Manuscript Blueprint: Conjecture 3 Partial-Resolution Package¶

Created: 2026-05-11

This blueprint organizes the current publishable Conjecture 3 package supported by the local graph. It is not a new proof node and does not add theorem claims beyond the cited nodes.

Strongest Publishable Package¶

The strongest package currently supported by the graph is a partial resolution centered on static 2D syndrome-depth lower bounds and their separator/cut-rank/compiler-native extensions:

A presentation-invariant static near-square 2D lower bound for theorem-level Quantum Tanner good families: any 2D-local syndrome-extraction circuit on \(m\) physical qubits has depth \(\Omega(n^{3/2}/m)\), hence \(\Omega(\sqrt n)\) in linear space.
A weighted-separator meta-theorem and fixed-minor-free hardware corollary: bounded-degree fixed-minor-free hardware gives depth \(\Omega(n/\sqrt N)\) in the local Clifford stabilizer-measurement model, hence \(\Omega(\sqrt n)\) in linear space.
A presentation-invariant stabilizer cut-rank functional \(\Xi(\mathcal S,G_{\mathrm{hw}})\) and computable rank formula for \(\chi_L(\mathcal S)\).
A canonical submodular compiler-demand object \(T_{\mathrm{sub}}(\mathcal S)\) and SWAP-only cut-congestion lower bound in the measurement-free SWAP-only regime.
Route-D hidden-variable realization and coordinate-toggle finite computations as appendix/frontier evidence only, not proof-critical material.

Title Options¶

Separator Barriers for Syndrome Extraction and Submodular Cut Rank
Static Two-Dimensional Barriers for Quantum Tanner Syndrome Extraction
Cut-Rank and Separator Lower Bounds for Local Quantum LDPC Syndrome Extraction

Preferred title:

Separator Barriers for Syndrome Extraction and Submodular Cut Rank

Abstract¶

We prove a static two-dimensional depth barrier for syndrome extraction of good quantum LDPC codes and package the result in a separator and cut-rank language suited to compiler lower bounds. For theorem-level Quantum Tanner code families with \(k=\Theta(n)\) and \(d=\Theta(n)\), the Baspin-Fawzi-Shayeghi syndrome-depth tradeoff gives depth \(\Omega(n^{3/2}/m)\) for any 2D-local implementation on \(m\) physical qubits, and therefore \(\Omega(\sqrt n)\) depth in linear space. We then isolate the separator mechanism behind this barrier: local Clifford stabilizer-measurement circuits for bounded-weight local-expander QLDPC codes on bounded-degree hardware with weighted separator profile \(s_{\mathrm{hw}}(N)\) require depth \(\Omega(n/(\Delta_{\mathrm{hw}}s_{\mathrm{hw}}(N)))\), yielding \(\Omega(n/\sqrt N)\) for fixed-minor-free hardware families. To connect these geometric statements to compiler-native quantities, we define an intrinsic stabilizer cut-rank functional \(\chi_L(\mathcal S)\), give its rank formula, and package it as a canonical symmetric submodular demand \(T_{\mathrm{sub}}(\mathcal S)\). In the measurement-free SWAP-only model, depth is lower-bounded by the associated submodular cut congestion. We close by separating these theorem-level results from open frontier questions about dense intrinsic tangle breadth and hidden-variable or packet-routing realizations.

Manuscript Thesis¶

The manuscript should present Conjecture 3 as partially resolved in the static 2D and separator regimes, while making clear that the full original congestion-dilation conjecture remains open. The main story is that separator geometry and intrinsic stabilizer cut rank already yield rigorous lower bounds strong enough to prove the intended near-square static 2D barrier.

Material Placement¶

Main Theorems¶

Theorem A: Static near-square 2D lower bound for theorem-level Quantum Tanner good families.
Theorem B: Weighted-separator syndrome-depth lower bound.
Corollary C: Fixed-minor-free hardware syndrome-depth barrier.
Theorem D: Stabilizer cut-rank functional lower-bounds Clifford syndrome-measurement depth.
Theorem E: Canonical submodular cut-congestion lower-bounds measurement-free SWAP-only depth.

