Exact Low-Deficiency Subfamilies Do Not Share Transferable Visible Template¶

Claim/Theorem¶

Keep the notation of [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]] and [[nu-saturation-yields-adapted-triangular-basis.md]]. On the current low-lambda balanced cuts of the explicit D_4, D_6, and D_8 Quantum Tanner quotient instances, exact low-deficiency subfamilies do exist, in fact exact zero-deficiency subfamilies with

\[ \rho_H(X;L)=w_H(X;L)=\nu_H(L)=\lambda_{M(H)}(L). \]

However, the currently visible construction-level labels on disk do not expose any transferable rule selecting such a subfamily.

More precisely, the obvious visible signatures already fail:

A fixed root-side profile is not transferable. Exact zero-deficiency witness families currently on disk have root-side counts

\[ D_4:(g,h)=(2,2),\qquad D_6:(g,h)=(1,5),\qquad D_8:(g,h)=(4,3). \]

So no rule of the form "take mainly g-rooted blocks", "take mainly h-rooted blocks", or "take a balanced number from each side" transfers across the three explicit low cuts.
A fixed local-dimension profile is not transferable either. The recorded exact zero-deficiency witness families have weight profiles

\[ D_4:(1,1,2,1),\qquad D_6:(2,2,2,2,2,2),\qquad D_8:(2,2,1,2,2,1,2). \]

Therefore:
- a max-dimension-only rule fails already on D_4 and D_8, which need 1-dimensional blocks;
- any rule forcing 1-dimensional participation fails on D_6, whose current exact witness uses only 2-dimensional blocks.
Even on one fixed cut, "one exact low-deficiency witness exists" is weaker than "the construction exposes a visible template." On the current D_4 low cut, multiple incompatible exact zero-deficiency families coexist:
- an all-g family
  
  \[ \{g=sr^2,\ g=sr,\ g=s,\ g=r^2\} \]
  
  with dimensions (2,1,1,1),
- and a mixed family
  
  \[ \{g=sr,\ g=sr^3,\ h=e,\ h=r\} \]
  
  with dimensions (1,1,2,1).
These are both exact zero-deficiency witness families, but they do not share one visible root-side or dimension template.
The same-cut nonuniqueness seen in [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]] shows the issue is not confined to one chosen witness. On the same low cuts there are many optimal families, and one-step exchange already fails among them, so the low-deficiency frontier does not collapse to a unique orbit-, side-, or dimension-type pattern.

Therefore the current frontier has sharpened again:

the sufficient low-deficiency inequality from [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]] is mathematically real;
exact low-deficiency subfamilies do exist on the explicit low cuts;
but no currently visible construction-level signature on block side, coarse type, or dimension profile selects them transferably.

The exact missing invariant can now be stated as

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

for every \beta-balanced cut L, there is a construction-controlled rule on the quotient-block arrangement that exposes a block subfamily X with

\[ w_H(X;L)=\Omega(|Q|) \]

and

\[ \rho_H(X;L)>\frac{1}{2}w_H(X;L), \]

without solving the full global optimization problem for \nu_H(L).

This strictly separates two statements:

existential low-deficiency subfamily:

\[ \exists X\ \text{with large }w_H(X;L)\text{ and }\rho_H(X;L)>\frac{1}{2}w_H(X;L), \]
transferable construction-controlled rule:

\[ \text{the construction itself reveals such an }X. \]

The explicit D_4, D_6, and D_8 evidence currently supports the first statement only after exact global optimization, not the second.

Dependencies¶

[[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]]
[[nu-saturation-yields-adapted-triangular-basis.md]]
[[no-transferable-selection-rule-is-visible-for-adapted-bases.md]]
[[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]]
[[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]

Conflicts/Gaps¶

This node is an obstruction to currently visible selection signatures, not a theorem that no transferable rule exists in principle.
The obstruction is grounded in the explicit low-cut D_4, D_6, and D_8 data already on disk; it does not yet cover larger explicit instances or all balanced cuts.
The node therefore sharpens the missing object but does not yet produce a positive construction-controlled rule.
What remains missing is a new global subfamily-selection invariant stronger than side counts, local dimension profile, or one chosen witness template.

Sources¶

10.48550/arXiv.2206.07571
10.48550/arXiv.2508.05095