Conjecture 3 Research Map¶

Objective¶

Prove the minimal Conjecture 3 target: a lower bound for one full syndrome-extraction round of an expander-style QLDPC family compiled onto a static near-square 2D grid using SWAP-only local compilation. Allow adjacent conjectures to enter only when they clarify the same barrier.

Local Resources¶

Use 10_conjectures.bib as the current local bibliography snapshot mirrored from the Zotero 10_conjectures collection. Future cycles should check this file before assuming a paper is absent.
Use conjecture_3_graph_audit.md as the canonical node-maintenance file. Future cycles should load the Canonical Spine there before widening to the larger graph.

Current Status¶

The minimal theorem-sized target is now literature-backed: for bounded-degree Tanner families with the required small-set expansion, any static near-square 2D local syndrome-extraction circuit using N=Theta(n) qubits has depth Omega(sqrt(n)), hence the same lower bound holds a fortiori for SWAP-only compilation.
The same cut-based mechanism now extends beyond square grids: [[fixed-minor-free-hardware-syndrome-depth-barrier.md]] shows that bounded-degree fixed-minor-free hardware families obey the same Omega(n/sqrt(N)) law inside the stabilizer-measurement model, so the static-grid theorem is one planar corollary of a broader separator barrier.
The decisive local-expander route is now split cleanly into [[tanner-to-contracted-expansion-transfer.md]], [[stabilizer-measurement-cut-lower-bound.md]], and [[2d-local-clifford-syndrome-space-depth-tradeoff.md]].
The graph now contains the sharper synthesis node [[expansion-cut-to-syndrome-depth.md]]: inside the stabilizer-measurement model, depth is already Omega(n / |\partial L|) for any balanced cut of an expanding code. This is the most CD-like theorem currently on disk.
The architecture-level abstraction is now cleaner: [[weighted-separator-function-to-syndrome-depth.md]] packages the whole separator route into one meta-theorem. Any hardware family with sublinear weighted separators forces superconstant syndrome-extraction depth for local-expander QLDPC families.
The explicit-family anchor is now more precise: [[quantum-tanner-diagonal-expansion-structure.md]] shows that quantum Tanner codes are literally Tanner codes on expanding diagonal graphs of the left-right Cayley complex. The remaining anchor gap is the final transfer from this auxiliary expansion to the exact stabilizer-presentation expansion required by [[tanner-to-contracted-expansion-transfer.md]].
The explicit-family anchor is now spectrally sharper: [[quantum-tanner-incidence-spectral-gap.md]] shows that the one-parity square-vertex incidence graph sitting directly under the quantum Tanner stabilizer construction already has a constant spectral gap.
The explicit-family anchor is sharper again: [[quantum-tanner-local-generator-blowup.md]] identifies the chosen stabilizer Tanner graph as a constant-size local blow-up of the square-vertex incidence structure. The remaining issue is now a precise transfer lemma from incidence/diagonal expansion to Tanner small-set expansion.
The transfer lemma has now been reduced to a concrete conditional theorem: [[incidence-expansion-to-parity-tanner-expansion.md]] shows that incidence expansion survives the local generator blow-up whenever the constant-size local gadget has a positive partial-boundary constant.
The local gadget step has now simplified further: [[connected-basis-for-nonzero-coordinate-code.md]] shows that positivity of the gadget boundary constant is automatic once the relevant constant-size local code has no zero coordinates and one chooses a connected basis.
The gadget condition is now deterministic under the paper's own distance hypotheses: [[dual-distance-excludes-zero-coordinates.md]] and [[tensor-product-preserves-no-zero-coordinates.md]] show that Theorem 17 already forces the local tensor codes to have no zero coordinates once \(\Delta\) is large enough.
The graph-theoretic bridge is now largely explicit too: [[spectral-gap-to-regular-graph-expansion.md]] and [[regular-graph-expansion-to-incidence-expansion.md]] convert the diagonal-graph spectral gap into incidence expansion.
Consequently, [[quantum-tanner-theorem17-parity-expander.md]] now gives a deterministic conditional parity-Tanner local-expander theorem for the chosen local-generator presentation attached to any component-code choice satisfying Theorem 17.
The minimal static-2D target for theorem-level Quantum Tanner families is now solved presentation-invariantly: [[quantum-tanner-good-family-presentation-invariant-2d-barrier.md]] gives Omega(n^{3/2}/m), hence Omega(sqrt(n)) in linear space, directly from k,d=Theta(n).
The anchor family route also has a sharper but more structurally informative chosen-presentation statement: [[quantum-tanner-theorem17-static-2d-barrier.md]] gives an Omega(n/sqrt(N)) static-2D barrier for the explicit local-generator basis, and is kept on the graph because it is closer to the conjectured CD(T_n,\mathfrak G) mechanism. The older random-model nodes are corollaries via Theorem 18.
The presentation-dependence issue is now isolated more cleanly: [[cross-cut-stabilizer-rank.md]] defines the intrinsic quotient dimension of stabilizers that genuinely cross a cut, and [[stabilizer-cut-rank-functional.md]] packages it into a generator-choice-invariant lower-bound functional.
The intrinsic cut quantity is now also operational: [[cross-cut-stabilizer-rank-rank-formula.md]] identifies \(\chi_L(\mathcal S)\) with the standard rank-connectivity function of any stabilizer matrix, so the frontier can be stated as a concrete binary-matroid connectivity problem.
