> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# NIOT-02 — Information Probe Calculus: Observation, Refinement, Composition, and Stopping Conditions for Numerical Information Objects

## Information Probe Calculus: Observation, Refinement, Composition, and Stopping Conditions for Numerical Information Objects

**Series:** Numerical Information Object Theory (NIOT) / Numerical Information Object Theory  
**Paper:** Paper 02  
**Author:** Neo.K  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Version:** v0.1  
**Date:** 2026-09-22  
**Document type:** Operator Foundations / Probe Calculus Definition Paper / Second Paper of the Series  
**Prerequisites:** NIOT-00, NIOT-01  
**Related theories:** ICNS, ISQL, Identity Calculus, Active Index Calculus, BCODR, MNVP, CAIR, bidirectional completion–projection semantics  
**Status:** DRAFT / OPERATOR FOUNDATION / FORMALIZATION-READY  
**Canonical Source:** UTF-8 Markdown  
**Mathematical source conventions:** Inline math uses only `$...$`, and display math uses only `$$...$$`

---

# Abstract

NIOT-01 has established the basic forms of the numerical information object $\mathfrak N$, the Projection Non-Identity Principle, the identity unit $\mathbf 1_{\mathfrak N}$, the value-equivalence fiber, and informational distinctions. If these definitions are to enter mathematical and engineering computation, the next unavoidable question is: **How can a system actually "look at" a numerical information object admissibly?**

This paper proposes **Information Probe Calculus** (IPC) as NIOT's system of first-class observation and refinement operators. An information probe is not merely reading metadata or an arbitrary function name. It is a constrained observation operator with a source domain, target domain, interpretation rule, identity constraints, error model, applicability domain, certificate state, and cost model:

$$
\boxed{
P_\lambda:
\mathfrak X
\rightharpoonup
\mathcal I_\lambda
}
$$

Here $\rightharpoonup$ indicates that a probe may be partially defined, while $\lambda$ denotes its type, scale, direction, semantic depth, frequency, time, or another observation parameter.

This paper divides the main functions of probes into six classes:

1. **Extractive Probe:** extracts existing addressable information;
2. **Derived Probe:** derives new structure according to rules;
3. **Comparative Probe:** compares multiple numerical information objects;
4. **Relational Probe:** returns relations or local neighborhoods;
5. **Spectral Probe:** produces multiscale information spectra along a parameter family;
6. **Bidirectional Probe:** localizes information responsibility from both support and exclusion, generation and constraints, and forward and backward directions.

This paper further treats a probe as an equivalence-class refinement operator. If:

$$
\sim_{\mathcal P_t}
$$

denotes numerical information objects indistinguishable under the current probe set $\mathcal P_t$, then after adding a new probe $P$:

$$
\boxed{
\sim_{\mathcal P_t\cup\{P\}}
\subseteq
\sim_{\mathcal P_t}.
}
$$

Thus "entering the microscopic world of numbers" can be formalized as continually shrinking the fiber that remains indistinguishable in the current observation language.

This paper also defines probe composition, commutation, redundancy, independence, resolution, information gain, certificates, cost, closure, and stopping conditions. Crucially, adding probes should not be blind, infinite expansion: the system must determine whether a new probe genuinely adds discriminating power and stop upon reaching task-relative closure. This supplies the operator foundation for the Unboundedly Expandable Finite Matrices of NIOT-03.

The final claim of this paper is:

$$
\boxed{
\text{The core of numerical information computation is not continually adding information, but increasing distinguishability with evidence.}
}
$$

**Keywords:** Information Probes, Probe Calculus, Equivalence-Class Refinement, Fibers, Information Gain, Probe Composition, Spectral Probes, Bidirectional Probes, Task Closure, Numerical Information Objects

---

# 1. The Problem: If a Numerical Object Is More Than a Scalar, How Can We Look at It Admissibly?

NIOT-01 has established that:

$$
\pi_v(\mathfrak N)=x
$$

does not mean:

$$
\mathfrak N=x.
$$

Yet once it is acknowledged that:

$$
\mathfrak N
$$

is richer than:

$$
x
$$

the following question must immediately be answered:

> How is that additional information actually discovered?

If the answer is merely:

> Add some more fields.

then NIOT is only metadata engineering.

If the answer is:

> Insert whatever any model happens to think of.

then NIOT loses verifiability.

Thus an intermediate layer is needed:

$$
\boxed{
\text{information probes}
}
$$

It is responsible for turning:

$$
\text{latent distinction}
$$

into:

$$
\text{observable distinction}.
$$

---

# 2. The Minimal Form of a Probe

## 2.1 Basic Definition

Let:

$$
\mathfrak X
$$

be the numerical information object space.

For a probe index:

$$
\lambda\in\Lambda,
$$

Define:

$$
\boxed{
P_\lambda:
\mathfrak X
\rightharpoonup
\mathcal I_\lambda.
}
$$

where:

- $\mathfrak X$: source numerical-information space;
- $\mathcal I_\lambda$: probe output space;
- $\rightharpoonup$: a partial function;
- $\lambda$: probe parameter / probe identity.

---

## 2.2 Why Use Partial Functions?

Not every probe applies to every numerical object.

For example:

- a phase probe need not apply to a general real number;
- a provenance probe may be meaningless for a purely abstract number;
- an uncertainty probe may be NOT_APPLICABLE to some exact symbolic objects;
- a spectral probe may require additional scale structure;
- a BCODR-style bidirectional probe needs at least two comparable fields.

Therefore:

$$
\boxed{
\operatorname{Dom}(P_\lambda)
\subseteq
\mathfrak X.
}
$$

---

# 3. Probe Object

## 3.1 A Probe Is More Than a Function Name

This paper defines a complete probe object:

$$
\boxed{
\mathfrak P_\lambda
=
\left\langle
\lambda,
D_\lambda,
R_\lambda,
F_\lambda,
E_\lambda,
C_\lambda,
K_\lambda,
G_\lambda
\right\rangle.
}
$$

where:

- $\lambda$: probe identity;
- $D_\lambda$: domain;
- $R_\lambda$: range;
- $F_\lambda$: evaluation rule;
- $E_\lambda$: error / uncertainty model;
- $C_\lambda$: certificate state;
- $K_\lambda$: cost model;
- $G_\lambda$: governance / applicability constraints.