Corollaries¶

Linear-space static near-square 2D hardware forces \(\Omega(\sqrt n)\) depth for theorem-level Quantum Tanner good families.
Bounded-degree fixed-minor-free hardware with \(N=\Theta(n)\) forces \(\Omega(\sqrt n)\) depth in the local Clifford stabilizer-measurement model.
Static grid lower bounds are instances of a broader weighted-separator obstruction, not isolated grid artifacts.
The SWAP-only compiler lower bound holds once the guest object is allowed to be the canonical submodular demand \(T_{\mathrm{sub}}(\mathcal S)\) rather than a packet-routing graph.

Architecture Examples¶

Near-square 2D grid with \(N=\Theta(n)\) hardware qubits.
Bounded-degree planar and fixed-minor-free hardware families, using the fixed-minor-free separator theorem already isolated on the graph.
General bounded-degree hardware families with a supplied sublinear weighted separator profile \(s_{\mathrm{hw}}(N)\).
Measurement-free SWAP-only compilation on a fixed hardware graph, stated through \(\operatorname{CD}_{\mathrm{sub}}\).

Proof Appendices¶

Parameter-only Quantum Tanner good-family proof using \(k,d=\Theta(n)\) and the 2D syndrome-depth theorem.
Weighted-separator to balanced data cut conversion.
Fixed-minor-free separator specialization.
Cross-cut stabilizer rank formula and submodularity of \(\lambda_{\mathcal S}\).
SWAP-only service lower bound and cut-capacity summation.
Model comparison and constant bookkeeping.

Frontier And Discussion Appendices¶

Dense intrinsic tangle breadth as the unresolved intrinsic route.
Route-D hidden-variable realization as frontier evidence, not a proof ingredient.
Coordinate-toggle finite branch demotion after Outcome D.
Classical packet/path routing semantics as an open realization question for \(T_{\mathrm{sub}}(\mathcal S)\).

Main Theorem Sequence¶

Theorem A: Static Near-Square 2D Lower Bound¶

Statement:

Let \(\{Q_n\}\) be an infinite theorem-level Quantum Tanner family with block length \(n\), dimension \(k_n=\Theta(n)\), and distance \(d_n=\Theta(n)\). Any 2D-local syndrome-extraction circuit for \(Q_n\) on \(m_n\) physical qubits has depth

\[ \Delta_n=\Omega\!\left(\frac{k_n\sqrt{d_n}}{m_n}\right) =\Omega\!\left(\frac{n^{3/2}}{m_n}\right). \]

In the linear-space regime \(m_n=\Theta(n)\), this gives

\[ \Delta_n=\Omega(\sqrt n). \]

Source nodes:

[[nodes/quantum-tanner-good-family-presentation-invariant-2d-barrier.md]]
[[nodes/2d-syndrome-depth-from-code-parameters.md]]

Required assumptions:

The Quantum Tanner family is one of the theorem-level good families with \(k=\Theta(n)\) and \(d=\Theta(n)\).
The circuit is 2D-local in the syndrome-extraction model covered by the parameter theorem.
The linear-space corollary assumes \(m=\Theta(n)\) physical qubits.

Exact conclusion:

Depth \(\Omega(n^{3/2}/m)\) for arbitrary \(m\).
Depth \(\Omega(\sqrt n)\) when \(m=\Theta(n)\).
The SWAP-only static near-square grid model is covered a fortiori as a special case of the stronger 2D-local model.

Proof dependencies:

Good-family parameters for Quantum Tanner codes.
Baspin-Fawzi-Shayeghi 2D syndrome-depth theorem.

Known limitations:

This is 2D-specific and parameter-based.
It does not identify the full conjectured congestion-dilation functional.
It does not construct a deterministic local component-code family beyond what is already certified by the source theorem.

