Skip to content

Conjecture 3 Progress Audit - 2026-03-31 (Update 8)

Executive Verdict

Minimal-target closure remains stable and high-confidence. Relative to update-7, the connected-regime hidden-vertex frontier is materially sharper: one-hidden realizability is now ruled out for the smallest connected witness M_{2,2,2} via pinned 5-ary shadows, while several natural global constructive ansatz families also fail exactly. Net effect: stronger lower-side obstruction maturity and improved disproof readiness, but full Conjecture 3 closure remains open because whole-family lift and compiler-native routing interpretation are still unresolved.

Contract Checklist (from conjecture statement)

  • 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.
  • Lower-side lift to full compiler-native CD(T_n, G) interpretation remains partial.
  • Upper-side D_emu <= c_2 * CD * polylog(n) construction in matching formal model remains open.
  • 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, canonical submodular demand lower bound, and a substantially strengthened connected hidden-vertex obstruction stack (including no one-hidden realization for M_{2,2,2}).
  3. Missing bridge: either (i) lift witness-level obstructions to a whole connected-family nonexpressibility theorem, or (ii) construct a genuinely global hidden-vertex realization for the connected family; then connect 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 statements.
  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: decomposition/tangle/connected-set reductions remain strong, and low-order (chi=1,2) regimes are now sharply separated from the connected chi>=3 frontier.
  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: 95-99
  • Full lower-bound side readiness: 74-86
  • Full upper-bound side readiness: 35-50
  • Full conjecture proof readiness overall: 61-72

Disproof Readiness Scores

  • Full disproof readiness: 32-48
  • Most plausible axis: promote connected-witness obstructions to a family-level hidden-vertex nonexpressibility theorem, or prove a compiler-interpretation mismatch even under connected realizability.
  • Counterexample maturity: improved versus update-7 because one-hidden realizability is now unconditionally excluded for M_{2,2,2}, but k=2 and fully less-symmetric k>=3 realizations remain open.

What Actually Changed Since Last Check

  • Node graph increased from 136 to 144 markdown nodes.
  • New theorem-level deltas on the connected hidden-vertex frontier:
  • [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]] shows the whole size-2 subfamily still satisfies direct higher-arity F_sep.
  • [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]] rules out k=1,2 realizations for M_{2,2,2} inside a large exact coefficient box.
  • [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]] rules out first natural orbit-symmetric global ansatz classes.
  • [[visible-symmetric-hidden-distinguished-ansatze-fail-on-m222.md]] rules out first hidden-symmetry-breaking visible-symmetric ansatz classes.
  • [[every-four-ary-pinning-minor-of-m222-is-one-hidden-representable.md]] shows all pinned 4-ary shadows are one-hidden representable.
  • [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]] gives the first exact global low-auxiliary obstruction at arity 5 and implies no one-hidden realization for full M_{2,2,2}.
  • Net effect: the local-shadow story is now split sharply (all pinned 4-ary positive, some pinned 5-ary obstructive), giving a clearer global obstruction signal while still not resolving whole-family realizability or the routing-semantic bridge.

Highest-Leverage Next 3 Moves

  1. Decide the k=2 boundary for M_{2,2,2} without coefficient-box restrictions, then test transfer criteria to the full connected family.
  2. Prove or refute liftability: establish a theorem that either transfers connected-witness obstructions to whole-family nonexpressibility, or proves such transfer impossible under current closure operations.
  3. Keep compiler-native closure aligned: whichever side wins (global realization or family-level obstruction), connect it to pivot-robust routing/load semantics needed for the CD(T_n, G) lower-bound interpretation.