What is RDM?
Recognition-Deductive Measurement (RDM) is an abstract measurement protocol that isolates a single missing positivity in a composed analytic object and supplies interchangeable closure front-ends. It's the meta-level "instrument" used to turn a deductive cascade into an empirical-style certificate within the Recognition Physics framework.
At its core, RDM bridges pure mathematics and physical measurement by providing rigorous positivity certificates for complex analytic functions that arise in the Recognition framework. These certificates are essential for proving that certain physical quantities (like probabilities and energy densities) remain positive as required by physics.
Core Definitions
An RDM instance consists of the data:
where:
- $\mathcal{A}: \HP \to \mathcal{B}(\mathcal{H})$ is holomorphic with $\mathcal{A}(s)$ Hilbert-Schmidt for each $s$
- $\det_2(I-\mathcal{A}(s))$ is the regularized Fredholm determinant
- $\mathcal{O}$ is an outer factor from uniformly-$\varepsilon$ bounded $L^1$ boundary data on $\bHP$
- $\xi$ is a fixed normalizer (e.g., the completed $\zeta$-factor)
The normalized ratio is:
The goal of an RDM instance is to certify:
Equivalent boundary forms (a.e. on $\bHP$):
- Phase cone: $\Arg \mathcal{J}(\frac{1}{2}+it) \in [-\pi/2, \pi/2]$
- Pick kernel PSD: The half-plane kernel
$$K_{\mathcal{J}}(s,w) := \frac{\mathcal{J}(s) + \overline{\mathcal{J}(w)}}{(s+\overline{w})-1}$$is positive semidefinite in the boundary limit $s \to \tfrac{1}{2}+it$, $w \to \tfrac{1}{2}+iu$.
The RDM Pipeline (R0–R6)
The RDM pipeline provides a systematic approach to establishing positivity certificates through six rigorous steps, with multiple equivalent closure options.
Verify $\mathcal{A}(\cdot)$ is HS-valued and holomorphic on $\HP$, so $\det_2(I-\mathcal{A}(s))$ is holomorphic on $\HP$.
From the uniform-$\varepsilon$ $L^1$ control on $\bHP$, form the outer $\mathcal{O}$ so that $\log|\mathcal{J}(\tfrac{1}{2}+it)| \in L^1_{\text{loc}}$ and $\mathcal{J}$ has no singular inner factor on $\HP$.
Construct finite-dimensional passive (positive-real) approximants $\mathcal{J}_n$ via Kalman-Yakubovich-Popov (KYP)/Nevanlinna-Pick synthesis so that $\mathcal{J}_n \to \mathcal{J}$ locally uniformly on $\HP$ and in $L^1_{\text{loc}}$ on $\bHP$.
Set $F(s) := 2\mathcal{J}(s)$. A Herglotz representation in the half-plane holds:
with finite positive measure $d\nu$ (the boundary distribution of $\Repart F$).
The boundary phase $\arg \mathcal{J}$ has phase-velocity density:
so $-w' \geq 0$ is the zero-side mass.
Choose A, B, or C as the final certificate:
With $\mu := \sum_{\beta>1/2} 2(\beta-\tfrac{1}{2})\delta_\gamma$, require for every finite $I=[T_1,T_2]$ the tent
to satisfy $\mu(\text{tent}(I)) \leq \pi/2$.
For every finite interval $I=[T_1,T_2]$:
For each nonnegative Schwartz $\psi$ and standard mollifier $\varphi_\varepsilon \geq 0$:
With $F = 2\mathcal{J}$ a Herglotz function on $\HP$, the Cayley map $$\Phi(s) := \frac{F(s)-1}{F(s)+1}$$ is Schur (contractive) on $\HP$. Therefore the half-plane Pick kernel $$K_{\mathcal{J}}(s,w) = \frac{F(s)+\overline{F(w)}}{(s+\overline{w})-1}$$ is PSD in the boundary limit, yielding $(P^+)$.
Pass to the limit $n \to \infty$ from passive approximants (using $L^1_{\text{loc}}$ boundary control from R1 and the chosen front-end in R4).
