User:Polygon/Page for analyses - Revision history

Polygon: /* Collatz-like? */ fixed typos and changed section heading

2026-10-18T12:51:41Z

Collatz-like?: fixed typos and changed section heading

← Older revision		Revision as of 12:51, 18 October 2026
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	</pre>		</pre>

	== ~~Collatz-like?~~ ==		== Reset behavior ==
	Let's restate the rules:		Let's restate the rules:

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	AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.		AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.

	The maximum ~~trajecture~~ following these rules is: AR6-2 --> AR1-2 --> AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt		The maximum trajectory following these rules is: AR6-2 --> AR1-2 --> AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

	This TM follows the ~~trajecture~~: AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt		This TM follows the trajectory: AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

	</div></div>		</div></div>

Polygon: /* Approximate Score */ im starting to think that I can't actually read...

2026-10-17T16:50:02Z

Approximate Score: im starting to think that I can't actually read...

← Older revision		Revision as of 16:50, 17 October 2026
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	<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>		<math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>

	<math>e_2 = f_2(\frac{~~e_1~~+7} 3, f_1^{2}(91))</math>		<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>

	<math>e_1 = 2^{95} \times 3 -5</math>		<math>e_1 = 2^{95} \times 3 -5</math>

Polygon: /* Trajectory */ thats e_2 and not e_3

2026-10-17T16:47:01Z

Trajectory: thats e_2 and not e_3

← Older revision		Revision as of 16:47, 17 October 2026
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	<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>		<math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>

	<math>D_1(1,0,~~e_3~~)</math>		<math>D_1(1,0,e_2)</math>

	<math>e_2 mod 3 = 1</math>		<math>e_2 mod 3 = 1</math>
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	--> <math>D_1(2e_2+7, 3, 1) --> D_1(2e_2+7, 0, f_1^{2}(91))</math>		--> <math>D_1(2e_2+7, 3, 1) --> D_1(2e_2+7, 0, f_1^{2}(91))</math>

	~~2e_3~~ + 7		2e_2 + 7

	Modulus: 2 + 7 --> 9 mod 3 = 0		Modulus: 2 + 7 --> 9 mod 3 = 0

Polygon: /* Collatz-like? */ AR2 --> AR2-2

2026-10-17T16:14:14Z

Collatz-like?: AR2 --> AR2-2

← Older revision		Revision as of 16:14, 17 October 2026
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	AR1-2: <math>D(0, 0, 3m+1) --> D(3m+2, 0, f_1^{3}(1)) --> D(2, 0, f_2(m,f_1^3(1)))</math> by rules AR1 and AR8. AR1-2 leads to AR3-2.		AR1-2: <math>D(0, 0, 3m+1) --> D(3m+2, 0, f_1^{3}(1)) --> D(2, 0, f_2(m,f_1^3(1)))</math> by rules AR1 and AR8. AR1-2 leads to AR3-2.

	AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.		AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2-2 halts.

	AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR5-2.		AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR5-2.

Polygon: /* Collatz-like? */ Completed section

2026-10-17T16:09:16Z

Collatz-like?: Completed section

← Older revision		Revision as of 16:09, 17 October 2026
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	AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.		AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.

			The maximum trajecture following these rules is: AR6-2 --> AR1-2 --> AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

			This TM follows the trajecture: AR3-2 --> AR5-2 --> AR4-2 --> AR2-2 --> halt

	</div></div>		</div></div>

Polygon: /* Collatz-like? */ AR5-2 not 4-2

2026-10-17T16:06:07Z

Collatz-like?: AR5-2 not 4-2

← Older revision		Revision as of 16:06, 17 October 2026
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	AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.		AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.

	AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to ~~AR4~~-2.		AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR5-2.

	AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2		AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2

Polygon: /* Collatz-like? */ fixed R25 D1 typo

2026-10-17T16:03:12Z

Collatz-like?: fixed R25 D1 typo

← Older revision		Revision as of 16:03, 17 October 2026
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	Rule 25:		Rule 25:
	* <math>D(2, 0, c) --> D(2c+8, 3, 1)</math>		* <math>D(2, 0, c) --> D_1(2c+8, 3, 1)</math>

	Rule 27:		Rule 27:
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	AR2: D(1, 0, c) --> halt with score 4c + 18 by rule 28.		AR2: D(1, 0, c) --> halt with score 4c + 18 by rule 28.

	AR3: <math>D(2, 0, c) --> D(2c+8, 3, 1) --> D(2c+8, 0, f_1^{3}(1))</math> by rules 25 and 30.		AR3: <math>D(2, 0, c) --> D_1(2c+8, 3, 1) --> D(2c+8, 0, f_1^{3}(1))</math> by rules 25 and 30.

