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Conjecture 3 Progress Audit - 2026-03-31 (Update 6)

Executive Verdict

Minimal-target closure remains stable. This round adds a meaningful frontier upgrade: the representability program now enters the first connected post-gluing regime (cuts with intrinsic rank >=3) and isolates a concrete selector bottleneck where the natural compressed route fails with up to two auxiliaries. This is substantial progress in bottleneck precision, but it is still not full closure of the two-sided Conjecture 3 contract.

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 full contract level remains partial.

Gap Matrix

  1. Obligation: Lower-bound side lift to full compiler-native CD interpretation.
  2. Current best support: generator-invariant stabilizer cut-rank functional, SWAP-only service lower bound, threshold-lift representability stack, and now a concrete connected chi>=3 test family with an explicit selector bottleneck.
  3. Missing bridge: either (i) a realization theorem for the connected family beyond the current compressed selector route, or (ii) a genuine nonrepresentability theorem in this connected regime, plus a theorem connecting the outcome to pivot-robust routing semantics.
  4. Risk: high.

  5. Horizon: medium.

  6. Obligation: Upper-bound side in full conjecture contract.

  7. Current best support: routing/separator machinery plus restricted-model tightness results.
  8. Missing bridge: broad constructive compiler theorem achieving CD * polylog in the same formal model as the lower side.
  9. Risk: high.

  10. Horizon: medium-long.

  11. Obligation: Threshold-consequence integration.

  12. Current best support: 2D-local overhead/noise-consequence stack.
  13. Missing bridge: direct derivation of the contract-level delta_eff * CD criterion from a completed two-sided theorem.
  14. Risk: medium-high.

  15. Horizon: medium-long.

  16. Obligation: Linear balanced-cut intrinsic rank for targeted parity-check spaces.

  17. Current best support: strong decomposition/tangle/connected-set reductions and clearer separation between low-order gluing and true high-width core.
  18. Missing bridge: theorem forcing linear balanced-cut intrinsic rank under final irreducibility assumptions.
  19. Risk: high.
  20. Horizon: medium-long.

Proof Readiness Scores

  • Minimal target proof readiness: 94-98
  • Full lower-bound side readiness: 70-81
  • Full upper-bound side readiness: 35-50
  • Full conjecture proof readiness overall: 58-68

Disproof Readiness Scores

  • Full disproof readiness: 20-34
  • Most plausible axis: prove that connected chi>=3 selector bottlenecks cannot be hidden-vertex representable at any bounded auxiliary complexity relevant to compiler-native semantics.
  • Counterexample maturity: improved relative to update-5 because the search is now concentrated on the first connected post-gluing family and a specific selected-OR gadget.

What Actually Changed Since Last Check

  • Node graph increased from 127 to 133 markdown nodes.
  • New structural deltas:
  • invariant characterization of the threshold-lift family as parallel extension of one circuit element;
  • direct-sum closure of threshold-lift pieces (strictly broader positive class, but disconnected);
  • explicit proof that this positive class never enters the connected chi>=3 core;
  • identification of the first natural connected multi-parallel-circuit family with cuts of rank >=3;
  • reduction of that family's compressed-state realization to one ternary selector gadget;
  • exact no-go for that selector with 0, 1, or 2 auxiliaries.
  • Net effect: the frontier moved from "is there any arity-6 obstruction?" to "does the first connected chi>=3 selector bottleneck admit higher-auxiliary/global realization, or is it a genuine obstruction?"

Highest-Leverage Next 3 Moves

  1. Push the selector barrier decisively: either prove nonrepresentability for the selected-OR gadget for all auxiliary counts, or construct the first explicit realization with minimal auxiliary complexity.
  2. If selector realization exists, lift it from compressed coordinates to a full connected-family hidden-vertex theorem and test pivot-robustness under basis changes.
  3. In parallel, continue intrinsic closure independent of representability: force linear balanced-cut intrinsic rank on target Quantum Tanner parity-check spaces so the lower-bound side does not hinge on one representability branch.