Key Results
For the normalized ratio $\mathcal{J}$ of Definition 1, each of A, B, or C implies $$\int_I (-w') dt \leq \pi/2$$ for every finite interval $I$, hence $(P^+)$.
A ⇒ phase-variation bound:
For a Poisson kernel $P_a(t-\gamma) = \frac{a}{(t-\gamma)^2+a^2}$ with $a = \beta - \frac{1}{2}$:
Summing over atoms of $\mu$ with $a \leq |I|/2$ (those inside tent$(I)$) gives $\int_I (-w') dt \leq \mu(\text{tent}(I)) \leq \pi/2$.
B ⇒ phase-variation bound:
This is the same inequality written explicitly by integrating $-w'$.
C ⇒ boundary positivity:
Let $\psi \geq 0$ approximate $\mathbf{1}_I$ from below and convolve with a standard mollifier. The de-smoothed positivity yields $\int_I \Repart F(\tfrac{1}{2}+it) dt \geq 0$ for all $I$, which is equivalent to $\int_I (-w') dt \leq \pi/2$ via the Hilbert transform/phase-velocity identity for outer-normalised ratios.
If $\{\mathcal{J}_n\}$ are passive (positive-real) and $\mathcal{J}_n \to \mathcal{J}$ locally uniformly on $\HP$ with $\Repart 2\mathcal{J}_n(\tfrac{1}{2}+i\cdot) \rightharpoonup \Repart 2\mathcal{J}(\tfrac{1}{2}+i\cdot)$ in $\mathcal{D}'(\mathbb{R})$, then the validity of A, B or C for $\mathcal{J}$ propagates from $\mathcal{J}_n$ to the limit, yielding $(P^+)$ for $\mathcal{J}$.
Implementation Guide
Implementer's Checklist for AI Agents
Given a powerful agent with perfect structural knowledge, the RDM instructions are:
- A: For every interval $I$, evaluate $\mu(\text{tent}(I))$ and check $\leq \pi/2$.
- B: Evaluate the arctan sum on $I$ and check $\leq \pi/2$.
- C: Verify de-smoothed positivity by testing $\langle \Repart 2\mathcal{J}(\tfrac{1}{2}+i\cdot), \psi * \varphi_\varepsilon \rangle \geq 0$ for a separating family of nonnegative Schwartz $\psi$, then pass $\varepsilon \downarrow 0$.
Notation Reference
| Symbol | Definition | Description |
|---|---|---|
| $\HP$ | $\mathbb{C}_{1/2} = \{s \in \mathbb{C} : \Repart(s) > 1/2\}$ | Right half-plane shifted by 1/2 |
| $\bHP$ | $\partial\mathbb{C}_{1/2} = \{s \in \mathbb{C} : \Repart(s) = 1/2\}$ | Boundary line (critical line) |
| $\mathcal{A}(s)$ | Hilbert-Schmidt operator | Analytic operator-valued function |
| $\det_2$ | Regularized Fredholm determinant | Well-defined for Hilbert-Schmidt operators |
| $\mathcal{O}(s)$ | Outer function | No zeros in $\HP$, boundary data in $L^1$ |
| $\mathcal{J}(s)$ | Normalized ratio | Central object of study |
| $(P^+)$ | Positivity condition | Target certificate to establish |
| KYP | Kalman-Yakubovich-Popov | Lemma/synthesis method |
Context Within Recognition Physics
RDM serves as the mathematical backbone for establishing rigorous positivity certificates in Recognition Physics. While the logical foundations establish what must be true from first principles, RDM provides the analytical machinery to verify that physical quantities derived from these principles satisfy necessary positivity conditions.
This formalism is particularly crucial when dealing with:
- Probability amplitudes: Ensuring quantum mechanical probabilities remain non-negative
- Energy densities: Verifying that energy conditions are satisfied
- Ledger balances: Confirming that the universal accounting system maintains positive costs
- Recognition events: Proving that measurement outcomes have well-defined positive probabilities
The interchangeable front-ends (A, B, C) provide flexibility in choosing the most appropriate certification method for different physical contexts, while maintaining mathematical rigor throughout.