Author: Brieuc de La Fournière
Independent researcher
We compute the Kaluza–Klein spectrum and harmonic cohomology of a compact twisted connected sum (TCS) 7-manifold K⁷ with Betti numbers (b₂, b₃) = (21, 77), using the explicit 169-parameter G₂ metric constructed in our companion paper [I]. All computations exploit the adiabatic decomposition K⁷ ≈ K3 × T² × I along the TCS neck, which we validate numerically to < 0.002% fiber variation across 5 independent tests.
The main results are: (i) what appears to be the first explicit numerical computation of scalar, 1-form and 2-form Kaluza–Klein spectra on a compact G₂ manifold, with spectral gap λ₁ = 0.1244 and Weyl exponent α = 1.998; (ii) spectral confirmation of all Betti numbers b₀ = 1, b₁ = 0, b₂ = 21, b₃ = 77; (iii) explicit construction of 21 harmonic 2-forms and 77 harmonic 3-forms with radial profiles, revealing a 2210× self-dual/anti-self-dual gap in the K3 intersection matrix; (iv) spectral democracy: the scalar, 1-form, and 2-form Laplacians share identical spectra to 10⁻⁴ precision, a consequence of the near-constant fiber metric; (v) stability of the spectral gap under singular perturbations of the K3 fiber.
Connections between these spectral data and quantities arising in particle physics are noted in Appendix E and developed systematically in the author’s companion framework (see Author’s Related Work).
All scripts and results are provided as a companion notebook (< 5 minutes total runtime, CPU-only) for independent verification.
The spectral geometry of compact Riemannian manifolds with special holonomy is a classical subject with rich connections to topology, representation theory, and mathematical physics [13, 14, 15]. While the existence of compact 7-manifolds with holonomy G₂ was established by Joyce [5, 6], and the twisted connected sum (TCS) construction of Kovalev [7] and Corti–Haskins–Nordström–Pacini [8] has produced large families of examples, essentially no numerical spectral data has been computed on any compact G₂ manifold: the existence theorems do not produce pointwise numerical values of g_{ij}(x).
In our companion paper [I], we constructed an explicit approximate metric on a compact TCS 7-manifold with (b₂, b₃) = (21, 77), given in closed form as a Chebyshev–Cholesky expansion with 169 parameters. A computational Newton–Kantorovich certificate establishes the existence of a unique torsion-free G₂ metric within relative distance 4.86 × 10⁻⁶ of our iterate.
This paper exploits the explicit metric to compute the spectral geometry of K⁷: Laplacian eigenvalues and harmonic forms across all relevant form degrees. The adiabatic structure of the TCS neck reduces all 7D spectral problems to families of 1D Sturm–Liouville ODEs, making comprehensive numerical computation tractable. Specifically, we compute:
We are not aware of comparable numerical spectral data on a compact G₂ manifold in the existing literature, though we do not claim an exhaustive survey. Substantial numerical work exists for non-compact G₂ manifolds (see e.g. [9, 10]), but compactness is essential for the Laplacian spectrum to be discrete.
The TCS structure of K⁷ provides a natural dimensional reduction. The manifold decomposes as
K⁷ = (M₁ × S¹) ∪_{K3 × T² × I} (M₂ × S¹)
where M₁, M₂ are asymptotically cylindrical Calabi–Yau threefolds, and the neck region is diffeomorphic to K3 × T² × I with I = [0, 1]. In the neck, the metric takes the block-diagonal form
ds² = g_{ss}(s) ds² + g_{θθ}(s) dθ² + g_{ψψ}(s) dψ² + g_{K3}(s)
where s ∈ [−2, 3] is the seam coordinate, (θ, ψ) parametrize the T² fiber, and g_{K3}(s) is a 4 × 4 metric on K3. This s-dependent block structure reduces 7-dimensional PDEs to families of 1-dimensional Sturm–Liouville ODEs, one for each fiber mode.
The adiabatic ansatz is not an approximation imposed a priori: it is a consequence of the TCS geometry, which we verify a posteriori to high precision. Five independent tests confirm that, for the present smooth metric, the fiber is numerically constant along the neck to high precision:
| Test | Observable | Measured | Expected | ||
|---|---|---|---|---|---|
| Fiber flatness | max s-variation of g_{K3}(s) eigenvalues | < 0.002% | 0 | ||
| Adiabatic additivity | Error in λ_{n,(m,q)} = λ_n^{(0,0)} + V_{(m,q)} | 0.003–0.023% | 0 | ||
| Weyl law | Eigenvalue growth exponent α | 1.998 | 2.0 | ||
| T² isotropy | Splitting | g^{θθ} − g^{ψψ} | 3 × 10⁻⁷ | 0 | |
| K3 near-roundness | Spread of K3 metric eigenvalues | < 0.1% | 0 |
These tests provide strong numerical evidence that the adiabatic decomposition is an excellent approximation for the present smooth metric. Whether this level of accuracy persists for singular K⁷ geometries or for quantities sensitive to fiber-dependent modes remains to be investigated.
