A research program exploring whether the constants of nature are counts: topological invariants of a compact geometry, with particles read as arithmetic-geometric objects.

Arithmon is the program: the durable hypothesis that the dimensionless constants of physics are arithmetic and topological invariants of geometry, rather than free parameters to be fitted. It is meant to outlive any single implementation: a direction of inquiry, in the spirit of a research program rather than one conjecture.

GIFT is its founding framework: the first concrete, dated realization of that hypothesis. A specific compact G₂-holonomy geometry: the manifold K₇, the (21, 77) topology, an E₈×E₈-motivated exceptional architecture. From it, a set of exact, parameter-free relations among Standard-Model and cosmological observables is derived, with a machine-checked formal core.

GIFT is the founding framework of the Arithmon program.

A framework can be revised, sharpened, or falsified. The program is the question it answers to.

• 0 free parameters
• 33 exact relations to observables
• 15 stated axioms (4 prediction-chain + 11 K3), 0 sorry in the Lean core
• Falsifiable: named, dated predictions (e.g. the δ_CP window, the θ₂₃ octant)
• Non-generic at set level (~10⁻⁶, assumption-free null model)
• Numerical precision is reported as a secondary figure, not the headline.

  Arithmon   the program                  →  are the constants of nature counts?
  GIFT       the first framework          →  one dated answer: K₇, (21, 77), 33 relations
  Sieve      the anti-numerology test     →  how surprising is that answer?
  Lean       the certified counting layer →  the arithmetic of the test, machine-checked
  Atlas      the map of neighbouring work →  who else stands near the question

• Program: the charter (hard core, heuristics, fair-play rules) and the open problems, with the confrontations scoreboard of frozen, dated predictions.
• Atlas: the annotated map of adjacent work, one precise delta per entry.
• Sieve: the methodology standard for the question how surprising is a claimed exact relation between mathematical invariants and measured constants? Inputs frozen and deposited before any search ran (DOI 10.5281/zenodo.20666879), four null models, calibration on historical verdicts (Eddington fails, the quantum Hall relation passes), and a scorecard reusable on any framework, including ours.
• Lean: the machine-checked formal layer of the Sieve. Certified expression-space counts (the haystack the trials factor divides by) and the in-framework-theorem rebate, in Lean 4.

Three ways in, by who you are.

• You do physics. Start with the Program (the hypothesis and the open problems) and the Sieve (whether the exact relations are more than coincidence). The physics itself, derived and dated, is in GIFT.
• You do mathematics. Start with Lean (the certified counts, 0 sorry) and the geometry entries in the Atlas (G₂ holonomy, K3, the (21, 77) topology). The full formal core is at gift-framework/core.
• You read with suspicion. Start with the Sieve: the test built to catch numerology, frozen before any search ran and calibrated so that Eddington fails and the quantum Hall relation passes. Then the confrontations scoreboard (named predictions against scheduled data) in the Program. Read these before you read any claim we make.

• Cited in Physics Letters B 878 (2026) 140566.

_{Program: arithmon.com · Founding framework: GIFT · Formal core verified with Lean 4.}