Universality proof question

[c0b0p0]

As it turns out, the cleanup is fairly difficult. The main problem now is the almost invincible T-shaped p2 composed of matter. The most promising pinwheel positions are below.

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x = 59, y = 99, rule = BTCA1
42.A$40.3A$41.3A$41.A15$57.A$14.A40.3A$14.3A39.3A$13.3A40.A2$54.A$54.
3A$53.3A$55.A4$35.B$35.3B$35.3B$3.A32.3B$.3A$2.3A$2.A4$47.A$47.3A$46.
3A$40.A7.A$40.3A$39.3A$41.A$2.A$3A$.3A$.A3$6.A$6.3A$5.3A$7.A2$7.A$7.
3A$6.3A$8.A7$16.A23.B$16.2A21.2B$15.4A19.4B$15.2A23.2B24$15.2B23.2A$
15.4B19.4A$16.2B21.2A$16.B23.A!

[knightlife]

Found a p108 gun:

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x = 45, y = 43, rule = BTCA1
27.B$27.3B$27.B2$34.B$34.3B$33.3B$35.B5$26.A$24.2AB2A$24.2AB3A$23.3AB
3A$18.3A6.3BA$19.2A.4A.3A$18.2AB5A.3A4.B7.A$20.A2.3A.A6.3B4.AB2A$19.
4A.2A7.3B4.AB2A$18.5A.A10.B5.3A$19.AB3AB2A14.A$17.6A.3A$17.3A.A2.A.A$
17.3A5$2.B$3B$.3B$.B6$10.B$10.3B$9.3B$11.B!

p1512 gun
14x period multiplier also converts big glider to small glider:

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x = 132, y = 175, rule = BTCA1
28.B$26.3B$28.B2$27.A7.B$25.4A4.3B$26.2A6.3B$26.A7.B24.B$59.B$53.B4.
3B$53.B$52.3B2$33.AB$32.3A$33.3A25.B$59.3B$60.3B$35.B24.B$33.3B$34.3B
$14.2A3.3A12.B$13.2ABA3.2A$14.3A.A2.A$16.A2.A2$18.3A$11.AB3A3.ABA$10.
3AB2A3.3A$11.3ABA4.A$.B10.3AB$.3B9.3A$3B11.A$2.B2$59.A5.3B$58.AB2A4.B
$46.B10.AB2A5.B$44.3B11.3A$11.B33.3B10.A$9.3B33.B22.B$10.3B55.3B$10.B
43.3B11.B$55.B$55.B$62.3B$63.B$63.B20$29.B$27.3B$29.B2$28.A7.B$26.4A
4.3B$27.2A6.3B$27.A7.B6$34.AB$33.3A$34.3A3$36.B$34.3B$35.3B$15.2A3.3A
12.B$14.2ABA3.2A$15.3A.A2.A$17.A2.A2$19.3A$12.AB3A3.ABA$11.3AB2A3.3A$
12.3ABA4.A$2.B10.3AB$2.3B9.3A$.3B11.A$3.B6$12.B$10.3B$11.3B$11.B27$
120.3B$121.B$114.3B4.B$115.B$115.B3$122.B$122.3B$121.3B$96.B26.B$96.
3B$95.3B$97.B13$128.B$128.B$127.3B$107.B$107.3B$106.3B$108.B22.B$117.
B11.3B$117.B13.B$116.3B$125.B$125.B$124.3B!

There is a shared pinwheel in the top pattern. The 14x multiplier has been separated from the p108 gun in the pattern below it.

This multiplier only works for p108

EDIT:
Actually, the extra pinwheel is not needed:

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x = 131, y = 108, rule = BTCA1
28.B$26.3B$28.B2$27.A7.B$25.4A4.3B$26.2A6.3B$26.A7.B6$33.AB$32.3A$33.
3A3$35.B$33.3B$34.3B$14.2A3.3A12.B$13.2ABA3.2A$14.3A.A2.A$16.A2.A2$
18.3A$11.AB3A3.ABA$10.3AB2A3.3A$11.3ABA4.A$.B10.3AB$.3B9.3A$3B11.A$2.
B6$11.B$9.3B$10.3B$10.B27$119.3B$120.B$113.3B4.B$114.B$114.B3$121.B$
121.3B$120.3B$122.B16$127.B$127.B$126.3B$106.B$106.3B$105.3B$107.B22.
B$116.B11.3B$116.B13.B$115.3B$124.B$124.B$123.3B!

[knightlife]

Found a simple 90 degree reflector!

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x = 33, y = 35, rule = BTCA1
26.A$24.3A$18.A6.3A$18.3A4.A$17.3A$19.A$32.A$30.3A$32.A4$28.A$28.3A$
27.3A$29.A5$24.A$22.3A$24.A8$.B$.3B$BA2B$.BA2B$2.B!