To maintain precision in what is and is not established, we follow the convention of [I] and classify all assertions:
| Level | What is claimed | How established | Section |
|---|---|---|---|
| Computed | KK spectrum: scalar, 1-form, 2-form Laplacians | Sturm–Liouville + FD convergence | §3–6 |
| Computed | Betti numbers b₀ = 1, b₁ = 0, b₂ = 21, b₃ = 77 | Spectral (near-zero eigenvalues, gap ratios) | §3–6 |
| Computed | 21 harmonic 2-forms, 77 harmonic 3-forms | PINN + algebraic assembly | §5 |
| Computed | Spectral democracy (p-form Laplacians share spectrum) | Operator comparison + numerics | §6 |
| Computed | Spectral stability under singular perturbations | Perturbative bump analysis | §7 |
| Not claimed | Full 7D fiber-dependent solution | All within 1D seam ansatz | §8 |
| Explored | Physical applications | Pointer to companion framework | App. E |
Section 2 summarizes the metric construction from [I]. Section 3 presents the scalar Laplacian spectrum. Section 4 treats the Hodge Laplacian on 2-forms and confirms b₂ = 21 spectrally. Section 5 constructs the harmonic forms and confirms b₃ = 77. Section 6 analyzes the 1-form Hodge Laplacian and documents spectral democracy across all form degrees. Section 7 examines the spectral stability under singular perturbations of the K3 fiber. Section 8 discusses limitations and compares with other approaches. Section 9 concludes. Appendices A–D contain solver details, PINN architecture, the full KK tower, and numerical tables. Appendix E notes connections to particle physics developed in the companion framework.
This section summarizes the metric construction of [I]; the reader is referred there for full details. We fix notation and record the key properties needed for the spectral analysis.
The metric g_{ij}(s) on K⁷ is parametrized as follows. At each point s along the seam coordinate, the 7 × 7 metric tensor is reconstructed from a lower-triangular Cholesky factor L(s) with entries given by Chebyshev polynomial expansions of order K = 5. With 7 × 7 = 49 independent entries (28 from the lower triangle), each expanded in 6 Chebyshev coefficients, plus normalizations and constraints, the total parameter count is 168 for the Chebyshev sector plus 1 for the asymptotically cylindrical (ACyl) decay rate γ = 5.81, yielding 169 parameters.
The metric is symmetric positive definite (SPD) by construction: the Cholesky factor guarantees g = LLᵀ > 0 everywhere. The determinant is fixed algebraically to det(g) = 65/32. The norm of the associative 3-form satisfies |φ|² = 42.
The companion paper [I] provides a multi-level certification chain:
| Property | Value | Method |
|---|---|---|
| Residual torsion | ‖T‖_{C⁰} = 2.98 × 10⁻⁵ | Spectral differentiation |
| NK contraction | h = 6.65 × 10⁻⁸ | Computational Newton–Kantorovich |
| Distance to exact | ≤ 4.86 × 10⁻⁶ | NK certificate |
| Holonomy | Hol(g*) = G₂ | Joyce [6, Prop. 11.2.3] |
| Torsion class | 99.6% τ₃, τ₁ = 0 exact | Decomposition into 4 classes |
| K3 fiber impact | ≤ 0.07% of 1D torsion | Fréchet derivative over 220k points |
The metric is evaluated on a grid of 80 nodes along s ∈ [−2, 3], with higher resolution (800 nodes) for spectral computations. The key features visible in the metric profile (Figure 1) are:
Figure 1. Metric component profiles g_{ss}(s), g_{θθ}(s), g_{K3}(s) along the seam coordinate s ∈ [−2, 3]. The neck region s ∈ [0, 1] is shaded. Note the near-constancy of the fiber components.
On K⁷ with the adiabatic metric g(s), a scalar eigenfunction f of the Laplace–Beltrami operator Δ₀ decomposes as
f(s, θ, ψ, x_{K3}) = e(s) × exp[i(mθ + nψ)] × η(x_{K3})
where (m, n) ∈ ℤ² labels the T² Fourier mode and η is an eigenfunction of the K3 Laplacian with eigenvalue μ. For each channel (m, n, μ), the radial function e(s) satisfies the Sturm–Liouville problem
-d/ds[p(s) de/ds] + q_{m,n,μ}(s) e(s) = λ w(s) e(s)
with coefficient functions determined by the metric:
| Coefficient | Definition | Physical meaning |
|---|---|---|
| p(s) | √det(g) × g^{ss} | Radial diffusion |
| w(s) | √det(g) | Volume weight |
| q(s) | √det(g) × [m²g^{θθ} + 2mn g^{θψ} + n²g^{ψψ} + μ g^{K3}] | Fiber potential |
Discretization. We use a uniform grid with N = 800 points on s ∈ [−2, 3],
yielding step size h = 6.26 × 10⁻³. The SL operator is discretized as a symmetric
tridiagonal matrix, and the generalized eigenvalue problem is solved using
scipy.linalg.eigh_tridiagonal. Convergence is verified by Richardson extrapolation
from 6 grid sizes (N = 200, 400, 600, 800, 1000, 1200).
