not_realizedDefectCollapseScalar_t1_bounded
plain-language theorem explainer
This theorem shows that the realized collapse scalar for any nonzero-charge defect sensor has T1 defect unbounded over the naturals. Researchers working on Recognition Science unification cite it to rule out single-scalar ledger closures. The proof is a direct contradiction: assume a uniform bound K, invoke the pre-established unboundedness of the collapse scalar defect, and obtain an immediate violation.
Claim. Let $s$ be a defect sensor with nonzero charge. There is no real $K$ such that the J-cost of the realized collapse scalar at $s$ and every natural $N$ is at most $K$.
background
The Unified RH module replaces earlier bounded total-cost assumptions with a three-component architecture: cost divergence (nonzero charge forces annular cost to grow without bound, proved from annular coercivity), Euler trace admissibility (convergent carrier with bounded logarithmic derivative), and physically realizable ledger (Euler carrier instance with scalar proxy). The defect functional equals the J-cost, which vanishes at unity and quantifies deviation from the fixed point. Upstream results supply the dimensionless bridge $K = phi^{1/2}$ and the canonical arithmetic object realized independently of any particular logic interpretation.
proof idea
The term proof proceeds by contradiction. Assume a bounding real $K$ exists for the defect sequence. Extract that $K$. Apply the lemma establishing that the realized collapse scalar defect exceeds any candidate bound at some natural $N$. Conclude via the negation of the less-than relation on the extracted inequality.
why it matters
This result is invoked directly by direct_t1_exclusion and single_scalar_obstruction, which constitute the one-scalar core of the RS ontological argument. It closes the step from Euler trace admissibility to the T1 boundary prohibition for realizable ledgers, as described in the module proof chain. It thereby blocks identification of the realizability proxy with the collapse observable and reinforces the T1 cost barrier within the eight-tick octave structure.
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