No Transferable Selection Rule Is Visible For Adapted Bases¶

Claim/Theorem¶

Keep the notation of [[nu-saturation-yields-adapted-triangular-basis.md]] and [[local-quotient-image-span-controls-rank-accumulation.md]]. For a qubit cut $L \sqcup R = Q$, let

\[ \{W_v(L)\}_v \]

be the local quotient images in $S/B(L)$, with

\[ \nu_H(L)\le \lambda_{M(H)}(L). \]

Consider a greedy direct-sum selector that scans blocks in some prescribed order and keeps a block $v$ only when it contributes its full local dimension beyond the current selected span:

\[ \dim\frac{U+W_v(L)}{U}=\dim W_v(L), \]

where $U$ is the span of the blocks already kept.

The current explicit Quantum Tanner evidence supports the following obstruction.

No currently visible construction-level ordering rule on the local quotient-image data transfers across the explicit D_4, D_6, and D_8 instances. In particular, the following tested rules all fail on some balanced cuts:
decreasing local quotient dimension \dim W_v(L),
V_1-rooted first, then decreasing \dim W_v(L),
V_0-rooted first, then decreasing \dim W_v(L),
decreasing natural-basis pivot count |\operatorname{Piv}_{\mathcal Q}(W_v(L))|, then decreasing \dim W_v(L).

This remains true both for static scan orders and for their basis-adaptive variants that re-evaluate admissibility at each step but still break ties only by those same visible local statistics.

A stronger basis-adaptive spanning heuristic can do better: if one greedily chooses a block with maximal marginal quotient-span gain

\[ g_U(v):=\dim\frac{U+W_v(L)}{U}, \]

then on the tested small instances one often spans the full quotient. But this is not yet the desired adapted-basis rule, because such a heuristic may keep blocks with

\[ 0<g_U(v)<\dim W_v(L), \]

so it need not output a direct-sum family and therefore need not directly certify \nu_H(L) or an adapted triangular basis.

Therefore the present small-instance evidence still does not determine a transferable construction-controlled rule that selects a saturating direct-sum family, even though existential adapted-basis recovery is already known whenever \nu_H(L)=\lambda_{M(H)}(L).
The exact missing invariant is now:

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

for every \beta-balanced cut L, the arrangement of local quotient images \{W_v(L)\}_v carries a construction-controlled shelling/score rule such that greedy acceptance of full-gain blocks yields a direct-sum family of total dimension \nu_H(L) (or at least \Omega(|Q|)).

Under H_{\mathrm{sel}}(\beta), one gets

\[ \lambda_{M(H)}(L)\ge \nu_H(L)=\Omega(|Q|), \]

and hence a linear balanced intrinsic cut-rank theorem.

Proof/evidence:

On sampled balanced cuts, the direct-sum success counts of the tested visible rules are:
static direct-sum scan:
- D_4: R_{\dim} succeeds on 1/10 cuts, R_{V_1} on 0/10, R_{V_0} on 9/10, and R_{\mathrm{piv}} on 1/10;
- D_6: 0/8, 0/8, 7/8, 0/8;
- D_8: 0/8, 0/8, 5/8, 0/8.
basis-adaptive direct-sum scan, where one re-tests full-gain admissibility at each step but still orders candidates by the same local statistics:
- D_4: R_{\dim} succeeds on 0/10 cuts, R_{V_1} on 0/10, R_{V_0} on 5/10, and R_{\mathrm{piv}} on 0/10;
- D_6: 0/8, 0/8, 6/8, 0/8;
- D_8: 0/8, 0/8, 6/8, 0/8.

So the best currently visible direct-sum rule, V_0-first, still fails repeatedly across all three families.

Concrete failure witnesses already appear on the distinguished low-lambda cuts or their nearest sampled analogues:
D_4: there is a balanced cut with

$$ \lambda=10 $$

on which V_0-first direct-sum selection achieves only rank 9;
D_6: there is a balanced cut with

$$ \lambda=18 $$

on which V_0-first achieves only 17;
D_8: on the current low-lambda witness one has

$$ \lambda=12, $$

while V_0-first achieves only 11.

So even the strongest currently visible rule on disk does not transfer uniformly.

The obstruction is not merely that one witness family was chosen badly. On the D_4 low-lambda cut, multiple incompatible exact saturating families coexist:
an all-V_0 family

$$ {g=sr^{2, g=sr, g=s, g=r}2} $$

with dimensions (2,1,1,1),
and the mixed family already recorded in [[nu-saturation-yields-adapted-triangular-basis.md]]

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

with dimensions (1,1,2,1).

Thus "a rule reproduces one optimized witness family" is strictly weaker than "the construction exposes a transferable selection rule."

A stronger boundary marker is the marginal-span greedy heuristic. On the same D_4 low-lambda cut, it reaches quotient rank 5 using the three blocks

\[ \{h=sr^2,\ g=sr^2,\ h=r^3\}, \]

whose local dimensions are (2,2,2). Since

\[ 2+2+2 > 5, \]

this spanning heuristic necessarily uses partial gains and is not a direct-sum selector. So even when a basis-adaptive rank-oracle heuristic spans the quotient, it still does not identify the missing adapted-basis rule.

This isolates the live construction-level obstruction exactly:

existential adapted-basis recovery is already available once \nu=\lambda;
fixed-basis pivot coverage is too rigid;
and currently visible local rules on block type, local dimension, or natural pivot statistics do not transfer.

What remains missing is a new global optimization invariant on the quotient-image arrangement: a shelling/elimination rule that is still construction-controlled, yet strong enough to recover a saturating direct-sum family uniformly across cuts and across the Quantum Tanner family.

Dependencies¶

[[nu-saturation-yields-adapted-triangular-basis.md]]
[[fixed-quotient-basis-pivot-coverage-is-too-rigid.md]]
[[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]
[[local-quotient-image-span-controls-rank-accumulation.md]]
[[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
[[quantum-tanner-local-generator-blowup.md]]

Conflicts/Gaps¶

This node does not prove that no transferable rule exists. It proves only that no such rule is currently visible among the natural local statistics tested on the explicit D_4, D_6, and D_8 instances.
The empirical success/failure counts are sampled, not exhaustive over all balanced cuts of those instances.
The marginal-span boundary marker is basis-adaptive and algebraically global; it is not yet tied to any source-grounded combinatorial invariant of the Quantum Tanner construction.
What remains missing is a source-grounded or construction-grounded invariant that predicts a saturating direct-sum shelling without first solving the exact packing problem on each cut.

Sources¶

10.48550/arXiv.2206.07571
10.48550/arXiv.2508.05095