Boundary conditions. For the (0, 0, 0) channel, we use both:
The comparison resolves the spectral gap unambiguously (§3.2).
Zero mode. The Neumann computation yields a lowest eigenvalue λ₀ = 3.47 × 10⁻¹³, consistent with machine-precision zero. The corresponding eigenfunction is constant to within numerical noise. This confirms b₀ = 1 spectrally: K⁷ is connected.
Spectral gap. The first nonzero eigenvalue is
λ₁ = 0.1244 ± 0.0001
converged via Richardson extrapolation:
| N | λ₁ (Neumann) |
|---|---|
| 200 | 0.12346 |
| 400 | 0.12409 |
| 600 | 0.12430 |
| 800 | 0.12440 |
| 1000 | 0.12446 |
| 1200 | 0.12450 |
The Dirichlet computation gives λ₁ = 0.1247, confirming that the first Dirichlet eigenvalue is the spectral gap (not a Dirichlet artifact of the zero mode). The Neumann spectrum inserts exactly one zero mode below, then matches the Dirichlet spectrum shifted by one index, approximately 0.25% lower (less restrictive BCs).
Kaluza–Klein scale. The spectral gap determines the KK mass scale:
m_{KK} = √λ₁ / L = 0.353 / L
where L is the characteristic length of the seam coordinate. Setting m_{KK} = M_{GUT} determines L.
Weyl law. For a Riemannian manifold of dimension d, Weyl’s asymptotic law predicts λ_n ~ C n^{2/d} as n → ∞. Our 1D effective problem should give λ_n ~ C n² (α = 2). Fitting the first 20 eigenvalues yields
α = 1.998, C = 0.125
The near-perfect agreement with α = 2 independently validates the metric construction and the adiabatic reduction.
Figure 2. Scalar eigenvalue staircase: λ_n vs n for the (0,0) channel. The Weyl law fit λ_n = 0.125 n² (dashed) is indistinguishable from the data.
Figure 3. First 5 scalar eigenfunctions e_n(s) for the (0,0) Neumann problem. The zero mode (n = 0) is constant; higher modes show increasing oscillation in the neck region.
The T² Fourier decomposition predicts that higher channels (m, n) ≠ (0, 0) should have eigenvalues shifted by a constant potential:
λ_{k,(m,n)} = λ_k^{(0,0)} + V_{(m,n)}
where V_{(m,n)} = m² g^{θθ} + 2mn g^{θψ} + n² g^{ψψ}. This additivity is a consequence of the fiber metric being independent of s.
9 channels computed (60 eigenvalues each, 540 total):
| Channel (m,n) | λ₁ | V_{(m,n)} | Theoretical V |
|---|---|---|---|
| (0,0) | 0.1247 | 0 | 0 |
| (1,0) | 0.9801 | 0.855 | g^{θθ} = 0.855 |
| (0,1) | 0.9801 | 0.855 | g^{ψψ} = 0.855 |
| (1,1) | 1.8356 | 1.711 | g^{θθ} + g^{ψψ} = 1.711 |
| (1,−1) | 1.8356 | 1.711 | g^{θθ} + g^{ψψ} = 1.711 |
| (2,0) | 3.5464 | 3.421 | 4 g^{θθ} = 3.421 |
| (0,2) | 3.5464 | 3.421 | 4 g^{ψψ} = 3.421 |
| (2,1) | 4.4018 | 4.277 | 4 g^{θθ} + g^{ψψ} = 4.277 |
| (1,2) | 4.4018 | 4.277 | g^{θθ} + 4 g^{ψψ} = 4.277 |
The adiabatic additivity is exact to within 2%, limited by the Dirichlet boundary conditions, not by physical s-dependence of the fiber. Key observations:
Figure 4. T² channel eigenvalue spectra. Each channel is shifted vertically by V_{(m,n)} = 0.855(m² + n²). The near-perfect overlap of the shifted spectra demonstrates adiabatic additivity.
The K3 fiber contributes additional eigenvalue channels through its own Laplacian eigenvalues μ_K3. These are treated analogously to the T² channels: each K3 eigenvalue μ adds a potential V = μ × g^{K3}(s) to the Sturm–Liouville problem.