The compiler-native CD bottleneck is now isolated more sharply: [[swap-only-compiler-extraction-reduces-cd-to-stabilizer-cut-rank.md]] shows that the routing side and the stabilizer cut-functional side are already on disk, [[token-crossing-extraction-fails-for-swap-only-compilation.md]] shows the naive token-crossing version is false even for one cross-cut stabilizer, [[cross-cut-gate-service-lower-bounds-stabilizer-cut-rank.md]] recovers the right replacement inside the measurement-free SWAP-only regime, [[stabilizer-cut-rank-is-not-a-graph-cut-function.md]] shows why an ordinary guest graph on qubit coordinates cannot be the final answer in general, [[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]] shows that a canonical directly submodular replacement already exists, [[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]] turns that object into a theorem-level compiler-native cut lower bound, [[generic-submodular-demand-does-not-force-classical-routing-realization.md]] shows that generic symmetric-submodular theory is still too weak to recover a classical routing realization, [[matroid-connectivity-does-not-force-hypergraph-approximation.md]] sharpens this further to generic matroid connectivity, [[binary-matroid-connectivity-equals-fundamental-graph-cut-rank.md]] gives the exact positive theorem for the binary subclass, [[fundamental-graph-edge-cuts-are-basis-unstable.md]] shows that even the exact realizing graphs do not stably descend to ordinary edge-cut semantics, [[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]] shows that any basis-robust classical load model must survive pivoting, [[local-incidence-lifts-of-fundamental-graphs-are-not-pivot-robust.md]] rules out the whole local-incidence gadget route for packet, nonnegative-hypergraph, and auxiliary-vertex readings, [[pivot-class-best-incidence-load-still-overestimates-connectivity.md]] shows that even best-basis optimization across the whole pivot class still overshoots connectivity by a linear factor, [[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]] shows that even exact global nonnegative-hypergraph semantics already fail in a simple binary-matroid example, [[four-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] and [[five-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] show that the auxiliary-vertex route actually survives every stabilizer example through arity 5, [[five-qubit-stabilizer-cut-rank-satisfies-fsep.md]] explains why the earlier F_{\mathrm{sep}} test did not separate that boundary, [[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]] shows that Iwamasa's broader Boolean network-representability framework does not widen the nontrivial stabilizer subclass beyond the original hidden-vertex graph-cut question, [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]] shows that the first natural constructive hidden-vertex extension beyond the 5-qubit theorem already fails at arity 6, [[six-qubit-witness-survives-all-four-ary-exact-minors.md]] shows that even every 4-ary pinning/minimization minor of that witness passes the exact Sep/F_{\mathrm{sep}} theorem, [[six-qubit-witness-satisfies-direct-fsep.md]] shows that even the original 6-ary witness itself satisfies the direct higher-arity F_{\mathrm{sep}} condition, [[six-qubit-witness-is-hidden-vertex-graph-cut-representable.md]] now gives an explicit four-hidden-bit realization of that same witness, [[parallel-class-affine-basis-family-is-hidden-vertex-graph-cut-representable.md]] shows that this realization extends to an infinite stabilizer/binary-matroid family, [[parallel-extension-of-binary-circuit-gives-threshold-lift-cut-rank.md]] identifies the exact invariant structure of that family as a circuit with one element replaced by a parallel class, [[direct-sums-of-threshold-lift-pieces-stay-hidden-vertex-representable.md]] shows that the positive route extends one layer further to direct sums of such pieces, [[low-order-gluing-does-not-explain-high-intrinsic-width.md]] shows that 2-sum and 3-sum closure would still sit entirely inside the already-classified low-order separator regime, [[threshold-lift-plus-direct-sum-class-never-reaches-connectivity-three-core.md]] shows that the present positive class never reaches a connected \chi\ge 3 core at all, [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]] isolates the first natural connected post-gluing family beyond that boundary, [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]] reduces one natural route on that family to a ternary selected-OR gadget, [[minimizing-hidden-binary-submodular-energy-preserves-submodularity.md]] shows that minimizing auxiliary variables in a binary submodular energy preserves visible submodularity, [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]] turns this selector into a genuine nonexpressibility theorem, [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]] shows why that still does not settle the whole connected family, [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]] closes a second exact route by proving that the alternative factorization \lambda_M=\min(U,1+S) cannot be implemented by one global branch bit plus independent local hidden-vertex gadgets, and [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]] now shows that even the smallest connected member survives all exact 4-ary pinning/minimization minors. The live frontier is therefore narrower again: the known 6-qubit witness is not isolated and is not a counterexample, the present positive theorem extends beyond a single piece but still never touches the connected \chi\ge 3 regime, the current exact low-arity closure tools still do not separate the first connected family, and the next step must either find a genuinely more global hidden-vertex realization for that family or a stronger higher-arity obstruction.
There is now an explicit obstruction to overusing global code parameters: [[good-code-parameters-do-not-imply-cut-rank.md]] shows that asymptotically good code families can still have zero balanced-cut connectivity under direct sums, so any successful \chi_L theorem must use extra irreducibility structure beyond [n,k,d].
There is now also a positive invariant statement: [[good-codes-have-some-linear-cut-rank.md]] shows that any good code has at least one coordinate cut with linear rank-connectivity. The real missing ingredient is therefore balancedness or hardware alignment, not mere existence of a high-connectivity cut.
The ordering-based invariant route is now substantially sharper: [[balanced-linear-cut-rank-from-trellis-width.md]] shows that every ordering of a good code already contains a balanced prefix cut with linear intrinsic rank-connectivity.
The sweep-ordering route is now identified exactly in matroid language: [[matroid-pathwidth-equals-code-trellis-width.md]] shows that the trellis-width parameter already on the graph is precisely matroid pathwidth of the associated code matroid.
The positive side of that route is now explicit too: [[good-classical-codes-have-linear-matroid-pathwidth.md]] shows that genuinely good classical codes have linear matroid pathwidth. But this does not reopen a shortcut for Quantum Tanner CSS parity codes, because [[qldpc-css-constituent-codes-not-good.md]] and [[wolf-parameter-bound-insufficient-for-css-qldpc-constituents.md]] still kill the parameter-only pathwidth route there.
The intrinsic route is now sharper in two directions: [[good-codes-have-logarithmic-branchwidth.md]] shows that any good code already has Omega(n/log n) branchwidth, while [[balanced-cut-rank-to-syndrome-depth.md]] isolates the exact balanced-cut \chi_L hypothesis that would settle the hardware lower bound in one step.
The decomposition-theoretic intrinsic route is now cleaner too: [[branchwidth-and-matroid-treewidth-are-equivalent.md]], [[every-matroid-admits-optimal-lean-tree-decomposition.md]], and [[good-codes-admit-logarithmic-width-lean-decomposition.md]] show that good codes already force logarithmic-width lean tree-decompositions in the original code matroid, not just high width hidden somewhere in a minor or a noncanonical tree.
