From mode spaces to a coupling
Explore the chosen collinear blow-up background: which modes exist, how a local residue couples them, and which missing action data still control the photon. The coupling shown here has rank at least three; it is not an observed mass matrix.
(0, 1, 2, −3, −1). Changing this representative can change the mode spaces.A net index does not count every mode
Move through the Hom powers. The blue bar includes paired modes that a chiral index alone cannot reveal.
Exact table and the cusp certificate
The first 13 canonical jet rows have determinant −18196056815164784640. Their full rank makes this cusp ordinary. Conditional on the frozen full modular-form basis, the dimensions follow from blow-up pushforwards and Serre duality.
| q | h⁰ | h¹ | h² | χ |
|---|
The blow-up creates usable jets
For L, local functions are reduced modulo I³. Six monomials remain before the one-dimensional global image is removed. Select a monomial.
z degree → · w degree ↑
w
The three highlighted classes are a selected subspace. The classes 1 and z² also survive the local quotient.
5 + 13 = 18
Five local jet classes plus thirteen base cohomology directions give h¹(L) = 18.
This is local algebra on a complex surface. The diagram does not depict matter inside physical spacetime.
Three independent pairings survive
The cubic pairs H¹(L) with H¹(L⁻³), using a direction in H⁰(KSL²). Click a matrix entry to inspect its residue.
Resz,w[p · n · φ(z) dz ∧ dw]
The leading coefficient is fixed to 1 in a holomorphic coordinate frame. Changing v₆ leaves the determinant −1. This is algebraic normalization, not a kinetic normalization.
The exact certificate chooses canonical coefficient a₄ = 1 and a₅ = 0. Since f(z)dz/z has canonical order 3 and the line frame contributes z², its relative coefficient begins at z⁵.
Gauge cycle and parent assumption
The oriented blocks (A1−L, B−A1, L−B) have line powers (1, −3, 2) and hypercharges (−½, +1, −½). Both sums vanish, and the matrix bracket is nonzero. The bulk cubic is conditional on the standard partially twisted eight-dimensional gauge action W ∝ ∫ Tr(F⁰˒² ∧ Φ²˒⁰). It does not by itself construct a complete global parent.
A flux projection needs an action
The traceless hypercharge Y has a zero linear flux projection and a nonzero quadratic projection. An added common center preserves adjoint charges but changes the possible axion shifts.
| Exact projection | Value |
|---|---|
| Tr[(Y+sI)c₁] | 0 |
| ∫ Tr[(Y+sI)c₁²] | 54 |
| Tr[Y+sI] | 0 |
The internal class has v² = −9. The quadratic integral is 54 − 144s, while the trace is 8s. A theta coefficient alone is not a vector mass.
Supply hypothetical independent rows
These inputs illustrate what a reduction must determine. They are not derived terms in MTFT's B3 action.