Reproduce

Run the code

Every number on this site comes from a function in a public repository. Here is how to get it and check it.

Quick start

# clone and install (Python 3.10+)
git clone https://github.com/jackalkahwati/BPR-Math-Spine.git
cd BPR-Math-Spine
pip install -e .

# run the test suite (about a minute on a laptop)
pytest -q

# derive constants from the two substrate integers
bpr predict --p 104761 --z 6
bpr constants --filter alpha

# verify field equations symbolically
bpr verify

The FEniCS finite-element dependency is optional. Most functionality, including everything on this site, runs without it. A Docker setup with FEniCS included is in the repository.

Reproduce the September 2026 corrections

python3 - <<'PY'
from bpr.flavor_foundations import compact_boson_weights, su_bundle_twisted_index, sphere_line_zero_modes
print(compact_boson_weights(1, 1))        # (4/3, 1/3): conformal spin mn = 1, an integer
print(su_bundle_twisted_index(3))          # 0: ordinary SU(3) bundle on S^2 has index zero
print(sphere_line_zero_modes(3).index)     # 3: only with an O(3) line twist, which is an input

from bpr.qcd_flavor import derive_l_modes
m = derive_l_modes(6, 3)
print(m["l_up"], m["derivation_status"]["l_up_2"], m["input_status"])
PY

Python API

from bpr.alpha_derivation import derive_alpha
r = derive_alpha(p=104761, z=6)
print(r.inv_alpha_0)                     # 137.039

from bpr.neutrino import PMNSMatrix
print(PMNSMatrix().mixing_angles())      # theta12 33.7, theta23 49.3, theta13 8.64

from bpr.qcd_flavor import CKMMatrix
print(CKMMatrix().mixing_angles())       # delta_CP 68.5 deg, J = 2.9e-5

from bpr.cosmology import BPRDarkEnergyEOS
print(BPRDarkEnergyEOS().desi_tension)   # w0 = -1.0, 2.9 sigma from DESI DR1

Papers and documents

The paper (PDF)

13 pages. Documents the BPR 1.0 derivation chain. The first page carries a status note explaining what has since been withdrawn. Read it as a historical record of the 1.0 claim, with the status page alongside.

The complete framework (PDF)

The long-form document with all 1.0 theories. Same caveat: parts of it describe claims that have been narrowed or withdrawn. The repository markdown version is kept current.

Negative-findings registry

Everything determined not to work, in one document. Start here if you are reviewing.

Derivation notes

One markdown note per derivation, including the withdrawn ones with their corrections.

A Wolfram Language port of the core equations is in wolfram/ for Mathematica users. A local REST API (uvicorn api.main:app) exists in the repository for scripting; there is no hosted API.

Reviewing, criticising, contributing

The most useful thing a reader can do is find the next invalid step. Two were found in September 2026 by an outside review and are now on record. Open an issue on GitHub, or write to jack@thestardrive.com. Pull requests that add a test for a claim, or that fail an existing one, are especially welcome.

Citation: Al-Kahwati, J. (2026). Boundary Phase Resonance: BPR-Math-Spine. GitHub repository, MIT license.