K3, χ=24
and the spectrum

1. Why a smooth quartic has χ=24

Take a smooth degree-four surface X in complex projective space ℂP³, for example the Fermat quartic:

z₀⁴ + z₁⁴ + z₂⁴ + z₃⁴ = 0

It is smooth: all partial derivatives vanish simultaneously only when every zᵢ is zero, which is not a projective point. Write H for the hyperplane class on X. The tangent-normal exact sequence gives:

c(TX) = (1+H)⁴ / (1+4H)
      = 1 + 6H²       (through complex degree 2)
∫X H² = 4
χ(X) = ∫X c₂(TX) = 6 · 4 = 24

Adjunction gives a trivial canonical bundle; the hyperplane theorem gives simple connectedness. Thus X is a K3 surface. This calculation explains 24 within this geometry. It does not yet select the geometry from all possibilities.

2. Further exact numbers and their meaning

With the complex orientation, the signature formula σ=(c₁²−2c₂)/3 gives:

σ = −16
b₂ = χ−2 = 22
b₂⁺ = 3,     b₂⁻ = 19
h¹,¹ = b₂−h²,⁰−h⁰,² = 22−1−1 = 20

Here b₁=b₃=0 and b₀=b₄=1. These numbers count topological and complex structures. Without a physical compactification model they do not count particles or observable spectral lines.

The local dimension of Ricci-flat K3 metric moduli at fixed volume is 3·19=57: positive three-planes in H²(X,ℝ) with signature (3,19). Including the volume scale gives 58. Additional fields or constraints change the physical parameter accounting.

Geometric background: Daniel Huybrechts, Lectures on K3 Surfaces.

3. From χ to the scalar heat kernel

Now take a smooth closed K3 with a Ricci-flat Riemannian metric. For the nonnegative scalar Laplacian Δ, using the full real tensor norm:

∫X |Riem|² dV = 32π²χ = 768π²

The integrated scalar heat coefficient is therefore:

a₄ = [1 / (360 · 16π²)] ∫X 2|Riem|² dV
   = χ/90 = 4/15

At volume 1, as t↓0:

Tr exp(−tΔ) = 1/(16π²t²) + 4/15 + O(t)

The general coefficient and tensor conventions appear in Vassilevich, Heat kernel expansion, equation 4.28. Other operators, such as a coupled Dirac², have different terms. Ricci terms cannot simply be discarded for an approximate metric that is not exactly Ricci-flat.

4. ζ(0) is exact; ζ′(0) needs the spectrum

Define ζΔ(s)=Σ_λ>0λ^−s for Re(s)>2, then analytically continue. A connected closed K3 has one scalar constant zero mode, so:

ζΔ(0) = a₄ − dim ker Δ = 4/15 − 1 = −11/15

This is not ζ′Δ(0). The regularised determinant is det′Δ=exp(−ζ′Δ(0)) and requires further spectral information. A topologically fixed ζ(0) does not fix its derivative.

The scale must also be explicit. For a positive factor c:

ζ(cΔ)(s) = c^(−s) ζΔ(s)
ζ′(cΔ)(0) = ζ′Δ(0) − (ln c)ζΔ(0)

Under g→r²g, λ→r⁻²λ and V→r⁴V. Thus λ√V is scale invariant in four real dimensions. This justifies the old scripts' volume convention, but does not validate their matrix assembly.

5. What currently blocks the spectrum calculation

The old k=3,…,6 files have nonzero first eigenvalues. An independent check locates a problem in the energy matrix of spectrum_matrices. In the basis φₐᵦ=sₐs̄ᵦ/D, the sum ΣGₐᵦφₐᵦ is exactly the constant function 1. Its Dirichlet energy must be zero, even for finite samples.

This is a matrix consistency error, not physical energy of a constant function. Independent derivatives were also checked with finite differences. The matrices differ by about 19.5% in relative Frobenius norm. This percentage is not an error estimate for individual eigenvalues.

The old precise eigenvalues, estimated ζ′(0) and predictions depending on them therefore do not return as verified results. The check uses a small diagnostic sample; it does not provide a converged replacement K3 spectrum.

6. What constant j fixes

For a minimal elliptic K3 Weierstrass model over ℂP¹, y²=x³+f x+g, f and g have degrees 8 and 12. With f=a h², g=b h³ and h of degree four:

j = 1728 · 4f³/(4f³+27g²)
  = 6912a³/(4a³+27b²)

j is constant over the base but still depends on a³/b². Taking h with four distinct zeros and a nonzero discriminant factor gives the generic four-I₀* family. This does not select a unique constant value of j.

1 + (10−1) + (47−1) = 56

For these families and orbifold examples: Justin Sawon, Isotrivial elliptic K3 surfaces and Lagrangian fibrations. An elliptic modulus or axiodilaton is not automatically the electromagnetic α; a physical dictionary is needed.

7. What the action principle still requires

Stationary action means δA=0, not necessarily a minimum. Critical points of the Einstein-Hilbert functional on closed manifolds at fixed volume are Einstein metrics Ric=λg, not necessarily Ricci-flat metrics. A round sphere illustrates the distinction. See Introduction to Einstein manifolds.

Nor is the real weight e^−S/ε the oscillatory quantum amplitude e^iA/ℏ without an additional construction. Geometry, fields, action, couplings and observations must form one explicit physical model.

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v0.24 · 16 September 2026