---

## 3.2 A Minimal Admissible Probe

A minimal probe must declare at least:

$$
\boxed{
D_\lambda,
\quad
R_\lambda,
\quad
F_\lambda.
}
$$

Without these three items, the system cannot even determine:

- which objects it acts on;
- what it outputs;
- how it produces the output.

---

# 4. Six Basic Probe Classes

## 4.1 Extractive Probe

Extract information already present in the canonical object.

For example:

$$
P_{\mathrm{precision}}(\mathfrak N)
=
q_{\mathrm{precision}}.
$$

Formally:

$$
\boxed{
P_{\mathrm{ext}}:
\mathfrak N
\mapsto
\text{stored substructure}.
}
$$

---

## 4.2 Derived Probe

Derive new results from existing information:

$$
\boxed{
P_{\mathrm{der}}(\mathfrak N)
=
F
\left(
P_1(\mathfrak N),
\dots,
P_k(\mathfrak N)
\right).
}
$$

A derived probe must retain:

- the derivation rule;
- the input probes;
- assumptions;
- a certificate.

---

## 4.3 Comparative Probe

Compare two or more objects:

$$
\boxed{
P_{\mathrm{cmp}}
:
\mathfrak X^n
\rightharpoonup
\mathcal C.
}
$$

For example:

$$
P_{\mathrm{cmp}}(\mathfrak N_1,\mathfrak N_2)
=
\text{same-value-different-provenance}.
$$

---

## 4.4 Relational Probe

Return a local neighborhood:

$$
\boxed{
P_{\mathrm{rel}}(\mathfrak N)
=
\mathcal N_k(\mathfrak N).
}
$$

where:

$$
\mathcal N_k
$$

may be:

- a graph neighborhood;
- a provenance neighborhood;
- a derivation neighborhood;
- a semantic neighborhood;
- a bridge neighborhood.

---

## 4.5 Spectral Probe

For a probe parameter:

$$
\lambda
$$

form:

$$
\boxed{
P_\lambda(\mathfrak N),
\qquad
\lambda\in\Lambda.
}
$$

Its output is not a single value, but:

$$
\boxed{
\mathcal S_{\mathfrak N}
=
\{
(\lambda,P_\lambda(\mathfrak N))
:
\lambda\in\Lambda
\}.
}
$$

---

## 4.6 Bidirectional Probe

Establish two directions:

$$
P_+,
\qquad
P_-,
$$

and compare:

$$
\boxed{
\mathcal O
=
P_+(\mathfrak N)
\cap
P_-(\mathfrak N).
}
$$

This probe class corresponds to BCODR-style:

- outward / inward;
- support / exclusion;
- candidate / certificate;
- generation / reconstruction;
- forward / backward.

---

# 5. Probe State

Every probe execution should produce a state.

This paper defines:

```text
OBSERVED
DERIVED
INFERRED
UNKNOWN
NOT_APPLICABLE
FAILED
WITHHELD
REVOKED
```

where:

$$
\boxed{
\text{UNKNOWN}
\neq
\text{FAILED}
\neq
\text{NOT_APPLICABLE}.
}
$$

---

# 6. Probe Result

A complete probe result is defined as:

$$
\boxed{
\mathfrak R_\lambda(\mathfrak N)
=
\left\langle
y,
s,
\varepsilon,
c,
k,
\tau
\right\rangle.
}
$$

where:

- $y$: result;
- $s$: state;
- $\varepsilon$: error / uncertainty;
- $c$: certificate;
- $k$: execution cost;
- $\tau$: time / epoch / version.

Therefore:

$$
P_\lambda(\mathfrak N)=y
$$

is merely shorthand.

---

# 7. Probe Refinement

## 7.1 The Current Distinguishability Relation

Given a probe set:

$$
\mathcal P_t
=
\{
P_1,\dots,P_n
\},
$$

Define:

$$
\boxed{
\mathfrak N_1
\sim_{\mathcal P_t}
\mathfrak N_2
}
$$

if and only if:

$$
P_i(\mathfrak N_1)
=
P_i(\mathfrak N_2)
$$

for every:

$$
P_i\in\mathcal P_t
$$

as displayed above.

---

## 7.2 Adding a New Probe

Add:

$$
P_{n+1}.
$$

then:

$$
\boxed{
\sim_{\mathcal P_t\cup\{P_{n+1}\}}
\subseteq
\sim_{\mathcal P_t}.
}
$$

The proof is immediate because the new relation adds an equivalence condition.

---

## 7.3 Refinement interpretation

This means that the essence of a probe can be understood as:

$$
\boxed{
\text{equivalence-class refinement}.
}
$$

We are not mysteriously "entering a number"; we progressively separate previously indistinguishable objects within an admissible observation language.

---

# 8. Probe Fiber

For a probe:

$$
P:
\mathfrak X
\rightharpoonup
Y
$$

Define:

$$
\boxed{
F^P_y
=
P^{-1}(y)
=
\{
\mathfrak N
:
P(\mathfrak N)=y
\}.
}
$$

If:

$$
|F^P_y|>1,
$$

the probe still cannot distinguish these objects sufficiently.

Thus probe quality can partly be measured through:

$$
\boxed{
\text{fiber contraction}
}
$$

the quantity above.

---

# 9. Probe Resolution

## 9.1 Definition

Given an object set:

$$
S\subseteq\mathfrak X,
$$

The resolution of a probe $P$ can be understood as the number of distinguishable classes into which it divides $S$.