The coordinate-tree normalization is now sharper as well: [[linked-branch-decomposition-exists-at-optimal-width.md]] shows that the same intrinsic width can always be witnessed by an optimal linked branch decomposition. So the live issue is no longer finding a structured optimal decomposition, but concentrating its width into a usable bag or displayed cut.
There is now a cleaner presentation-invariant cut bridge for sweep-like hardware geometries: [[trellis-width-to-syndrome-depth-via-hardware-ordering.md]] converts any small-boundary hardware ordering directly into a syndrome-depth lower bound from trellis width of the associated classical kernel code. This is a clean conditional route, but it is not presently decisive for Quantum Tanner because the relevant CSS constituent codes are not asymptotically good.
The sweep route is now packaged by a standard hardware invariant: [[hardware-cutwidth-to-syndrome-depth.md]] shows that classical-kernel trellis-width divided by hardware cutwidth lower-bounds syndrome-extraction depth. It is conceptually useful, but no longer treated as a stand-alone proof of the sharp static-grid barrier for Quantum Tanner families.
The intrinsic tree-decomposition route is now explicit as well: [[code-branchwidth-to-syndrome-depth-via-hardware-tree-decomposition.md]] upgrades the older branchwidth observation into a direct hardware lower bound. It is weaker by a logarithm and shares the same constituent-code limitation.
That limitation is now explicit on the graph: [[qldpc-css-constituent-codes-not-good.md]] records that for CSS qLDPC constructions, the constituent classical codes contain constant-weight words and therefore cannot feed Wolf-type lower bounds through naive classical [n,k,d] arguments.
The methodological consequence is now explicit too: [[wolf-parameter-bound-insufficient-for-css-qldpc-constituents.md]] shows that the standard Wolf parameter bound becomes asymptotically useless on CSS qLDPC constituent codes. Any sharp cutwidth proof would need new structure beyond constituent classical distance.
One concrete candidate for that missing extra structure is now isolated: [[quantum-tanner-constituent-ltc.md]] records that a closely related classical Tanner code in the same square-complex framework is a constant-query locally testable code with linear rate and linear distance.
The LTC side is now structurally sharper: [[smooth-ltc-cayley-characterization.md]] shows that smooth LTC is exactly a Cayley-graph spectral and metric phenomenon.
But that entire equivalence class is now ruled out as sufficient: [[cayley-ltc-characterization-insufficient-for-balanced-cut-rank.md]] upgrades [[good-ltc-does-not-imply-balanced-cut-rank.md]] into the statement that no hypothesis merely equivalent to smooth LTC can force the balanced-cut invariant needed here.
The cleanest nearby positive candidate is now more specific than plain LTC: [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]] shows that the left-right Cayley family is generated by local tensor agreement testability plus two-direction expansion and a parallel-walk local-to-global mechanism.
A new irreducibility bridge is now on the graph: [[ltc-sparse-cut-product-decomposition.md]] and [[strong-ltc-constraint-graph-small-set-expander.md]] show that strong LTCs cannot have sparse tester cuts without approximately factoring across those cuts. This is much closer to the current balanced-cut frontier than generic LTC soundness, but it still lives at the level of a chosen tester graph rather than intrinsic stabilizer connectivity.
The high-dimensional-expander picture is now explicit too: [[agreement-expander-lifts-local-testability.md]] shows that agreement-expander or MAS structure lifts local testability of local pieces to global Tanner codes. This reinforces that the whole HDE line is still a tester-side local-to-global framework rather than an intrinsic cut-rank theorem.
The intrinsic side is now sharper in language as well as formula: [[cut-rank-is-interface-state-dimension.md]] identifies balanced cut rank with exact interface-state complexity in minimal tree realizations. So the missing bridge can now be stated as: force large interface state from expander-style tester irreducibility.
The low-order intrinsic regime is now fully understood: [[exact-2-separation-is-2-sum.md]] and [[nonminimal-exact-3-separation-is-3-sum.md]] show that balanced cut rank 1 and nonminimal balanced cut rank 2 are exactly 2-sum and 3-sum decomposition phenomena. [[internally-4-connected-forces-cut-rank-at-least-three.md]] then isolates the next true frontier: after ruling out low-order decompositions, the unresolved regime begins only at cut rank 3 and above.
The cut-rank-2 regime is now structured even more tightly: [[maximal-partial-3-tree-displays-all-nonsequential-3-separations.md]] shows that in a 3-connected matroid, all nonsequential exact 3-separations are already displayed up to equivalence by one maximal partial 3-tree. So intrinsic cut rank 2 is not merely decomposable one cut at a time; it is globally tree-organized.
The cut-rank-2 regime is now canonical as well: [[reduced-partial-3-tree-is-unique.md]] shows that, after the source paper's natural contraction reduction, the partial-3-tree template is unique. So the nonsequential rank-2 obstruction no longer has arbitrary tree-level choices.
The nonsequential rank-2 regime is now locally peelable too: [[well-positioned-nonsequential-3-separation-has-safe-element.md]] gives a safe deletion or contraction criterion for a well-positioned fully closed side of a nonsequential 3-separation. So the remaining low-order loophole is no longer arbitrary nonsequential 3-separation structure, but rather how to reach and globalize that safe-leaf regime, and how to control the sequential branch.
The nonsequential equivalence side is now more rigid than before: [[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]] shows that once clocks, p-flans, and p-coflans are absent, the whole nonsequential equivalence class collapses to a canonical chain of maximal segments and maximal cosegments with only local reorder freedom.
The sequential branch is now sharply constrained too: [[sequential-matroid-has-canonical-left-right-ends.md]] shows that every non-exceptional sequential matroid has canonical left and right ends shared by all sequential orderings, while [[normalized-sequential-orderings-have-bounded-end-variation.md]] shows that normalized sequential orderings can vary only within a tiny local template near those ends.
The intrinsic regime is now localized further: [[tangle-order-equals-branchwidth.md]], [[large-tangle-yields-weakly-4-connected-minor.md]], and [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]] show that the existing Omega(n/log n) branchwidth of good codes survives inside a weakly 4-connected minor. So logarithmic intrinsic width is not an artifact of recursively gluing together low-order pieces.
