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Carréphylic ancestors
Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that x = a2y ⇔ √x = a√y.
Since y is carréphylic, the approach of √x should pose no practical problems.
Here's how to find the 's' and 'p' of the rational approach of x:
s2-xp2 = 1 ⇔ s2-y(ap)2 = 1.
The fraction s/ap is the ath fraction of the set of sp-blocks of the rational approach of y, starting from the trivial block.
It is therefore the first fraction of the base-a series accelleration of the rational approach of y.
The corresponding factor '2s' is Fa or the factor of y 'base-a'.
Since √x = a√y the sp-fraction of x is a(s/ap) = s/p.

Example carréphylic ancestry
The position of a fraction indicates whether it is over or under the root-value.

The approach of √7 - carréphylic
 1 0 1 2 3 5 8 21 29 37 45 82 127 336 463 590 717 1307 2024 5355 7379 9403 11427 20830 32257 85344 117601 149858 182115 331973 514088 1360149 1874237 2388325 2902413 5290738 8193151 21677040 29870191 38063342 46256493 84319835 130576328 345472491 ... 0 1 1 1 1 2 3 8 11 14 17 31 48 127 175 223 271 494 765 2024 2789 3554 4319 7873 12192 32257 44449 56641 68833 125474 194307 514088 708395 902702 1097009 1999711 3096720 8193151 11289871 14386591 17483311 31869902 49353213 130576328 ...
 Accellerations numerator denominator factor base-2 U(1,127) U(0,48) 254 base-3 U(1,2024) U(0,765) 4048 base-4 U(1,32257) U(0,12192) 64514

The approach of √28=2√7 - carréphobic
 1 0 1 2 3 4 5 11 16 21 37 90 127 672 799 926 1053 1180 1307 2741 4048 5355 9403 22854 32257 170688 202945 235202 267459 299716 331973 696203 1028176 1360149 2388325 5804826 8193151 43354080 ... 0 1 1 1 1 1 1 2 3 4 7 17 24 127 151 175 199 223 247 518 765 1012 1777 4319 6096 32257 38353 44449 50545 56641 62737 131570 194307 257044 451351 1097009 1548360 8193151 ...

Using the above data we find that the first non trivial sp-fraction of 28 is twice the second non-trivial fraction of 7, or 127/24.
The corresponding factor 2*127 is the base-2 accelleration of the factor of 7: 162-2.

The approach of √63=3√7 - carréphylic
 1 0 1 2 3 4 5 6 7 8 63 71 79 87 95 103 111 119 127 1008 1135 1262 1389 1516 1643 1770 1897 2024 16065 18089 20113 22137 24161 26185 28209 30233 32257 256032 ... 0 1 1 1 1 1 1 1 1 1 8 9 10 11 12 13 14 15 16 127 143 159 175 191 207 223 239 255 2024 2279 2534 2789 3044 3299 3554 3809 4064 32257 ...

63 is one less than a square, so the exception mentioned in on root approach applies: 127 and 16, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 8 and 1 because 82-63*12 = 1 satisfies the diophantine equation.
Nevertheless, trice the third non-trivial fraction of 7 is 2024/255 and the resulting series is the base-3 accelleration of the sp-series of √63.
The corresponding factor 2*2024 is the base-3 accelleration of the factor of 7: 163-3*16.

The approach of √112=4√7 - carréphobic
As it happens 112 falls outside the first 100 non-squares, so I didn't do a search, but using the above data the first non trivial sp-fraction of 112 is four times the fourth non-trivial fraction of 7, or 32257/3048.
The corresponding factor 2*32257 is the base-4 accelleration of the factor of 7: 164-4*162+2.

Carréphylic descendants
Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that a2x = y ⇔ a√x = √y.
Here's how to find the 's' and 'p' of the rational approach of x:
For y the first non-trivial 's' and 'p' satisfy: s2-yp2 = 1 ⇔ s2-x(ap)2 = 1 ⇔ 's' and 'ap' are the first non-trivial element of the set of sp-blocks of the rational approach of x. The corresponding factor is the same.

Example carréphylic descendency

The approach of √13 - carréphobic
The fractional approach of root 13, the first carréphobic number, as provided by Ed's straightforward program, looks like this:
 1 0 1 2 3 4 7 11 18 83 101 119 137 256 393 649 2340 2989 3638 4287 4936 9223 14159 23382 107687 131069 154451 177833 332284 510117 842401 3037320 3879721 4722122 5564523 6406924 11971447 18378371 30349818 139777643 170127461 200477279 230827097 431304376 662131473 1093435849 3942439020 ... 0 1 1 1 1 1 2 3 5 23 28 33 38 71 109 180 649 829 1009 1189 1369 2558 3927 6485 29867 36352 42837 49322 92159 141481 233640 842401 1076041 1309681 1543321 1776961 3320282 5097243 8417525 38767343 47184868 55602393 64019918 119622311 183642229 303264540 1093435849 ...

As it happens, 52*13 = 325 = 182-1 and thus carréphylic. The formula renders 649/36 as the first sp-block of the approach of 5√13, making 649/180 the first sp-block of √13 with the same factor 1298.
Easy as fruitpie.