If:

$$
\mathcal Q_P(S)
$$

denotes the partition induced by $P$, then:

$$
\boxed{
\operatorname{Res}(P;S)
=
|\mathcal Q_P(S)|.
}
$$

---

## 9.2 normalized resolution

If:

$$
|S|<\infty,
$$

one can define:

$$
\boxed{
\operatorname{NRes}(P;S)
=
\frac{
|\mathcal Q_P(S)|
}{
|S|
}.
}
$$

where:

$$
0<
\operatorname{NRes}(P;S)
\le
1.
$$

---

# 10. Probe Redundancy

If, after adding a probe:

$$
P
$$

one has:

$$
\sim_{\mathcal P\cup\{P\}}
=
\sim_{\mathcal P},
$$

then $P$ is called, relative to the current target set:

$$
\boxed{
\text{refinement-redundant}.
}
$$

Note that this does not mean that $P$ never has value.

It means only that:

$$
\boxed{
P
\text{ it adds no discriminating power in the current data and task.}
}
$$

---

# 11. Probe Independence

## 11.1 Weak Independence

If:

$$
\sim_{\mathcal P\cup\{P\}}
\subsetneq
\sim_{\mathcal P},
$$

then $P$ is called, relative to $\mathcal P$:

$$
\boxed{
\text{weakly independent}.
}
$$

---

## 11.2 Strong Independence

If there is no function:

$$
F
$$

such that:

$$
P(\mathfrak N)
=
F
\left(
P_1(\mathfrak N),\dots,P_n(\mathfrak N)
\right)
$$

holds for every admissible $\mathfrak N$, then:

$$
\boxed{
P
\text{ is strongly non-derivable from }
\mathcal P.
}
$$

This is stronger than weak independence.

---

# 12. Probe Composition

## 12.1 Sequential composition

If:

$$
P:
\mathfrak X
\rightharpoonup
Y
$$

and:

$$
Q:
Y
\rightharpoonup
Z,
$$

Define:

$$
\boxed{
Q\circ P:
\mathfrak X
\rightharpoonup
Z.
}
$$

---

## 12.2 Parallel composition

If:

$$
P_i:
\mathfrak X
\rightharpoonup
Y_i,
$$

then:

$$
\boxed{
P_1\oplus\cdots\oplus P_n
:
\mathfrak X
\rightharpoonup
Y_1\times\cdots\times Y_n.
}
$$

---

## 12.3 Conditional composition

If:

$$
C(\mathfrak N)
$$

is a condition:

$$
\boxed{
P_{\mathrm{cond}}(\mathfrak N)
=
\begin{cases}
P_1(\mathfrak N), & C(\mathfrak N)=1,\\
P_2(\mathfrak N), & C(\mathfrak N)=0.
\end{cases}
}
$$

This allows adaptive probe routing.

---

# 13. Probe Commutation

## 13.1 Definition

If:

$$
P_i\circ P_j
=
P_j\circ P_i
$$

holds on the common domain, write:

$$
\boxed{
[P_i,P_j]=0.
}
$$

where:

$$
[P_i,P_j]
=
P_i\circ P_j
-
P_j\circ P_i
$$

is the formal probe commutator.

---

## 13.2 The Meaning of Noncommutation

If:

$$
[P_i,P_j]\neq0,
$$

this may mean:

- observation order changes the state;
- the probe has side effects;
- an earlier probe changes subsequent applicability domains;
- indices / representations are rewritten;
- information extraction itself is not passive.

This provides an entry point for subsequent "information curvature."

---

# 14. Passive and Active Probes

## 14.1 Passive Probe

Does not modify the canonical object:

$$
\boxed{
P_{\mathrm{passive}}
:
\mathfrak N
\mapsto
y.
}
$$

---

## 14.2 Active Probe

May change:

- indices;
- caches;
- expansion states;
- local representations;
- routing;
- selected neighborhoods.

Therefore:

$$
\boxed{
P_{\mathrm{active}}
:
\mathfrak N_t
\mapsto
(\mathfrak N_{t+1},y).
}
$$

---

## 14.3 Identity preservation

An active probe must explicitly declare:

$$
\boxed{
\iota(\mathfrak N_{t+1})
\overset{?}{=}
\iota(\mathfrak N_t).
}
$$

If identity is not preserved, this is a state transition rather than mere observation.

---

# 15. Probe Cost

Define a cost for each probe:

$$
\boxed{
K(P)
=
\left\langle
k_{\mathrm{time}},
k_{\mathrm{memory}},
k_{\mathrm{compute}},
k_{\mathrm{external}},
k_{\mathrm{risk}}
\right\rangle.
}
$$

Thus probe selection seeks more than maximal discriminating power.

The actual objective is:

$$
\boxed{
\text{information gain}
\quad
\text{vs}
\quad
\text{cost}.
}
$$

---

# 16. Information Gain

## 16.1 Partition gain

If:

$$
\mathcal Q_{\mathcal P}(S)
$$

is the current partition,

then after adding $P$:

$$
\mathcal Q_{\mathcal P\cup\{P\}}(S).
$$

one can define:

$$
\boxed{
G_{\mathrm{part}}(P)
=
|\mathcal Q_{\mathcal P\cup\{P\}}(S)|
-
|\mathcal Q_{\mathcal P}(S)|.
}
$$

---

## 16.2 Entropic gain

If a probability model exists:

$$
p(C),
$$

one can define:

$$
\boxed{
G_H(P)
=
H(\mathcal Q_{\mathcal P})
-
H(\mathcal Q_{\mathcal P\cup\{P\}}).
}
$$

This is only an optional definition; not every probe system is required to use Shannon entropy.

---

# 17. Probe Utility

Define task-relative utility:

$$
\boxed{
U(P\mid\mathcal T)
=
\alpha G(P\mid\mathcal T)
-
\beta K(P)
-
\gamma R(P),
}
$$

where:

- $G$: information gain;
- $K$: cost;
- $R$: risk / uncertainty.

This may be an initial objective function for an AI probe selector.

---

# 18. Probe Certificate

Each high-value probe should produce a certificate:

$$
\boxed{
\mathcal C_P
=
\left\langle
\text{rule},
\text{input},
\text{version},
\text{error},
\text{status},
\text{replay}
\right\rangle.
}
$$

It must answer at least:

1. How was the result computed?
2. Which data were used?
3. Which version?
4. What errors are present?
5. Is the result observed, derived, or inferred?
6. Can it be replayed?

