Champions: Difference between revisions

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Line 52: Line 52:
|<math>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|<math>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|Jeff Ketchersid and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|-
|-
Line 58: Line 58:
|<math>> f_\omega^3(12)</math>
|<math>> f_\omega^3(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|Jeff Ketchersid and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|-
|-
Line 64: Line 64:
|<math>> f_\omega^7(12)</math>
|<math>> f_\omega^7(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|Jeff Ketchersid and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|-
|-
Line 70: Line 70:
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (claimed)]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in Oct 2026 (claimed)]
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|-
|-
Line 76: Line 76:
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|<span style="word-break:break-all">{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}</span>
|<span style="word-break:break-all">{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|Jeff Ketchersid and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|-
|-
Line 82: Line 82:
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}</span>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|Jeff Ketchersid and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|-
|-
Line 88: Line 88:
|<math>> f_{\omega+2}^6(f_{\omega+1}^2(3))</math>
|<math>> f_{\omega+2}^6(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH|halt}}</span>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB13 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB13 Claimed analysis by Claude] never verified.
|-
|-
Line 94: Line 94:
|<math>> f_{\omega+3}(30)</math>
|<math>> f_{\omega+3}(30)</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_0RN1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_0LK0LH_1LM1RZ|halt}}</span>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_0RN1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_0LK0LH_1LM1RZ|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|-
|-
Line 100: Line 100:
|<math>> f_{\omega \times 2}(13)</math>
|<math>> f_{\omega \times 2}(13)</math>
|<span style="word-break:break-all">{{TM|1LH0LG_0LI1RC_0LD1LA_0RE1LD_0RH1RL_1RB1RF_1LA1LG_1RF0RO_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LA0RN_0LA1RO_1RZ1RM|halt}}</span>
|<span style="word-break:break-all">{{TM|1LH0LG_0LI1RC_0LD1LA_0RE1LD_0RH1RL_1RB1RF_1LA1LG_1RF0RO_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LA0RN_0LA1RO_1RZ1RM|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB15 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB15 Claimed analysis by Claude] never verified.
|-
|-
Line 106: Line 106:
|<math>> f_{\omega \times 2}(f_\omega(14))</math>
|<math>> f_{\omega \times 2}(f_\omega(14))</math>
|<span style="word-break:break-all">{{TM|1LG1LA_0LI1RC_0LD1LG_0RE1LD_0RM1RL_1RB1RF_1LH0LA_1RF0RN_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LG0RN_0RP1RO_1RZ1RM_1LB1LI|halt}}</span>
|<span style="word-break:break-all">{{TM|1LG1LA_0LI1RC_0LD1LG_0RE1LD_0RM1RL_1RB1RF_1LH0LA_1RF0RN_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LG0RN_0RP1RO_1RZ1RM_1LB1LI|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.
|-
|-
Line 118: Line 118:
|<math>> f_{\omega \times 2}(f_\omega^9(35))</math>
|<math>> f_{\omega \times 2}(f_\omega^9(35))</math>
|<span style="word-break:break-all">{{TM|0LB1LF_0RC1LB_0RD1RM_1LE0RN_1LF1LE_1LG0LE_1RH1RP_1RI1RH_0LJ1RA_1LK1LJ_0LL1LJ_1RM1RC_0RM0RH_1LE1RO_1RZ1RD_0LF0RQ_0LH1RR_0LF1RP|halt}}</span>
