[zxv]

Result

An exactly factored dyadic causal cut has the full tensor product of its saturated prefix/suffix eligible modules. Let $L$ consist of the vectors $v$ in $F$ satisfying $vA\in G$. Arbitrary linear combinations across tensor coordinates add no further eligible channel:

$$ L=\bigotimes_i L_i, \qquad K=\bigotimes_i(L_{0,i}R_{0,i}). $$

The proof establishes the module premise behind Nick Petalas's PR #5882, whose numerical primewise capacity formula and sharp envelope are credited rather than repeated as new results.

For a fixed catalogue of $q$ defective cuts, with rational ranks $r_j$ and unit capacities $c_j<r_j$, define

$$ \theta=\max_{1\leq j\leq q}\frac{c_j}{r_j}<1. $$

Every sequence $j_1,\ldots,j_n$ of length $n$ then satisfies

$$ \boxed{\displaystyle \frac{c\left(\bigotimes_{\ell=1}^{n}K_{j_\ell}\right)} {\prod_{\ell=1}^{n}r_{j_\ell}} \leq \theta^{\lceil n/q\rceil}. } $$

This is uniform over arbitrary switching. Complementary-prime finite gains remain possible; a fixed defective catalogue still retains at most an exponentially vanishing fraction of rational channels.

Explicit arbitrary-dirty reversible scalar words realize diagonal pairings with entries $(1,3)$ and $(1,5)$, with complete identity-word tariffs of 16 and 32 signed ADDs. These are causal calibration circuits, not selected P104 native suppliers. Strict chart nesting, tensor carriers, delivery, productive Gaussian targets and the complete caller bill remain open.

Significance for researchers

Searching more general vectors inside the same pure product support cannot evade its odd-prime obstruction. A useful escape needs a changed support, prefix, full-capacity type or growing catalogue, with its costs accounted for. The theorem is about unit-channel density and supplies no universal $\kappa$ ceiling or new $\kappa$.

Validation

CPython 3.12.8 registered normal and optimized replays pass seven independently recomputed full product cuts, bidirectional dyadic module-basis checks, all 8,190 two-type schedules through length 12, complete scalar restoration and rejection controls. The pinned saturated-kernel routine from Ali Romman's PR #5615 is reused with separate Fraction elimination and module-coordinate checks. Committed bytes pass. No full selected replay or formal-kernel check ran; the optional full-history check is unavailable in this shallow checkout. The subsequent formatting change only replaces plaintext mathematics with rendered inline and display equations; metadata/pins and whitespace checks pass.

The navigation repair from PR #6050 is included to pass shared planning after the selection of PR #6030. Related proof jobs in PR #6047 and PR #6049 now pass, while unrelated archived input-pin jobs triggered by those navigation files fail. Frozen packages are not modified here.

Open next questions

Extract a legal native product cut with exact supports and simultaneous timing, then price its tensor carrier and delivery. Alternatively, specify one added nonproduct cut coordinate and test its complete signed invoice before expanding the construction.

Prepared by zxv with substantial OpenAI Codex assistance. Predecessor code, proofs and attribution remain unchanged. No independent human or proof-assistant review is claimed.

Final export validation at a3595fe3e1c4db6777347d2387b246b4a97d0eea: official committed export, integrity and declared-input coverage pass. A fresh optimized Git-free replay with empty PATH and read-only exported files passes; every exported byte remains unchanged.