A research program exploring whether the constants of nature are counts: topological invariants of a compact geometry.

The dimensionless constants of physics are arithmetic and topological invariants of a compact geometry. None of them is a free parameter.

That sentence is the hard core, and it is the only thing that does not move. The name says the wager in three syllables: from arithmos (number) and -on (particle), the number as particle, the particle as number.

K₇, formerly GIFT, is the founding framework: the first concrete, dated answer. A specific compact G₂-holonomy geometry, the manifold K₇, the (21, 77) topology, an E₈×E₈-motivated exceptional architecture, and from it a set of exact, parameter-free relations with a machine-checked formal core. An answer can be revised or refuted. The question remains.

  Arithmon   the program                  →  are the constants of nature counts?
  K7         the first framework          →  one dated answer: K₇, (21, 77), 33 relations (formerly GIFT)
  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

• 0 free parameters
• 33 exact relations to observables
• 15 stated axioms (4 prediction-chain + 11 K3), 0 sorry in the Lean core
• Non-generic at set level (about 10⁻⁶, assumption-free null model)
• Numerical precision is reported as a secondary figure, not the headline.

Frozen and dated before the data that judges them, never revised after the fact.

In 2025 the δ_CP best fit moved away and the prediction sat outside 1σ; in late 2025 it moved back. The prediction did not move between those two entries, the data did. Both verdicts are recorded, and so is the θ₂₃ one. The full scoreboard is in Program.

A refuted route is a theorem about the territory, so the program publishes its own negative results rather than burying them.

• (21, 77) is not arithmetically privileged at fitting the frozen constants: in a blind sweep over 242,064 Betti pairs, about 64% beat it on the discriminating axis. One candidate selection principle, closed.
• The K3-block coefficient 64/77 is a rational approximation (0.41 percent, basin-dependent), not an exact spectral value. Demoted by the program's own interval certification.
• b₂ + b₃ = 98 = dim K₇ × dim G₂ remains underived. Until it is derived it is an observation, not a principle, and must not be used as one.

• K7: the founding framework itself, the physics, the papers and the falsification tests.
• K7-Lean: its machine-checked formal core in Lean 4 (formerly gift-framework/core).
• Program: the charter (hard core, heuristics, six fair-play rules, chief among them never fit and never revise a frozen prediction), the eight open problems across five axes, and the confrontations scoreboard.
• Sieve: the methodology standard for the question how surprising is a claimed exact relation between mathematical invariants and measured constants? 28 dimensionless constants and the expression grammars were frozen and deposited on 2026-06-12, with a dated DOI, before any search ran (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 and including one built to beat it.
• 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.
• Atlas: the annotated map of adjacent work, one precise delta per entry, written so its author would sign it.

• You do physics. The bets above, then the open problems in Program.
• You do mathematics. The certified counts in Lean and the geometry entries in the Atlas.
• You read with suspicion. The Sieve first, then the charter, which states what is never done. Read these before you read any claim we make.

Everything above, at full length and on one page, is at arithmon.com.

The founding paper is under journal review, a numerical G₂ dataset from this work is cited in Physics Letters B 878 (2026) 140566, and the certified K3 result has a presubmission inquiry pending. The author publishes solo and without institutional affiliation, which makes the usual channels slower; that is a constraint of the territory, not an excuse. Independent reproduction or refutation is the thing the program most needs and least controls.

Invited remote participant, "DANGER: Data, Numbers, and Geometry" workshop, Banff International Research Station, April 5 to 10, 2026.

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