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

Executive Verdict

Minimal-target closure remains stable. The newly added nodes again do not close the full Conjecture 3 statement, but they sharpen the bottleneck substantially: the hidden-vertex route now survives through arity 5, while a first natural constructive extension already fails at arity 6. This is meaningful frontier localization, not full closure.

Contract Checklist

  • Full two-sided Conjecture-3 inequality in the general contract is still open.
  • Minimal static-2D lower-bound target remains closed 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: intrinsic stabilizer cut-rank functional + SWAP-only service replacement + strengthened representability frontier.
  3. Missing bridge: globally robust routing/load semantics matching intrinsic connectivity across basis changes and allowed compilation maps.
  4. Risk: high.

  5. Horizon: medium.

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

  7. Current best support: strong routing/separator machinery and known model-level tightness in restricted regimes.
  8. Missing bridge: broad constructive compiler theorem achieving CD * polylog in the same formal model.
  9. Risk: high.

  10. Horizon: medium-long.

  11. Obligation: Threshold-consequence integration.

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

  15. Horizon: medium-long.

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

  17. Current best support: reductions, obstructions, and connected-set/tangle routes remain strong and focused.
  18. Missing bridge: final theorem proving linear balanced-cut intrinsic rank under the right structural assumptions.
  19. Risk: high.
  20. Horizon: medium-long.

Proof Readiness Scores

  • Minimal target proof readiness: 93-97
  • Full lower-bound side readiness: 65-75
  • Full upper-bound side readiness: 35-50
  • Full conjecture proof readiness overall: 53-63

Disproof Readiness Scores

  • Full disproof readiness: 28-43
  • Most plausible axis: impossibility of any globally robust compiler-native routing interpretation compatible with intrinsic cut-rank demand.
  • Counterexample maturity: improved by new arity boundary evidence, but still incomplete for full disproof.

What Actually Changed Since Last Check

  • Node graph increased from 119 to 123.
  • New bottleneck refinements added:
  • hidden-vertex graph-cut representability now explicitly survives arity 5,
  • F_sep explanation node clarifies why prior separator-style test did not split that boundary,
  • Boolean-network generalization is shown not to add non-monotone expressive power on this subclass,
  • first natural constructive hidden-vertex extension beyond the modular-plus-fan cone fails at arity 6.
  • Net delta: no full closure, but stronger localization of the surviving route and better calibration of where to search for either theorem or obstruction.

Highest-Leverage Next 3 Moves

  1. Either prove a genuine hidden-vertex representability theorem beyond arity 5 or build a principled obstruction family starting at arity 6+.
  2. Convert representability (if available) into compiler-native routing semantics that remain stable under pivot/basis changes.
  3. In parallel, push the intrinsic side: prove linear balanced-cut intrinsic rank for the targeted Quantum Tanner parity-check spaces or isolate a structural counterexample class that blocks it.