Theorem B: Weighted-Separator Syndrome-Depth Lower Bound¶

Statement:

Let \(G_{\mathrm{hw}}\) be an \(N\)-vertex hardware graph with maximum degree \(\Delta_{\mathrm{hw}}\) and weighted separator function \(s_{\mathrm{hw}}(N)\). Suppose a local Clifford stabilizer-measurement circuit on \(G_{\mathrm{hw}}\) measures a bounded-weight local-expander QLDPC code with \(n\) data qubits. If \(s_{\mathrm{hw}}(N)=o(n)\), then

\[ \operatorname{depth}(C) =\Omega\!\left(\frac{n}{\Delta_{\mathrm{hw}}s_{\mathrm{hw}}(N)}\right). \]

Source nodes:

[[nodes/weighted-separator-function-to-syndrome-depth.md]]

Required assumptions:

Local Clifford stabilizer-measurement circuit model.
Bounded-weight local-expander QLDPC family.
Hardware graph has bounded maximum degree parameter \(\Delta_{\mathrm{hw}}\).
Hardware admits weighted separators of size \(s_{\mathrm{hw}}(N)\).
The useful balanced-cut regime requires \(s_{\mathrm{hw}}(N)=o(n)\).

Exact conclusion:

There exists a balanced data cut with edge boundary \(O(\Delta_{\mathrm{hw}}s_{\mathrm{hw}}(N))\).
The circuit depth is \(\Omega(n/(\Delta_{\mathrm{hw}}s_{\mathrm{hw}}(N)))\).

Proof dependencies:

Weighted separator theorem for the hardware family.
Expansion-cut-to-syndrome-depth theorem for local-expander QLDPC stabilizer measurement.
Degree conversion from vertex separator size to edge-boundary capacity.

Known limitations:

This is not an arbitrary compiler theorem.
It depends on local Clifford stabilizer-measurement structure.
If \(s_{\mathrm{hw}}(N)\) is not sublinear relative to \(n\), the separator may absorb too much data mass.

Corollary C: Fixed-Minor-Free Hardware Barrier¶

Statement:

Let \(G_{\mathrm{hw}}\) be an \(N\)-vertex hardware graph of maximum degree \(\Delta_{\mathrm{hw}}\) excluding a fixed \(h\)-vertex minor. If a local Clifford stabilizer-measurement circuit on \(G_{\mathrm{hw}}\) measures a bounded-weight local-expander QLDPC code with \(n\) data qubits and \(h^{3/2}\sqrt N=o(n)\), then

\[ \operatorname{depth}(C) =\Omega\!\left(\frac{n}{\Delta_{\mathrm{hw}}h^{3/2}\sqrt N}\right). \]

For bounded-degree fixed-minor-free hardware with fixed \(h\), this becomes

\[ \operatorname{depth}(C)=\Omega\!\left(\frac{n}{\sqrt N}\right), \]

and in linear space \(N=\Theta(n)\) it gives \(\Omega(\sqrt n)\).

Source nodes:

[[nodes/fixed-minor-free-hardware-syndrome-depth-barrier.md]]
[[nodes/weighted-separator-function-to-syndrome-depth.md]]

Required assumptions:

Hardware excludes a fixed \(h\)-vertex minor.
Hardware maximum degree is at most \(\Delta_{\mathrm{hw}}\).
Same circuit and code assumptions as Theorem B.
The clean balanced-cut statement assumes \(h^{3/2}\sqrt N=o(n)\).

Exact conclusion:

Depth \(\Omega(n/(\Delta_{\mathrm{hw}}h^{3/2}\sqrt N))\).
Depth \(\Omega(n/\sqrt N)\) for bounded-degree fixed-minor-free hardware with fixed excluded minor.
Depth \(\Omega(\sqrt n)\) for linear-space implementations.

Proof dependencies:

Weighted excluded-minor separator theorem.
Theorem B.