|<span style="word-break:break-all">{{TM|0LB1LF_0RC1LB_0RD1RM_1LE0RN_1LF1LE_1LG0LE_1RH1RP_1RI1RH_0LJ1RA_1LK1LJ_0LL1LJ_1RM1RC_0RM0RH_1LE1RO_1RZ1RD_0LF0RQ_0LH1RR_0LF1RP|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.
|-
|-
Line 144: Line 144:
|<math>> f_{\omega^2}^5(4 \uparrow\uparrow 341)</math>
|<math>> f_{\omega^2}^5(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LV_0LH0LW_1LH0LX_1LH1LY_0LE1LR|halt}}</span>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LV_0LH0LW_1LH0LX_1LH1LY_0LE1LR|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB25 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB25 Claimed analysis by Claude] never verified.
|-
|-
Line 150: Line 150:
|<math>> f_{\omega^\omega}(3)</math>
|<math>> f_{\omega^\omega}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|mxdys in 10/2026 (claimed)
|mxdys in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|-
|-
Line 156: Line 156:
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|mxdys in 10/2026 (claimed)
|mxdys in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|-
|-
Line 162: Line 162:
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|mxdys in 10/2026 (claimed)
|mxdys in Oct 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|-
|-
Line 251: Line 251:
|<math>> f_{\omega+1}^4(f_\omega^3(14)) > g_{64}</math>
|<math>> f_{\omega+1}^4(f_\omega^3(14)) > g_{64}</math>
|<span style="word-break:break-all">{{TM|2LH1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD0RH_1RC2LE1LG_0LA0LD1RZ_1LD2RA2LG_2RD1LF1LG|halt}}</span>
|<span style="word-break:break-all">{{TM|2LH1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD0RH_1RC2LE1LG_0LA0LD1RZ_1LD2RA2LG_2RD1LF1LG|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 257: Line 257:
|<math>> f_{\omega+2}^3(f_{\omega+1}^2(f_\omega(14)))</math>
|<math>> f_{\omega+2}^3(f_{\omega+1}^2(f_\omega(14)))</math>
|<span style="word-break:break-all">{{TM|2LI1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD2RA_1RC2LE1LG_0LA0LD1RH_1LD2RA2LG_1RZ2LA2LI_1RD1LI2LH|halt}}</span>
|<span style="word-break:break-all">{{TM|2LI1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD2RA_1RC2LE1LG_0LA0LD1RH_1LD2RA2LG_1RZ2LA2LI_1RD1LI2LH|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 263: Line 263:
|<math>> f_{\omega \times 2}(11)</math>
|<math>> f_{\omega \times 2}(11)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF1LA0RI|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF1LA0RI|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 269: Line 269:
|<math>> f_{\omega \times 2}(46)</math>
|<math>> f_{\omega \times 2}(46)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF2LK0RI_2RG2RK1LK|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF2LK0RI_2RG2RK1LK|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 275: Line 275:
|<math>> f_{\omega \times 2}(f_{\omega+2}^2(4))</math>
|<math>> f_{\omega \times 2}(f_{\omega+2}^2(4))</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0RL_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LA_2LL0LE2LB|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0RL_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LA_2LL0LE2LB|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 281: Line 281:
|<math>> f_{\omega \times 2}^3(f_{\omega+1}(f_\omega^2(5)))</math>
|<math>> f_{\omega \times 2}^3(f_{\omega+1}(f_\omega^2(5)))</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0LM_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LB_2RG2RE2RL_1LL0RL0LK|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0LM_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LB_2RG2RE2RL_1LL0RL0LK|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 323: Line 323:
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|[https://github.com/jeff87654/busy-beaver/tree/main/BB44 Claimed analysis by MiniMax] Not independently verified.
|[https://github.com/jeff87654/busy-beaver/tree/main/BB44 Claimed analysis by MiniMax] Not independently verified.
|-
|-
|BB(5,4)
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
|<math>> f_\omega^3(2 \uparrow^{15} 5)</math>
|<span style="word-break:break-all">{{TM|1LD3LB---2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_3RB---3LE3LD_------1LD1RZ|halt}}</span>
|<span style="word-break:break-all">{{TM|1RB3LB1RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LE2LE3LE1RZ_3RC2LE1LE3LE|halt}}</span>
|Jacobzheng in 2026
|Big_numbers in Nov 2026
|[https://github.com/jeff87654/busy-beaver/tree/main/BB54 Claimed analysis by MiniMax] Not independently verified.
|[https://discord.com/channels/960643023006490684/1084047886494470185/1569814397801468005 Discord] Not independently verified.
|-
|-
|BB(6,4)
|BB(6,4)
Line 341: Line 341:
|<math>> f_{\omega+3}(21)</math>
|<math>> f_{\omega+3}(21)</math>
|<span style="word-break:break-all">{{TM|3RD1LE2RA1LF_0LE2LE2RB2RD_1RF1RC2RB0LB_0RC0LD3RC2RA_1RC2LE1LG2RF_0LA0LD3LG1RZ_1LD2RA2LG0RB|halt}}</span>
|<span style="word-break:break-all">{{TM|3RD1LE2RA1LF_0LE2LE2RB2RD_1RF1RC2RB0LB_0RC0LD3RC2RA_1RC2LE1LG2RF_0LA0LD3LG1RZ_1LD2RA2LG0RB|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 347: Line 347:
|<math>> f_{\omega+8}^2(4)</math>
|<math>> f_{\omega+8}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA---_2LE2LH2RB1LF_2LA1RC2RB---_0RC1RD0LG1RH_0LF1LE1LE3LB_1RG1RZ3RD3LF_0RC1LH2RG0LC_1RC2LH1LA2LB|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA---_2LE2LH2RB1LF_2LA1RC2RB---_0RC1RD0LG1RH_0LF1LE1LE3LB_1RG1RZ3RD3LF_0RC1LH2RG0LC_1RC2LH1LA2LB|halt}}</span>
|[[User:Loader3229]] in 2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
|BB(9,4)
|BB(9,4)
|
|<math>> f_{\omega^2}(18)</math>
|
|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_1LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0RC3LI1RI0RI|halt}}</span>
|
|[[User:Loader3229]] in Nov 2026
|
|Not independently verified.
|-
|-
|BB(10,4)
|BB(10,4)
|
|<math>> f_{\omega^2}(58)</math>
|
|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_1RI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LJ3RJ3LI2RI_3LD2RI3LI3RI|halt}}</span>
|
|[[User:Loader3229]] in Nov 2026
|
|-
|BB(11,4)
|<math>> f_{\omega \times 2}(2 \uparrow^{11} 6)</math>
|<span style="word-break:break-all">{{TM|1RK3LB1RZ2RA_0LG1LE1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1RJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_0LB0LJ3LJ3RK_2LC0LD2RA3RK|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(12,4)
|<math>> f_{\omega \times 2}^2(2 \uparrow^{6} 3)</math>
|<span style="word-break:break-all">{{TM|1LH---1LD2RB_2LL3LA1RZ2RB_3RA1LA3LC2RC_3RE---3LC2RC_2LI3RE3LH2LF_3RF2RF2LE1LF_1LL3LH2LJ2RG_2LC3RH1LI2RG_3RH1LH3LI2RI_------3LJ3RK_------2RB3RK_3RB---1LL3LL|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 406: Line 394:
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 412: Line 400:
|<math>> f_{\omega+1}(f_\omega^4(6))</math>
|<math>> f_{\omega+1}(f_\omega^4(6))</math>
|{{TM|1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC|halt}}
|{{TM|1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC|halt}}
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 418: Line 406:
|<math>> f_{\omega+3}(11)</math>
|<math>> f_{\omega+3}(11)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF|halt}}</span>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF|halt}}</span>
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 424: Line 412:
|<math>> f_{\omega+3}^2(10)</math>
