rungCost
plain-language theorem explainer
rungCost assigns the J-cost to each selected rung in a Balanced J-Subset Sum instance. Researchers modeling RS-native subset-sum problems with phi-ladder costs would reference this when bounding total expense. It is defined directly as the application of the J-cost function to the exponential phi-rung value.
Claim. Given a BJSS instance $inst$ and index $i$, the cost of rung $i$ is $J( e^{r_i log phi} )$, where $r_i$ is the rung of item $i$ and $J$ is the Recognition Science cost function.
background
The 8-Balanced J-Subset Sum module defines a finite instance with $n$ items, each carrying an integer weight, a residue in ZMod 8, and a phi-rung, together with a target integer sum and a real cost bound. A witness is a subset of items whose weights sum to the target, whose residues sum to zero modulo 8, and whose total J-cost stays below the bound. rungValue maps an integer rung $k$ to the real $exp(k log phi)$, placing the item on the phi-ladder. The J-cost function, imported from the foundational cost module, then converts this value into the contribution to the total expense. This construction sits inside the broader Recognition Science effort to derive discrete structures from the forcing chain and the Recognition Composition Law.
proof idea
The definition is a direct one-line application of Cost.Jcost to rungValue applied to the rung field of the instance at index $i$.
why it matters
rungCost supplies the per-item term inside totalJCost, which aggregates the cost over a witness support. It is invoked in the nonnegativity theorem and in the reduction showing that ordinary subset-sum solutions yield zero-cost BJSS witnesses. The definition therefore embeds the phi-ladder (T6) and J-uniqueness (T5) into the cost bound of the cryptography candidate, without introducing new hypotheses.
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