Known limitations:

Fixed-minor-free only; stronger bounded-genus constants are not isolated in the current graph.
Not an arbitrary-compiler or full CD theorem.
Large ancilla regimes weaken the clean \(\Omega(\sqrt n)\) form.

Theorem D: Stabilizer Cut-Rank Functional¶

Statement:

Let \(\mathcal S\) be the stabilizer space measured by a Clifford syndrome-extraction circuit on hardware graph \(G_{\mathrm{hw}}\). For each cut \(L\), let \(\chi_L(\mathcal S)\) denote the intrinsic cross-cut stabilizer rank, and define

\[ \Xi(\mathcal S,G_{\mathrm{hw}}) = \max_{L\subseteq V(G_{\mathrm{hw}}),\,|\partial L|>0} \frac{\chi_L(\mathcal S)}{|\partial L|}. \]

Then every Clifford circuit measuring a generating family of \(\mathcal S\) obeys

\[ \operatorname{depth}(C)\ge \frac{1}{64}\Xi(\mathcal S,G_{\mathrm{hw}}). \]

Moreover, if \(H=[H_L\mid H_R]\) is any full-row-rank binary stabilizer matrix with row space \(\mathcal S\), then

\[ \chi_L(\mathcal S) = \operatorname{rank}(H_L)+\operatorname{rank}(H_R)-\operatorname{rank}(H). \]

Source nodes:

[[nodes/stabilizer-cut-rank-functional.md]]
[[nodes/cross-cut-stabilizer-rank-rank-formula.md]]

Required assumptions:

Clifford circuit measuring commuting Pauli stabilizers.
Hardware graph supplies the cut boundary \(|\partial L|\).
\(\mathcal S\) is treated as a stabilizer space independent of generator presentation.

Exact conclusion:

A generator-choice-invariant cut-rank functional lower-bounds depth.
\(\chi_L(\mathcal S)\) is computable as a rank-connectivity function of the stabilizer matrix column matroid.

Proof dependencies:

Cross-cut stabilizer rank definition.
Stabilizer-measurement cut lower bound.
Rank formula for \(\chi_L(\mathcal S)\).

Known limitations:

Does not by itself prove large \(\chi_L\) for Quantum Tanner parity-check matroids.
Cut-based rather than path-based.
Applies to Clifford stabilizer-measurement circuits, not arbitrary non-measurement compilers.

Theorem E: SWAP-Only Submodular Cut-Congestion Lower Bound¶

Statement:

Let \(\mathcal S\) be a stabilizer space on qubit set \(Q\), and define

\[ \lambda_{\mathcal S}(L)=\chi_L(\mathcal S), \qquad T_{\mathrm{sub}}(\mathcal S)=(Q,\lambda_{\mathcal S}). \]

The function \(\lambda_{\mathcal S}\) is integer-valued, symmetric, and submodular. For a hardware graph \(G_{\mathrm{hw}}\) on the same qubit locations, define

\[ \operatorname{CD}_{\mathrm{sub}}(T_{\mathrm{sub}}(\mathcal S),G_{\mathrm{hw}}) = \max_{L\subseteq Q,\,|\partial_{G_{\mathrm{hw}}}L|>0} \frac{\lambda_{\mathcal S}(L)}{|\partial_{G_{\mathrm{hw}}}L|}. \]

Every measurement-free SWAP-only syndrome-extraction circuit \(C\) for \(\mathcal S\) on \(G_{\mathrm{hw}}\) obeys

\[ \operatorname{depth}(C) \ge \frac{1}{16} \operatorname{CD}_{\mathrm{sub}}(T_{\mathrm{sub}}(\mathcal S),G_{\mathrm{hw}}). \]

Source nodes:

[[nodes/stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]]
[[nodes/submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]]
[[nodes/cross-cut-stabilizer-rank-rank-formula.md]]

Required assumptions:

Measurement-free SWAP-only syndrome-extraction circuit with final local measurements.
Hardware graph on the same qubit locations.
Stabilizer space \(\mathcal S\) supplies the canonical submodular demand function.