K3 adiabatic additivity is verified on 8 test channels (with K3 eigenvalues ranging from μ₁ to μ₈), with gap errors of 0.003–0.023%. The small but nonzero errors reflect the K3 eigenvalues’ slight s-dependence.
Effective fiber metric. The inverse metric on the 6D fiber K3 × T² is block-diagonal:
g^{fiber} = diag(0.855, 0.855, 1.208, 1.208, 1.208, 1.208)
────T²──── ──────────K3──────────
The K3 block is nearly round: its eigenvalues are [1.2075, 1.2087], with spread < 0.1%.
Three-scale hierarchy. The first eigenvalue in each fiber sector determines a mass scale:
| Sector | First eigenvalue | Scale (1/L units) |
|---|---|---|
| Neck (radial) | λ₁ = 0.125 | √λ₁ = 0.353 |
| T² | V_{(1,0)} = 0.855 | √V = 0.925 |
| K3 | V_{K3,1} = 1.208 | √V = 1.099 |
The ordering neck < T² < K3 reflects the geometric size hierarchy of the three sectors.
Complete KK tower. Enumerating all states below λ = 20 using the additive decomposition yields:
1744 distinct eigenvalues (4460 states with multiplicities)
The multiplicities arise from T² channel degeneracies and K3 eigenvalue multiplicities. The tower is sparse at low energies (dominated by neck modes) and dense at high energies (as T² and K3 channels overlap).
The Hodge Laplacian Δ₂ = dδ + δd acts on 2-forms on K⁷. Under the adiabatic decomposition, 2-forms on K⁷ fall into natural channels based on their index structure:
| Type | Local form | Origin |
|---|---|---|
| I | f(s) ds ∧ dθ | Neck–T² mixed |
| II | f(s) ds ∧ dψ | Neck–T² mixed |
| III | f(s) dθ ∧ dψ | T² sector |
| IV | f(s) ω_I(x_{K3}) | K3 2-forms |
Each type leads to a separate Sturm–Liouville problem with coefficient functions involving the appropriate metric components.
The computation reveals a clear spectral structure:
The gap ratio exceeds 10⁴, making the identification of b₂ = 21 unambiguous. The 21 harmonic 2-forms decompose as 11 from the N₁ building block and 10 from N₂, consistent with the Mayer–Vietoris computation b₂ = rk(N₁) + rk(N₂) = 11 + 10 = 21.
Figure 5. Eigenvalue spectrum of Δ₂ on 2-forms. The first 21 eigenvalues cluster near zero (< 10⁻⁸); a gap of 4 orders of magnitude separates them from the 22nd eigenvalue at 0.12. This confirms b₂ = 21 spectrally.
The K3 harmonic 2-forms carry an intersection form Q₂₂ defined by
Q_{IJ} = ∫_{K3} ω_I ∧ *ω_J
where * is the K3 Hodge star. The SU(2) holonomy of K3 splits 2-forms into self-dual (SD, *ω = +ω) and anti-self-dual (ASD, *ω = −ω) sectors, corresponding to the decomposition b₂(K3) = b⁺₂ + b⁻₂ = 3 + 19.
On our K3 fiber, the Q₂₂ eigenvalues separate cleanly:
| Sector | Eigenvalues | Count |
|---|---|---|
| SD | 4.863, 5.499, 7.795 | 3 |
| ASD | −0.00423 to −0.00219 | 19 |
The SD/ASD gap is
|Q_{SD}| / |Q_{ASD}| ≈ 2210
This gap is geometric (arising from the Hodge star on 2-forms), not spectral (the L² norms G_L2 ≈ I are approximately uniform across all forms). Its origin is:
Physical interpretations of this gap are explored in Appendix E.
The K3 surface has b₂(K3) = 22. The Kähler form ω_{K3} and the real and imaginary parts of the holomorphic (2,0)-form account for 2 of these; the remaining 20 (1,1)-forms are computed independently.
We compute these 20 harmonic 2-forms using a Physics-Informed Neural Network (PINN) on the K3 metric obtained from cymyc [10]. The PINN enforces the harmonic condition Δ₂ ω = 0 as a loss function, with L² orthonormalization at each training step.
The resulting Gram matrix is nearly diagonal:
G_{L2}(I, J) = ∫_{K3} ω_I ∧ *ω_J ≈ δ_{IJ}
with diagonal entries = 1.0 and max off-diagonal = 0.012. The intersection form Q has signature (1, 19), consistent with the K3 lattice of signature (3, 19) restricted to the (1,1)-sector.
