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Packed Quotient Images Already Attain Global Cut Rank On Small Quantum Tanner Instances

Claim/Theorem

Keep the notation of [[local-quotient-image-span-controls-rank-accumulation.md]]. For a qubit cut \(L \sqcup R = Q\) in the chosen local-generator presentation of a Quantum Tanner code, write

\[ W_v(L)=\frac{S(v)+B(L)}{B(L)} \le \frac{S}{B(L)}, \]

and define

\[ \nu_H(L) := \max\Big\{ \sum_{v\in F}\dim W_v(L) : \sum_{v\in F} W_v(L)\ \text{is a direct sum} \Big\}. \]

Then the exact small-instance quotient-image behavior is sharper than the previous bottleneck statement:

  1. On the currently lowest sampled balanced cuts of the explicit D_4, D_6, and D_8 Quantum Tanner instances reconstructed from Appendix C / Table V of 10.48550/arXiv.2508.05095, one has

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

  2. Therefore, on these explicit low-cut witnesses, the main algebraic collapse is not an additional post-packing loss

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

    but only the earlier losses

    \[ \mu_H(L)\rightsquigarrow \sigma_H(L)\rightsquigarrow \lambda_{M(H)}(L)=\nu_H(L). \]

  3. In particular, the concrete missing family-level invariant is not merely "some direct-sum packing might exist." The right missing construction-level statement is a linear-size quotient-pivot or triangular-witness property forcing

    \[ \nu_H(L)=\Omega(|Q|) \]

    for every balanced cut.

Equivalently, the live family theorem now needed is:

\[ H'_{\mathrm{pivot}}(\beta): \quad \forall\,L\subseteq Q,\ |L|\in[\beta|Q|,(1-\beta)|Q|], \]

there exists a family of local quotient images \(W_v(L)\) containing a direct-sum subfamily of total dimension \(\Omega(|Q|)\).

Since [[local-quotient-image-span-controls-rank-accumulation.md]] already proves

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

the hypothesis \(H'_{\mathrm{pivot}}(\beta)\) would immediately imply linear balanced intrinsic cut rank.

Proof sketch:

  1. For each sampled balanced cut \(L\), compute the exact quotient

    \[ \frac{S}{B(L)} \]

    and the exact local images \(W_v(L)\) in a fixed quotient basis.

  2. Compute \(\nu_H(L)\) exactly by dynamic programming on reachable quotient subspaces: from a current span \(U\), add a block image \(W_v(L)\) only when

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

    so the addition is direct. The maximum reachable dimension is exactly \(\nu_H(L)\).

  3. Compare the exact output to the global cut rank

    \[ \lambda_{M(H)}(L)=\dim\frac{S}{B(L)}. \]

    For the current low-cut witnesses, the maximum reachable direct-sum dimension equals the full quotient dimension in all three cases.

Exact low-cut data from the present cycle:

  • D_4 / [36,8,3]:

    \[ \lambda=5,\qquad \nu=5,\qquad \sigma=16,\qquad \mu=19. \]

    The local quotient-image dimensions have histogram

    \[ \{0:4,\ 1:8,\ 2:4\}. \]

  • D_6 / [54,11,4]:

    \[ \lambda=12,\qquad \nu=12,\qquad \sigma=35,\qquad \mu=35. \]

    The local quotient-image dimensions have histogram

    \[ \{1:13,\ 2:11\}. \]

  • D_8 / [72,14,4]:

    \[ \lambda=12,\qquad \nu=12,\qquad \sigma=34,\qquad \mu=48. \]

    The local quotient-image dimensions have histogram

    \[ \{0:9,\ 1:12,\ 2:11\}. \]

These exact computations isolate the dominant failure mode:

  • in D_4 and D_8, some local images vanish after quotienting, and many surviving images are redundant inside the same low-dimensional quotient span;
  • in D_6, there is essentially no survival loss at all on the sampled low cut, but there is still heavy redundancy because many surviving images occupy the same 12-dimensional quotient space;
  • in all three cases, once one passes to the best direct-sum packed family, no further gap remains: the packed family already spans the full quotient.

So the explicit-family evidence now points to quotient-pivot extraction, not another obstruction to packing, as the correct next invariant.

Dependencies

  • [[local-quotient-image-span-controls-rank-accumulation.md]]
  • [[quantum-tanner-needs-balanced-local-block-rank-accumulation.md]]
  • [[quantum-tanner-local-generator-blowup.md]]
  • [[stabilizer-cut-rank-functional.md]]
  • [[cut-rank-is-interface-state-dimension.md]]

Conflicts/Gaps

  • This node is still small-instance evidence, not a theorem over the whole Quantum Tanner family.
  • The balanced cuts used here are the currently lowest sampled cuts in the reconstructed D_4, D_6, and D_8 instances, not provably extremal cuts.
  • The node does not yet prove that \(\nu_H(L)=\lambda_{M(H)}(L)\) for all balanced cuts, even in these explicit small instances.
  • What is now isolated more sharply is the exact missing invariant: a family theorem that produces linearly many independent quotient-image pivots, or equivalently a linear lower bound on \(\nu_H(L)\).

Sources

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