Conjecture 3 Progress Audit - 2026-03-30 (Update 4)¶

Executive Verdict¶

Minimal-target closure remains stable. The newest nodes sharpen the compiler-native frontier again: the known 6-qubit witness that previously looked like a plausible obstruction is now explicitly hidden-vertex graph-cut representable, so that line is not a disproof route by itself. This is real progress in bottleneck localization, but the full two-sided Conjecture 3 contract is still open.

Contract Checklist¶

Full two-sided Conjecture-3 inequality in the general contract remains open.
Minimal static-2D lower-bound target remains closed at theorem level in the map.
Compiler-native routing interpretation bridge remains the dominant unresolved step.
Threshold-consequence integration at the full contract level remains partial.

Gap Matrix¶

Obligation: Lower-bound side lift to full compiler-native CD interpretation.
Current best support: generator-invariant stabilizer cut-rank functional, SWAP-only service lower bound, and strengthened submodular/matroid realizability diagnostics.
Missing bridge: a globally pivot-robust routing/load semantics matching intrinsic demand across allowed compilation maps.
Risk: high.
Horizon: medium.
Obligation: Upper-bound side in full conjecture contract.
Current best support: strong routing/separator machinery and tightness nodes in restricted local models.
Missing bridge: broad constructive compiler theorem achieving CD * polylog in the same formal model as the lower side.
Risk: high.
Horizon: medium-long.
Obligation: Threshold-consequence integration.
Current best support: 2D-local overhead/noise-consequence nodes.
Missing bridge: direct derivation of the contract-level delta_eff * CD criterion from a completed two-sided theorem stack.
Risk: medium-high.
Horizon: medium-long.
Obligation: Linear balanced-cut intrinsic rank for targeted parity-check spaces.
Current best support: improved decomposition/tangle/connected-set reductions and clearer low-order separation classification.
Missing bridge: final theorem forcing linear balanced-cut intrinsic rank under the right irreducibility assumptions.
Risk: high.
Horizon: medium-long.

Proof Readiness Scores¶

Minimal target proof readiness: 94-98
Full lower-bound side readiness: 67-77
Full upper-bound side readiness: 35-50
Full conjecture proof readiness overall: 55-65

Disproof Readiness Scores¶

Full disproof readiness: 22-36
Most plausible axis: prove that even hidden-vertex representability (where it exists) cannot induce a routing-style compiler-native semantics with required robustness.
Counterexample maturity: decreased on the prior arity-6 witness line, because that witness is now explicitly representable.

What Actually Changed Since Last Check¶

Node graph increased from 123 to 126 markdown nodes.
The former leading arity-6 obstruction candidate was neutralized: the known 6-qubit witness now has an explicit hidden-vertex graph-cut realization.
This flips the immediate interpretation of the representability frontier:
not "we found the 6-ary blocker",
but "the first 6-ary candidate survives too," so either a different obstruction is needed, or a broader positive theorem is plausible.
The map also keeps sharpening why representability alone may still be insufficient for Conjecture 3: routing/native semantics must remain robust under basis/pivot effects.

Highest-Leverage Next 3 Moves¶

Push arity >= 7 search in a theorem-guided way: either find a genuine hidden-vertex obstruction family or extract a reusable constructive pattern extending the positive side.
Prove a semantics bridge theorem: when hidden-vertex representations exist, characterize exactly when they do or do not induce pivot-robust compiler-native routing/load interpretations.
Continue intrinsic closure in parallel: force linear balanced-cut intrinsic rank for the target Quantum Tanner parity-check spaces (or isolate a structural counterexample class with all current assumptions preserved).