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 abstract intrinsic missing hypothesis has now been sharpened to a concrete qubit-matrix bottleneck: [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]] shows that the actual Quantum-Tanner construction already exposes the code as translated local tensor-code row blocks on original qubits, and that the exact missing family-level invariant is no longer generic dense breadth but a linear accumulation theorem from local crossing mass to global balanced cut rank.
The rank-accumulation bottleneck is now sharper again: [[local-quotient-image-span-controls-rank-accumulation.md]] shows that the real obstruction is not merely support overlap among crossed root neighborhoods. The global cut rank is exactly the span of the local block images in the quotient \(S/(S_L+S_R)\), so the live missing invariant is linear independence of those quotient images, or equivalently a linear lower bound for the packed surrogate \nu_H(L).
The intrinsic balanced-cut-rank side is now organized by [[dense-tangle-breadth-is-the-canonical-remaining-intrinsic-target.md]]: within the current graph, dense tangle breadth is the canonical remaining intrinsic invariant, while dense \(k\)-connected sets, dense large-rank lean bags, and local quotient-image accumulation are best viewed as mechanisms that would have to feed into that same dense original-matroid concentration theorem. Taken together with [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]], [[local-quotient-image-span-controls-rank-accumulation.md]], [[dense-tangle-breadth-forces-balanced-cut-rank.md]], [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]], and [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]], the intrinsic unresolved task remains to derive dense connectivity in the original qubit parity-check matroid of the target Quantum Tanner / left-right-Cayley family.
The best external literature still does not close the tangle-orientation subgap. [[best-external-near-bridges-still-stop-before-robust-balanced-cut-orientation.md]] records that Clark 2013 / 2016 only organize nonsequential separations after robustness and nonsequentiality are already known, that Brettell et al. remains the best breadth-side near-bridge but still only reaches weakly 4-connected minors or minor-side connected mass, and that the LTC / agreement / Cayley / connectivity-graph package remains tester-side or otherwise non-closing. So the exact missing theorem remains H_{\mathrm{ns}}^{\beta} from [[robust-nonsequentiality-route-stops-before-tangle-orientation-of-balanced-cuts.md]].
The robustness prerequisite is now sharper too: [[source-package-stops-before-robust-high-order-original-matroid-tangles.md]] shows that the current source-grounded family package stops before robustness itself, because it does not yet produce the required linear-order tangles, equivalently linear branchwidth, in the original qubit parity-check matroid. So for the Clark--Whittle route the exact earlier missing theorem is now H_{\mathrm{tan}}^{\Omega} before any putative robustness upgrade H_{\mathrm{rob}}^{\Omega}.
The robust/nonsequential Route B is now sharper on the post-orientation side as well. [[generic-robust-flower-control-still-stops-before-dense-tangle-breadth.md]] shows that even if one granted the missing balanced-cut orientation theorem into one robust high-order tangle, the generic Clark--Whittle plus Aikin--Oxley flower package is still too permissive to force dense breadth: every admissible local-connectivity flower template occurs with arbitrary petal count. So beyond H_{\mathrm{ns}}^{\beta}, the exact next missing theorem is a family-specific petal-mass exclusion or concentration statement on \beta-balanced cuts.
The branchwidth subroute now has one explicit positive conditional reduction: [[sufficiently-dense-k-connected-set-forces-branchwidth.md]] shows that if the original qubit parity-check matroid contains a \(k_n\)-connected set \(Z_n\) with \(k_n=\Omega(|Q_n|)\) and \(|Z_n| > 2|Q_n|/3 + k_n - 2\), then every branch decomposition has a 1/3-balanced displayed cut of rank at least \(k_n-1\), hence the matroid has linear branchwidth and linear-order tangles. So for the chosen mechanism branchwidth-from dense connected-set concentration, the exact missing theorem is now a sufficiently dense linear \(k\)-connected-set theorem in the original qubit matroid.