|<math>> f_{\omega+3}^2(10)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG|halt}}</span>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG|halt}}</span>
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|Not independently verified.
|Not independently verified.
|-
|-
|BB(8,5)
|BB(8,5)
|<math>> f_{\omega \times 2+1}(93)</math>
|<math>> f_{\omega^2}(13)</math>
|<span style="word-break:break-all">{{TM|2RG3LB4LC2RA4LB_1RZ---1LD2RA4LD_3RB1LB3LC2RC4LC_3RE---3LC2RC3LC_---3RE4LE4LH4RB_4LE3LG3LB2RF4LG_2LH3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}</span>
|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG---_4LC2LH2RB2RF0RB_4LE1RC2RB3LG4RB_0RC1RC1LD3RD3LA_3RD1LE1LE3LE4RE_1RZ3LD2LE3LE4RD_1RG1LH2RG3LG3LF_1RC2LH1LA1LE---|halt}}</span>
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 465: Line 453:
|<math>> f_\omega^3(15)</math>
|<math>> f_\omega^3(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|[[User:Loader3229]] in 10/2026
|[[User:Loader3229]] in Oct 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 471: Line 459:
|<math>> f_{\omega+2}(29)</math>
|<math>> f_{\omega+2}(29)</math>
|{{TM|5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD|halt}}
|{{TM|5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD|halt}}
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 477: Line 465:
|<math>> f_{\omega+3}(31)</math>
|<math>> f_{\omega+3}(31)</math>
|<span style="word-break:break-all">{{TM|1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE|halt}}</span>
|<span style="word-break:break-all">{{TM|1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE|halt}}</span>
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 483: Line 471:
|<math>> f_{\omega \times 2}(89)</math>
|<math>> f_{\omega \times 2}(89)</math>
|<span style="word-break:break-all">{{TM|4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB|halt}}</span>
|<span style="word-break:break-all">{{TM|4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB|halt}}</span>
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 489: Line 477:
|<math>> f_{\omega^2}(110)</math>
|<math>> f_{\omega^2}(110)</math>
|<span style="word-break:break-all">{{TM|5LG3LB---2RA4LB0LC_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LE4LC3LG5LE_5RB5RB2LB1LF---5LF_5LF---0LG1LG---5LG|halt}}</span>
|<span style="word-break:break-all">{{TM|5LG3LB---2RA4LB0LC_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LE4LC3LG5LE_5RB5RB2LB1LF---5LF_5LF---0LG1LG---5LG|halt}}</span>
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 495: Line 483:
|<math>> f_{\omega^2+1}(f_\omega(31))</math>
|<math>> f_{\omega^2+1}(f_\omega(31))</math>
|<span style="word-break:break-all">{{TM|5LH3LB2LH2RA4LB1RZ_2RC3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_0LC---1LH1LH---5RG|halt}}</span>
|<span style="word-break:break-all">{{TM|5LH3LB2LH2RA4LB1RZ_2RC3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_0LC---1LH1LH---5RG|halt}}</span>
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 524: Line 512:
|<math>> f_{\omega+1}(117) > g_{64}</math>
|<math>> f_{\omega+1}(117) > g_{64}</math>
|{{TM|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA|halt}}
|{{TM|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA|halt}}
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 530: Line 518:
|<math>> f_{\omega+3}(34)</math>
|<math>> f_{\omega+3}(34)</math>
|{{TM|5LC3LB6LA2RA4LB2LD3RB_1RZ3RB1LC2RA3RD2LC4LC_3RB1LB3LC2RC4LC1RA5LD_3RB5LD6LC4LD4LC4LD6LC|halt}}
|{{TM|5LC3LB6LA2RA4LB2LD3RB_1RZ3RB1LC2RA3RD2LC4LC_3RB1LB3LC2RC4LC1RA5LD_3RB5LD6LC4LD4LC4LD6LC|halt}}
|[[User:Loader3229]] in 11/2026
|[[User:Loader3229]] in Nov 2026
|Not independently verified.
|Not independently verified.
|-
|-
Line 657: Line 645:


== References ==
== References ==
[[Category:Individual machines]]
[[Category:Zoology]]
[[Category:Zoology]]

Latest revision as of 23:19, 10 November 2026

Busy Beaver Champions are the current record holding Turing machines which maximize a Busy Beaver function. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for Busy Beaver champions and the History of Previous Champions. 1-state domains are omitted as BB(1,m) = 1 for m > 1.

Trivial Champions

BB(n,1) = n

BB(1,m) = 1

2-Symbol TMs

Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the fx(n) used in the lower bounds represent the Fast-Growing Hierarchy while ↑ represents Knuth's up-arrow notation; see measuring large numbers for more information. Machines below 6 states are listed in TNF format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.


Domain Runtime Champions Discovered by Verification
BB(2) 6 1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch) Tibor Radó Direct Simulation
BB(3) 21 1RB1RZ_1LB0RC_1LC1LA (bbch) Proven by Shen Lin Direct Simulation
BB(4) 107 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch) Allen Brady Direct Simulation
BB(5) 47176870 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) Heiner Marxen & Jürgen Buntrock in 1989 Direct Simulation
BB(6) >10↑↑10↑↑10↑↑8 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch) mxdys in 2025 See mxdys's analysis on the TM page
BB(7) >3↑203>fω(20) 1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB (bbch) Jeff Ketchersid and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(8) >fω3(12) 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ (bbch) Jeff Ketchersid and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(9) >fω7(12) 1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC (bbch) Jeff Ketchersid and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(10) >fω+1(2↑363)>g64 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC (bbch) Jeff Ketchersid and Claude in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(11) >fω+1(fω2(5)) 1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE (bbch) Jeff Ketchersid and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(12) >fω+2(fω+12(3)) 1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK (bbch) Jeff Ketchersid and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(13) >fω+26(fω+12(3)) 1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH (bbch) Jeff Ketchersid, Claude and Gemini in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(14) >fω+3(30) 1RB1RA_1RI1RC_0LD1LG_0RE1LD_0RN1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_0LK0LH_1LM1RZ (bbch) Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed) Claimed analysis by Claude never verified.
BB(15) >fω×2(13) 1LH0LG_0LI1RC_0LD1LA_0RE1LD_0RH1RL_1RB1RF_1LA1LG_1RF0RO_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LA0RN_0LA1RO_1RZ1RM (bbch) Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed) Claimed analysis by Claude never verified.
BB(16) >fω×2(fω(14)) 1LG1LA_0LI1RC_0LD1LG_0RE1LD_0RM1RL_1RB1RF_1LH0LA_1RF0RN_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LG0RN_0RP1RO_1RZ1RM_1LB1LI (bbch) Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed) Claimed analysis by Claude never verified.
BB(17)
BB(18) >fω×2(fω9(35)) 0LB1LF_0RC1LB_0RD1RM_1LE0RN_1LF1LE_1LG0LE_1RH1RP_1RI1RH_0LJ1RA_1LK1LJ_0LL1LJ_1RM1RC_0RM0RH_1LE1RO_1RZ1RD_0LF0RQ_0LH1RR_0LF1RP (bbch) Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed) Claimed analysis by Claude never verified.
BB(19)
BB(20) >fω×2(fω+22(f10(2))) 1LH0LG_0LI1RC_0LD1LA_0RE1LD_1RQ1RL_1RB1RF_1LA1LG_1RF1RM_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LN1RM_0RF0LO_0RF1LP_0RF1LN_1LR1RQ_1RZ1LS_0LF1LT_0LF0LR (bbch) Jacobzheng in 2026 Claimed analysis by Claude never verified.
BB(21) >fω23(4↑↑341) 0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR (bbch) Racheline in 2024 Claimed analysis by Claude never verified.
...
BB(25) >fω25(4↑↑341) 0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LV_0LH0LW_1LH0LX_1LH1LY_0LE1LR (bbch) Jeff Ketchersid, Claude and Gemini in Nov 2026 (claimed) Claimed analysis by Claude never verified.
BB(29) >fωω(3) [1] mxdys in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(31) >fω↑↑4(3) [2] mxdys in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(33) >fω↑↑6(3) [3] mxdys in Oct 2026 (claimed) Claimed analysis by Claude never verified.
BB(48) >fε1(3) [4] roromachine14 and Gemini in 2026 Not independently verified.
BB(150) >flim(BMS)(10↑↑15) too large to show Patcail in 2025[1] Not independently verified.
BB(1015) >D5(99) = Loader's Number CatsAreFluffy in 2024[2] Not independently verified.

Conjectured

Domain Runtime Champions Discovered by Notes
BB(67) >q81(q(5)−2) [5] racheline in 2025 q(n) is derived from Laver tables. From racheline: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."

If q(5) doesn't exist (this can only happen if ZFC+I3 is inconsistent), then this machine doesn't halt.