---

# 19. Missing Is Not Pass

Follow the governance principle of BCODR-LUC:

$$
\boxed{
\text{missing result}
\neq
\text{successful probe}.
}
$$

If:

$$
P(\mathfrak N)
$$

cannot execute, its state should be:

```text
FAILED
```

If there are no data:

```text
UNKNOWN
```

If it is inapplicable:

```text
NOT_APPLICABLE
```

Never default to:

```text
PASS
```

---

# 20. Spectral Probe Calculus

## 20.1 Probe family

Let:

$$
\{P_\lambda\}_{\lambda\in\Lambda}
$$

be a family of probes.

Define:

$$
\boxed{
\mathcal S_{\mathfrak N}(\lambda)
=
P_\lambda(\mathfrak N).
}
$$

This is the numerical information spectrum.

---

## 20.2 Spectral derivative

If:

$$
\lambda
$$

is continuous and the probe output is differentiable, define:

$$
\boxed{
\frac{d}{d\lambda}
P_\lambda(\mathfrak N).
}
$$

It describes:

> how the information representation changes as the observation scale changes.

---

## 20.3 Spectral discontinuity

If:

$$
\lim_{\lambda\to\lambda_0^-}
P_\lambda(\mathfrak N)
\neq
\lim_{\lambda\to\lambda_0^+}
P_\lambda(\mathfrak N),
$$

then:

$$
\boxed{
\lambda_0
\text{ is a probe spectral transition point}.
}
$$

This may correspond to:

- a representation transition;
- a topology change;
- an active-set change;
- a resolution threshold;
- a certificate boundary.

---

# 21. Probe Topology

For a probe family:

$$
\mathcal P
$$

one can construct a graph:

$$
G_{\mathcal P}
=
(V,E),
$$

where:

$$
V=\mathcal P
$$

An edge may represent:

- dependency;
- composition;
- redundancy;
- conflict;
- mutual refinement;
- a shared source;
- a bridge relation.

Thus a probe system itself can become a computable space.

---

# 22. Probe Conflict

If:

$$
P_i(\mathfrak N)
$$

and:

$$
P_j(\mathfrak N)
$$

are incompatible on the same semantic question, one cannot simply choose either arbitrarily.

Establish:

$$
\boxed{
\operatorname{Conflict}(P_i,P_j,\mathfrak N).
}
$$

then check:

- a domain mismatch;
- a representation mismatch;
- a stale version;
- overlapping error intervals;
- a semantic-schema disagreement;
- an actual contradiction.

---

# 23. Probe Consensus

If multiple independent probes agree on the same invariant:

$$
P_1(\mathfrak N)
=
\cdots
=
P_k(\mathfrak N),
$$

one can establish:

$$
\boxed{
\operatorname{Consensus}_k.
}
$$

Yet consensus is not truth.

It represents only:

$$
\boxed{
\text{multi-probe agreement under declared assumptions}.
}
$$

---

# 24. Probe Depth

Define probe depth:

$$
d(P)
$$

as the number of derivation layers from the canonical source to the result.

For example:

$$
d(P_{\mathrm{raw}})=0,
$$

$$
d(P_{\mathrm{derived}})=1,
$$

For a deeper composite probe:

$$
d(Q\circ P)
=
d(P)+d(Q).
$$

This aids:

- auditing;
- confidence;
- cost;
- debugging;
- explanation.

---

# 25. Probe Locality

Let:

$$
U\subseteq\mathfrak N
$$

denote a local information domain.

Define:

$$
\boxed{
P_{\lambda,U}(\mathfrak N).
}
$$

This probe observes only:

$$
U.
$$

Therefore:

$$
\boxed{
\text{local probe}
}
$$

can avoid global expansion.

---

# 26. Probe Radius

A radius can be introduced for a relation graph or information geometry:

$$
r.
$$

Define:

$$
\boxed{
P_r(\mathfrak N)
=
\mathcal N_r(\mathfrak N).
}
$$

If:

$$
r_1<r_2,
$$

usually:

$$
P_{r_1}(\mathfrak N)
\subseteq
P_{r_2}(\mathfrak N).
$$

This supplies a natural multiscale probe family.

---

# 27. Bidirectional Probe Calculus

## 27.1 Two Directions

Define:

$$
P^+,
\qquad
P^-.
$$

representing, respectively:

- forward / backward;
- support / exclusion;
- candidate / certificate;
- generation / reconstruction.

---

## 27.2 Overlap

Define:

$$
\boxed{
\mathcal O(\mathfrak N)
=
P^+(\mathfrak N)
\cap
P^-(\mathfrak N).
}
$$

---

## 27.3 Atomic overlap

If an overlap contains different membership / provenance / responsibility, it must be divided into:

$$
\boxed{
\mathcal O
=
\bigsqcup_i
\mathcal O_i.
}
$$

where each:

$$
\mathcal O_i
$$

has a stable responsibility signature.

---

## 27.4 Structural alignment

Two overlap regions should not be judged identical solely by their coordinates.

One must compare:

$$
\boxed{
\text{source}
+
\text{support}
+
\text{active set}
+
\text{topology}
+
\text{certificate provenance}.
}
$$

This directly inherits the structural-alignment principle of BCODR-LUC.

---

# 28. Probe Selection

## 28.1 Random Probing Is Insufficient

With many probe candidates:

$$
|\Lambda|
\gg1,
$$

they cannot be executed blindly one by one.

Thus one needs:

$$
\boxed{
\operatorname{SelectProbe}
(
\mathfrak N,
\mathcal P_t,
\mathcal T
).
}
$$

---

## 28.2 Information gap

Define:

$$
\boxed{
G_t
=
\mathcal G
(
\mathfrak N,
\mathcal P_t,
\mathcal T
).
}
$$

$G_t$ describes:

- which invariants remain undecided;
- which fibers are too large;
- which conflicts remain unresolved;
- which certificates are missing;
- which uncertainties exceed the threshold.

---

## 28.3 The Next Probe

Select:

$$
\boxed{
P_{t+1}
=
\arg\max_P
U(P\mid G_t,\mathcal T).
}
$$

This is the simplest greedy probe policy.