The dense connected-set route now reaches the canonical breadth target more directly. [[large-k-connected-set-generates-order-k-tangle.md]] extracts an order-k tangle from any k-connected set of size at least 3k-5, with tangle breadth at least the size of that set. Consequently [[sufficiently-dense-k-connected-set-yields-dense-tangle-breadth.md]] shows that a sufficiently dense k-connected set in the original qubit matroid already implies dense tangle breadth itself, not merely balanced cut rank or branchwidth. In particular, the earlier 1/3-balanced threshold from [[sufficiently-dense-k-connected-set-forces-branchwidth.md]] automatically satisfies the 3k-5 side condition and therefore factors through the canonical dense-breadth target.
The decomposition-concentration route is now reduced to one explicit bag-density target. [[dense-large-rank-lean-bag-yields-dense-tangle-breadth.md]] shows that if an optimal lean tree-decomposition of the original matroid contains a bag whose rank is itself dense enough to beat the hardware balance threshold, then a basis of that bag is already a sufficiently dense k-connected set and therefore yields dense tangle breadth. So the lean-bag mechanism is now theorem-sized, but its remaining gap is severe and precise: current source theorems provide lean decompositions and rank-connected bags, not dense large-rank bag concentration.
The lean/linked decomposition route is now sharper on the negative side too. [[lean-linked-decomposition-route-stops-before-dense-bag-rank-concentration.md]] isolates the exact mismatch: Hlineny--Whittle width is a sum of branch rank defects, Erde lean bags become useful only after bag rank is already large, and linked branch decompositions control only edge widths and pathwise linkage of displayed cuts. So all currently sourced decomposition theorems still stop before any theorem concentrating intrinsic width into one dense bag rank, or into any source-faithful bounded adjacent-bag merger strong enough to feed the canonical dense-breadth route.
The whole decomposition-concentration branch is now demoted from default-active status. [[decomposition-concentration-branch-is-demoted-before-subtree-union-theorem.md]] records that even the last plausible variant, source-faithful bounded adjacent-bag merger, is not presently supported by the source base: leanness is bag-local or path-local, tree-width is a node/branch-defect quantity, and linkedness is an edge-path property. None of these currently upgrades to a theorem on subtree unions or merged adjacent-bag connected mass. Future runs should load the decomposition package only if they introduce a genuinely new subtree-union invariant.
The surviving tangle-concentration spine is now sharper too. [[tangle-concentration-spine-stops-before-order-to-density-theorem.md]] compares the three remaining candidates on that spine and shows that the direct order-to-density route is strongest, but still quantitatively non-closing: the current Geelen--van Zwam / Brettell line extracts connected mass with a fixed 20-to-1 loss from tangle order, so even maximal feasible order on an n-element matroid only guarantees O(n/20) connected mass, far short of the dense >(1-\beta)n regime required by the canonical dense-breadth route.
The tangle-side Route A candidate is now sharper in a new way. [[large-binary-tangles-dominate-grid-minors-but-grid-domination-stops-before-dense-breadth.md]] records that, for binary matroids, sufficiently large-order tangles already dominate arbitrarily large grid minors in the tangle-matroid sense of Geelen--Gerards--Whittle. So the strongest currently sourced invariant strictly above order alone is not just a weakly 4-connected minor, but large grid domination. The route still stops one theorem later, however: the graph has no domination-to-density lift back to dense original-matroid concentration.
The tester-side Route C is now stress-tested just as sharply. [[tester-side-irreducibility-still-stops-before-original-qubit-matroid-connectivity.md]] shows that the strongest currently sourced left-right-Cayley package does yield linear balanced prefix-cut interface dimension for the auxiliary square code and irreducibility for chosen tester graphs, but it still has no presentation-invariant lift to exact connectivity in the original qubit parity-check matroid. So the exact missing theorem is now H_{\mathrm{tester}\to\mathrm{matroid}}^{\beta}.
The chosen-presentation local-block route still stops one theorem earlier. [[local-block-route-stops-before-cut-uniform-dense-k-connected-set.md]] shows that the current local quantities \(\mu_H(L)\), \(\sigma_H(L)\), \(\nu_H(L)\), and \(W_v(L)\) are all indexed by a single cut \(L\), so even a strong cutwise rank-accumulation theorem would not yet produce one fixed dense carrier set \(Z_n\) with the universal connectivity inequality required by [[sufficiently-dense-k-connected-set-forces-branchwidth.md]]. So this subordinate route still lacks a cut-uniform densification theorem from local-block data to a dense linear \(k\)-connected set.