The 21 harmonic 2-forms on K⁷ are assembled from the K3 harmonic forms using the TCS structure. Each form has the product structure
ω_I(s, x) = f_I(s) × ω̃_I(x_{K3})
where the radial profile f_I(s) solves the harmonic ODE
d/ds[√g × g^{ss} × f_I'] = 0
yielding a monotone function that interpolates between 1 on one ACyl end and 0 on the other. The 11 forms from N₁ decay from left to right; the 10 forms from N₂ decay from right to left.
The assembled forms satisfy 11 verification checks:
| Check | Result |
|---|---|
| Number of forms | 21 (= b₂) |
| L² orthonormality | G_{L2} diagonal ≈ 1.0, off-diag max = 0.012 |
| Q₂₂ signature | (3, 19) |
| SD eigenvalue count | 3 |
| ASD eigenvalue count | 19 |
| SD/ASD gap | 2210× |
| Cross-block coupling | ‖Q_{SD,ASD}‖_F = 0.018 |
| Profile monotonicity | All f_I monotone |
| Boundary values | f_I → 1 or 0 at each end |
| N₁/N₂ assignment | 11 + 10 = 21 |
| Consistency with Δ₂ | Same count as Hodge Laplacian (§4) |
Figure 6. Harmonic 2-form profiles f_I(s) across the seam. The 11 N₁-type forms (blue, solid) decay from left to right; the 10 N₂-type forms (red, dashed) decay from right to left. The neck region s ∈ [0, 1] is shaded.
The harmonic 3-forms on K⁷ decompose into three types, following the Mayer–Vietoris analysis of [8]:
| Type | Local form | Count | Origin |
|---|---|---|---|
| Constant | Ω_α(x_{K3}) ∧ ds (or pure K3 3-forms) | 35 | b₃(M₁) + b₃(M₂) − rk corrections |
| dθ-fiber | dθ ∧ ω_I(s, x_{K3}) | 21 | T² fiber × K3 2-forms |
| dψ-fiber | dψ ∧ ω_I(s, x_{K3}) | 21 | T² fiber × K3 2-forms |
Total: 35 + 21 + 21 = 77 = b₃.
The computation confirms:
We are not aware of a prior explicit spectral confirmation of b₃ = 77 on a compact G₂ manifold. Combined with §3.2 (b₀ = 1), §4.2 (b₂ = 21), and §6 (b₁ = 0), all Betti numbers of K⁷ are confirmed:
(b₀, b₁, b₂, b₃, b₄, b₅, b₆, b₇) = (1, 0, 21, 77, 77, 21, 0, 1)
where b_k = b_{7−k} by Poincaré duality.
The Hodge Laplacian Δ₁ = dδ + δd acts on 1-forms on K⁷. On a Ricci-flat manifold, the Weitzenböck identity gives Δ₁ = ∇*∇ (the rough Laplacian), since the Ricci curvature contribution vanishes.
Theorem. On K⁷ with diagonal s-dependent metric, for a 1-form α = f(s) dθ:
Δ₁(f dθ) = Δ₀(f) · dθ
That is, the 1-form Laplacian on transverse modes reduces exactly to the scalar Laplacian.
Proof. Since f depends only on s and θ is a coordinate (df/dθ = 0):
Therefore Δ₁(f dθ) = (dδ + δd)(f dθ) = 0 + δd(f dθ) = Δ₀(f) · dθ. □
The same argument applies to f(s) dψ. The dθ and dψ channels therefore have identical eigenvalue spectra to the scalar Laplacian Δ₀.
ds-channel. For longitudinal 1-forms α = f(s) ds, the operator takes a different form: Δ₁(f ds) = −[(Pf)’/W]’ ds, where P = √g · g^{ss} and W = √g. This leads to a Sturm–Liouville problem with modified coefficient functions p_{eff} = 1/W and w_{eff} = g_{ss}/W.
Spectral democracy. Despite the different operator, the ds-channel eigenvalues are identical to the scalar eigenvalues to 10⁻⁴ precision. This occurs because the fiber metric is flat in s to < 0.002%: when the fiber metric is exactly s-independent, all p-form Laplacians have the same spectrum (up to overall normalization by metric components that are constant). The slight s-dependence produces splitting at the 10⁻⁴ level.
b₁ = 0 confirmed. No genuine zero modes appear in any 1-form channel. The near-zero Dirichlet artifact from the (0, 0) scalar channel does not correspond to a harmonic 1-form: the T² circle forms dθ, dψ are not globally well-defined on the TCS because the hyper-Kähler rotation at the neck mixes the two S¹ factors. By the Seifert–van Kampen argument (§2 of [I]), π₁(K⁷) = {1}, so b₁ = 0.
The smooth K₇ produces only abelian gauge groups. To obtain the Standard Model gauge group SU(3) × SU(2) × U(1), one must consider singular limits of the K3 fiber [1, 11, 12]. An orbifold singularity of type A_{n-1} on K3 produces an SU(n) gauge group via the McKay correspondence.
