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Claim. Fix an integer $m\ge2$ and put $\alpha=2^{1/m}$. For every real $t>0$, every integer $L\ge0$ and every finite set $F$ of integers, every sufficiently large integer is a sum of distinct positive values in $\{\lfloor t\alpha^n\rfloor:n\ge L\}$ outside $F$. Taking $L=1$ and $F$ empty, and choosing one index for each value used, every sufficiently large integer is $\sum_{n\ge1}\varepsilon_n\lfloor t\alpha^n\rfloor$ with $\varepsilon_n\in\{0,1\}$ and finitely many $\varepsilon_n=1$. Thus the sequence $\lfloor t\alpha^n\rfloor$, $n\ge1$, is complete for every $t>0$ at each base $2^{1/m}$, $m\ge2$, and stays complete after any finite deletion of values.

Covers. Every pair $(t,2^{1/m})$ with $t>0$ and $m\ge2$ of the modified Statement of Problem 349. These bases, $\sqrt2,2^{1/3},2^{1/4},\ldots$, lie strictly between $1$ and the golden ratio. Because the sums use distinct values, the same pairs are complete under the erdosproblems.com wording as well. The claim says nothing about any other base, gives no completeness threshold uniform in $t$, $m$, $L$ or $F$, and leaves open the rest of the classification of the pairs with $1<\alpha<(1+\sqrt5)/2$.

Credit. The input is the accepted indexed two-ray theorem at base two of Liam Kruer and Jensen Kohlmeyer's Lean proof: for $u,v>0$ with $u/v$ irrational, every sufficiently large integer is a sum of terms $\lfloor u2^r\rfloor$ and $\lfloor v2^s\rfloor$ over finite sets of exponents. Plasma AI supplies the deduction: in a late tail where the values are positive, strictly increasing and outside $F$, the exponents in two consecutive residue classes modulo $m$ form two such rays with coefficient ratio $2^{-1/m}$, which is irrational. No novelty of the deduction is asserted.

Standing. The native claim is at tier 1. The independent whole-conclusion examination and its separate passing grade, both dated 2026-10-14, return refutation-failed for the single-ray theorem, including an arbitrary first index, finite deletion and the transfer from indexed sums to distinct values. The record retains a disclosed exposure of the reviewer to source-proof and acceptance prose, which the grader ruled immaterial to these conclusions. The imported base-two theorem was read at claims-checked depth: its statement, indexed-sum convention and recorded acceptance were checked, and its long proof was not re-examined. No Lean proof of this deduction is asserted. Catalog acceptance is pending.

Depends on. L33 (tier 1), whose only external premise is the accepted indexed base-two theorem.