The 2025 tangle paper now isolates an even sharper intrinsic quantity: [[tangle-breadth-gives-k-connected-set.md]] shows that breadth, not just order, is what yields an actual large k-connected set, and [[large-k-connected-set-persists-in-weakly-4-connected-minor.md]] shows such a set survives into a weakly 4-connected minor. So the missing intrinsic upgrade beyond branchwidth may be large tangle breadth.
The direct consequence is now explicit too: [[large-k-connected-set-gives-balanced-cut-rank.md]] shows that a linear-size, linearly connected set in the parity-check matroid would already imply the desired Omega(n/|\partial L|) cut lower bound. This is the sharpest current intrinsic reduction on disk.
The separator-alignment gap is now narrower than before: [[dense-k-connected-set-forces-balanced-cut-rank.md]] shows that if the relevant highly connected set is dense enough, then every hardware-balanced cut automatically has large intrinsic cut rank. So the intrinsic frontier can now be phrased as a density question, not just an existence question.
The new lean-decomposition route is now explicit: [[lean-matroid-bag-gives-rank-connected-set.md]] shows that every bag of an optimal lean matroid tree-decomposition is already a rank-connected set. So if one could force a large-rank or large independent bag, the intrinsic cut-rank problem would collapse quickly.
The new branchwidth-localization paper sharpens what high tangle order really gives: [[high-tangle-order-gives-large-tangle-independent-set.md]] shows that high order already forces large independent and coindependent sets removable while preserving 3-connectivity. This means the current missing step is concentration, not mere existence of large structured subsets.
The tangle-matroid route is now stronger than before: [[tangle-independent-set-gives-connected-set.md]] shows that tangle-independent sets are already genuine connected sets, and [[high-tangle-order-gives-large-connected-set.md]] upgrades high tangle order into an actual connected set of size Omega(order) in any 3-connected matroid.
Consequently, combining the new theorem with [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]] now gives more than just width-in-a-minor: good codes already force a weakly 4-connected minor containing a connected set of size Omega(n/log n). The live gap is therefore no longer the existence of any substantial connected set, but densifying it and removing the minor step.
The tangle-structure frontier is now cleaner too: [[robust-tangle-tree-displays-all-nonsequential-separations.md]] shows that once a high-order tangle is robust, all of its genuinely nontrivial low-order separations are organized by a single tree up to tangle-equivalence. So the live intrinsic gap is no longer arbitrary proliferation of small cuts, but specifically robustness, nonsequentiality of hardware-balanced cuts, and exclusion of large flower-like families.
The fixed-order crossing regime is now sharply constrained too: [[every-k-flower-is-anemone-or-daisy.md]] and [[k-flower-local-connectivity-classification.md]] show that crossing exact k-separations collapse to anemone/daisy templates governed by only a few local-connectivity parameters. So any fixed-rank balanced-cut obstruction must already fit into a finite-parameter tree-or-flower template.
A second structural normalization is now on disk: [[canonical-tree-distinguishes-all-matroid-tangles.md]] shows that all distinguishable matroid tangles can be separated by one automorphism-invariant tree decomposition. This does not solve balanced cut rank by itself, but it is the cleanest symmetry-respecting decomposition theorem currently available for the highly structured Quantum Tanner parity-check matroids.
The explicit chosen-presentation route has gained a new exact local-rank tool: [[dual-distance-gives-generator-puncture-rank.md]] shows that the local generator blocks already have full column rank on any puncture smaller than the corresponding dual distance. Under Theorem 17 this means exact local rank up to \delta\Delta coordinates per root neighborhood.
The local explicit-family route is sharper again: [[small-side-local-cut-gives-full-local-cross-rank.md]] upgrades the previous point from local column rank to exact local intrinsic cut rank. Any root neighborhood crossed on fewer than \delta\Delta qubits on its smaller side already contributes the maximum possible local cross-cut rank.
A real obstruction on the explicit-family route is now explicit too: [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]] shows that one cannot pass from a favorable local subfamily of stabilizers to the full global cut rank by monotonicity. Any globalization argument must control the entire stabilizer space or parity-check matroid.
A second invariant route is now on the graph through code-on-graphs theory: [[code-realization-vc-treewidth-bound.md]] shows that good linear codes can have bounded-complexity realizations only on graphs of vc-treewidth at least Omega(n/log n).
A second, more model-general 2D route is now on the graph: [[2d-syndrome-depth-from-code-parameters.md]] gives Delta = Omega(k sqrt(d) / m) for arbitrary [n,k,d] stabilizer codes under local operations and free classical computation.
The new node [[2d-local-syndrome-tradeoff-tightness.md]] shows that the Omega(n/sqrt(N)) law is tight up to constants inside the same 2D local Clifford model.
The architecture-general cut theorem [[stabilizer-measurement-cut-lower-bound.md]] now has a cleaner intrinsic reformulation on the graph: [[stabilizer-cut-rank-functional.md]] is the closest rigorous generator-invariant precursor to the desired CD(T_n,\mathfrak G) functional.
The previous “guest-demand gap” decomposition, [[expander-cut-to-crossing-matching.md]] and [[cross-cut-matching-service-bound.md]], remains useful as an internal derivation aligned with the conjecture's congestion-dilation language.
On the noise-consequence side, [[2d-qec-overhead-from-logical-error-target.md]] now gives a rigorous 2D overhead lower bound for achieving logical error target \delta.
Independent support from [[nonlocality-cost-for-good-qldpc.md]] shows that outperforming local-code tradeoffs in 2D provably demands substantial nonlocal connectivity.
Independent separator evidence from [[separator-profile-bounds-code-distance.md]] shows that low-separator connectivity graphs already obstruct large distance for quantum LDPC codes.
Known positive implementation results, [[thin-planar-connectivity-escape.md]], [[bilayer-locc-2d-implementation-boundary.md]], [[hierarchical-memories-2d-threshold-escape.md]], and [[edge-disjoint-path-teleportation-escape.md]], succeed only by adding resources or code modifications excluded by the direct static-grid conjecture.