The direct local-block route now has its own exact conditional closure criterion. [[lightly-crossed-direct-sum-local-blocks-force-balanced-cut-rank.md]] shows that direct linear balanced cut rank would follow if every hardware-balanced cut contained a linear-mass family of lightly crossed local blocks whose full local cross rank survives in the global quotient and whose quotient images form a direct sum. So the local-block package is no longer just vague evidence: the precise remaining theorem is now a uniform no-survival-loss plus no-overlap-loss packing statement on balanced cuts.
The first ingredient of that direct criterion is now isolated sharply too. [[local-geometry-stops-before-uniform-linear-light-side-mass.md]] shows that the current sourced Quantum Tanner local geometry gives constant-size root neighborhoods, bounded overlap, and auxiliary incidence expansion, but still does not control the minority-load profile \(m_v(L)\) across neighborhoods for arbitrary balanced qubit cuts. So the exact earlier missing theorem is now a uniform linear light-side-mass statement on lightly crossed local blocks, not merely a count of crossed neighborhoods.
That incidence/local-geometry subroute is now demoted as a primary branch in its plain form. [[plain-incidence-local-geometry-route-is-exhausted-before-minority-load-anti-concentration.md]] records that, once the exact missing light-side-mass theorem has been isolated, re-running incidence expansion plus bounded overlap in isolation no longer advances the graph: the next reopening would have to add a genuinely new qubit-side minority-load anti-concentration invariant on balanced cuts. The canonical frontier therefore stays with dense tangle breadth, while plain incidence/local geometry is boundary evidence only.
The explicit quotient-block witness line should now be treated as a secondary boundary branch. [[explicit-quotient-block-witness-branch-is-boundary-not-primary-frontier.md]] packages what this branch has actually established: on the current D_4, D_6, and D_8 low cuts one has \nu_H(L)=\lambda_{M(H)}(L), exact low-deficiency witnesses exist, the objective reduces to represented-matroid block packing or weighted linear matroid parity, but pairwise proxies, visible templates, common-core rules, and local-repair rules all fail. Future prompts should cite this package only as falsification evidence against proposed selector classes unless a genuinely new family-level invariant is specified.
The stronger quotient-packing subroute is now boundary-labeled as well. [[quotient-packing-route-is-demoted-before-construction-controlled-low-deficiency-theorem.md]] records that fixed-basis pivots are too rigid, adapted-basis shellings are non-transferable, and the exact block-packing/parity reduction is only an optimization-class identification. The strongest remaining bridge on that branch is the low-deficiency union-rank criterion from [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]], but no source-grounded Quantum Tanner theorem currently reaches it on balanced cuts. So quotient-packing beyond the diagnostic quotient-span node should now be loaded only on demand unless a genuinely new construction-controlled low-deficiency invariant is proposed.
The intrinsic macro-frontier is now compressed into one explicit closure package. [[all-intrinsic-macro-routes-now-reduce-to-three-family-specific-lifts.md]] records that, after stress-testing A, B, and C in the current graph, every surviving intrinsic route reduces to one of three family-specific lifts: H_{\mathrm{grid}\to\mathrm{dense}}(\beta), H_{\mathrm{flower}}^{\mathrm{dense}}(\beta), or H_{\mathrm{tester}\to\mathrm{matroid}}(\beta). Relative to the present source base, further generic intrinsic widening is exhausted; unless a new family-specific original-matroid theorem is introduced, the compiler-native CD route becomes the best remaining global frontier while dense tangle breadth remains the canonical intrinsic target if that side reopens.
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.
The Route-D semantic side is now sharper than the pure realizability side. [[connected-hidden-vertex-realizability-still-fails-terminal-routing-semantics.md]] shows that there is already a connected 5-qubit stabilizer cut-rank function that is exactly hidden-vertex graph-cut representable while still failing every exact terminal nonnegative-hypergraph semantics, hence every exact terminal graph/packet/path routing semantics on the physical qubit set. So auxiliary-variable expressibility is strictly weaker than usable routing-style compiler meaning, even on connected binary examples.