---

# 29. Adaptive Probe Loop

The complete loop:

$$
\boxed{
\mathfrak N
\rightarrow
\mathcal P_0
\rightarrow
G_0
\rightarrow
P_1
\rightarrow
\mathcal P_1
\rightarrow
G_1
\rightarrow
\cdots
}
$$

until:

$$
\boxed{
\operatorname{Close}_{\mathcal T}
(
\mathfrak N,
\mathcal P_t
)
=
1.
}
$$

---

# 30. Task-relative Closure

## 30.1 Definition

For a task:

$$
\mathcal T,
$$

if the current probe set suffices to determine all task-required invariants, then:

$$
\boxed{
\operatorname{Close}_{\mathcal T}=1.
}
$$

---

## 30.2 Closure conditions

The first version may require:

1. required invariants determined;
2. uncertainty below the threshold;
3. no unresolved critical conflict;
4. certificate obligations satisfied;
5. next-probe expected utility below the threshold.

Formally:

$$
\boxed{
G_t
\le
\epsilon_{\mathcal T}.
}
$$

---

# 31. Stop Rule

Define:

$$
\boxed{
\operatorname{Stop}
=
\left[
\operatorname{Close}_{\mathcal T}=1
\right]
\lor
\left[
\max_P U(P)\le\tau
\right]
\lor
\left[
K_{\mathrm{used}}\ge K_{\max}
\right].
}
$$

Thus an unbounded probe space does not mean actual infinite probing.

---

# 32. Probe Minimality

For a task $\mathcal T$, define a minimal sufficient probe set:

$$
\boxed{
\mathcal P^\star_{\mathcal T}
=
\arg\min_{\mathcal P}
|\mathcal P|
}
$$

subject to:

$$
\operatorname{Close}_{\mathcal T}
(
\mathfrak N,
\mathcal P
)
=
1.
$$

This supplies:

$$
\boxed{
\text{minimal sufficient numerical observability}.
}
$$

---

# 33. Probe Basis

If there is a probe set:

$$
\mathcal B
=
\{P_1,\dots,P_k\}
$$

such that all important results of the target probe family can be generated or derived from:

$$
\mathcal B
$$

then:

$$
\boxed{
\mathcal B
\text{ is a task-relative probe basis}.
}
$$

---

# 34. Dynamic Probe Basis

Different tasks may need different bases:

$$
\mathcal B_{\mathcal T_1}
\neq
\mathcal B_{\mathcal T_2}.
$$

Therefore:

$$
\boxed{
\text{There need not be a unique universal finite probe basis}.
}
$$

This is consistent with NIOT's unbounded extensibility.

---

# 35. Probe Algebra

Define a probe algebra:

$$
\boxed{
\mathfrak A_P
=
\langle
\mathcal P,
\circ,
\oplus,
[\cdot,\cdot],
\operatorname{Refine},
\operatorname{Restrict}
\rangle.
}
$$

where:

- $\circ$: sequential composition;
- $\oplus$: parallel composition;
- commutator: order sensitivity;
- Refine: increasing discriminating power;
- Restrict: localizing the domain.

This advances probes from a list of tools to an operator system that can be studied.

---

# 36. Probe Category

One can further construct a category:

$$
\boxed{
\mathbf{Probe}
}
$$

Objects:

$$
\mathfrak X_i
$$

Morphisms:

$$
P:
\mathfrak X_i
\rightharpoonup
\mathfrak X_j.
$$

Restricting to total identity-preserving probes gives a subcategory.

This direction is not fully developed in the present paper.

---

# 37. Interface with Identity Calculus

Identity Calculus gives:

$$
d_I(\mathfrak N)
=
\{
(\mathfrak N,i)
\}.
$$

Probe Calculus gives:

$$
P_i(\mathfrak N).
$$

The two can be connected as:

$$
\boxed{
P_i
\leftrightarrow
\text{indexed observation along }i.
}
$$

Thus an index can become a probe coordinate.

---

# 38. Interface with Active Index Calculus

An active index transition:

$$
\widehat{\mathcal E}_{i\to j}
:
(\mathfrak N,i)
\mapsto
(\mathfrak N,j)
$$

can control:

$$
\boxed{
\text{the information position from which the next probe enters.}
}
$$

Thus AIC supplies a geometry for probe navigation.

---

# 39. Interface with ISQL

ISQL can supply:

- probe addresses;
- semantic anchors;
- a probe-result registry;
- relation links;
- history links;
- runtime paths.

Therefore:

$$
\boxed{
\mathrm{ISQL}
\approx
\text{the semantic addressing substrate of Probe Calculus}.
}
$$

Yet this paper does not require probes to be implemented using ISQL.

---

# 40. Interface with CAIR

Probe execution may produce a candidate update:

$$
\Delta\mathfrak N.
$$

If the canonical state is governed, then:

$$
\boxed{
P
\rightarrow
\text{candidate observation}
\rightarrow
\text{validation}
\rightarrow
\text{canonical commit}.
}
$$

This directly connects to CAIR's candidate-diff pattern.

---

# 41. Interface with MNVP

A probe result:

$$
\mathfrak R_\lambda
$$

need not be directly suitable for human reading.

MNVP can provide:

$$
\boxed{
\mathfrak R_\lambda
\xrightarrow{\mathrm{MNVP}}
\text{perceptual view}.
}
$$

Thus Probe Calculus does not handle final UI presentation.

---

# 42. Interface with ICNS

If a probe input or output is a numeral token, one needs:

$$
\boxed{
\mathrm{ICNS}
}
$$

first to ensure that its:

- domain;
- grammar;
- comparison;
- normalization;
- successor;
- resolution;

have been sufficiently defined.

Therefore:

$$
\boxed{
\mathrm{ICNS}
\rightarrow
\mathrm{Probe}
}
$$

is one numeral-entry pathway.

