Route-D Post-Fan-Cone Next Target Is Fan-Certificate-To-Polymorphism Lift¶

Claim/Theorem¶

After [[source-all-arity-expressive-power-stops-before-fan-certificate-lift.md]] and [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]], the post-fan-cone Route-D frontier no longer has two symmetric next tasks.

The obstruction side is now the better-supported theorem target.

The reason is not that the obstruction is solved. It is that the negative side already has:

an exact primary-source schema, namely the all-arity Theorem-16 / Conjecture-20 characterization;
a concrete family-level fan-cone separation for every connected size-2 member;
a direct F_{\mathrm{sep}} pass theorem showing exactly why the older necessary condition is too weak;
a family-lifting mechanism by pinning extra classes to the split state (1,0).

By contrast, the constructive side has no surviving source generator after excluding:

terminal-side and broader Boolean-network semantics;
fixed-prefix-state recursion;
modular-plus-fan closure;
separable selected-threshold architectures;
the low-budget and symmetry-preserving ansatze already on disk.

Therefore the exact next theorem-level task is the following family-specific bridge:

Starting from the integer fan-cone separating certificate for f_3, construct a weighted-polymorphism or multimorphism certificate that is valid for all of \langle\Gamma_{\mathrm{sub},2}\rangle but violated by f_3; then use the existing pinning lift to obtain nonexpressibility for all connected size-2 multi-parallel functions f_t, t\ge 3.

Equivalently, prove the Conjecture-20 implication

\[ f_t\notin\operatorname{Cone}(\Gamma_{\mathrm{fans},2t+1}) \quad\Longrightarrow\quad f_t\notin \langle\Gamma_{\mathrm{sub},2}\rangle \]

for this one selected-threshold family, without proving the full all-arity conjecture.

If that bridge fails, the failure should be explicit: it would mean the existing fan-cone certificate is not an invariant certificate for hidden-vertex expressibility, and the constructive side should then pivot to a nonseparable selected-threshold global energy rather than to another fan-like gadget search.

So the canonical post-fan-cone Route-D target is now:

\[ \text{fan-cone certificate} \longrightarrow \text{weighted-polymorphism obstruction} \]

for the connected size-2 multi-parallel family.

Dependencies¶

[[source-all-arity-expressive-power-stops-before-fan-certificate-lift.md]]
[[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]]
[[route-d-now-stops-at-conjecture20-vs-beyond-fan-cone-global-construction.md]]
[[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]]
[[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]]
[[higher-arity-theorem16-would-close-negative-side-on-connected-size-2-family.md]]

Conflicts/Gaps¶

This node is a continuation theorem target, not a completed nonexpressibility proof.
It promotes the obstruction-side bridge because it is closer to the loaded source theorems than any currently surviving construction class, not because Conjecture 20 is known.
The exact remaining gap is a family-specific Farkas/fan-certificate-to-weighted-polymorphism lift. Without it, fan-cone escape remains only a constructive-class barrier.
A positive nonseparable hidden-vertex construction would supersede this ranking immediately.

Sources¶

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