formal-conjectures · paper · ams-5
If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be
divided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by
fusing the n + 1st successor of x with a fusible number.
This formalization differs from Conjecture 7.1 in the paper in four ways:
(1) it is obtained from Conjecture 7.1 by plugging in n + 1 into n, which simplifies the expressions
and removes the need to assume n ≥ 1;
(2) the n + 1st successor s^(n+1)(x) is replaced by the explicit value x + (2 - 1 / 2 ^ n) * m;
(3) instead of defining y to be the successor of x, we assert that there is no fusible number
strictly between x and y;
(4) instead of using ∃ z, IsFusible z ∧ q = s^(n+1)(x) ~ z we use the value of z determined by the equality,
namely z = 2 * q - 1 - s^(n+1)(x), and it is easy to see z ∈ [x + 1 - m / 2 ^ n, x + 1) as required.
Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.
Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/2003.14342
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://doi.org/10.46298/lmcs-18%283%3A6%292022
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Paper/FusibleNumber.lean
Local target: Corpus.PaperFusibleNumber.conj_7_1
Source SHA-256: cf58c928c7d62dc1af452cfe02fc7aefd24f83e9cabf896ae10c3511e1032799
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.
Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.
If `x` is a fusible number and `y` is its successor, then the interval `[x + 1, y + 1)` can be divided into intervals `[ℓₙ, ℓₙ₊₁)`, such that the fusible numbers in `[ℓₙ, ℓₙ₊₁)` are obtained by fusing the `n + 1`st successor of `x` with a fusible number.
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