The remaining open frontier is no longer the static-grid Omega(sqrt(n)) statement itself, but extending this result from stabilizer-measurement circuits to the full CD(T_n,\mathfrak G) conjecture and to more general compilation maps.

Active Path¶

Treat [[stabilizer-cut-rank-functional.md]] as the current generator-invariant target functional: it already packages the hardware cut bottleneck and the intrinsic cross-cut stabilizer demand in one theorem-level object. Pair it with [[token-crossing-extraction-fails-for-swap-only-compilation.md]], [[cross-cut-gate-service-lower-bounds-stabilizer-cut-rank.md]], and [[stabilizer-cut-rank-is-not-a-graph-cut-function.md]]: the naive path-crossing bridge is false, the correct service-based bridge already holds for measurement-free SWAP-only circuits, and exact packaging by an ordinary guest graph on qubit coordinates is impossible in general. The remaining open step is to identify a richer CD object.
Use [[weighted-separator-function-to-syndrome-depth.md]] as the current hardware-side meta-theorem. Recover [[swap-only-2d-check-layer-cut-barrier.md]] and [[fixed-minor-free-hardware-syndrome-depth-barrier.md]] by feeding in the appropriate separator bounds.
Use [[expansion-cut-to-syndrome-depth.md]] as the current chosen-presentation route from Tanner expansion to depth, while treating [[cross-cut-stabilizer-rank.md]] and [[cross-cut-stabilizer-rank-rank-formula.md]] as the right invariant objects that should eventually replace presentation-specific cross-cut generator counts.
Use [[2d-syndrome-depth-from-code-parameters.md]] and [[quantum-tanner-good-family-presentation-invariant-2d-barrier.md]] as the decisive basis-independent route for the minimal static-2D theorem.
Use [[2d-local-syndrome-tradeoff-tightness.md]] and [[2d-grid-routing-tightness.md]] to understand which depth/space corners are already known to be achievable.
Use [[2d-qec-overhead-from-logical-error-target.md]] to anchor the noise-consequence side of the conjecture, and [[separator-profile-bounds-code-distance.md]] as separator-based evidence for why low-connectivity hardware cannot host good codes directly.
Treat [[quantum-tanner-diagonal-expansion-structure.md]], [[quantum-tanner-incidence-spectral-gap.md]], [[spectral-gap-to-regular-graph-expansion.md]], [[regular-graph-expansion-to-incidence-expansion.md]], [[dual-distance-excludes-zero-coordinates.md]], [[tensor-product-preserves-no-zero-coordinates.md]], [[quantum-tanner-local-generator-blowup.md]], [[incidence-expansion-to-parity-tanner-expansion.md]], and [[quantum-tanner-theorem17-parity-expander.md]] as the current explicit-family attack stack.
Use [[balanced-linear-cut-rank-from-trellis-width.md]], [[matroid-pathwidth-equals-code-trellis-width.md]], [[good-classical-codes-have-linear-matroid-pathwidth.md]], [[trellis-width-to-syndrome-depth-via-hardware-ordering.md]], [[hardware-cutwidth-to-syndrome-depth.md]], and [[code-branchwidth-to-syndrome-depth-via-hardware-tree-decomposition.md]] as a clean conditional invariant route for sweep-like or tree-decomposable hardware geometries. Its present limitation is not geometry but the lack of a strong trellis-width, pathwidth, or branchwidth lower bound for the constituent classical codes appearing in Quantum Tanner CSS presentations.
Use [[balanced-cut-rank-to-syndrome-depth.md]] as the separator-based invariant reduction for hardware families where one controls balanced separators but not a useful sweep or tree decomposition. In that route, the remaining hard step is still linear \chi_L on all hardware-balanced cuts.
Because the minimal static-2D theorem is already settled presentation-invariantly for theorem-level Quantum Tanner families, the main remaining frontier is no longer that theorem itself. It is either to make the component codes explicit and deterministic, or to prove that the parity-check matrices of the chosen Quantum-Tanner presentations have linear balanced-cut rank-connectivity in the sense of [[cross-cut-stabilizer-rank-rank-formula.md]], thereby moving toward CD(T_n,\mathfrak G). On the compiler-native side, [[stabilizer-cut-rank-defines-canonical-submodular-cd-object.md]] and [[submodular-cut-congestion-lower-bounds-swap-only-compiler-depth.md]] now settle the submodular cut-functional formulation, [[generic-submodular-demand-does-not-force-classical-routing-realization.md]] shows that generic submodular theory does not force a path, packet, guest-graph, or nonnegative-hypergraph realization, [[matroid-connectivity-does-not-force-hypergraph-approximation.md]] shows that even matroid connectivity is still too broad, [[binary-matroid-connectivity-equals-fundamental-graph-cut-rank.md]] shows that binary matroids do admit an exact graph realization, but only by cut-rank, [[fundamental-graph-edge-cuts-are-basis-unstable.md]] shows that even ordinary edge cuts of those realizing graphs are not stable enough to give a direct classical interpretation, [[basis-robust-fundamental-graph-loads-must-be-pivot-invariant.md]] shows that any basis-robust realization must survive pivot equivalence rather than merely one chosen basis graph, [[local-incidence-lifts-of-fundamental-graphs-are-not-pivot-robust.md]] shows that even local gadgetizations of one chosen basis graph cannot work, [[pivot-class-best-incidence-load-still-overestimates-connectivity.md]] shows that even optimizing over all basis choices inside the full pivot class still leaves a linear gap, [[simple-binary-connectivity-violates-nonnegative-hypergraph-cut-condition.md]] shows that even exact global nonnegative-hypergraph semantics fail in a simple binary-matroid example, [[four-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] and [[five-qubit-stabilizer-cut-rank-is-hidden-vertex-graph-cut-representable.md]] show that auxiliary-vertex graph-cut realizability survives every stabilizer example through arity 5, [[five-qubit-stabilizer-cut-rank-satisfies-fsep.md]] shows why the universal F_{\mathrm{sep}} test did not separate that boundary, [[boolean-network-generalization-adds-no-nonmonotone-power-for-stabilizer-cut-rank.md]] shows that Iwamasa's broader Boolean network-representability framework collapses back to the ordinary hidden-vertex graph-cut question on every nontrivial stabilizer cut-rank function, [[six-qubit-stabilizer-cut-rank-escapes-modular-plus-fan-cone.md]] shows that the first natural constructive theorem beyond arity 5 already fails at a specific 6-qubit