# Exact calculations for the exceptional varieties

All indices, signs, coefficient variables, and LP normalizers are those of the
manuscript. Frozen coefficients remain independent; no numerical specialization
is used in the polynomial identities or rank calculations.

## Check the mirror calculations

Run these commands from this directory, using Python 3:

```sh
python verify_certificates.py
python remaining_e7_contacts.py
```

These two scripts use only the Python standard library. They check:

- Eight Cayley-plane coefficient systems, their normalized kernels, and maximal
  minors equal to 1 or -1; the coefficient 2 in the specified boundary function.
- Eighteen Freudenthal boundary coefficient systems.
- The full 28-dimensional integral marking, determinant 1, all 87 terms of the
  source exchanges, and twelve coefficient determinations by initial forms.
- One hundred nonmonomial support-subset factorizations, eighty irreducible
  factors, and all rational inequalities used in the Freudenthal divisor proof.
  Seventy cases give bound zero and ten give bound one; the largest denominator
  of a rational certificate is 13.
- Eighty first-step valuation comparisons for the eight remaining mixed-contact
  directions. They establish the limitation of those initial-form calculations,
  not the mixed-contact incidence counts themselves.

## Check the Freudenthal source

```sh
python check_freudenthal_quadrics.py
python check_quadric_rank_certificate.py
python check_freudenthal_source.py
python check_freudenthal_primes.py
```

The first script uses only the standard library. The other two require SymPy.
All arithmetic is exact over the rational numbers.

`check_freudenthal_quadrics.py` constructs the 144 cyclic translates of the eight
printed quadrics and computes their rank, 133.

`freudenthal_source_coordinates.json` gives all 56 coordinate functions as
Laurent expressions in the ten auxiliary mutable variables and eighteen
polynomial frozen variables. `check_freudenthal_source.py` checks all 144
quadrics, the ten inverse coordinate identities, all ten first-mutation
identities, and 204 negative-slice divisibilities in the adjacent Laurent rings.
The highest pole order in those tests is two. It also checks that no frozen
variable occurs in a denominator.

`check_freudenthal_primes.py` reconstructs the eight-dimensional grading from the
quadrics and checks the homogeneous factorizations used in the source proof.
It tests all 21 possible degree-one factors of the three nonlinear initial
variables and all ten possible divisibilities of a first mutation by its old
variable. Each test uses the complete homogeneous weight space, spanned by all
coordinate monomials of the required degree and weight. The largest spanning
set has six members.

The report files are:

- `verification_report.json`
- `remaining_e7_contacts.json`
- `freudenthal_quadrics_report.json`
- `freudenthal_source_report.json`
- `freudenthal_source_divisibilities.json`
- `freudenthal_primes_report.json`

## Reconstruct the mirror certificate files

The stored mirror certificates can also be regenerated from the Laurent formulas:

```sh
python e6_recognition.py
python e7_calibration.py
python e7_boundary_recognition.py
python e7_divisors.py
python verify_certificates.py
```

Generation uses SymPy and, for the divisor multipliers, SciPy.
`e7_boundary_recognition.py` reads the fixed Laurent formulas from
`e7_divisors.py`, so keep the scripts together. The numerical optimizer only
suggests multipliers; the saved identities are rational and the standard-library
verifier checks them without trusting floating-point feasibility.

## Scope

These are algebraic checks in the printed coordinates. The geometric support
bounds, local counting arguments, and the eight mixed-contact calculations are
proved in the manuscript and are not certified by these scripts. No script
asserts that every Freudenthal linear coordinate is a cluster variable, or that
the generated Freudenthal LP algebra equals its upper LP algebra.

## Static rank certificate

`freudenthal_quadric_rank_certificate.json` contains the 144 original sparse
rows, 133 normalized echelon rows with distinct leading monomials, and exact
change-of-span identities in both directions. All indices are zero-based.
`check_quadric_rank_certificate.py` checks every identity and confirms that the
original rows are the cyclic translates of the printed equations.