Skip to content

Route-D Now Stops At Conjecture20 Versus Beyond-Fan-Cone Global Construction

Claim/Theorem

Keep the notation of [[higher-arity-theorem16-would-close-negative-side-on-connected-size-2-family.md]], [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]], and [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]].

After the current dichotomy-closure cycle, the surviving Route-D frontier is sharper than the earlier

higher-arity obstruction versus non-prefix global construction

stop point.

The exact remaining dichotomy is now:

  1. R1: prove an all-arity Theorem-16-type obstruction, most naturally Zivný's Conjecture 20 or an equivalent higher-arity weighted-polymorphism theorem, that upgrades the new fan-cone escape theorem into ordinary-hidden-vertex nonexpressibility for the connected size-2 family;

  2. R2: build an ordinary hidden-vertex realization for that family using genuinely global couplings beyond both:

    • the fixed-prefix-state recursive route already ruled out by [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]];
    • the modular-plus-fan constructive closure now ruled out on the whole connected size-2 family by [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]].

So the Route-D comparison now changes as follows.

  • On the negative side, the missing theorem is no longer a vague higher-arity obstruction; it is an explicit Conjecture-20-type characterization target.
  • On the positive side, the remaining constructive space is no longer “some non-prefix gadget.” It must beat both fixed local state-update recursion and the whole modular-plus-fan closure.

Hence the exact next Route-D target is now:

\[ \text{Conjecture 20 / all-arity Theorem 16} \quad\text{versus}\quad \text{a beyond-fan-cone global hidden-vertex construction.} \]

Relative to that sharpened stop point, the earlier D1 and D3 continuations remain secondary.

Dependencies

  • [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]]
  • [[higher-arity-theorem16-would-close-negative-side-on-connected-size-2-family.md]]
  • [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]]
  • [[after-current-m1-and-m2-cycles-route-d-stops-at-higher-arity-obstruction-versus-nonprefix-global-construction.md]]

Conflicts/Gaps

  • This node does not resolve the ordinary-hidden-vertex theorem on the connected size-2 family.
  • It records the strongest theorem-level compression currently supported by the source base plus the new family fan-cone obstruction.
  • A future beyond-fan-cone realization or a future proof of Conjecture 20 would supersede this stop point.

Sources

  • 10.1016/j.dam.2009.07.001
  • 10.1007/s10878-017-0136-y
  • 10.48550/arXiv.2109.14599