---

# 43. A First Simple Example

Consider:

$$
x=0.5.
$$

There are two objects:

$$
\mathfrak N_A,
\qquad
\mathfrak N_B.
$$

Scalar probe:

$$
P_v(\mathfrak N_A)
=
P_v(\mathfrak N_B)
=
0.5.
$$

Thus under:

$$
\mathcal P_0
=
\{P_v\}
$$

the probe set displayed above:

$$
\mathfrak N_A
\sim_{\mathcal P_0}
\mathfrak N_B.
$$

Add a provenance probe:

$$
P_p(\mathfrak N_A)
=
\text{exact rational},
$$

$$
P_p(\mathfrak N_B)
=
\text{sensor estimate}.
$$

then:

$$
\boxed{
\mathfrak N_A
\not\sim_{\mathcal P_1}
\mathfrak N_B.
}
$$

where:

$$
\mathcal P_1
=
\{P_v,P_p\}.
$$

This is a minimal probe refinement.

---

# 44. A Second Example: Spectral Discrimination

Suppose:

$$
P_\lambda
$$

denotes an uncertainty probe at different scales.

If:

$$
P_{\lambda_1}(\mathfrak N)
=
0.01,
$$

$$
P_{\lambda_2}(\mathfrak N)
=
0.2,
$$

then one cannot retain only:

$$
\text{uncertainty}=0.01.
$$

Fuller information is:

$$
\boxed{
\mathcal S_{\mathfrak N}
=
\{
(\lambda_1,0.01),
(\lambda_2,0.2),
\dots
\}.
}
$$

This shows that uncertainty itself may be a spectral object.

---

# 45. A Third Example: Bidirectional Responsibility Localization

Suppose:

$$
P_+
$$

returns a provenance region supporting a candidate state, while:

$$
P_-
$$

returns a certificate region excluding that candidate state.

If:

$$
\mathcal O
=
P_+(\mathfrak N)
\cap
P_-(\mathfrak N)
\neq
\varnothing,
$$

one should not immediately declare a contradiction.

One must further ask:

- Is it the same position?
- Is it the same time?
- Is it the same representation?
- Is it the same structural mechanism?
- Is it merely container overlap?

This is precisely the warning supplied by BCODR-LUC.

---

# 46. Probe Error Propagation

If:

$$
Q\circ P
$$

is a composite probe, its errors cannot be ignored.

If:

$$
\varepsilon_P,
\qquad
\varepsilon_Q
$$

are their respective error models, one needs:

$$
\boxed{
\varepsilon_{Q\circ P}
=
\mathcal E
(
\varepsilon_P,
\varepsilon_Q,
Q,P
).
}
$$

The specific form depends on the probe types.

---

# 47. Probe Provenance

Every probe result should carry:

$$
\boxed{
\operatorname{Prov}
(
\mathfrak R_\lambda
).
}
$$

It should at least be traceable to:

- the source object;
- the source version;
- the probe version;
- input probes;
- the execution environment;
- a certificate.

---

# 48. Probe Replayability

A high-value probe should satisfy:

$$
\boxed{
\operatorname{Replay}
(
\mathfrak P_\lambda,
\mathfrak N,
\text{environment}
)
\approx
\mathfrak R_\lambda.
}
$$

If a black-box model can generate it only once and it cannot be reproduced, its certificate strength must be downgraded.

---

# 49. Probe Trust Levels

The first version may define:

```text
T0 — unverified
T1 — replayable
T2 — independently reproduced
T3 — formally verified rule
T4 — formally verified + independently reproduced
```

This is only a draft and may be finalized in a subsequent Runtime paper.

---

# 50. Probe Governance

Not every probe result should immediately enter the canonical object.

One can adopt:

$$
\boxed{
\text{probe result}
\rightarrow
\text{candidate state}
\rightarrow
\text{review / validate}
\rightarrow
\text{commit}.
}
$$

This avoids the error:

$$
\boxed{
\text{observation}
=
\text{truth}
}
$$

shown above.

---

# 51. Probe Expansion and Compression

As the number of probes grows:

$$
|\mathcal P_t|
\rightarrow
\text{large},
$$

compression must be allowed.

Define:

$$
\boxed{
\operatorname{Compress}
(
\mathcal P_t
)
\rightarrow
\widehat{\mathcal P}_t.
}
$$

Require:

$$
\operatorname{Close}_{\mathcal T}
(
\widehat{\mathcal P}_t
)
=
\operatorname{Close}_{\mathcal T}
(
\mathcal P_t
)
$$

equivalence at least for the target task.

---

# 52. Probe Garbage Collection

If some probes are:

- persistently redundant;
- certificate-revoked;
- source-expired;
- no longer relevant;
- replaced by stronger probes;

they can enter:

$$
\boxed{
\text{probe GC}.
}
$$

Yet provenance must be retained before deletion.

---

# 53. Probe Versioning

A probe rule itself may be updated:

$$
P^{(1)}
\rightarrow
P^{(2)}.
$$

Thus probe identity must include its version:

$$
\boxed{
\lambda
=
(\text{name},\text{version}).
}
$$

Results from different versions cannot be assumed to be the same observation.

---

# 54. Probe Equivalence

If two probes:

$$
P,
Q
$$

satisfy, on a specified domain:

$$
P(\mathfrak N)
=
Q(\mathfrak N)
$$

for every:

$$
\mathfrak N\in D
$$

as displayed above, then:

$$
\boxed{
P\equiv_D Q.
}
$$

This can be used for probe deduplication.

---

# 55. Probe Dominance

If:

$$
\sim_{\mathcal P\cup\{P\}}
\subseteq
\sim_{\mathcal P\cup\{Q\}}
$$

and:

$$
K(P)\le K(Q),
$$

one can say:

$$
\boxed{
P
\text{ dominates }Q
}
$$

for the current task / domain.

---

# 56. Probe Frontier

There may be a Pareto frontier for information gain and cost:

$$
\boxed{
\mathcal F_P
=
\operatorname{Pareto}
(
G(P),K(P)
).
}
$$

An AI probe selector need not find a single best probe; it can select on the frontier.

---

# 57. Probe Curriculum

For a complex numerical object, one can design:

$$
\boxed{
P_1
\rightarrow
P_2
\rightarrow
\cdots
\rightarrow
P_n
}
$$

progressing from low cost and resolution to high cost and resolution.

