = 3 gives three chiral modes. So does q = 4 give four. Topology does not select three. Implementation with exact rational arithmetic: bpr/flavor_foundations.py. Full note: color_bundle_index.md.

6b. "Exactly three generations" from the compact boson

The claim. The c = 1 compact boson at R² = 3 was said to contain exactly one SO(3) triplet of spin-½ single-particle operators, hence three families, with a fourth excluded by the fusion ring.

Why it fails. With the standard normalisation p_L,R = m/R ± nR and h = p_L²/4, the conformal spin of the momentum-winding lattice is

$$s = h - \bar h = mn \in \mathbb{Z},$$

an integer for every (m, n) and every radius. The untwisted lattice has no spin-½ operators. Dressing with an unspecified weight-1/16 field is not a construction. The 3 + 1 decomposition of four labels was asserted without any group action on the states. The fusion V_(1,1)V_(1,1) → V_(2,2) produces a primary, not a descendant, so it excludes nothing. And 10/3 + 1/16 = 163/48, not 167/48 as written.

Consequence. n_gen = 3 is an empirical input. No fourth-generation exclusion follows from BPR topology. Full note: generations_from_CFT.md.

7. The Casimir prediction

The one laboratory-scale prediction that has been in the framework since the beginning is a deviation in the Casimir force at sub-micron separation:

$$\frac{\Delta F}{F_{\text{Casimir}}} = \delta\left(\frac{\lambda_{\text{eff}}}{R}\right)^2,\qquad \delta = 2\Delta_\phi = 2$$

The exponent δ = 2 is derived under Postulate 0c (a quasicrystalline projection with unit Pisot inflation, Δ_φ = 1). An earlier fitted value δ ≈ 1.37 is superseded and would now count as a refutation if measured. The phonon-collective channel brings the predicted coupling to about 10⁻⁸, one to two orders of magnitude from current MEMS sensitivity. Implementation: bpr/recursive_boundary.py.

8. The condensate result, and its limits

For the frozen ring Hamiltonian with hopping C > 0 and repulsive quartic g > 0 at fixed norm N,

$$H \ge -2CN + \frac{gN^2}{2p},$$

with equality for the uniform, phase-aligned state. This is an exact classical statement, and gradient-flow tests confirm it. It does not compute the quantum vacuum of the interacting substrate. The characteristic temperature T* ≈ C/ln p on the two-sphere comes from Rayleigh–Jeans occupation on an assumed dispersion, cutoff, and density. The substrate temperature is not derived. Coherence is therefore conditional, not established. Implementation: bpr/condensate_regime.py.