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Conjecture 3 Progress Audit - 2026-03-30 (Update 2)

Executive Verdict

Minimal-target closure remains stable and strong. The newly added nodes materially sharpen the full-conjecture bottleneck by narrowing the surviving routing-interpretation loophole to genuinely global constructions (now effectively arity-5-and-up in the auxiliary-vertex direction), but they do not yet complete either full proof or disproof of Conjecture 3.

Contract Checklist

  • Full two-sided Conjecture-3 inequality in the general statement remains open.
  • Minimal static-2D lower-bound target remains closed at theorem level in the map.
  • Compiler-native interpretation gap remains the dominant unresolved bridge.
  • Threshold-consequence layer remains partial and not fully integrated with a completed two-sided CD theorem.

Gap Matrix

  1. Obligation: Lower-bound side lift to full compiler-native CD interpretation.
  2. Best support: stabilizer-cut-rank functional, service-based SWAP-only replacement, and expanded obstruction stack against naive realizations.
  3. Missing bridge: globally pivot-robust routing/load semantics that coincide with intrinsic connectivity demand.
  4. Risk: high.

  5. Horizon: medium.

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

  7. Best support: strong routing/separator machinery and model-level tightness nodes.
  8. Missing bridge: theorem-level constructive compiler achieving CD * polylog in the same broad model.
  9. Risk: high.

  10. Horizon: medium-long.

  11. Obligation: Full threshold-consequence integration.

  12. Best support: overhead/noise consequence nodes in 2D-local regimes.
  13. Missing bridge: direct derivation of the contract-level delta_eff * CD condition from a completed two-sided theorem.
  14. Risk: medium-high.

  15. Horizon: medium-long.

  16. Obligation: Balanced-cut intrinsic rank at required scale for targeted parity-check spaces.

  17. Best support: multiple reductions/obstructions and sharpened connected-set/tangle routes.
  18. Missing bridge: linear balanced-cut intrinsic rank theorem under the final structural assumptions.
  19. Risk: high.
  20. Horizon: medium-long.

Proof Readiness Scores

  • Minimal target proof readiness: 93-97
  • Full lower-bound side readiness: 64-74
  • Full upper-bound side readiness: 35-50
  • Full conjecture proof readiness overall: 52-62

Disproof Readiness Scores

  • Full disproof readiness: 24-38
  • Most plausible disproof axis: impossibility of any global pivot-robust classical realization compatible with intrinsic connectivity demands.
  • Counterexample maturity: improved but still not near complete disproof.

What Actually Changed Since Last Check

  • Node graph increased from 112 to 119 nodes.
  • New results in the compiler-native bottleneck stack tightened the frontier from generic representability failure toward a narrower surviving loophole:
  • pivot-class best incidence load still overestimates connectivity,
  • simple binary connectivity violates nonnegative-hypergraph cut condition,
  • auxiliary-vertex graph-cut representability survives all 4-qubit stabilizer cases.
  • Net effect: no closure of full conjecture, but significantly better localization of where closure/disproof must occur.

Highest-Leverage Next 3 Moves

  1. Prove or refute a global auxiliary-vertex realization theorem beyond arity 4 (starting with arity 5), under pivot-robustness constraints.
  2. Build a direct bridge theorem from intrinsic cut-rank demand to a compiler-native routing/load formalism that survives basis changes.
  3. Either prove linear balanced-cut intrinsic rank for the targeted Quantum Tanner parity-check spaces, or construct a structured family that blocks it while preserving other map assumptions.