EDIT:
This 90 degree reflector converts big glider to small glider and is even simpler:

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x = 35, y = 34, rule = BTCA1
32.A$32.3A$31.3A$21.A11.A$19.3A$21.A10.A$30.3A$32.A22$.B$.3B$BA2B$.BA
2B$2.B!

[period54]

First oscillator with period 3 in this rule:

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x = 9, y = 11, rule = BTCA1
3.2B$3.2B2$4.A$2B.3A$2B2.A2.2B$3.3A.2B$4.A2$4.2B$4.2B!

[c0b0p0]

Here is a start towards a p2 glider duplicator. The main issue is rebuilding the antipinwheel.

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x = 41, y = 41, rule = BTCA1
38.A$38.3A$37.3A$39.A22$.AB30.A$A3B29.3A$3B29.3A$.B32.A9$20.B$20.3B$
19.3B$21.B!

[knightlife]

Here is a 180 degree p2 reflector:

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x = 83, y = 85, rule = BTCA1
7.A$7.A6.A$6.3A5.A$13.3A4$.A$.3A$3A$2.A4$4.A$2.3A$3.3A$3.A24.A$26.3A$
27.3A$10.A16.A$10.3A$10.A8$16.3A$17.A$17.A19$50.B$49.BA2B$48.BA2B$49.
3B12.3A$49.B15.A$65.A7$71.A$71.3A$54.A16.A$54.3A$53.3A$55.A22.A$78.3A
$77.3A$79.A4$81.A$79.3A$80.3A$80.A3$68.A$68.A6.A$67.3A5.A$74.3A!

Two reflectors can make a p564 + 6n oscillator (where n=0,1,2...).
The phase of the reflector can be the same if moved diagonally by an even # of cells, otherwise the other phase is needed.

[knightlife]

A backrake (needs cleanup) turns into a clean p72 rake by adding 2 gliders to the floatilla:

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x = 48, y = 87, rule = BTCA1
29.BA14.A.A$29.3A14.2A$30.A4.A.A4.B2.3A$36.2A3.2B.B$35.3A5.2B$42.3B$
39.A.A$40.2A$39.3A40$.A$AB2A$.2A$.A8$17.A$16.AB2A$17.2A$17.A16$8.BA
14.A.A$8.3A14.2A$9.A4.A.A4.B2.3A$15.2A3.2B.B$14.3A5.2B$21.3B$18.A.A$
19.2A$18.3A!

[knightlife]

Found a true p2 glider duplicator:

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x = 44, y = 34, rule = BTCA1
11.B$10.BA2B$9.BA2B$10.3B$10.B2$28.A$28.3A$27.3A$5.A23.A$5.3A$5.A26.A
$30.3A$31.3A$31.A$3A$.A$.A$38.A$38.3A$37.3A$39.A3$13.A28.A$13.3A24.3A
$12.3A9.A16.3A$14.A7.3A16.A$23.3A$23.A$29.A$29.A6.A$28.3A5.A$35.3A!

One output is an exact 180 reflection and the other is 90 degree reflection.
Unfortunately the 180 reflection interferes with the next incoming glider, but still a variable high period gun can be made:

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x = 96, y = 95, rule = BTCA1
7.A$7.A6.A$6.3A5.A$13.3A2$19.A$19.3A$.A16.3A9.A$.3A16.A7.3A$3A26.3A$
2.A26.A$35.A$33.3A$5.A28.3A$3.3A28.A$4.3A$4.A4$11.A$11.3A$10.3A$12.A
2$15.A$13.3A$14.3A$14.A33$63.B$62.BA2B$61.BA2B$62.3B$62.B2$80.A$80.3A
$79.3A$57.A23.A$57.3A$57.A26.A$82.3A$83.3A$83.A$52.3A$53.A$53.A$90.A$
90.3A$89.3A$91.A3$65.A28.A$65.3A24.3A$64.3A9.A16.3A$66.A7.3A16.A$75.
3A$75.A$81.A$81.A6.A$80.3A5.A$87.3A!

The period of this gun (p678 as posted here) can vary by multiples of 6 (like the previously posted oscillator).
The minimum period of this gun is 438.