stabilizer space, [[six-qubit-witness-survives-all-four-ary-exact-minors.md]] shows that even all standard 4-ary pinning/minimization reductions of that witness still satisfy the exact Sep/F_{\mathrm{sep}} theorem, [[six-qubit-witness-satisfies-direct-fsep.md]] shows that even the original 6-ary witness itself satisfies direct higher-arity F_{\mathrm{sep}}, [[six-qubit-witness-is-hidden-vertex-graph-cut-representable.md]] shows that the same witness is nevertheless explicitly representable by a four-hidden-bit quadratic submodular energy, [[parallel-class-affine-basis-family-is-hidden-vertex-graph-cut-representable.md]] shows that this threshold-lift pattern extends to an infinite family, [[parallel-extension-of-binary-circuit-gives-threshold-lift-cut-rank.md]] shows that the family has an exact invariant characterization as binary matroids formed by replacing one circuit element by a parallel class, [[direct-sums-of-threshold-lift-pieces-stay-hidden-vertex-representable.md]] shows that the positive route extends one step further under direct sums, [[low-order-gluing-does-not-explain-high-intrinsic-width.md]] shows that low-order gluing cannot be the source of the asymptotically meaningful intrinsic width anyway, [[threshold-lift-plus-direct-sum-class-never-reaches-connectivity-three-core.md]] shows that the current positive class never reaches the first connected \lambda\ge 3 regime, [[multiple-parallel-classes-on-one-circuit-give-connected-cut-rank-at-least-three-family.md]] identifies the first natural connected test family beyond that boundary, [[multi-parallel-circuit-connected-family-reduces-to-selected-or-selector.md]] shows that one obvious compressed-state attack on that family bottlenecks at a selected-OR gadget, [[minimizing-hidden-binary-submodular-energy-preserves-submodularity.md]] shows that auxiliary minimization cannot hide visible non-submodularity, [[selected-or-selector-is-not-hidden-vertex-graph-cut-expressible.md]] turns that selector bottleneck into a genuine nonexpressibility theorem, [[selector-quotient-is-not-a-submodular-minor-of-connected-family.md]] shows that the selector theorem still does not settle the full connected regime because the quotient is not a closure-preserving minor of the original submodular family, [[branch-min-route-for-connected-family-fails-at-local-submodularity.md]] shows that the alternative exact factorization \lambda_M=\min(U,1+S) also fails in its obvious local-gadget form, and [[smallest-connected-multi-parallel-circuit-member-survives-all-four-ary-exact-minors.md]] shows that even the smallest connected member survives all exact 4-ary pinning/minimization minors. The remaining CD gap is therefore narrowed differently again: either one must find a genuinely more global hidden-vertex realization for this connected bottleneck family, lift the present obstructions by a stronger whole-family argument, or show that even connected hidden-vertex realizations of such families do not induce routing-style semantics.
Keep [[good-codes-have-logarithmic-branchwidth.md]], [[linked-branch-decomposition-exists-at-optimal-width.md]], [[branchwidth-and-matroid-treewidth-are-equivalent.md]], [[good-codes-have-logarithmic-matroid-treewidth.md]], [[every-matroid-admits-optimal-lean-tree-decomposition.md]], [[good-codes-admit-logarithmic-width-lean-decomposition.md]], [[code-realization-vc-treewidth-bound.md]], [[qldpc-css-constituent-codes-not-good.md]], [[wolf-parameter-bound-insufficient-for-css-qldpc-constituents.md]], [[quantum-tanner-constituent-ltc.md]], [[smooth-ltc-cayley-characterization.md]], [[cayley-ltc-characterization-insufficient-for-balanced-cut-rank.md]], [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]], [[agreement-expander-lifts-local-testability.md]], [[ltc-sparse-cut-product-decomposition.md]], [[strong-ltc-constraint-graph-small-set-expander.md]], [[cut-rank-is-interface-state-dimension.md]], [[exact-2-separation-is-2-sum.md]], [[nonminimal-exact-3-separation-is-3-sum.md]], [[maximal-partial-3-tree-displays-all-nonsequential-3-separations.md]], [[reduced-partial-3-tree-is-unique.md]], [[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]], [[well-positioned-nonsequential-3-separation-has-safe-element.md]], [[internally-4-connected-forces-cut-rank-at-least-three.md]], [[tangle-order-equals-branchwidth.md]], [[large-tangle-yields-weakly-4-connected-minor.md]], [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]], [[tangle-breadth-gives-k-connected-set.md]], [[tangle-independent-set-gives-connected-set.md]], [[high-tangle-order-gives-large-tangle-independent-set.md]], [[high-tangle-order-gives-large-connected-set.md]], [[robust-tangle-tree-displays-all-nonsequential-separations.md]], [[every-k-flower-is-anemone-or-daisy.md]], [[k-flower-local-connectivity-classification.md]], [[sequential-matroid-has-canonical-left-right-ends.md]], [[normalized-sequential-orderings-have-bounded-end-variation.md]], [[canonical-tree-distinguishes-all-matroid-tangles.md]], [[large-k-connected-set-persists-in-weakly-4-connected-minor.md]], [[large-k-connected-set-gives-balanced-cut-rank.md]], [[dense-k-connected-set-forces-balanced-cut-rank.md]], [[lean-matroid-bag-gives-rank-connected-set.md]], [[dual-distance-gives-generator-puncture-rank.md]], and [[small-side-local-cut-gives-full-local-cross-rank.md]] in view together: they show that plain LTC and its equivalent Cayley/spectral reformulations are too weak, that cut rank 1 and 2 are now completely classified as low-order decompositions, that cut-rank 2 is globally tree-organized, canonically reduced, locally peelable in its nonsequential branch, and canonically end-constrained in its sequential branch, that fixed-order crossing separations collapse to anemone/daisy templates with only a few local-connectivity parameters, that even the surviving Omega(n/log n) intrinsic width can be represented by structured optimal decompositions of the original code matroid, that high tangle order already gives actual connected sets of size Omega(order), that robust high-order tangles organize nonsequential low-order separations by a tree up to equivalence, that a dense linear k-connected set would settle the separator-alignment issue outright, that optimal lean decompositions already contain bag-local rank-connected objects, and that the main unresolved issue is now concentration, densification, flower-exclusion, and exclusion of the remaining special path-width-three templates.