	AR4: <math>D_1(0, 0, c) --> D(2c+5, 3, 1) --> D(2c+5, 0, f_1^{3}(1))</math> by rules 27 and 30.		AR4: <math>D_1(0, 0, c) --> D(2c+5, 3, 1) --> D(2c+5, 0, f_1^{3}(1))</math> by rules 27 and 30.
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	AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.		AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.

	AR3-2: <math>D(2, 0, 3m+1) --> D(6m+10, 0, f_1^{3}(1)) = D(3(2m+3)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to ~~AR2~~-2.		AR3-2: <math>D(2, 0, 3m+1) --> D_1(6m+10, 0, f_1^{3}(1)) = D_1(3(2m+3)+1, 0, f_1^{3}(1)) --> D_1(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR4-2.

	AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2		AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2

Polygon: /* Functions */ Rule 25 typo

2026-10-17T16:01:48Z

Functions: Rule 25 typo

← Older revision		Revision as of 16:01, 17 October 2026
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	Rule 25 becomes:		Rule 25 becomes:
	* <math>D(2, 0, c) --> D(2c+8, 3, 1)</math>		* <math>D(2, 0, c) --> D_1(2c+8, 3, 1)</math>

	Rule 26 becomes:		Rule 26 becomes:

Polygon: /* Collatz-like? */ expanded

2026-10-17T15:56:35Z

Collatz-like?: expanded

← Older revision		Revision as of 15:56, 17 October 2026
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	Every one of these rules will set c to either f_1(n), f_1^a(n) or f_2(a,b) with a ≠ 0, thus c mod 3 = 1 after the first rule application.		Every one of these rules will set c to either f_1(n), f_1^a(n) or f_2(a,b) with a ≠ 0, thus c mod 3 = 1 after the first rule application.

			Thus, c can be considered to be of the form 3m+1:


			AR1-2: <math>D(0, 0, 3m+1) --> D(3m+2, 0, f_1^{3}(1)) --> D(2, 0, f_2(m,f_1^3(1)))</math> by rules AR1 and AR8. AR1-2 leads to AR3-2.

			AR2-2: D(1, 0, 3m+1) --> halt with score 12m + 22 by rule AR2. AR2 halts.

			AR3-2: <math>D(2, 0, 3m+1) --> D(6m+10, 0, f_1^{3}(1)) = D(3(2m+3)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+3,f_1^3(1)))</math> by AR3 and AR8. AR3-2 leads to AR2-2.

			AR4-2: <math>D_1(0, 0, 3m+1) --> D(6m+7, 0, f_1^{3}(1)) = D(3(2m+2)+1, 0, f_1^{3}(1)) --> D(1, 0, f_2(2m+2,f_1^3(1)))</math> by rules AR4 and AR8. AR4-2 leads to AR2-2

			AR5-2: <math>D_1(1, 0, 3m+1) --> D_1(6m+9, 0, f_1^{3}(1)) = D_1(3(2m+3), 0, f_1^{3}(1)) --> D_1(0, 0, f_2(2m+3,f_1^3(1)))</math> by rules AR5 and AR8. AR5-2 leads to AR4-2.

			AR6-2: <math>D_1(2, 0, 3m+1) --> D(6m+12, 0, f_1^{2}(1)) = D(3(2m+4), 0, f_1^{2}(1)) --> D(0, 0, f_2(2m+4,f_1^2(1)))</math> by rules AR6 and AR8. AR6-2 leads to AR1-2.

	</div></div>		</div></div>

Polygon: /* Collatz-like? */ expand

2026-10-17T15:32:29Z

Collatz-like?: expand

← Older revision		Revision as of 15:32, 17 October 2026
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	AR6: <math>D_1(2, 0, c) --> D(2c+10, 2, 1) --> D(2c+10, 0, f_1^{2}(1))</math> by rules 29 and 30.		AR6: <math>D_1(2, 0, c) --> D(2c+10, 2, 1) --> D(2c+10, 0, f_1^{2}(1))</math> by rules 29 and 30.

			Together with:

			AR7 by rule 30:
			* <math>D(a, b, c) --> D(a, 0, f_1^{b}(c))</math>
			* <math>D_1(a, b, c) --> D_1(a, 0, f_1^{b}(c))</math>

			AR8 by rule 32 for k ≠ 0:
			* <math>D(3k+d, 0, c) --> D(d, 0, f_2(k, c))</math>
			* <math>D_1(3k+d, 0, c) --> D_1(d, 0, f_2(k, c))</math>

			Let's also consider the following which was shown earlier:
			<pre>
			f_1(n) mod 3 = 1
			f_1^a(n) mod 3 = 1
			f_2(a,b) mod 3 = 1 for a ≠ 0
			</pre>

			Every one of these rules will set c to either f_1(n), f_1^a(n) or f_2(a,b) with a ≠ 0, thus c mod 3 = 1 after the first rule application.
	</div></div>		</div></div>