We model singularities as localized metric perturbations, bump functions centered at position s₀ along the neck:
g_{K3}(s) → g_{K3}(s) × [1 + A × exp(−(s − s₀)²/(2σ²))]
with amplitude A and width σ. This is a phenomenological model; a rigorous treatment requires resolving the orbifold singularity.
Position sensitivity. The mass hierarchy is sensitive to the singularity position:
| s₀ | m₁/m₂ | m₁/m₃ | Comment |
|---|---|---|---|
| 0.0 | 12.3 | 1140 | Too mild |
| 0.35 | 16.5 | 3400 | Best match |
| 0.5 | 14.1 | 2700 | Near neck midpoint |
| 1.0 | 17.9 | 5100 | Too strong |
The optimal position (s₀ ≈ 0.35) lies in the neck region, where the two ACyl halves begin to merge. The m₁/m₂ ratio varies by 38% across the neck; m₁/m₃ varies by a factor ∼ 3.
Spectral gap stability. The scalar Laplacian spectral gap is robust:
No spectral collapse occurs: the hierarchy survives singularities. The eigenfunction localization increases near the bump, concentrating the wavefunction around the singularity.
A realistic G₂-MSSM requires [2, 3]:
The current smooth metric provides the starting point for such constructions. Computing the spectrum and Yukawa couplings on a genuinely singular K₇ is left for future work. The perturbative bump analysis above suggests that the qualitative features (spectral gap, mass hierarchy, Betti number structure) are stable under singular deformations.
The computations in this paper rest on the adiabatic decomposition of K⁷ along the TCS neck. This decomposition is validated by five independent tests (§1.2), all confirming that the fiber metric varies by < 0.002% along s. The spectral computations are converged (Richardson extrapolation, §3.2), and the harmonic form constructions pass all verification checks (§5.2, §5.3).
The key results are summarized:
| Observable | Computed | Status |
|---|---|---|
| Spectral gap λ₁ | 0.1244 (converged) | Computed |
| Weyl exponent α | 1.998 | Computed |
| b₀, b₁, b₂, b₃ | 1, 0, 21, 77 | Computed |
| b₂ gap ratio | 14,635 | Computed |
| SD/ASD gap | 2210 | Computed |
| Spectral democracy | 10⁻⁴ splitting | Computed |
| Spectral stability | λ₁ shift ≤ 36% (all A ≤ 10) | Computed |
The key numerical identities (Q₂₂ signature (3, 19) with 3 SD forms matching the generation count, SD/ASD gap exceeding 2000×, and Betti number arithmetic) are formally verified in Lean 4 as part of a modular certificate system (150 files, 48 published axioms, all proofs closed). The certificate is available in the companion code repository.
1D seam ansatz. All computations use the adiabatic reduction to 1D Sturm–Liouville problems. While this is validated to < 0.002% on the smooth metric, singular K₇ geometries with strong fiber variation would require a full 7D treatment.
NK certificate. The Newton–Kantorovich certificate of [I] uses computational operator norms (not interval arithmetic). Full interval verification is left for future work.
Smooth K₇ gauge group. The smooth K₇ with abelian U(1)²¹ gauge group does not contain the Standard Model gauge group. Realistic model building requires singular limits (§7), which have not been computed on our explicit metric.
G₂-MSSM program (Acharya, Kane et al. [2, 3, 4]): develops the phenomenological framework for G₂ compactifications using abstract TCS properties (Betti numbers, moduli spaces). Our work is complementary: it provides an explicit metric instance from which spectral quantities can be computed numerically.
F-theory [12]: compactifies on Calabi–Yau 4-folds with 7-branes, yielding different phenomenological structures. F-theory naturally produces non-abelian gauge groups from brane stacks, while G₂ compactification uses singularities. The two frameworks address related but distinct questions.
String landscape: our approach does not address the vacuum selection problem. We study one specific TCS manifold in detail, computing spectral quantities to numerical precision, without claiming it is preferred over other vacua.
We have computed the spectral geometry accessible within the adiabatic seam reduction from an explicit G₂ metric on a compact 7-manifold. Starting from the 169-parameter Chebyshev–Cholesky metric of [I], we have computed the scalar, 1-form, and 2-form Laplacian spectra; confirmed all Betti numbers spectrally; constructed all harmonic forms with explicit profiles; characterized the self-dual/anti-self-dual structure of the K3 intersection form; documented spectral democracy across form degrees; and tested spectral stability under singular perturbations.
The central technical finding is that the adiabatic decomposition of the TCS neck reduces all 7D spectral problems to 1D Sturm–Liouville equations, validated to < 0.002% fiber variation by the explicit metric.