Exact conclusion:

The submodular CD object \(T_{\mathrm{sub}}(\mathcal S)\) is canonical and presentation-invariant.
SWAP-only depth is at least a constant times the submodular cut congestion.

Proof dependencies:

Rank formula identifies \(\lambda_{\mathcal S}\) with matroid connectivity.
Matroid connectivity gives symmetry and submodularity.
Cross-cut gate service lower-bounds stabilizer cut rank.
One SWAP-only depth layer can serve at most \(|\partial_{G_{\mathrm{hw}}}L|\) across cut \(L\).

Known limitations:

This is not a packet-routing theorem.
This is not an ordinary guest-graph realization of \(\operatorname{CD}\).
The hidden-variable and auxiliary-realization questions remain frontier material.

Section-By-Section Outline¶

1. Introduction¶

State the motivating obstruction: static local hardware cannot perform one full syndrome-extraction round for good QLDPC families at constant depth in the near-square 2D linear-space regime.
Present the main result as a separator and cut-rank package, not as a full proof of the original CD conjecture.
Preview the four theorem layers: static 2D, weighted separators, stabilizer cut rank, and SWAP-only submodular cut congestion.

2. Models And Scope¶

Define syndrome-extraction circuits, 2D-local circuits, local Clifford stabilizer-measurement circuits, and measurement-free SWAP-only circuits.
Define hardware graph size \(N\), data size \(n\), ancilla count, and depth.
State explicitly which theorems allow arbitrary local operations and which require Clifford or SWAP-only restrictions.

3. Parameter-Only Static 2D Barrier¶

Prove Theorem A from the 2D syndrome-depth theorem and Quantum Tanner good-family parameters.
Emphasize presentation independence.
State the linear-space \(\Omega(\sqrt n)\) corollary.

4. Separator Barriers Beyond Grids¶

Develop the weighted-separator meta-theorem.
Prove Theorem B using data-weighted balanced separators and expansion-cut-to-depth.
Specialize to fixed-minor-free hardware and prove Corollary C.
Explain that square-grid behavior is one instance of a broader separator law.

5. Intrinsic Stabilizer Cut Rank¶

Define \(\chi_L(\mathcal S)\) and \(\Xi(\mathcal S,G_{\mathrm{hw}})\).
Prove the rank formula.
Prove Theorem D.
Explain how this avoids generator-presentation dependence while exposing the remaining matroid-connectivity problem.

6. Compiler-Native Submodular Cut Congestion¶

Define \(\lambda_{\mathcal S}\) and \(T_{\mathrm{sub}}(\mathcal S)\).
Prove symmetry and submodularity.
Define \(\operatorname{CD}_{\mathrm{sub}}\).
Prove Theorem E for measurement-free SWAP-only circuits.
Explain what this resolves and what it deliberately does not resolve.

7. Relation To The Original Conjecture 3 Program¶

Map the main theorem package onto the original static near-square 2D target.
Explain why the solved package is enough for the intended static-grid lower bound.
Separate the solved submodular cut-congestion theorem from unresolved packet/path routing semantics.
Explain that dense intrinsic tangle breadth remains a possible future route but is not proof-critical here.

8. Discussion And Frontier Evidence¶

Summarize Route-D hidden-variable realization as exploratory frontier evidence.
Record that the coordinate-toggle finite branch is unresolved and demoted to appendix evidence after Outcome D.
State concrete reopening conditions without making the finite branch part of the main proof.

Appendix Plan¶

Appendix A: Source Theorems And Parameter Substitution¶

State the 2D syndrome-depth theorem used for Theorem A.
Record the Quantum Tanner good-family parameter input \(k,d=\Theta(n)\).
Track the \(m\)-qubit and linear-space specializations.

