Improve the lower bound on the sixth Busy Beaver value S(6)/Sigma(6) for 2-symbol Turing machines

For $n$-state, 2-symbol Turing machines (single bi-infinite tape, starting on the all-blank tape), let $S(n)$ be the maximum number of steps taken by any such machine that halts, and $\Sigma(n)$ the maximum number of 1s left on the tape by a halting machine (the Busy Beaver functions, Rado 1962). The values are known exactly through $n=5$: $S(5)=47{,}176{,}870$ and $\Sigma(5)=4098$ (proved 2024). For $n=6$ they are unknown and astronomically large; the current record lower bound (2025) is $S(6) > \Sigma(6) > 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 9$ (Knuth up-arrow / tetration). Exhibit a 6-state 2-symbol Turing machine that provably halts on the blank tape and whose halting step-count (or final 1-count) exceeds the current champion, thereby improving the lower bound on $S(6)$ (resp. $\Sigma(6)$).

Acceptance. FULLY (advances the frontier): a 6-state 2-symbol Turing machine (given as its transition table) with a proof that (i) it halts when started on the all-blank tape and (ii) its step count (resp. final number of 1s) exceeds the current record. Because the record is far beyond direct simulation, the halting proof must be an analytic/machine-checked argument (e.g. an accelerated 'macro-machine' / translated-cycler / closed-form certificate as used by the Busy Beaver Challenge deciders). A modest but directly-simulable improvement is acceptable if it beats a stated, simulable sub-record. PARTIAL: a rigorous new lower bound on $S(6)$ or $\Sigma(6)$ with a checkable certificate, or a machine plus a verified lower bound on its runtime. Provide the transition table and the halting/step-count certificate.