Three results may be new, though we do not claim an exhaustive literature survey:
Many important questions remain open: the singular limit for non-abelian gauge groups (§7); the extension to a full 7D fiber-dependent treatment; and the mathematical status of the connections to particle physics recorded in Appendix E. The companion notebook allows independent verification of all computed results in < 5 minutes on a standard CPU.
All scripts, numerical results, and the trained PINN weights used in this paper are deposited as a Zenodo data package:
de La Fournière, B. (2026). GIFT Spectral Physics: Data and Code. Zenodo. DOI: 10.5281/zenodo.18920368.
The companion notebook (12 scripts, < 5 minutes total, CPU-only) reproduces every figure and table. The explicit metric coefficients from [I] and the K3 harmonic forms are included for self-contained verification.
This work was developed through sustained collaboration between the author and several AI systems, primarily Claude (Anthropic), with contributions from GPT (OpenAI) for specific mathematical insights. The computational framework and key derivations emerged from iterative dialogue sessions over several months. This collaboration follows a transparent crediting approach for AI-assisted mathematical research. The value of any proposal depends on mathematical coherence and empirical accuracy, not origin. Mathematics is evaluated on results, not résumés.
This paper is the second in a series. The spectral computations reported here build on the explicit metric constructed in [I]. Physical interpretations of the spectral data are developed systematically in [II].
[I] de La Fournière, B. (2026). An explicit approximate G₂ metric on a compact TCS 7-manifold with certified torsion-free completion. Zenodo. DOI: 10.5281/zenodo.18860358.
[II] de La Fournière, B. (2026). GIFT: Geometric Information Field Theory (v3.3). Zenodo. DOI: 10.5281/zenodo.18837071.
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The Sturm–Liouville problem
-d/ds[p(s) de/ds] + q(s) e = λ w(s) e
is discretized on a uniform grid s_j = s_min + j h, j = 0, …, N−1, with h = (s_max − s_min)/(N − 1). The stiffness matrix A and mass matrix B are symmetric tridiagonal:
A_{j,j} = (p_{j+1/2} + p_{j-1/2})/h² + q_j
A_{j,j+1} = A_{j+1,j} = -p_{j+1/2}/h²
B_{j,j} = w_j
where p_{j+1/2} = (p_j + p_{j+1})/2 (arithmetic mean at half-grid points).
Dirichlet (e = 0): rows 0 and N−1 are removed, yielding an (N−2) × (N−2) system.
Neumann (de/ds = 0): the full N × N system is used. Boundary rows are modified:
This preserves symmetry and correctly captures the zero mode.
Richardson extrapolation from 6 grid sizes (N = 200 to 1200) yields the converged spectral gap λ₁ = 0.1244 with uncertainty ±0.0001 (dominated by the discretization error, not by machine precision).
The dense solver scipy.linalg.eigh is used for Neumann problems (where the mass
matrix has zero eigenvalues that prevent shift-invert). The sparse solver
scipy.sparse.linalg.eigsh with shift-invert (σ = 0) is used for Dirichlet
problems.
Each harmonic 2-form ω_I is represented as a neural network mapping K3 coordinates to 2-form components. The network has 4 hidden layers with 64 neurons each and tanh activation. Training minimizes
L = ‖Δ₂ ω_I‖² + μ × ∑_{J<I} |⟨ω_I, ω_J⟩_{L²}|²
where the first term enforces harmonicity and the second enforces L² orthogonality to previously trained forms.
After training all 20 forms, a Gram–Schmidt procedure is applied using the L² inner product to ensure ⟨ω_I, ω_J⟩ = δ_{IJ}. The resulting Gram matrix has diagonal entries 1.000 and off-diagonal entries < 0.012.
The intersection form Q_{IJ} = ∫ ω_I ∧ *ω_J is computed by numerical integration over the K3 fiber. The signature (1, 19) is consistent with the K3 lattice restricted to the algebraic (1,1) sector.
The KK tower below λ = 20 contains 1744 distinct eigenvalues (4460 states with multiplicities). The table below lists the first 30 states:
| n | λ_n | Channel | Multiplicity |
|---|---|---|---|
| 0 | 0 | (0,0,0) | 1 |
| 1 | 0.125 | (0,0,0) | 1 |
| 2 | 0.499 | (0,0,0) | 1 |
| 3 | 0.855 | (1,0,0) | 2 |
| 4 | 0.980 | (1,0,0) | 2 |
| 5 | 1.122 | (0,0,0) | 1 |
| 6 | 1.208 | (0,0,K3₁) | 4 |
| … | … | … | … |
The full table is provided in the companion notebook data files.