Appendix B: Separator Machinery¶

State the weighted separator profile definition.
Prove the data-weighted balanced-cut lemma.
Convert vertex separators to edge boundaries using \(\Delta_{\mathrm{hw}}\).
Specialize to fixed-minor-free hardware.

Appendix C: Local-Expander Cut-To-Depth Step¶

Record the exact expansion-cut-to-syndrome-depth dependency.
Separate chosen-presentation local-expander claims from the parameter-only theorem.
Keep constants and bounded-weight hypotheses explicit.

Appendix D: Stabilizer Cut Rank¶

Prove the quotient definition of \(\chi_L(\mathcal S)\).
Prove the rank formula.
Prove symmetry and submodularity of \(\lambda_{\mathcal S}\) through matroid connectivity.

Appendix E: SWAP-Only Service Bound¶

Prove the service lower bound cut by cut.
Prove the layer-capacity upper bound.
Derive the \(\operatorname{CD}_{\mathrm{sub}}\) lower bound.

Appendix F: Frontier Evidence And Demoted Route-D Computations¶

Summarize hidden-variable realization only as non-critical context.
Include the coordinate-toggle finite branch demotion.
State that no sparse dual certificate, toggle decomposition, or sorter-completeness theorem was obtained.
List reopening criteria: closed resumable column generation, complete tail-conditioned sorter theorem, or external valued-CSP/multimorphism theorem.

Proof-Polish Checklist¶

Keep theorem labels aligned with model assumptions: arbitrary 2D-local, local Clifford stabilizer measurement, and measurement-free SWAP-only are distinct.
Ensure every asymptotic statement specifies whether the size parameter is data qubits \(n\), total hardware qubits \(N\), or total physical qubits \(m\).
Do not mix the parameter-only Quantum Tanner proof with the chosen-presentation local-expander proof.
Verify the separator proof uses weighted separators and does not assume unweighted balance over all hardware qubits.
Keep the bounded-degree dependence \(\Delta_{\mathrm{hw}}\) visible in separator statements.
Keep the fixed-minor-free dependence \(h^{3/2}\) visible unless specializing to fixed \(h\).
State constants such as 1/64 and 1/16 only where supported by the nodes; otherwise use asymptotic language.
Prove \(\chi_L(\mathcal S)\) is generator-choice invariant before using it as a manuscript-level object.
When introducing \(T_{\mathrm{sub}}(\mathcal S)\), say explicitly that it is a submodular cut-demand object, not a classical routing instance.
Keep Route-D finite computations out of the main theorem chain.

Claims To Avoid¶

Do not claim a full proof of the original CD conjecture.
Do not claim a classical packet-routing, path-routing, or guest-graph realization of \(\operatorname{CD}_{\mathrm{sub}}\).
Do not claim that the hidden-variable or auxiliary-vertex realization theorem is resolved.
Do not claim that the coordinate-toggle finite branch gives an impossibility theorem.
Do not claim a resolved dense intrinsic tangle-breadth theorem.
Do not claim a presentation-invariant lower bound on Quantum Tanner balanced cut rank beyond the parameter-based 2D theorem.
Do not claim arbitrary compiler lower bounds from the local Clifford or SWAP-only theorems.
Do not claim bounded-genus refinements unless a separate sourced node is added.

Route-D Treatment¶

Route-D should appear only in Appendix F and in a short discussion paragraph. The text should say that hidden-variable and auxiliary-realization attempts were stress-tested to the coordinate-toggle finite branch, but that the latest finite implementation returned Outcome D: the branch is unresolved, not disproved, and the corrected sparse-pricing implementation did not close the 750-row support. This supports the manuscript's scoping decision but is not proof-critical.

Current Active Frontier After This Blueprint¶

The active manuscript frontier is proof-polishing and packaging of the solved static 2D, separator, stabilizer cut-rank, and SWAP-only submodular cut-congestion theorem chain. The active proof-search frontier is not reopened here. Dense intrinsic tangle breadth and Route-D hidden-variable realization remain frontier discussion material only.