This curriculum can substantially reduce probe explosion.

---

# 58. Probe Horizon

Define:

$$
H_P
$$

as the maximum number of probe steps considered ahead in a planning round.

Local greedy:

$$
H_P=1.
$$

Long-range planning:

$$
H_P>1.
$$

This leaves a planning interface for the AI-Native Runtime.

---

# 59. Probe Search Space

All probe candidates form:

$$
\boxed{
\mathcal P_{\mathrm{all}}.
}
$$

The actually available set:

$$
\boxed{
\mathcal P_{\mathrm{avail}}(\mathfrak N,\mathcal T)
\subseteq
\mathcal P_{\mathrm{all}}.
}
$$

is jointly constrained by:

- domain;
- permissions;
- cost;
- data availability;
- runtime;
- safety;
- certificates;

as listed above.

---

# 60. The Central Proposition of Probe Calculus

This paper condenses Probe Calculus into the following proposition:

$$
\boxed{
\text{Probe}
=
\text{admissibly refining indistinguishability.}
}
$$

More fully:

$$
\boxed{
\mathfrak X
\big/
\sim_{\mathcal P_t}
\quad
\longrightarrow
\quad
\mathfrak X
\big/
\sim_{\mathcal P_{t+1}}.
}
$$

where:

$$
\sim_{\mathcal P_{t+1}}
\subseteq
\sim_{\mathcal P_t}.
$$

---

# 61. Difference from Traditional Calculus

Traditional calculus concerns:

$$
\frac{dx}{dt}
$$

or:

$$
\int f(x)\,dx.
$$

Information Probe Calculus concerns:

$$
\boxed{
\frac{
\text{change in distinguishable information structure}
}{
\text{change in probe parameters / probe sets}
}.
}
$$

Thus it is not a replacement for calculus, but another analytical layer.

---

# 62. Probe Differential

If a probe family:

$$
P_\lambda
$$

is differentiable, define:

$$
\boxed{
d_P\mathfrak N
=
\frac{d}{d\lambda}
P_\lambda(\mathfrak N)
\,d\lambda.
}
$$

This is an observation-space differential.

---

# 63. Probe Integral

If probe outputs can be aggregated over $\lambda$, one can define:

$$
\boxed{
\int_{\Lambda}
P_\lambda(\mathfrak N)
\,d\mu(\lambda).
}
$$

Yet this denotes only probe-space aggregation.

It must not be confused with the forgetful integration of Identity Calculus.

---

# 64. Two Kinds of "Integration" Must Be Distinguished

This paper specifically distinguishes:

### Observation aggregation

$$
\int_\Lambda
P_\lambda(\mathfrak N)
\,d\mu(\lambda).
$$

### Identity-forgetting integration

$$
\int_{\mathrm{id}}
(\mathfrak N,i)
=
\mathfrak N.
$$

These are not the same operator.

---

# 65. An Introduction to Probe Curvature

If three probes:

$$
P_i,P_j,P_k
$$

give different results when composed in different orders:

$$
P_k\circ P_j\circ P_i
\neq
P_i\circ P_j\circ P_k,
$$

one may need to define a kind of:

$$
\boxed{
\text{probe-space holonomy / curvature}.
}
$$

This paper proposes only the direction, not its complete form.

---

# 66. Probe Completeness

Given a task $\mathcal T$, if a probe set $\mathcal P$ satisfies:

$$
\boxed{
\forall
\mathfrak N_1,\mathfrak N_2
\in
S_{\mathcal T},
}
$$

whenever:

$$
\mathfrak N_1
$$

and:

$$
\mathfrak N_2
$$

differ in task-relevant invariants, there is:

$$
P\in\mathcal P
$$

such that:

$$
P(\mathfrak N_1)
\neq
P(\mathfrak N_2),
$$

then:

$$
\boxed{
\mathcal P
\text{ is task-complete for }\mathcal T\text{}.
}
$$

---

# 67. Probe Incompleteness

If there exist:

$$
\mathfrak N_1\neq\mathfrak N_2
$$

differing in a task-relevant invariant, but every:

$$
P\in\mathcal P
$$

cannot distinguish them, then:

$$
\boxed{
\mathcal P
\text{ task-incomplete}.
}
$$

---

# 68. Probe Discovery

Not every valuable probe is known in advance.

Thus one can define a meta-process:

$$
\boxed{
\operatorname{DiscoverProbe}
(
\mathfrak N,
G_t
)
\rightarrow
P_{\mathrm{new}}.
}
$$

This differs from merely using SelectProbe:

- SelectProbe: selects from a known probe library;
- DiscoverProbe: proposes a new observation direction.

---

# 69. AI's Role in Probe Calculus

AI can handle:

1. information-gap diagnosis;
2. candidate probe selection;
3. new probe proposals;
4. probe composition;
5. conflict triage;
6. certificate checking;
7. stopping decisions;
8. compression planning.

Yet AI should not:

$$
\boxed{
\text{write unverified inference directly as a canonical fact}.
}
$$

---

# 70. A Minimal Runtime Data Model

The first version may use:

```json
{
  "probe_id": "probe:provenance:v1",
  "domain": "niot:numerical-object",
  "range": "niot:provenance-node",
  "mode": "extractive",
  "state": "OBSERVED",
  "value": {},
  "error": null,
  "certificate": {},
  "cost": {},
  "dependencies": [],
  "timestamp": null
}
```

---

# 71. Probe Registry

All probes can enter a:

$$
\boxed{
\mathcal R_P
}
$$

registry.

Each entry records at least:

- probe ID;
- version;
- domain;
- range;
- rule;
- dependency;
- trust level;
- cost;
- compatibility;
- deprecation state.

---

# 72. Probe DAG

Derived probes naturally form a DAG:

$$
\boxed{
G_D
=
(V_D,E_D).
}
$$

where:

- node = probe;
- edge = dependency.

This supplies:

- incremental recomputation;
- cache invalidation;
- provenance;
- debugging.

---

# 73. Probe Cache

If:

$$
P(\mathfrak N)
$$

is costly and the source state has not changed, it can be cached.

