How to define a named constant with evaluation rules similar to GoldenRatio?

Question

I want to define a named constant standing for a more complicated symbolic expression, for example, PlasticConstant for the plastic constant:

((9 - √69)^(1/3) + (9 + √69)^(1/3))/(2^(1/3) 3^(2/3))

and I want it to have evaluation rules similar to GoldenRatio:

• it remains in its named form during normal evaluation (so, simple Set does not solve my problem);
• it can be numerically evaluated to an arbitrary precision using N;
• it can be converted to the underlying expression in radicals by FunctionExpand, can be converted to a Root object by RootReduce;
• FullSimplify is able to replace it with its underlying expression when it helps reducing overall complexity of a containing expression, but leaves it in its named form otherwise;

Equality and inequality operators between the named constant and its underlying expression should automatically evaluate to Boolean values:

PlasticConstant == ((9 - √69)^(1/3) + (9 + √69)^(1/3))/(2^(1/3) 3^(2/3))
(* True *)

similar to how it happens here:

GoldenRatio == (1 + √5)/2
(* True *)

How can we do it?

Answer 1

Rather long for a comment:

I think the OP's goals are going to be hard to achieve. GoldenRatio is embedded in the system in ways I don't know. Here is some examples of the embedding that come from an internal Trace of FunctionExpand@GoldenRation:

GoldenRatio /. SimplifyDump`PositiveRules
SimplifyDump`$FSTab /@ %
(*
  {901}
  {GoldenRatio :> 1/2 (1 + Sqrt[5])}
*)

Some of the OP's goals can be handled programmatically. For instance:

ClearAll[OneThird];
SetAttributes[OneThird, {Constant}];
N[OneThird, p_: MachinePrecision] := N[1/3, p];

OneThird // N[#, 22] &
(*  0.3333333333333333333333  *)

However, automatic conversion to its numerical value does not happen for OneThird as it does for GoldenRatio:

3. OneThird
3. GoldenRatio
(*
  3. OneThird
  4.8541
*)

Maybe someone else will know how to do it. An internal trace reveals nothing.