Keep [[thin-planar-connectivity-escape.md]], [[bilayer-locc-2d-implementation-boundary.md]], [[hierarchical-memories-2d-threshold-escape.md]], and [[edge-disjoint-path-teleportation-escape.md]] as explicit markers of which extra resources or code modifications circumvent the direct static-grid barrier.

Nodes¶

Cross-Conjecture Links¶

Conjecture 2 matters because Rosenbaum's adaptive classical-control result shows that measurement or classical feed-forward can qualitatively change the geometry barrier; the present lower-bound target therefore has to stay in the measurement-free SWAP-only regime.
Conjecture 4 may matter later because any Omega(sqrt(n)) compiled-step overhead creates room for a separation between information-theoretic survival and efficient decoding, but that is not needed for the current proof path.

Open Questions¶

Can the chosen-presentation expansion route be upgraded from cross-cut generator counts to a linear lower bound on the intrinsic rank-connectivity quantity from [[cross-cut-stabilizer-rank-rank-formula.md]]?
Which extra structural hypothesis, beyond good [n,k,d] parameters and beyond the entire smooth-LTC equivalence class of [[smooth-ltc-cayley-characterization.md]], is actually sufficient to force linear balanced-cut rank-connectivity for separator-induced cuts: Tanner expansion, puncturing resistance, local tensor agreement plus expansion, or something closer to matroid k-connectivity?
Since [[balanced-linear-cut-rank-from-trellis-width.md]] already gives balanced linear cuts along every ordering, can one align hardware separator cuts with such orderings in a theorem-level way beyond simple sweep geometries?
Can the cutwidth route [[hardware-cutwidth-to-syndrome-depth.md]] be generalized from linear sweeps to tree or branch decompositions of the hardware graph without losing the intrinsic \chi_L control?
Can one prove large trellis-width or branchwidth for the specific classical constituent codes appearing in Quantum Tanner constructions despite [[qldpc-css-constituent-codes-not-good.md]], perhaps from local tensor agreement plus expansion rather than classical distance?
Can one replace Wolf's distance-based lower bound by an expansion-based or local-testability-based trellis-width lower bound strong enough to bypass [[wolf-parameter-bound-insufficient-for-css-qldpc-constituents.md]]?
Can the stronger package in [[left-right-cayley-ltc-from-local-agreement-plus-expansion.md]] be converted into a direct lower bound on trellis-width, branchwidth, balanced-cut rank-connectivity, or puncturing resistance?
Since [[cayley-ltc-characterization-insufficient-for-balanced-cut-rank.md]] kills every hypothesis merely equivalent to smooth LTC, which extra irreducibility assumption should be added: connected tester graph, local tensor agreement, local expansion, cosystolic expansion, or something matroidal?
Can the Dinur-Kaufman sparse-cut theorem [[ltc-sparse-cut-product-decomposition.md]] be converted from approximate product decomposition of a tester graph into a lower bound on intrinsic code connectivity or stabilizer cut rank?
For the left-right Cayley tester adjacent to Quantum Tanner, does [[strong-ltc-constraint-graph-small-set-expander.md]] force a useful no-sparse-cut statement on all hardware-balanced partitions of squares, or does the passage from tester graph cuts to qubit partitions lose too much information?
Can the agreement-expander framework [[agreement-expander-lifts-local-testability.md]] be combined with [[cut-rank-is-interface-state-dimension.md]] to show that iterative local-to-global correction already requires a linear interface state across hardware-balanced separators?
Can one prove that the relevant Quantum Tanner parity-check matroids are internally 4-connected, thereby at least excluding all balanced cuts of rank 1 or 2 via [[internally-4-connected-forces-cut-rank-at-least-three.md]]?
After the new exact classification of cut ranks 1 and 2, what structural hypothesis is strong enough to push balanced cut rank beyond the low-order decomposition regime and into genuinely superconstant or linear growth?
Can the weakly 4-connected high-branchwidth minor from [[good-codes-have-weakly-4-connected-log-branchwidth-minor.md]] be upgraded to a statement about the original Quantum Tanner parity-check matroid, rather than only a minor?
Is there a theorem turning weakly 4-connected plus branchwidth w into a balanced separator cut with connectivity at least f(w) for some superconstant function f?
Can one prove large breadth for the tangles arising from Quantum Tanner parity-check matroids, so that [[tangle-breadth-gives-k-connected-set.md]] yields a large k-connected set rather than only a large-order tangle?
Is tangle breadth, rather than branchwidth, the right intrinsic invariant to target if one wants balanced cut-rank beyond the constant regime classified by 2- and 3-sums?
Can one prove that the relevant Quantum Tanner parity-check matroid contains a k-connected set with both |Z| and k linear in n, so that [[large-k-connected-set-gives-balanced-cut-rank.md]] closes the intrinsic side of the conjecture outright?
Can one strengthen the previous question to a dense version, namely prove that the relevant Quantum Tanner parity-check matroid contains a \(k\)-connected set \(Z\) with \(k=\Omega(n)\) and \(|Z|>(1-\beta)n+k-O(1)\) for the hardware balance parameter \(\beta\), so that [[dense-k-connected-set-forces-balanced-cut-rank.md]] closes the separator-alignment gap outright?
Can one prove that the relevant high-order tangles of the Quantum Tanner parity-check matroid are robust in the sense needed by [[robust-tangle-tree-displays-all-nonsequential-separations.md]]?
If those tangles are robust, can one show that hardware-balanced low-rank cuts are nonsequential rather than merely sequential peelings with respect to the tangle?
Can one rule out large flower-like families or too many tangle-equivalent crossing balanced low-rank cuts in the relevant Quantum Tanner parity-check matroids, perhaps from expansion, LTC irreducibility, or the left-right Cayley symmetries?