Yet the cache key must include at least:

$$
\boxed{
\iota(\mathfrak N),
\text{version}(\mathfrak N),
\lambda,
\text{version}(P).
}
$$

Otherwise a stale observation may result.

---

# 74. Probe Invalidation

If the source:

$$
\mathfrak N
$$

is updated, find all dependent probes:

$$
\operatorname{Descendants}_{G_D}(\mathfrak N)
$$

and perform:

$$
\boxed{
\text{invalidate / recompute}.
}
$$

---

# 75. Probe Security

Active or external probes may cause:

- source mutation;
- data leakage;
- untrusted output;
- side effects;
- resource exhaustion.

Thus Probe Calculus needs:

$$
\boxed{
\text{permission-aware execution}.
}
$$

This is left to the Runtime paper for elaboration.

---

# 76. Probe Boundary

Each probe must declare:

$$
\boxed{
\text{what it can conclude}
}
$$

and:

$$
\boxed{
\text{what it cannot conclude}.
}
$$

For example:

$$
P_{\mathrm{proximity}}
$$

can prove only proximity; it cannot automatically prove:

$$
\text{structural identity}.
$$

This directly inherits the conceptual correction of BCODR-LUC.

---

# 77. Probe Responsibility

Every conclusion should answer:

$$
\boxed{
\text{which probes are responsible for this conclusion?}
}
$$

Thus a conclusion object:

$$
\mathfrak C
$$

should carry:

$$
\operatorname{Resp}(\mathfrak C)
=
\{P_{i_1},\dots,P_{i_k}\}.
$$

---

# 78. Probe Explanation

A minimal explanation:

```text
CONCLUSION:
    X

SUPPORTED_BY:
    P1
    P4
    P7

DERIVED_THROUGH:
    F

UNCERTAINTY:
    ε

OPEN:
    P9 pending
```

This allows numerical information computation to be audited.

---

# 79. Probe Closure Certificate

When:

$$
\operatorname{Close}_{\mathcal T}=1,
$$

the system should output:

$$
\boxed{
\mathcal C_{\mathrm{close}}
}
$$

including at least:

- task;
- probe set;
- unresolved items;
- uncertainty;
- stopping reason;
- certificate hash / version.

---

# 80. The Seven Central Principles of This Paper

## Principle 1

$$
\boxed{
\text{Probe is an operator, not a label.}
}
$$

## Principle 2

$$
\boxed{
\text{Probe must refine or justify its existence.}
}
$$

## Principle 3

$$
\boxed{
\text{More probes do not automatically mean more truth.}
}
$$

## Principle 4

$$
\boxed{
\text{Unknown is not pass.}
}
$$

## Principle 5

$$
\boxed{
\text{Projection overlap is weaker than structural alignment.}
}
$$

## Principle 6

$$
\boxed{
\text{Probe expansion must remain task-relative.}
}
$$

## Principle 7

$$
\boxed{
\text{Every high-value probe result requires provenance and certificate.}
}
$$

---

# 81. Direct Interface with NIOT-03

NIOT-03 will no longer treat a matrix as an arbitrary data table.

It will use:

$$
\boxed{
\mathcal P_t
}
$$

as the generation source of rows / columns / coordinates.

Therefore:

$$
M_t
$$

is a:

$$
\boxed{
\text{probe-activated finite information state}.
}
$$

rather than "putting all information into a matrix."

---

# 82. Prerequisite Propositions for NIOT-03

The next paper must build on the following results:

1. At any epoch, a probe set is a finite executable set;
2. A probe family can theoretically expand without bound;
3. Probe refinement refines equivalence classes;
4. Redundant probes can be compressed;
5. Active probes can reindex;
6. Spectral probes can add continuous parameter dimensions;
7. Bidirectional probes can establish responsibility overlaps;
8. Closure can be task-relative rather than ontology-complete.

---

# 83. Conclusion

This paper elevates "information probes" from an intuitive concept to first-class NIOT operators.

For a numerical information object:

$$
\mathfrak N,
$$

we no longer ask:

> How much information does it still conceal?

Instead, we ask:

$$
\boxed{
\text{Under which probes
can which differences be distinguished admissibly?}
}
$$

Thus probes do not invent information out of nothing; they establish:

$$
\boxed{
\text{observable distinction}.
}
$$

When a probe set expands from:

$$
\mathcal P_t
$$

to:

$$
\mathcal P_{t+1},
$$

what actually occurs is:

$$
\boxed{
\sim_{\mathcal P_{t+1}}
\subseteq
\sim_{\mathcal P_t}.
}
$$

That is:

$$
\boxed{
\text{further separation of previously indistinguishable numerical objects.}
}
$$

This is what this paper calls:

$$
\boxed{
\textbf{Information Probe Calculus.}
}
$$

It also supplies:

- probe definitions;
- probe types;
- composition;
- commutation;
- redundancy;
- independence;
- fibers;
- spectral families;
- bidirectional localization;
- certificates;
- cost;
- utility;
- adaptive selection;
- closure;
- stopping conditions.

Thus the next step naturally enters:

$$
\boxed{
\textbf{NIOT-03 — Unboundedly Expandable Finite Matrices.}
}
$$

Its core will be:

$$
\boxed{
\text{How can every actual state remain finite
while its probe-induced information dimension continues to expand without bound?}
}
$$

---

# Canonical Source Declaration

This file is the UTF-8 Markdown canonical source for `NIOT-02-v0.1`.

The only mathematical delimiters are `$...$` and `$$...$$`.

This paper defines:
- probes;
- probe objects;
- probe results;
- refinement relations;
- probe fibers;
- redundancy;
- independence;
- composition;
- commutation;
- spectral probes;
- bidirectional probes;
- task-relative closure;
- stopping conditions.

This paper does not claim that:
- every probe family is differentiable;
- every probe is invertible;
- every probe result is a true value;
- every probe applies to all numerical objects;
- task closure equals ontology completion;
- all current theorems have been Lean / Coq verified.

Subsequent NIOT-03 should use this Probe Calculus as the prerequisite semantics for matrix generation and dynamic expansion.

===END===