3-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch) Allen Brady in 1988 Direct Simulation
BB(3,3) >1017 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch) Terry & Shawn Ligocki in 2007 Analysis by Pascal Michel
BB(4,3) >10↑44 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) Pavel Kropitz in 2024 See Polygon's analysis on the TM page
BB(5,3)
BB(6,3) >fω2(10↑↑↑6) 1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE (bbch) Jacobzheng in 2026 Not independently verified.
BB(7,3)
BB(8,3) >fω+14(fω3(14))>g64 2LH1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD0RH_1RC2LE1LG_0LA0LD1RZ_1LD2RA2LG_2RD1LF1LG (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(9,3) >fω+23(fω+12(fω(14))) 2LI1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD2RA_1RC2LE1LG_0LA0LD1RH_1LD2RA2LG_1RZ2LA2LI_1RD1LI2LH (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(10,3) >fω×2(11) 1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF1LA0RI (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(11,3) >fω×2(46) 1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF2LK0RI_2RG2RK1LK (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(12,3) >fω×2(fω+22(4)) 1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0RL_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LA_2LL0LE2LB (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(13,3) >fω×23(fω+1(fω2(5))) 1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0LM_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LB_2RG2RE2RL_1LL0RL0LK (bbch) User:Loader3229 in Nov 2026 Not independently verified.
...
BB(36,3) >fε0(7) Wythagoras in 2014[3] Not independently verified.
BB(37,3) >fε0(373676378) Wythagoras in 2014[4] Not independently verified.

4-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,4) 3932964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch) Terry & Shawn Ligocki in 2005 Pascal Michel, Heiner Marxen, Allen Brady
BB(3,4) >2↑155 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch) Pavel Kropitz in 2024 Analysis by Shawn Ligocki
BB(4,4) >2↑5093>fω(508) 0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA (bbch) User:Loader3229 in Oct 2026 Claimed analysis by MiniMax Not independently verified.
BB(5,4) >fω3(2↑155) 1RB3LB1RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LE2LE3LE1RZ_3RC2LE1LE3LE (bbch) Big_numbers in Nov 2026 Discord Not independently verified.
BB(6,4) >fω+1(10↑↑65533)>g64 1LF2RA3RB---_3RD2LB3RB1LC_2RA1LB3RC2LC_1LD2RE0LA1LD_1RZ0LE1LF---_2RA---2LF2LF (bbch) Jacobzheng in 2026 Not independently verified.
BB(7,4) >fω+3(21) 3RD1LE2RA1LF_0LE2LE2RB2RD_1RF1RC2RB0LB_0RC0LD3RC2RA_1RC2LE1LG2RF_0LA0LD3LG1RZ_1LD2RA2LG0RB (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(8,4) >fω+82(4) 1LD2RG2LA---_2LE2LH2RB1LF_2LA1RC2RB---_0RC1RD0LG1RH_0LF1LE1LE3LB_1RG1RZ3RD3LF_0RC1LH2RG0LC_1RC2LH1LA2LB (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(9,4) >fω2(18) 1LD2RG2LA3LG_1LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0RC3LI1RI0RI (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(10,4) >fω2(58) 1LD2RG2LA3LG_1RI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LJ3RJ3LI2RI_3LD2RI3LI3RI (bbch) User:Loader3229 in Nov 2026 Not independently verified.
...
BB(31,4) >fε0(373676378) Wythagoras in 2014[5] Not independently verified.

5-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch) Daniel Yuan in 2024 mxdys in Rocq
BB(3,5) >fω(2↑155)>fω2(15) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch) Racheline in 2024 Not independently verified.
BB(4,5) >fω+1(27041)>g64 1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC (bbch) User:Loader3229 in Oct 2026 Not independently verified.
BB(5,5) >fω+1(fω4(6)) 1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC (bbch) User:Loader3229 in Oct 2026 Not independently verified.
BB(6,5) >fω+3(11) 2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF (bbch) User:Loader3229 in Oct 2026 Not independently verified.
BB(7,5) >fω+32(10) 2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG (bbch) User:Loader3229 in Oct 2026 Not independently verified.
BB(8,5) >fω2(13) 1LD2RG2LA3LG---_4LC2LH2RB2RF0RB_4LE1RC2RB3LG4RB_0RC1RC1LD3RD3LA_3RD1LE1LE3LE4RE_1RZ3LD2LE3LE4RD_1RG1LH2RG3LG3LF_1RC2LH1LA1LE--- (bbch) User:Loader3229 in Nov 2026 Not independently verified.
...
BB(19,5) >fε0(1.7⋅10352) 1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS (bbch) Wythagoras in 2014[6] Not independently verified.
BB(94,5) >fψ0(Ω2)(1.7⋅10352) Wythagoras in 2014[7] Not independently verified.

