charge_zero_cost_scalar_t1_bounded
For a defect sensor with zero charge, the T1 defect of the Euler ledger scalar state is bounded by some constant K for all natural numbers N, in fact equaling zero. This result is cited in the ontological dichotomy that equates zero charge with physical realizability in the Recognition Science framework. The proof is a one-line term wrapper that refines the bound to zero and simplifies using the state definition together with the zero-charge hypothesis and the fact that defect vanishes at unity.
claimLet $s$ be a defect sensor with charge zero. Then there exists a real number $K$ such that for every natural number $n$, the defect of the scalar state of the Euler ledger for $s$ at step $n$ is at most $K$.
background
The defect functional equals the J-cost on positive reals and vanishes at unity. The module replaces an earlier total-cost claim with a three-component architecture: cost divergence for nonzero charge, Euler trace admissibility (convergent carrier with bounded logarithmic derivative), and physically realizable ledgers whose scalar proxy states remain T1-bounded. Upstream, the theorem defect at unity is zero follows directly from the definition of defect as J, while K is defined as the dimensionless bridge ratio phi to the power one-half.
proof idea
This is a term-mode proof. It refines the existential quantifier to the constant zero and then applies simplification on the Euler ledger scalar state definition, the zero-charge hypothesis, and the theorem that defect at unity equals zero.
why it matters in Recognition Science
The theorem is invoked by the ontological_dichotomy result, which states the full equivalence between zero charge and uniform T1-bounded defect and presents this as the Recognition Science ontological argument for the Riemann hypothesis. It completes the charge-zero half of the realizability architecture in the module, linking bounded defect to physical existence while the nonzero-charge half is handled by cost divergence. The argument sits inside the T1-bounded realizability component of the unified RH chain and touches the framework's J-uniqueness and eight-tick structure.
scope and limits
- Does not treat the nonzero-charge case or prove divergence.
- Does not establish the converse implication without the dichotomy theorem.
- Does not identify an explicit numerical value for the bound K.
- Does not address boundary transport or the external bridge hypothesis in the module.
Lean usage
exact charge_zero_cost_scalar_t1_bounded sensor hzeroformal statement (Lean)
859theorem charge_zero_cost_scalar_t1_bounded (sensor : DefectSensor)
860 (hzero : sensor.charge = 0) :
861 ∃ K : ℝ, ∀ N : ℕ,
862 IndisputableMonolith.Foundation.LawOfExistence.defect
863 (eulerLedgerScalarState sensor N) ≤ K := by
proof body
Term-mode proof.
864 refine ⟨0, fun N => ?_⟩
865 simp [eulerLedgerScalarState, hzero,
866 IndisputableMonolith.Foundation.LawOfExistence.defect_at_one]
867
868/-- The full ontological dichotomy: the cost scalar is T1-bounded iff
869charge `= 0`. This IS the RS ontological argument for RH:
870
871* Charge `0` ↔ bounded T1 defect ↔ physically realizable ↔ exists
872* Charge `≠ 0` ↔ unbounded T1 defect ↔ not realizable ↔ does not exist -/