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Exact Low-Deficiency Witnesses Have No Transferable Core Or Local Repair

Claim/Theorem

Keep the notation of [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]] and [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]]. The current explicit Quantum Tanner low-cut data do not expose any transferable finer support/dependency rule of either of the following two common kinds:

  1. a large common-core rule:

    \[ H_{\mathrm{core}}: \]

    every exact low-deficiency witness family on a cut contains one canonical large block core;

  2. a local-repair rule:

    \[ H_{\mathrm{repair}}: \]

    any two exact low-deficiency witness families differ only by a short sequence of local support-preserving repairs.

More precisely:

  1. The common-core idea already fails on the current D_4 low cut. Two exact zero-deficiency witness families on that cut are

    \[ A=\{g=sr^2,\ g=sr,\ g=s,\ g=r^2\}, \]

    and

    \[ B=\{g=sr,\ g=sr^3,\ h=e,\ h=r\}. \]

    Their intersection is

    \[ A\cap B=\{g=sr\}, \]

    so they share only one block. Thus no large common-core description is visible even on one explicit low cut where both families are exact zero-deficiency witnesses.

  2. The size of this shared core is not even stable across the current explicit low-cut witnesses. Existing exact-optimal witness pairs already on disk have the following intersection sizes:

    • D_4: |A\cap B|=1 for the two zero-deficiency witnesses above;
    • D_4: |A\cap B|=2 for the same-cardinality optimal pair in [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]];
    • D_6: |A\cap B|=4 for the recorded optimal 6-block pair there;
    • D_8: |A\cap B|=5 for the recorded optimal 7-block pair there.

    So the current explicit data do not exhibit any transferable support-core size or core-shape pattern across cuts and instances.

  3. The local-repair idea already fails at the same time. The same D_4, D_6, and D_8 pairs from [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]] witness failure of one-step same-cardinality exchange. Therefore even when two exact witness families share a large overlap, the remaining difference is not repaired by one local swap that preserves optimality.

  4. Combining the two phenomena gives the sharper obstruction:

    • some exact low-deficiency witnesses on the same cut can have very small overlap, so there is no visible large forced core;
    • other exact witness pairs can have substantial overlap, yet still fail local repair;
    • hence the missing selection object is neither a common core nor a bounded local repair graph on visible supports.

Therefore the current frontier has already moved beyond support-level templates:

  • coarse visible labels failed in [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]];
  • finer common-core and local-repair explanations also fail on the explicit witness pairs already on disk;
  • the remaining missing object is a genuinely global selection invariant on the arrangement of exact low-deficiency witness families themselves.

The exact missing invariant can now be stated as

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

for every \beta-balanced cut L, there is a construction-controlled global invariant of the quotient-block arrangement that identifies or certifies a block subfamily X with

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

and

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

without relying on a visible common core, a fixed profile, or a bounded local repair rule.

This sharpens the distinction among three levels:

  • one witness satisfies a pattern;
  • all witnesses on one cut share a pattern;
  • a transferable construction-controlled rule persists across cuts and instances.

The current D_4, D_6, and D_8 evidence supports only the first level.

Dependencies

  • [[exact-low-deficiency-subfamilies-do-not-share-transferable-visible-template.md]]
  • [[exact-optimal-quotient-families-do-not-share-transferable-shelling-law.md]]
  • [[subfamily-union-rank-deficiency-gives-parity-lower-bound.md]]
  • [[nu-saturation-yields-adapted-triangular-basis.md]]
  • [[packed-quotient-images-already-attain-global-cut-rank-on-small-quantum-tanner-instances.md]]

Conflicts/Gaps

  • This node rules out currently visible common-core and local-repair explanations only at the level of the explicit witness pairs already on disk; it does not prove that no deeper transferable rule exists.
  • The intersection statistics used here come from explicit witness families, not from exhaustive classification of all low-deficiency witnesses on every cut.
  • So the node sharpens the obstruction but does not yet replace exact global optimization by a new positive invariant.
  • What remains missing is a genuinely global selection invariant for exact low-deficiency witness arrangements.

Sources

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