6-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,6) >10↑↑10↑↑1010115 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch) Pavel Kropitz in 2023 Analysis by Shawn Ligocki
BB(3,6) >fω3(15) 1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC (bbch) User:Loader3229 in Oct 2026 Not independently verified.
BB(4,6) >fω+2(29) 5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(5,6) >fω+3(31) 1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(6,6) >fω×2(89) 4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(7,6) >fω2(110) 5LG3LB---2RA4LB0LC_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LE4LC3LG5LE_5RB5RB2LB1LF---5LF_5LF---0LG1LG---5LG (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(8,6) >fω2+1(fω(31)) 5LH3LB2LH2RA4LB1RZ_2RC3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_0LC---1LH1LH---5RG (bbch) User:Loader3229 in Nov 2026 Not independently verified.
...
BB(61,6) >fψ0(Ω2)(1.7⋅10352) Wythagoras in 2014[8] Not independently verified.

7-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,7) >10↑↑10↑↑10↑↑155 1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB (bbch) roromachine14 in 2026 Not independently verified.
BB(3,7) >fω+1(117)>g64 6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA (bbch) User:Loader3229 in Nov 2026 Not independently verified.
BB(4,7) >fω+3(34) 5LC3LB6LA2RA4LB2LD3RB_1RZ3RB1LC2RA3RD2LC4LC_3RB1LB3LC2RC4LC1RA5LD_3RB5LD6LC4LD4LC4LD6LC (bbch) User:Loader3229 in Nov 2026 Not independently verified.
...
BB(18,7) >fε0(1.7⋅10352) 1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------ (bbch) Wythagoras in 2014[9] Not independently verified.
BB(46,7) >fψ0(Ω2)(1.7⋅10352) Wythagoras in 2014[10] Not independently verified.
BB(134,7) >fψ0(Ωω)(2050) LittlePeng9+Wythagoras in 2014[11] Not independently verified.

8-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,8) >10↑↑10↑↑10↑↑10↑↑155 1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA (bbch) roromachine14 in 2026 Not independently verified.
...
BB(17,8) >fε0(1.7⋅10352) 1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ--------- (bbch) Wythagoras in 2014[12] Not independently verified.

9-Symbol TMs

Domain Runtime Champions Discovered by Verification
BB(2,9) >10↑↑10↑↑10↑↑10↑↑10↑↑155 1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA (bbch) roromachine14 in 2026 Not independently verified.
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BB(42,9) >fψ0(Ω2)(1.7⋅10352) Wythagoras in 2014[13] Not independently verified.

Zoology

There are many types of champions.

Note: the scale here is not formally defined. It mostly serves to estimate the average values produced by a type of champions.

Classification Description Examples Scale
Trivial The simplest champions that can exist. They mostly appear in some BB-adjacent functions like BBf or BBCS. O(n)
Chaotic Have a chaotic behavior with repeating patterns that go back and forth.
  • 1RB1LB_1LA1RZ (bbch)
  • 1RB1RZ_1LB0RC_1LC1LA (bbch)
  • 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
 O(2n)
Countdown Compute a number then "count down" (usually while bouncing) until reaching 0. They are common in some BB-adjacent functions like BBf.
  • 1RB2LB1RZ_2LA2RB1LB (bbch)
 O(n2)
Collatz-like Compute a Collatz-like function. Repeatedly multiply and add a number depending of its modulo until reaching a number with a certain modulo. O(2n)
Random walk Compute a Collatz-like function with a random walk. O(22n)
Tetrational O(2↑↑n)
Ackermannian O(2↑n3)

References

  1. ↑ https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345
  2. ↑ https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe
  3. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)
  4. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)
  5. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)
  6. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)
  7. ↑ https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines
  8. ↑ https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines
  9. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)
  10. ↑ https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines
  11. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines
  12. ↑ https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)
  13. ↑ https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines
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