[zxv]

Result

At a fixed dyadic prefix, suffix and helper boundary, adding $d$ eligible cut coordinates raises unit-channel capacity by at most $2d$:

$$ 0\leq c(K')-c(K)\leq2d. $$

The bound is sharp for every positive $d$. The causal fixture in Ali Romman's PR #5615 has old pairing [5], hence zero unit capacity. Making its third coordinate eligible gives two unit channels. Compatible extended bases yield the bordered matrix [[5,-2],[-2,1]], whose determinant is one. Independent blocks attain every even partial gain, using a complete arbitrary-dirty identity word of 12 signed unit ADDs per block.

The proof extends full dyadic module bases using the injection into the added coordinate quotient, then applies the single-cut prime-rank criterion. The numerical tensor formula from Nick Petalas's PR #5882 also gives a necessary coordinate budget for a fixed-prefix nonproduct rescue of a defective tensor cut.

Significance for researchers

A zero-capacity restricted cut can become useful after one changed eligibility contract; each added coordinate can unlock two channels. Conversely, the sharp bound screens how much new eligible support a fixed-program escape requires beyond PR #6054. It is a coordinate requirement, not a paid-job tariff. The added coordinate may already exist, while legal common charts, timing, channel delivery and the complete native caller bill remain unresolved. No productive-target credit or new κ is claimed.

Validation

CPython 3.12.8 normal and optimized registered replays pass all 126 support enlargements through six independent blocks, complete scalar-word restoration, saturated-kernel and literal unit-block checks, modular upper bounds, 706 bordered-matrix controls and semantic rejection controls. Committed bytes pass. The retained kernel implementation is pinned and unchanged; no selected replay or independent human/formal-kernel review is claimed.

The navigation-only repair from PR #6050 is included. The optional full-history preflight is unavailable in this shallow repository; declared pins use its reachable base. Frozen packages and the selected record remain unchanged.

Fresh read-only Git-free export replay also passes at head 5fd1fdedddf32307db31416958a1f03a8a67aa16, under optimized isolated CPython with empty PATH/PYTHONPATH; exported file hashes remain unchanged. The package’s hosted scientific replay and planning job pass at this head. Three other replay jobs report archived-source pin drift after the included navigation repair; those frozen packages are unchanged. The canonical investigation and remaining obligations are in Discussion #6060.

Open next questions

Can an actual native cut admit one outside coordinate at a complete chart/timing/delivery cost that the extra channels fund? Extract its fixed maps and support embedding before implementing a larger supplier. Separately, a changed prefix or helper endpoint is outside this theorem and needs its own full invoice.

Prepared by zxv with substantial OpenAI Codex assistance. Predecessor source, fixture and attribution remain intact.