The Route-D ranking is therefore now explicit. [[route-d-semantic-separation-now-dominates-the-remaining-cd-frontier.md]] records that D1 currently stops at terminal and basis-robust impossibility results without excluding all global auxiliary constructions, D2 is the strongest remaining frontier because it asks whether any such auxiliary realization can ever recover compiler meaning, and D3 is secondary until a genuinely new connected positive building block appears beyond the current multi-parallel-circuit impasse.
The D2 witness-level separation has now been lifted to an infinite connected family. [[even-threshold-lift-connected-family-separates-hidden-vertex-from-terminal-routing-semantics.md]] shows that the connected threshold-lift family with even parallel-class size already has exact hidden-vertex realizations while failing every exact terminal graph/packet/path/nonnegative-hypergraph semantics on the physical terminals. So the semantic-separation frontier is no longer witness-level.
The exact next D2 bottleneck is now sharper too. [[route-d2-family-lift-now-stops-at-first-connected-family-beyond-threshold-lift.md]] records that the family-lift problem is solved on the last connected family with an exact hidden-vertex theorem currently on disk, and that the next unresolved D2 step begins at the first connected \chi\ge 3 family beyond threshold-lift, typified by the multi-parallel-circuit family.
The first connected post-threshold family has now also been compressed on the terminal side. [[odd-two-element-multi-parallel-family-violates-terminal-hypergraph-cut-condition.md]] shows that the odd size-2 multi-parallel-circuit subfamily already fails every exact terminal graph/packet/path/nonnegative-hypergraph semantics on the physical terminals, even though this family had previously survived the direct F_{\mathrm{sep}} test and other low-arity necessary conditions.
So the multi-parallel boundary is no longer a diffuse “is there some family lift?” question. [[multi-parallel-hidden-vertex-boundary-now-reduces-to-global-coupling-or-auxiliary-growth.md]] isolates the exact remaining theorem shape: either a genuinely global hidden-vertex realization with nontrivial auxiliary resources, or a stronger whole-family nonexpressibility theorem beyond the current selector/shadow/symmetry obstructions.
Consequently the global Route-D frontier has moved again. [[route-d-terminal-side-semantics-now-stop-before-nonterminal-auxiliary-frontier.md]] records that terminal-side compiler semantics are now largely exhausted on the connected families currently on disk, so the remaining canonical Route-D question is the semantic value of nonterminal auxiliary-variable constructions.
The nonterminal Boolean-network escape hatch is now exhausted too. [[multi-parallel-boundary-has-no-broader-boolean-network-positive-route-than-hidden-vertex.md]] shows that on the connected multi-parallel boundary, Iwamasa's broader Boolean network framework collapses back to the original hidden-vertex question, so there is no larger sourced positive route hiding there.
The broader semantic stop point is correspondingly sharper. [[current-sourced-classical-auxiliary-semantics-still-add-no-new-compiler-meaning-beyond-cdsub.md]] records that, on the connected family stock currently on the graph, sourced classical auxiliary semantics beyond CD_{\mathrm{sub}} either fail on connected families or collapse to ordinary hidden-vertex expressibility.
So the remaining Route-D frontier now centers on one exact theorem. [[route-d-now-centers-on-ordinary-hidden-vertex-theorem-at-first-connected-boundary.md]] records that the best remaining target is an ordinary hidden-vertex realization or nonexpressibility theorem for the first connected \chi\ge 3 multi-parallel boundary itself, with genuinely global auxiliary coupling and auxiliary-budget growth.
That remaining ordinary hidden-vertex frontier is now itself methodologically compressed. [[connected-multi-parallel-ordinary-hidden-vertex-boundary-now-requires-deeper-multimorphism-or-explicit-global-gadget.md]] shows that, on current sources, only two methodology types survive there: a deeper expressive-power obstruction than the present Sep/F_{\mathrm{sep}} package, or an explicit global hidden-vertex gadget construction with cross-class coupling and possibly growing auxiliary budget.
Consequently the global Route-D frontier is now as narrow as the current source base supports. [[route-d-now-reduces-to-two-ordinary-hidden-vertex-methodologies-on-the-first-connected-family.md]] records that all sourced classical semantics beyond CD_{\mathrm{sub}} have been compressed away, leaving one ordinary hidden-vertex theorem on the connected multi-parallel boundary with exactly two surviving methodology templates.
