eulerLedgerScalarState_pos
The theorem establishes that the Euler ledger scalar state is strictly positive for any defect sensor and natural number N. Researchers modeling T1-bounded realizability under the Euler carrier in the Unified RH architecture would cite this when confirming ledger positivity. The proof proceeds by case split on sensor charge, with direct simplification when charge vanishes and reduction to the realized defect collapse scalar positivity lemma otherwise.
claimFor any defect sensor $s$ and natural number $N$, the Euler ledger scalar state satisfies $0 < $ Euler ledger scalar state of $s$ at $N$.
background
The Unified RH module organizes T1-bounded realizability into cost divergence, Euler trace admissibility, physically realizable ledgers, and boundary transport. The Euler ledger scalar state serves as the scalar proxy whose T1 defect remains uniformly bounded for the Euler carrier instance. Upstream carrier definitions from engineering modules fix the frequency at $5 phi$ Hz, while is theorems establish collision-free or tautological properties for simplicial edges and mechanism structures. The module proof chain runs from euler trace admissible through physically realizable ledger to the T1 boundary prohibition.
proof idea
The tactic proof opens with by_cases on sensor.charge = 0. The zero branch applies simp using the definition of eulerLedgerScalarState. The nonzero branch applies simpa using the same definition and then invokes realizedDefectCollapseScalar_pos.
why it matters in Recognition Science
This positivity result closes a gap in the physically realizable ledger component of the T1 architecture. It supports the downstream chain that prevents boundary approach for realizable Euler ledgers and aligns with the eight-tick octave and phi-ladder structure. The declaration fills the scalar positivity step required before boundary transport can be applied.
scope and limits
- Does not establish positivity for carriers outside the Euler trace.
- Does not bound the magnitude of the scalar state.
- Does not address annular cost divergence.
- Does not discharge the external bridge hypothesis for divergence witnesses.
formal statement (Lean)
467theorem eulerLedgerScalarState_pos (sensor : DefectSensor) (N : ℕ) :
468 0 < eulerLedgerScalarState sensor N := by
proof body
Tactic-mode proof.
469 by_cases hzero : sensor.charge = 0
470 · simp [eulerLedgerScalarState, hzero]
471 · simpa [eulerLedgerScalarState, hzero] using
472 realizedDefectCollapseScalar_pos sensor hzero N
473
474/-- The Euler carrier is physically realizable in the T1-bounded sense with
475realizability scalar given by the bounded carrier-derived proxy
476`eulerScalarProxy`. -/