User:Wythagoras/Rado's sigma function

This page displays all known bounds for Rado's sigma function. All bounds with a linked (see machine) are discovered by me.

My results so far
This section briefly gives my most important results on the Busy Beaver
 * Reducing the number of states to beat G from 25 to 23, using Deedlit11's earlier results.
 * Giving a general bound for \(\Sigma(n)\) using Deedlit11's earlier results.
 * Proving four other bounds for \(\Sigma(n)\) - including a bound for \(\Sigma(7)\).
 * Proving 17 bounds for multicolored TMs.
 * Giving a definition and 10 bounds for multi-headed TMs.
 * Proving 7 HNRs to loop

One dimension, one head, absolute movement, Sigma
Further,

\(\Sigma(1,n) = 1\) for any \(n\)

\(\Sigma(2,6) > 1.9 \cdot 10^{4933}\)

\(\Sigma(81,10) > f_{\vartheta(\varepsilon_{\Omega+1})}(2050)\)

\(\Sigma(134,7) > f_{\vartheta(\Omega_\omega)}(2050)\)

\(\Sigma(150,7) > f_{\vartheta(\varepsilon_{\Omega_\omega+1})}(683)\)

One dimension, one head, absolute movement, S
Further:

\(S(n,m) ≥ \Sigma(n,m)\) for any \(n, m\).

\(S(1,m) = 1\) for any \(m\)

\(S(2,6) > 2.4 \cdot 10^{9866}\)

One dimension, two heads, absolute movement, Sigma
The small machines can be seen in the subpage for Multi-Headed TMs.

One dimension, three heads, absolute movement, Sigma and S
\(\Sigma(1,2,3) ≥ 6\)

\(S(1,2,3) ≥ 10\)

Subpages
This page gives a definition for Oracle TMs, and some bounds. Comments and bounds based on LittlePeng9's hydra machines. This page describes the progress on the 42 Hardly Non-Regular TMs, and thus the progress on proving \(\Sigma(5) = 4098\).
 * Oracle TMs
 * Hydra Machines
 * Progress on the 42 Hardly Non-Regular TMs
 * Multi-Headed TMs