The negative methodology is now pinned down more sharply. [[current-expressive-power-package-stops-before-higher-arity-family-obstruction-on-connected-multi-parallel.md]] shows that the current expressive-power machinery already fails at one exact point: there is no sourced higher-arity obstruction beyond the Sep/F_{\mathrm{sep}} package that separates the connected multi-parallel family.
The strongest remaining recursive constructive route is now also pinned down. [[fixed-prefix-state-construction-route-for-size-2-multi-parallel-family-fails-at-transition-and-readout-submodularity.md]] shows that a fixed-prefix-state family construction for the size-2 connected multi-parallel family fails because its required transition and readout gadgets are themselves non-submodular.
The constructive side now has a genuine family-level barrier beyond that recursive failure. [[connected-size-2-multi-parallel-family-escapes-modular-plus-fan-cone.md]] shows that every connected member of the size-2 multi-parallel family already lies outside the modular-plus-fan closure that succeeded at arity 5, so any positive theorem here must use binary-submodular objects beyond the fan cone.
The obstruction side is now sharper too. [[higher-arity-theorem16-would-close-negative-side-on-connected-size-2-family.md]] shows that the exact negative-side target is no longer a generic higher-arity multimorphism search but an all-arity Theorem-16 / Conjecture-20 type characterization.
So the best exact stop point now is the refined dichotomy recorded in [[route-d-now-stops-at-conjecture20-vs-beyond-fan-cone-global-construction.md]]: either an all-arity Theorem-16-style obstruction, or a genuinely beyond-fan-cone global hidden-state construction.
The all-arity obstruction side has now been source-specialized. [[source-all-arity-expressive-power-stops-before-fan-certificate-lift.md]] records that the loaded Zivny / Iwamasa source package still stops before converting the connected size-2 fan-cone certificate into a weighted-polymorphism or multimorphism obstruction for \langle\Gamma_{\mathrm{sub},2}\rangle.
The beyond-fan constructive side has also been narrowed. [[beyond-fan-cone-construction-requires-nonseparable-selected-threshold-energy.md]] shows that any positive construction can no longer be a separable realization of the two global 00 and 11 threshold events followed by a selector; it must use one genuinely nonseparable global hidden energy coupling u and all parallel classes at once.
Therefore the post-fan-cone Route-D frontier now has one promoted exact theorem target: [[route-d-post-fan-cone-next-target-is-fan-certificate-to-polymorphism-lift.md]] asks for a family-specific lift from the existing fan-cone separating certificate for f_3 to a weighted-polymorphism obstruction, with the existing pinning argument then extending the result to all connected size-2 members.
The certificate-level attack now has a reusable finite pipeline. [[f3-fan-certificate-passes-necessary-multimorphism-tests-but-not-operation-lift.md]] reconstructs the f_3 integer certificate, verifies the modular and fan inequalities, and shows that the certificate passes the first conservative/Hamming necessary checks for a multimorphism reading, but still lacks a total operation or weighted-polymorphism decomposition.
The same run also closes the generic lift shortcut. [[fan-cone-certificates-are-not-hidden-vertex-invariants.md]] uses the already representable 6-qubit witness to show that fan-cone certificates are not hidden-vertex invariants by themselves; cone duality must be upgraded to operation-level structure.
The current Route-D certificate frontier is therefore [[certificate-lift-frontier-is-operation-extension-not-cone-duality.md]]: decide whether the signed support of the reconstructed f_3 certificate extends to a conservative Hamming-nonincreasing operation or to a weighted-polymorphism decomposition. If yes, the connected size-2 family is nonexpressible by pinning; if no, the obstruction side needs a different certificate and the constructive side remains the nonseparable selected-threshold energy.
The operation-extension target now has an exact finite model. [[conservative-hamming-operation-extension-is-rank-order-lattice-extension.md]] proves that Boolean conservative Hamming-nonincreasing operations are precisely rank-preserving order-preserving maps of the Boolean lattice, so the signed f_3 support asks for a rank-order extension from the seven positive-support columns to the seven negative-support columns.