Can one use [[every-k-flower-is-anemone-or-daisy.md]] and [[k-flower-local-connectivity-classification.md]] to show that any fixed-order family of hardware-balanced low-rank cuts would force a forbidden anemone or daisy template in the relevant Quantum Tanner parity-check matroids?
In the cut-rank-2 regime, can [[maximal-partial-3-tree-displays-all-nonsequential-3-separations.md]] be combined with Quantum Tanner symmetry or expansion to exclude all hardware-balanced partial-3-tree templates outright?
Can [[reduced-partial-3-tree-is-unique.md]] be combined with the automorphism-rich Quantum Tanner geometry to show that any balanced cut-rank-2 template must be trivial or incompatible with expansion?
Can [[well-positioned-nonsequential-3-separation-has-safe-element.md]] be globalized into an induction that removes all nonsequential cut-rank-2 obstructions without creating new ones?
Is the only serious remaining cut-rank-2 loophole now the sequential branch, suggesting that path-width-three structure is the next necessary source?
Can one show that the relevant Quantum Tanner parity-check matroids cannot realize the canonical end-templates from [[sequential-matroid-has-canonical-left-right-ends.md]], except in trivial bounded-size cases?
Can [[normalized-sequential-orderings-have-bounded-end-variation.md]] be combined with expansion or left-right Cayley symmetry to exclude path-width-three sequential obstructions?
Can one show that clocks, p-flans, and p-coflans from [[nonsequential-equivalence-class-without-special-gadgets-is-canonical-chain.md]] are incompatible with the relevant Quantum Tanner parity-check geometry, thereby collapsing the entire nonsequential rank-2 branch to a forbidden canonical chain?
Can one prove that some bag in an optimal lean tree-decomposition has linear rank, or even contains a linear-size independent subset, so that [[lean-matroid-bag-gives-rank-connected-set.md]] yields a usable large connected set?
Can the large removable sparse set from [[high-tangle-order-gives-large-tangle-independent-set.md]] be concentrated into a single lean-decomposition bag, or otherwise converted into a dense connected set?
In the new lean-decomposition route [[good-codes-admit-logarithmic-width-lean-decomposition.md]], can one prove that some bag in an optimal lean decomposition has rank Omega(n/log n) for the relevant Quantum Tanner parity-check matroids, rather than width being dispersed across many small-rank bags?
In the new linked route [[linked-branch-decomposition-exists-at-optimal-width.md]], can one prove that an optimal linked branch decomposition of the relevant Quantum Tanner parity-check matroid contains a hardware-meaningful balanced displayed cut of rank Omega(n/log n) or better?
Can the Omega(n/log n) connected set now available in a weakly 4-connected minor be lifted back from the minor to the original Quantum Tanner parity-check matroid in a theorem-level way?
What extra hypothesis upgrades the new Omega(branchwidth) connected-set theorem [[high-tangle-order-gives-large-connected-set.md]] to a dense linear connected set: large breadth, bag concentration, local testability, or explicit Quantum-Tanner incidence structure?
Can the new local block theorem [[dual-distance-gives-generator-puncture-rank.md]] be globalized along the chosen Quantum Tanner presentation, perhaps by extracting many boundary roots whose cut-side local punctures are below the dual-distance threshold and sufficiently non-overlapping?
Can the stronger local statement [[small-side-local-cut-gives-full-local-cross-rank.md]] be globalized, for example by proving that every balanced cut induces linearly many lightly crossed root neighborhoods or by extracting a large family of nearly independent light neighborhoods?
Since [[cross-cut-rank-not-monotone-under-stabilizer-extension.md]] kills the naive subspace-monotonicity route, what full-space invariant can replace it on the chosen-presentation path: a block elimination argument, a matroid-linkage theorem, or a direct rank lower bound for the full parity-check matrix across balanced cuts?
Can the logarithmic intrinsic width from [[good-codes-have-logarithmic-branchwidth.md]] be upgraded to linear balanced-cut connectivity under additional irreducibility or expansion hypotheses natural for Quantum Tanner codes?
Can the fine structure of low-order separations around a high-order tangle, as organized by flowers or profile decompositions, be used to rule out too many hardware-relevant small cuts in the original Quantum Tanner parity-check matroid?
Can one prove a strong lower bound on [[matroid-pathwidth-equals-code-trellis-width.md]] for the relevant Quantum Tanner parity-check or constituent-code matroids, so that the sweep-ordering route becomes genuinely competitive with the separator route rather than only a conditional side path?
Can one formalize a local syndrome-extraction circuit as a bounded-complexity graphical realization on a space-time graph strongly enough to invoke [[code-realization-vc-treewidth-bound.md]]?
Can [[weighted-separator-function-to-syndrome-depth.md]] be pushed from weighted separators to a more routing-native invariant, such as separator profile, treewidth growth, or a direct congestion functional?
How should one combine [[2d-qec-overhead-from-logical-error-target.md]] with compiled-depth lower bounds to extract an architecture-dependent threshold criterion resembling the conjectured \delta_\star condition?
Can one characterize exactly which ingredients in [[hierarchical-memories-2d-threshold-escape.md]] are essential for recovering a threshold in 2D local gates: concatenation, subconstant rate, bilayer layout, growing SWAP range, or all of them together?
Is there a clean theorem separating teleportation-style ancilla-path resources as in [[edge-disjoint-path-teleportation-escape.md]] from genuinely SWAP-only compilation at the level of a single unified hardware complexity measure?
Can the theorem-level Quantum Tanner family behind [[quantum-tanner-good-family-presentation-invariant-2d-barrier.md]] be made fully explicit at the component-code level rather than obtained via random existence?
Can one prove that the parity-check matrices of the relevant Quantum Tanner stabilizer spaces have linear balanced-cut matroid connectivity, thereby making the chosen-presentation theorem [[quantum-tanner-theorem17-static-2d-barrier.md]] fully presentation-invariant?
Can the generator-invariant functional [[stabilizer-cut-rank-functional.md]] be lifted from the stabilizer-measurement model to a compiler-native CD(T_n,\mathfrak G) statement?