Exact Optimal Quotient Families Do Not Share Transferable Shelling Law

Claim/Theorem

Keep the notation of [[no-transferable-selection-rule-is-visible-for-adapted-bases.md]] and [[nu-saturation-yields-adapted-triangular-basis.md]]. For a qubit cut L \sqcup R = Q, define the exact optimal direct-sum family set

\[ \mathcal F_H^{\mathrm{opt}}(L) := \Big\{ F : \sum_{v\in F} W_v(L)\ \text{direct} \ \text{and}\ \sum_{v\in F}\dim W_v(L)=\nu_H(L) \Big\}. \]

On the current low-lambda balanced cuts of the explicit D_4, D_6, and D_8 Quantum Tanner instances, where [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]] gives

\[ \nu_H(L)=\lambda_{M(H)}(L), \]

the exact optimal families themselves still do not exhibit any currently visible transferable shelling/exchange law strong enough to be construction-controlled.

More precisely:

1. Pure accessibility is vacuous here. Every F\in \mathcal F_H^{\mathrm{opt}}(L) is already a direct-sum family, so every ordering of F has all prefixes feasible. Therefore accessibility does not distinguish the correct optimal family and cannot be the missing invariant.

2. A one-for-one basis-exchange law already fails inside fixed-cardinality strata of \mathcal F_H^{\mathrm{opt}}(L). There exist optimal families A,B of the same cardinality and an element a\in A\setminus B such that no b\in B\setminus A makes (A\setminus\{a\})\cup\{b\} optimal.

3. Multi-step one-for-one swap connectivity also fails globally, because the optimal families already occupy several different cardinality strata. Since each one-for-one swap preserves cardinality, no chain of such swaps can connect optimal families lying in different strata.

4. The strongest natural shelling candidate tested on the exact optimal families, namely global max-gain shellability, also fails for most of them. Here "global max-gain shellable" means that the family admits an ordering in which each chosen block has maximal marginal quotient gain among all currently unused blocks in the whole arrangement.

5. Allowing later pruning from a global max-gain spanning path does not rescue this on the low cuts of D_4 and D_8: exactly the same minority of optimal families that are globally max-gain shellable are obtainable as subsets of such max-gain spanning paths.

Therefore the current frontier really has collapsed to a new global optimization invariant on the arrangement \{W_v(L)\}_v, not to a visible shelling/exchange law shared by all exact optimal families.

The exact missing invariant can now be stated as

\[ H_{\mathrm{opt}}(\beta): \]

for every \beta-balanced cut L, the family \mathcal F_H^{\mathrm{opt}}(L) is controlled by a construction-level optimization invariant that either:

• canonically selects at least one member of \mathcal F_H^{\mathrm{opt}}(L), or
• certifies a linear lower bound on \nu_H(L) without enumerating or globally optimizing over all candidates.

This is strictly stronger than accessibility and strictly different from the shelling/exchange candidates tested below.

Proof/evidence:

1. Exact low-cut enumeration gives:

2. D_4: |\mathcal F_H^{\mathrm{opt}}(L)|=44;

3. D_6: |\mathcal F_H^{\mathrm{opt}}(L)|=33066;

4. D_8: |\mathcal F_H^{\mathrm{opt}}(L)|=150.

5. The optimal-family cardinality distributions are:

6. D_4: 3, 4, and 5 blocks, with counts

   $$ {3:6, 4:23, 5:15}; $$

7. D_6: 6, 7, 8, 9, 10, and 11 blocks, with counts

   $$ {6:68, 7:3095, 8:13939, 9:13288, 10:2608, 11:68}; $$

8. D_8: 6, 7, and 8 blocks, with counts

   $$ {6:4, 7:58, 8:88}. $$

Hence multi-step one-for-one swap exchange already fails to organize the whole optimal-family space, because different cardinalities are disconnected from one another under such moves.

1. Same-cardinality one-step exchange nevertheless fails even before that global obstruction:

2. D_4: the two optimal 4-block families

   $$ A={g=sr, g=sr^{3, h=e, h=r}2}, $$

   $$ B={g=s, g=sr^{3, h=r}2, h=r^3} $$

   violate one-step exchange at a=h=e: no b\in B\setminus A gives another optimal 4-block family.

3. D_6: the two optimal 6-block families

   $$ A={g=sr^{4, h=s, h=sr, h=sr}2, h=sr^{3, h=sr}5}, $$

   $$ B={g=sr^{4, h=r}4, h=sr, h=sr^{3, h=sr}4, h=sr^5} $$

   violate one-step exchange at a=h=s.

4. D_8: the two optimal 7-block families

   $$ A={g=r, g=r^{3, g=r}5, g=sr^{5, h=r}4, h=r^{7, h=sr}7}, $$

   $$ B={g=r, g=r^{3, g=r}5, g=sr^{5, g=sr}7, h=r^{4, h=sr}4} $$

   violate one-step exchange at a=h=sr^7.

5. Global max-gain shellability is rare, not universal:

6. D_4: only 6/44 optimal families are globally max-gain shellable;

7. D_6: only 156/33066;
8. D_8: only 36/150.

Explicit non-shellable optimal families include:

• D_4:

  $$ {g=sr, g=sr^{3, h=e, h=r}2}; $$

• D_6:

  $$ {g=sr^{4, h=r, h=r}2, h=s, h=sr, h=sr^{3, h=sr}4}; $$

• D_8:

  $$ {g=r, g=r^{3, g=r}5, g=sr^{5, h=r}4, h=r^{7, h=sr}7}. $$

• On the low cuts of D_4 and D_8, pruning from global max-gain spanning families does not enlarge the shellable class:

• D_4: exactly 6/44 optimal families are subsets of some global max-gain spanning path;

• D_8: exactly 36/150.

So in these two fully checked cases, "obtainable by pruning a max-gain spanning family" collapses to ordinary global max-gain shellability and still misses most exact optimal families.

1. Additional exact balanced-cut evidence on D_4 shows the shelling gap is not confined to one low-cut witness:

2. on a sampled cut with \lambda=9, one gets 32 exact optimal families, of which only 26 are globally max-gain shellable;

3. on a sampled cut with \lambda=6, one gets 19 exact optimal families, of which only 11 are globally max-gain shellable.

This isolates the frontier more sharply than the previous node:

• one exact optimal family satisfying a shelling rule is easy;
• all exact optimal families sharing a transferable shelling/exchange law is what fails;
• and the failure persists even after moving from heuristic direct-sum selectors to the exact optimal-family space itself.

So the unresolved object is now genuinely a global optimization invariant for the quotient-image arrangement, not another currently visible shelling heuristic.

Dependencies

• [[no-transferable-selection-rule-is-visible-for-adapted-bases.md]]
• [[nu-saturation-yields-adapted-triangular-basis.md]]
• [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]
• [[local-quotient-image-span-controls-rank-accumulation.md]]
• [[stabilizer-cut-rank-functional.md]]

Conflicts/Gaps

• The obstruction is still empirical on explicit small instances; it does not prove that no transferable shelling/exchange law exists for the full Quantum Tanner family.
• The D_6 pruning-from-max-gain check was not exhaustively completed because the optimal-family count is already 33066; the node therefore states the fully checked pruning failure only for D_4 and D_8.
• Accessibility remains mathematically true for every direct-sum family, but precisely for that reason it is too weak to help; the node uses it only as a boundary marker.
• What remains missing is a source-grounded or construction-grounded invariant that explains why the right optimal family should exist or be detectable without exact global optimization on each cut.

Sources

• 10.48550/arXiv.2206.07571
• 10.48550/arXiv.2508.05095