[knightlife]

This glider duplicator can duplicate glider streams:

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x = 86, y = 52, rule = BTCA1
34.AB$33.A3B$33.3B$34.B4$13.A$11.3A$12.3A$12.A2$19.A$17.3A6.A$18.3A5.
3A$18.A6.3A$27.A$.A30.A$.3A26.3A28.A$3A28.3A25.3A$2.A28.A28.3A4.A$60.
A4.3A$66.3A$66.A5.A$38.A31.3A$36.3A32.3A$37.3A31.A$37.A2$74.A$72.3A$
73.3A$73.A2$15.A$13.3A$14.3A63.A$14.A9.A55.3A$8.A13.3A17.A36.3A$2.A5.
3A12.3A14.3A38.A$3A4.3A13.A18.A$.3A5.A21.A$.A27.3A27.A24.A$30.3A24.3A
22.3A$30.A27.3A5.A16.3A$58.A5.3A16.A$65.3A$65.A$71.A$71.A6.A$70.3A5.A
$77.3A!

There is enough room to redirect the glider streams (p280 minimum period):

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x = 145, y = 151, rule = BTCA1
2.B$.BA2B$BA2B$.3B$.B67$40.A$40.A$39.3A$47.A$45.3A$46.3A$34.A11.A$34.
3A$33.3A18.A$35.A18.A$53.3A3$37.A$35.3A$36.3A$36.A7$95.AB$94.A3B$42.A
51.3B$42.A52.B$41.3A$49.A22.A$47.3A20.3A$48.3A20.3A$36.A11.A22.A$36.
3A$35.3A18.A21.A$37.A18.A19.3A6.A$55.3A19.3A5.3A$77.A6.3A$86.A$39.A
20.A30.A$37.3A20.3A26.3A28.A$38.3A18.3A28.3A25.3A$38.A22.A28.A28.3A4.
A$119.A4.3A$125.3A$125.A5.A$97.A31.3A$95.3A32.3A$96.3A31.A$96.A2$133.
A$131.3A$132.3A$132.A2$74.A$72.3A$17.A55.3A63.A$17.3A53.A9.A55.3A$17.
A49.A13.3A17.A36.3A$61.A5.3A12.3A14.3A38.A$59.3A4.3A13.A18.A$60.3A5.A
21.A$60.A27.3A27.A24.A$12.A76.3A24.3A22.3A$12.3A74.A27.3A5.A16.3A$11.
3A103.A5.3A16.A$13.A110.3A$124.A$130.A$130.A6.A$9.A119.3A5.A$9.3A124.
3A$9.A$22.A$22.3A$16.A4.3A$14.3A6.A$15.3A$15.A!

[knightlife]

Universality can be reached with the following gates:

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x = 162, y = 163, rule = BTCA1
33.BA92.B$3.B28.3BA90.BA2B$2.BA2B27.3B89.BA2B30.B$.BA2B29.B91.3B28.2B
AB$2.3B121.B31.2BAB$2.B155.3B$160.B6$29.A$27.3A$28.3A$28.A9$13.A$13.
3A8.A$12.3A7.3A107.A$14.A8.3A104.3A$23.A107.3A$131.A36$127.B$126.BA2B
$125.BA2B$126.3B$2.B123.B$.BA2B$BA2B$.3B$.B7$28.A$26.3A$27.3A$27.A8$
132.A$12.A117.3A$12.3A8.A107.3A$11.3A7.3A107.A$13.A8.3A$22.A37$158.B$
34.BA120.2BAB$33.3BA120.2BAB$34.3B120.3B$35.B123.B9$30.A$28.3A$29.3A$
29.A8$131.A$14.A114.3A$14.3A8.A104.3A$13.3A7.3A104.A$15.A8.3A$24.A!

p2 AND gate on the left and "A AND NOT B" gate on the right, also based on a p2 pinwheel.

Since the glider has three phases, it might be necessary to find phase shifters to build complex logic easily, but the p2 glider duplicator and the p2 90 degree reflector posted previously combined with these logic gates can be the basis for universal computation.

EDIT:
Much simpler 180 degree reflector than previous:

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x = 26, y = 27, rule = BTCA1
.AB$A3B$3B$.B12$23.A$23.3A$22.3A$24.A3$14.A$12.3A$14.A4.A$17.3A$18.3A
$18.A!

EDIT2:
Two more:
More useful 180 degree reflector can accept glider streams:

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x = 26, y = 29, rule = BTCA1
.AB$A3B$3B$.B12$7.A15.A$5.3A15.3A$6.3A13.3A$6.A17.A3$12.A$12.3A$11.3A
5.A$13.A3.3A$18.3A$2.A15.A$2.3A$2.A!

Symmetry allows slight mod to make shifter:

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x = 26, y = 28, rule = BTCA1
.AB$A3B$3B$.B12$7.A15.A$5.3A15.3A$6.3A13.3A$6.A17.A3$12.A$12.3A$11.3A
5.A$13.A3.3A$2.A15.3A$2.A15.A$.3A!