Figure 7. Histogram of KK states as a function of eigenvalue. The density increases with λ, consistent with the Weyl law prediction for effective dimension d_{eff} = 1 + 2 + 4 = 7.
| n | λ_n |
|---|---|
| 0 | 3.47 × 10⁻¹³ |
| 1 | 0.12440 |
| 2 | 0.49760 |
| 3 | 1.11958 |
| 4 | 1.99035 |
| 5 | 3.10989 |
| 6 | 4.47818 |
| 7 | 6.09520 |
| 8 | 7.96092 |
| 9 | 10.07532 |
| 10 | 12.43850 |
| 11 | 15.05003 |
| 12 | 17.90975 |
| 13 | 21.01753 |
| 14 | 24.37324 |
| 15 | 27.97681 |
| 16 | 31.82828 |
| 17 | 35.92753 |
| 18 | 40.27446 |
| 19 | 44.86896 |
The first 21 eigenvalues are < 10⁻⁸ (harmonic forms). The 22nd is 0.12 (first massive mode). See companion notebook for full data.
The 77 × 77 moduli metric G_{IJ} has eigenvalues in [1.66, 12.69]. The full spectrum
is provided in effective_lagrangian_4d_results.json.
The 22 × 22 Yukawa matrix Y(I,J) and the 3 × 3 mass matrix M(i,j) at optimal
positions are provided in wilson_line_3gen_results.json.
The spectral data computed in this paper bear several striking numerical relationships to quantities arising in particle physics. We record the most direct observation (the mass hierarchy from the SD/ASD structure) and summarize the key physical quantities extracted from the metric. Further connections, including topological coupling formulas, gauge running, and falsifiable predictions, are developed in the author’s companion framework.
In M-theory compactified on K⁷, the Yukawa couplings arise from the triple overlap integral of harmonic forms [2, 3]:
Y_{IJK} = ∫_{K⁷} ω_I ∧ ω_J ∧ φ
Under the adiabatic decomposition, this reduces to a 1D integral involving the harmonic 2-form profiles f_I(s) from §5.2 and the intersection form Q_{IJ}(s).
The Yukawa matrix Y has effective rank 3, but the mass matrix M = ψ Y ψᵀ has rank 2 in the adiabatic limit: the T² circle profiles ψ₁, ψ₂ are degenerate by isotropy. The third generation requires a non-adiabatic correction: a potential V_I(s) = c × Q_{II} that breaks the profile degeneracy for any c > 0. The effective coupling c and the matter localization positions (s₁, s₂, s₃) are correlated parameters. At optimized positions, the mass ratios are:
| Ratio | Computed | Experimental (τ/μ/e) | Deviation |
|---|---|---|---|
| m₁/m₂ | 16.5 | 16.82 | 1.9% |
| m₁/m₃ | 3400 | 3477 | 2.2% |
| m₂/m₃ | 206 | 206.7 | 0.3% |
The mechanism has a clear geometric origin in the SD/ASD decomposition of K3 harmonic 2-forms. The SD sector (3 eigenvalues Q ∼ 5–8) provides 100% of the τ and μ masses through 2 independent contributions (the third SD profile is exactly degenerate). The ASD sector (19 eigenvalues |Q| < 0.005) independently generates the electron mass as a residual. The Q-value hierarchy maps directly to fermion masses: Q = (7.16, 5.50, 0.003) → (τ, μ, e). The 2210× SD/ASD gap (§4.3) is the geometric origin of the τ/e ∼ 3500 ratio. The qualitative structure (2 heavy + 1 light, correct ordering) is robust under every Q₂₂ deformation tested (40/40 configurations).
Figure 8. Mass eigenvalues of the 3 × 3 Yukawa matrix as a function of the non-adiabatic coupling c. At c = 0 (adiabatic), only 2 masses are nonzero; for any c > 0, all 3 masses are nonzero.
The effective coupling c_{eff} ≈ 0.18 is ∼ 10× larger than the physical estimate c_{phys} ≈ 0.017 from the smooth metric’s K3 s-derivative. This enhancement factor is genuine (not a normalization artifact; see main text §4.3) and remains an open question.
Dimensional reduction of 11D supergravity on smooth K⁷ yields a 4D N = 1 supergravity theory with gauge group U(1)²¹, 77 chiral multiplets, and 99 total massless fields (1 + 21 + 77). The 77 × 77 moduli Kähler metric has full rank, condition number 7.7, and eigenvalues in [1.66, 12.69]. The gauge kinetic function f ∈ [2.21, 2.32] yields α_{GUT}⁻¹ ≈ 27. The canonically normalized Yukawa coupling from the dominant SD sector is λ_{phys} = 1.36, comparable to the top quark Yukawa.
With SU(8) gaugino condensation: m_{3/2} = 166 GeV (electroweak scale), m_{moduli} = 3.2 TeV. The KK mass scale from the spectral gap is m_{KK} = √λ₁ / L = 0.353 / L. Non-abelian gauge groups require ADE singularities on the K3 fiber (§7); computing the spectrum on a genuinely singular K₇ is left for future work.