The first exact checker does not refute that model. [[f3-certificate-passes-pairwise-rank-order-extension-but-not-full-closure.md]] shows that the f_3 certificate passes observed-support and pairwise rank-order closure tests, while the representable six-qubit control passes the same tests; full meet/join closure, not pairwise Hamming, is the remaining finite obstruction target.
The fallback certificate search also sharpens negatively without closing. [[strict-hamming-replacement-and-meet-join-decomposition-do-not-close-certificate-lift.md]] shows that fractional strict-Hamming replacement certificates exist, integer strict replacements remain unresolved in the searched boxes, and the current y is not a conic combination of ordinary meet/join submodularity inequalities.
Thus the compressed Route-D frontier is now [[route-d-operation-extension-frontier-now-reduces-to-full-rank-order-or-new-generator.md]]: solve the full rank-order lattice extension, find a stronger higher-arity weighted-polymorphism generator/replacement certificate, or return to a genuinely unrestricted nonseparable hidden-energy theorem.
The old full-closure cap is now removed. [[full-rank-order-closure-reduces-to-pattern-upset-lattice.md]] proves that the closure generated by the observed support columns is exactly the upset lattice of the realized support-pattern poset; for f_3 this closure has 2,849,631 elements.
The full-closure diagnostic now reaches every closure element for the interval/Hall test. [[full-closure-hall-and-critical-ilp-do-not-refute-f3-operation-extension.md]] finds no f_3 interval or forced-propagation obstruction, and its 123-point critical-subposet ILP and LP are feasible; the same pipeline rejects the six-qubit fan certificate, so the test is nontrivial but still not decisive for f_3.
The exact remaining operation-lift target is now [[route-d-full-rank-order-frontier-is-monotone-count-decomposition.md]]: decompose the closure rank function into 50 monotone Boolean label functions with prescribed generator boundary values, or extract a dual/Hall obstruction for that monotone count-decomposition instance.
The monotone count-decomposition instance has now been compressed further. [[f3-count-decomposition-is-coordinate-zero-toggle-boundary.md]] proves computationally that the 50 negative support patterns are exactly the positive support patterns with coordinate 0 toggled, so the live boundary is to toggle generator 0 while preserving generators 1,\ldots,6.
The flow/cut attempt is now classified. [[monotone-count-decomposition-is-column-generation-not-single-flow.md]] records that the exact formulation is a column-generation integer covering problem: single-label pricing is a min-weight closure problem, but the 50 label-upsets are coupled by every rank-sum row. The same run finds no Hall/Farkas obstruction and rules out only direct interval-fill and rank-greedy constructive subclasses.
Consequently the current Route-D operation-lift frontier is [[route-d-count-decomposition-now-requires-higher-arity-toggle-generator.md]]: either find a full column-generation dual certificate, or produce a finite higher-arity conservative generator basis that decides the coordinate-0 toggle boundary. The unrestricted nonseparable hidden-energy construction route remains secondary because the obstruction route has not failed theorem-level.
The coordinate-toggle attack now has a sharper stop point. [[coordinate-zero-toggle-is-not-single-interface-transport.md]] proves from the exact support data that the toggle is not a pure A_0 interface transport problem, because the six preserved generator coordinates impose same-slice tail constraints. [[toggle-branch-price-sparse-dual-formulation.md]] gives the exact sparse row-supported dual/pricing theorem and records a 750-row branch-price run that added 78 unseen columns but did not certify infeasibility. [[tail-conditioned-toggle-interface-3-sorter-is-next-generator-basis-target.md]] names the next concrete finite generator class, and [[route-d-toggle-frontier-sharpens-to-sparse-pricing-or-tail-sorter-completeness.md]] keeps the global Route-D frontier at sparse-pricing obstruction versus tail-sorter completeness.
The coordinate-toggle finite branch is now demoted. [[coordinate-toggle-finite-branch-demoted-to-computational-frontier.md]] records Outcome D: after correcting the sparse-pricing dual direction, the 750-row run still had positive restricted slack and final pricing still found positive dual activity, so no sparse dual certificate, feasible toggle decomposition, or complete tail-sorter theorem is available. Treat this branch as appendix/computational frontier evidence unless a genuinely new column-generation certificate, sorter-completeness theorem, or external valued-CSP theorem appears.
