# DRCT v2 — Closure-Defect Geometry and Observable Synthesis

## Purpose

DRCT v1 answered:

> When is a projected state dynamically sufficient, how much predictive
> information is missing, and can path/history repair it?

DRCT v2 answers the next inverse problem:

> Given the exact pattern of predictive distinctions destroyed by the current
> descriptor, what must a new observable distinguish, and how can an ideal or
> library-constrained measurement be synthesized?

The results below are mathematical theorems for finite state spaces. They do
not imply that an arbitrary physical sensor implementing the ideal observable
exists.

---

# 1. Closure-defect graph

Let \(Z:\Omega\to\mathcal Z\) be the descriptor currently available and let
\(C_\*:\Omega\to\mathcal C\) identify canonical predictive closure classes.

Define the **closure-defect graph**

\[
\boxed{
G_Z=(\Omega,E_Z)
}
\]

with an edge

\[
\boxed{
\{x,y\}\in E_Z
\iff
Z(x)=Z(y)
\;\land\;
C_\*(x)\neq C_\*(y).
}
\]

Each edge is therefore an explicit witness to a distinction erased by \(Z\)
that matters to observable future behavior.

The graph is empty iff \(Z\) is predictively sufficient.

---

# 2. Exact repair as graph coloring

An auxiliary observable

\[
Q:\Omega\to\mathcal Q
\]

repairs \(Z\) when the pair \((Z,Q)\) identifies the predictive class:

\[
H(C_\*\mid Z,Q)=0
\]

in the probabilistic formulation, or equivalently in the deterministic finite
form,

\[
(Z(x),Q(x))=(Z(y),Q(y))
\Rightarrow
C_\*(x)=C_\*(y).
\]

## Theorem 14 — Defect-Coloring Theorem

\[
\boxed{
Q\text{ repairs }Z
\iff
Q\text{ is a proper coloring of }G_Z.
}
\]

### Proof

If \(Q\) repairs \(Z\), take a defect edge \(\{x,y\}\). By definition,
\(Z(x)=Z(y)\) but \(C_\*(x)\neq C_\*(y)\). If \(Q(x)=Q(y)\), the combined
descriptor would still merge two predictive classes, contradicting repair.
Thus every defect edge receives different \(Q\) values: \(Q\) is a proper
coloring.

Conversely, assume \(Q\) properly colors \(G_Z\). If two states satisfy
\(Z(x)=Z(y)\) and \(Q(x)=Q(y)\), they cannot be joined by a defect edge.
Since their \(Z\) values agree, absence of a defect edge implies
\(C_\*(x)=C_\*(y)\). Thus \((Z,Q)\) identifies \(C_\*\). QED.

---

# 3. Geometry of the defect graph

Fix a descriptor value \(z\). Its fiber

\[
V_z=Z^{-1}(z)
\]

is partitioned by the predictive classes \(C_\*\).

Let

\[
m_z
=
\#\{C_\*(x):x\in V_z\}.
\]

Within \(V_z\), two vertices are adjacent exactly when they lie in different
predictive subclasses.

Therefore:

## Theorem 15 — Multipartite Defect Geometry

Every induced fiber graph \(G_Z[V_z]\) is a complete \(m_z\)-partite graph.
Different \(Z\)-fibers are disconnected.

Hence

\[
\boxed{
\chi(G_Z)
=
\max_z m_z.
}
\]

### Proof

Inside one fiber, vertices from the same predictive class have no edge and
vertices from different predictive classes have all possible cross edges.
That is exactly a complete multipartite graph. Its chromatic number equals the
number of nonempty parts. The chromatic number of a disjoint union is the
maximum chromatic number of its components. QED.

This recovers the v1 minimum auxiliary alphabet theorem geometrically:

\[
\boxed{
|\mathcal Q|_{\min}
=
\chi(G_Z)
=
\max_z m_z.
}
\]

---

# 4. Exact binary sensor budget

If an unconstrained auxiliary measurement may consist of \(b\) Boolean
channels,

\[
Q=(Q_1,\ldots,Q_b)\in\{0,1\}^b,
\]

then it can encode at most \(2^b\) colors.

## Corollary 15.1

\[
\boxed{
b_{\min}
=
\left\lceil\log_2\chi(G_Z)\right\rceil.
}
\]

This is achievable by coloring the defect graph with the local predictive
subclass label and binary-encoding that color.

Important: this constructs the **information content** of an ideal observable.
It does not tell us whether the corresponding physical quantity is directly
measurable.

