All Intrinsic Macro-Routes Now Reduce To Three Family-Specific Lifts

Claim/Theorem

Keep the notation of [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]], [[large-binary-tangles-dominate-grid-minors-but-grid-domination-stops-before-dense-breadth.md]], [[generic-robust-flower-control-still-stops-before-dense-tangle-breadth.md]], and [[tester-side-irreducibility-still-stops-before-original-qubit-matroid-connectivity.md]].

After the current run has explicitly stress-tested Routes A, B, and C, the intrinsic Conjecture-3 macro-frontier reduces to three exact family-specific lift theorems, with no broader generic theorem from the present source base closing any of them.

More precisely:

1. Route A reduces to the tangle-side lift

   \[ H_{\mathrm{grid}\to\mathrm{dense}}(\beta): \quad \text{dominated large grid structure in a binary original-matroid tangle} \Longrightarrow \text{dense original-matroid concentration.} \]

   By [[large-binary-tangles-dominate-grid-minors-but-grid-domination-stops-before-dense-breadth.md]], current sources reach large grid domination but stop before any domination-to-density lift.

2. Route B reduces to the flower-side lift

   \[ H_{\mathrm{flower}}^{\mathrm{dense}}(\beta): \quad \text{in the target family, a }\beta\text{-balanced sublinear-rank cut cannot persist inside an admissible robust flower template} \]

   unless that template already yields dense original-matroid concentration.

   By [[generic-robust-flower-control-still-stops-before-dense-tangle-breadth.md]], generic robust/nonsequential/flower theory remains too permissive even after granting the earlier orientation gap.

3. Route C reduces to the tester-lift

   \[ H_{\mathrm{tester}\to\mathrm{matroid}}(\beta): \quad \text{the left-right-Cayley / local-agreement / tester-irreducibility package} \Longrightarrow \text{exact original-qubit matroid connectivity.} \]

   By [[tester-side-irreducibility-still-stops-before-original-qubit-matroid-connectivity.md]], current sources reach only auxiliary-code balanced prefix cuts and chosen-tester irreducibility, not intrinsic qubit-side connectivity.

4. These are all family-specific original-matroid lifts. None is implied by the generic matroid-tangle theorems, generic robust-flower theorems, or generic LTC/tester theorems already on disk.

5. Therefore the intrinsic macro-frontier is now compressed more sharply than before:

   • future intrinsic progress must introduce one genuinely new family-specific theorem of the three forms above;
   • further generic widening along A, B, or C would mostly repeat already exhausted patterns in the current graph.

6. Consequently, unless a new intrinsic family theorem is proposed first, the best remaining macro-route is Route D, the compiler-native CD side built from [[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]] and its realization obstructions.

This does not change the intrinsic target itself. If a future run reopens intrinsic work, dense tangle breadth remains the canonical intrinsic closure target by [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]. What changes here is the macro-ranking: after exhausting A, B, and C relative to current sources, Route D becomes the canonical remaining global frontier.

Dependencies

• [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]
• [[large-binary-tangles-dominate-grid-minors-but-grid-domination-stops-before-dense-breadth.md]]
• [[generic-robust-flower-control-still-stops-before-dense-tangle-breadth.md]]
• [[tester-side-irreducibility-still-stops-before-original-qubit-matroid-connectivity.md]]
• [[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]]
• [[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]]
• [[generic-submodular-demand-does-not-force-classical-routing-realization.md]]
• [[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]]

Conflicts/Gaps

• This is a frontier-compression theorem for the current graph state, not an impossibility theorem against future intrinsic progress.
• It does not prove that Route D will close the full conjecture. It only records that A, B, and C now require genuinely new family-specific lifts not present in the current source base.
• A future family-specific theorem proving any one of \(H_{\mathrm{grid}\to\mathrm{dense}}(\beta)\), \(H_{\mathrm{flower}}^{\mathrm{dense}}(\beta)\), or \(H_{\mathrm{tester}\to\mathrm{matroid}}(\beta)\) would immediately reopen the intrinsic side and restore dense tangle breadth as the canonical target there.

Sources

• 10.1016/j.jctb.2007.10.008
• 10.1016/j.jctb.2013.03.002
• 10.1016/j.aam.2007.05.004
• 10.1109/FOCS54457.2022.00117
• 10.1145/3519935.3520024
• 10.48550/arXiv.2005.01045
• 10.37236/12467
• 10.48550/arXiv.2109.14599