The active path should now pivot toward manuscript packaging. [[static-2d-separator-cut-rank-manuscript-package.md]] organizes the publishable core around the solved static near-square 2D lower bound, fixed-minor-free and weighted-separator extensions, the stabilizer cut-rank functional, the SWAP-only submodular cut-congestion theorem, and Route-D hidden-variable realization as appendix/frontier evidence.
The first connected multi-parallel-circuit regime is now sharper on the negative-testing side too: [[two-element-multi-parallel-circuit-family-satisfies-direct-fsep.md]] shows that the whole size-2 subfamily still satisfies the direct higher-arity F_{\mathrm{sep}} condition for every number of parallel classes. So the selector quotient route, the one-branch-bit local route, the exact 4-ary closure route, and the direct F_{\mathrm{sep}} route all fail to separate the first connected family currently on disk.
The smallest connected member now resists a more global exact bounded-budget search as well: [[smallest-connected-multi-parallel-circuit-member-has-no-1-or-2-auxiliary-realization-in-large-coefficient-box.md]] shows that no hidden-vertex realization of M_{2,2,2} exists with 1 or 2 auxiliaries inside the large coefficient box [-64,64]. So any positive construction there, if it exists, already needs a less trivial auxiliary budget.
Even the first symmetry-respecting global hidden architectures now fail: [[natural-orbit-symmetric-hidden-vertex-ansatze-fail-on-m222.md]] rules out both the one-hidden-bit-per-class ansatz and the one-global-plus-one-per-class ansatz for M_{2,2,2} inside the same large coefficient box. So a future constructive theorem must either break the obvious family symmetries or use a richer hidden coupling pattern.
The first hidden-symmetry-breaking extension now fails too: [[visible-symmetric-hidden-distinguished-ansatze-fail-on-m222.md]] rules out the visible-symmetric but hidden-distinguished k=3 and k=4 quadratic ansatz classes for M_{2,2,2}. So even the first global symmetry-breaking constructive route still does not reach a realization.
On the other hand, the local shadows are already extremely positive: [[every-four-ary-pinning-minor-of-m222-is-one-hidden-representable.md]] shows that every 4-ary pinning minor of M_{2,2,2} already has a one-hidden-bit realization. So any exact k<=2 impossibility theorem for the full connected witness must be genuinely global rather than visible in pinned 4-terminal views.
That first genuinely global low-auxiliary obstruction now appears one arity higher: [[pinned-five-ary-shadows-of-m222-obstruct-one-hidden-realizability.md]] shows that pinned 5-ary shadows already split into one-hidden feasible and infeasible classes. In particular, M_{2,2,2} has no one-hidden realization at all, while k=2 and fully less-symmetric k>=3 remain open.
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 current intrinsic route now has its cleanest exact closure criterion: [[dense-tangle-breadth-forces-balanced-cut-rank.md]] shows that dense tangle breadth already forces large balanced stabilizer cut rank, interface-state dimension, and SWAP-only cut congestion. So the live frontier is no longer "find some width" but "prove dense intrinsically connected mass."
The obstruction side is now equally sharp: [[quantum-tanner-ltc-package-still-misses-dense-intrinsic-connectivity.md]] shows that the present Quantum Tanner / left-right-Cayley package still stops at tester expansion and local agreement. It does not yet produce dense intrinsic connectivity in the parity-check matroid, which is why the bridge should currently be stated first in matroid language and only then in interface-state language.
The family-contact bottleneck is now source-grounded at theorem level: [[quantum-tanner-left-right-cayley-source-package-stops-at-tester-side-structure.md]] isolates the exact target-family theorems on the table, namely Leverrier--Zemor Lemma 4 and Theorem 17 and Dinur--Evra--Livne--Lubotzky--Mozes Theorem 4.5 and Theorem 1.1, and shows that all of them still stop at chosen Tanner or tester structure rather than the original qubit parity-check matroid. So the previous bottleneck was only shifted upward: one still needs a new qubit-side hypothesis, with dense tangle breadth the cleanest currently sufficient choice.
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?