zeroCompositionWitness_forces_zero_defect
A witness to the zero composition law at a complex point forces the associated zero-location defect to vanish. Researchers exploring alternate Vector C routes to the Riemann hypothesis would cite this interface result. The proof is a one-line application of the equivalence between vanishing defect and critical-line membership to the prior result that any witness forces the critical line.
claimLet $w$ be a ZeroCompositionWitness at a point $ρ ∈ ℂ$. Then the zero-location defect of $ρ$ is zero.
background
The Zero Composition Interface module isolates an abstract interface that a completed Vector C construction would need to turn a zero-location observable into a critical-line forcing theorem. It is classified as an alternate path, not on the primary route to unconditional RH, and requires genuinely extra Euler or Hadamard analytic input beyond pure functional-equation symmetry and RCL doubling data. A ZeroCompositionWitness at $ρ$ is a structure consisting of a ZeroCompositionLaw together with the assertion that the law's H function evaluates to 1 at the zero deviation of $ρ$. The zeroDefect function attaches the RS defect (via LawOfExistence on the exponential of the deviation) to any complex point. The upstream theorem zeroDefect_zero_iff_on_critical_line states that this defect vanishes if and only if the point lies on the critical line.
proof idea
The proof is a one-line wrapper. It applies the right-to-left direction of the equivalence zeroDefect_zero_iff_on_critical_line at $ρ$ to the already-established fact that the witness forces the point onto the critical line.
why it matters in Recognition Science
This theorem completes the local interface by showing that any concrete Vector C witness automatically produces a vanishing zero-location defect. It sits inside the alternate classification of the module, which explicitly notes that pure FE symmetry plus RCL data cannot produce such a witness and that any successful path must import the same extra analytic data used by the primary routes. No downstream results currently depend on it.
scope and limits
- Does not prove existence of a ZeroCompositionWitness for any zeta zero.
- Does not supply Euler or Hadamard data needed to instantiate the interface.
- Does not reduce the EulerBoundaryBridgeAssumption or HonestPhaseCostBridge.
- Does not place the result on the primary forcing chain to unconditional RH.
formal statement (Lean)
97theorem zeroCompositionWitness_forces_zero_defect
98 {ρ : ℂ} (w : ZeroCompositionWitness ρ) :
99 zeroDefect ρ = 0 := by
proof body
Term-mode proof.
100 exact (zeroDefect_zero_iff_on_critical_line ρ).mpr
101 (zeroCompositionWitness_forces_on_critical_line w)
102
103end
104
105end NumberTheory
106end IndisputableMonolith