The Golden Shell Hofstadter Recurrence
Abstract
The golden shell s(n)=floor((n+1)/phi) satisfies the Hofstadter G recurrence s(n)=n-s(s(n-1)).
Theorem 1.1 (The golden shell satisfies the Hofstadter G self-referential recurrence).
Proof. Machine-checked in Lean as D5/S1/Phase/SelfReference/GoldenShellRecurrence.golden_shell_recurrence (✓ std3). ∎
Source. Repository-derived.
Commentary.
For the golden shell function s(n) = floor((n+1)/phi) — the Zeckendorf/Beatty golden shift with slope 1/phi (approximately 0.618, not phi itself) — the theorem proves the Hofstadter G self-referential recurrence s(n) = n - s(s(n-1)) for every n >= 1, equivalently the additive core s(m+1) + s(s(m)) = m+1. Thus the ledger’s golden shift and Douglas Hofstadter’s self-referential G sequence (OEIS A005206) are the same function.
The recurrence reduces to the additive core. Writing tau = 1/phi = phi - 1 (so tau^2 = 1 - tau and tau*(1+tau) = 1), the real value x = (m+1)*tau has floor A = s(m). The fractional part f = x - A is never exactly tau^2 (because (m+2)*tau is irrational), so a case split on f < tau^2 versus f > tau^2 evaluates both nested floors, floor(x + tau) = s(m+1) and floor((A+1)*tau) = s(s(m)), via floor bracket bounds; in each branch the two contributions cancel exactly to m+1.
Mathlib supplies the golden ratio and floor arithmetic but no Hofstadter G recurrence, so this is a genuine construction rather than a library restatement; it also upgrades the source observation’s numerical check (no exception for n <= 10^5) to a proof for all n. Only this G-identity (part one of the observation) is recorded; the separate MIU invariant (part two, that the reachable theorem strings have I-number congruent to 1 or 2 modulo 3) is not covered.
References
- Truth anchor:
D5/S1/Phase/SelfReference/GoldenShellRecurrence.golden_shell_recurrence