Zendo, play by forum

[namida]

I wrote my own code to do it (that also checks the rule for me too, although I'm double-checking it by hand too).

White Koans
  POTATO       

Black Koans
* BROCCOLI      * POT           
* CAR           * ROT           
  CARROT        * TOMATO       

[Silken Healer]

AKHTBN iff it is not ok to eat raw.

[Silken Healer]

AKHTBN iff it is not ok to eat raw, but would be ok to eat cooked.

[namida]

AKHTBN iff it is not ok to eat raw.

Aside from that this probably relies too much on outside knowledge to really define as a proper Zendo rule.

Putting this aside, your counterexample is CHICKEN, which would be white by your guess, but is black.

White Koans
  POTATO       

Black Koans
  BROCCOLI        POT           
  CAR             ROT           
  CARROT          TOMATO       
* CHICKEN       

[Silken Healer]

RAPTOR
ROTOR
MOTOR
ROTATE
CART
OR
TOR
P

[namida]

White Koans
* MOTOR           POTATO       

Black Koans
  BROCCOLI        POT           
  CAR           * RAPTOR       
  CARROT          ROT           
* CART          * ROTATE       
  CHICKEN       * ROTOR         
* OR              TOMATO       
* P             * TOR           

[Silken Healer]

AKHTBN iff, removing the first character, the word can be turned into an anagram by removing one character, and the word is not originally an anagram.

[Proxima]

NOTOR
OOTOR
POTOR

[namida]

AKHTBN iff, removing the first character, the word can be turned into an anagram by removing one character, and the word is not originally an anagram.

A word can't be an anagram on its own; it must be an anagram of some specific other word (or sequence of characters). I think you might be confusing the term "anagram" with something else. As currently worded, this rule doesn't mean anything and can't be meaningfully judged. (It crosses my mind that maybe you meant palindrome, but all three-letter koans as well as ROTOR and POTATO are already counterexamples to that.)

White Koans
  MOTOR           POTATO       
* NOTOR         * POTOR         

Black Koans
  BROCCOLI        POT           
  CAR             RAPTOR       
  CARROT          ROT           
  CART            ROTATE       
  CHICKEN         ROTOR         
* OOTOR           TOMATO       
  OR              TOR           
  P             

[Silken Healer]

AKHTBN iff, removing the first character, the koan can be turned into a palindrome by removing one character, the koan, with no chatacters removed, is not originally a palindrome, and the koan is more than one character long.

[namida]

AKHTBN iff, removing the first character, the koan can be turned into a palindrome by removing one character, the koan, with no chatacters removed, is not originally a palindrome, and the koan is more than one character long.

As mentioned in my edit to the previous post, there are already multiple counterexamples that disprove this guess.

[Proxima]

CARAT
TOROT
TOTOR

[namida]

White Koans
* CARAT           POTATO       
  MOTOR           POTOR         
  NOTOR         

Black Koans
  BROCCOLI        RAPTOR       
  CAR             ROT           
  CARROT          ROTATE       
  CART            ROTOR         
  CHICKEN         TOMATO       
  OOTOR           TOR           
  OR            * TOROT         
  P             * TOTOR         
  POT           

[Proxima]

Guess: A koan has the Buddha-nature if it contains exactly four distinct letters and its first letter is not repeated.

[namida]

Guess: A koan has the Buddha-nature if it contains exactly four distinct letters and its first letter is not repeated.

Counterexample: ASSASSINATION (HTBN but has more than four distinct letters *and* the first letter is repeated)

White Koans
* ASSASSINATION        NOTOR             
  CARAT                POTATO             
  MOTOR                POTOR             

Black Koans
  BROCCOLI             RAPTOR             
  CAR                  ROT               
  CARROT               ROTATE             
  CART                 ROTOR             
  CHICKEN              TOMATO             
  OOTOR                TOR               
  OR                   TOROT             
  P                    TOTOR             
  POT