euler_collapse_boundary_transport
plain-language theorem explainer
The theorem shows that for a nonzero-charge defect sensor, the realized collapse boundary approach condition directly forces the Euler ledger scalar state to approach zero. Researchers tracing the T1-bounded realizability architecture in Recognition Science would cite it when examining the boundary transport step. The proof is a one-line term wrapper that obtains the witness from the boundary predicate and transfers it by simplification under the definitions of the scalar state and nonzero charge.
Claim. Let $S$ be a defect sensor with nonzero charge. If the realized defect collapse scalar of $S$ approaches zero (i.e., for every $ε>0$ there exists $N$ such that the scalar is less than $ε$), then the Euler ledger scalar state of $S$ also approaches zero.
background
The module replaces an earlier total-cost assertion with a three-component architecture for T1-bounded realizability. Cost divergence forces annular cost to grow unbounded for nonzero charge. Euler trace admissibility establishes convergence and bounded logarithmic derivative for the Euler carrier. Physically realizable ledger carries a scalar proxy whose T1 defect remains bounded. Boundary transport is the remaining external bridge hypothesis: if a realizable Euler ledger is cost-divergent, its scalar proxy is forced toward the T1 boundary at zero. The defect functional equals the J-cost $J(x)$ for positive $x$, with $J$ satisfying the recognition composition law.
proof idea
The proof is a term-mode wrapper. It introduces the collapse hypothesis and the positive epsilon, obtains the natural-number witness directly from the boundary predicate, and applies simpa using the definitions of the Euler ledger scalar state together with the nonzero-charge assumption to conclude the required inequality.
why it matters
This theorem fills the boundary-transport slot in the Unified RH proof chain, linking the realized collapse observable to the Euler ledger scalar. It supports the diagnostic that forced identification of the two scalars cannot satisfy the T1-bounded realizability interface, as shown by the impossibility result above it. The declaration touches the open external bridge hypothesis that would discharge the remaining step without contradicting cost divergence or the eight-tick octave structure.
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