---

# 5. The missing observable as a measurement specification

The defect graph itself is an exact specification:

\[
\boxed{
\forall\{x,y\}\in E_Z:
\quad Q(x)\neq Q(y).
}
\]

Thus DRCT can output more than "two bits are missing."

It outputs:

> These are the exact pairs of currently indistinguishable states that any
> successful new measurement system must separate.

This distinction matters because multiple physical sensors may implement the
same abstract separation constraints.

---

# 6. Candidate sensor libraries

Suppose physics/engineering supplies a finite candidate library

\[
\mathcal L=\{Q_1,\ldots,Q_M\}.
\]

Each sensor separates a subset of defect edges:

\[
\boxed{
S_i
=
\{
\{x,y\}\in E_Z:
Q_i(x)\neq Q_i(y)
\}.
}
\]

A sensor subset \(A\subseteq\{1,\ldots,M\}\) is used jointly.

## Theorem 16 — Sensor-Cover Theorem

\[
\boxed{
A\text{ repairs }Z
\iff
\bigcup_{i\in A}S_i=E_Z.
}
\]

### Proof

The joint sensor vector distinguishes a defect pair iff at least one selected
sensor assigns different values to its endpoints. By Theorem 14, exact repair
requires this for every defect edge. QED.

Therefore minimum-cost exact repair from a supplied sensor library is the
weighted set-cover problem

\[
\boxed{
\min_A\sum_{i\in A}c_i
\quad
\text{s.t.}
\quad
\bigcup_{i\in A}S_i=E_Z.
}
\]

This is not a claim that DRCT invented set cover. Sensor selection and
observability design already have extensive prior work, including formulations
as set cover.

---

# 7. Library insufficiency certificate

Let

\[
S_{\rm all}
=
\bigcup_{i=1}^M S_i.
\]

## Theorem 17 — Uncovered-Edge Certificate

If

\[
E_Z\setminus S_{\rm all}\neq\varnothing,
\]

then no combination of observables from the candidate library can produce
predictive closure.

Every uncovered edge

\[
\{x,y\}
\]

is a certificate:

\[
Z(x)=Z(y),\qquad
C_\*(x)\neq C_\*(y),
\]

and

\[
Q_i(x)=Q_i(y)
\quad\forall i.
\]

Thus any successful enlarged measurement library must contain some new channel
that separates each remaining uncovered defect edge.

This is the exact "which dimension do we need next?" output of the finite
theory.

---

# 8. Ideal observable synthesis

Because each descriptor fiber is a complete multipartite defect graph, an
optimal unconstrained observable can be synthesized directly:

1. fix a descriptor fiber \(Z^{-1}(z)\);
2. enumerate the predictive subclasses contained in that fiber;
3. assign them local labels \(0,\ldots,m_z-1\);
4. reuse labels across different \(Z\)-fibers.

Call this observable

\[
Q^\dagger.
\]

## Theorem 18 — Optimal Abstract Observable

\(Q^\dagger\) repairs the descriptor and uses exactly

\[
\max_zm_z
\]

symbols. Therefore it is alphabet-optimal.

This is constructive, not merely existential.

Again, \(Q^\dagger\) is an abstract target measurement. Translating its class
labels into a physically realizable sensor is a separate inverse problem.

---

# 9. Finite-horizon observable synthesis

For an unknown or partially observed system we may not know the complete
infinite-future closure class.

Define

\[
C_n(x)
=
(P(x),P(Fx),\ldots,P(F^n x)).
\]

This is the finite-horizon future-word class from DRCT v1.

Build a horizon-\(n\) defect graph using \(C_n\) instead of \(C_\*\).

Let

\[
m_z^{(n)}
=
\#\{C_n(x):Z(x)=z\}
\]

and define

\[
\boxed{
B_n
=
\left\lceil
\log_2
\max_z m_z^{(n)}
\right\rceil.
}
\]

## Theorem 19 — Measurement Demand Monotonicity

\[
\boxed{
B_0\le B_1\le B_2\le\cdots.
}
\]

For a finite deterministic system, the sequence eventually stabilizes at

\[
\boxed{
B_\*
=
\left\lceil
\log_2
\max_z m_z^\*
\right\rceil.
}
\]

### Proof

\(C_{n+1}\) refines \(C_n\), so the number of future subclasses inside any
fixed descriptor fiber cannot decrease. Therefore each
\(m_z^{(n+1)}\ge m_z^{(n)}\), and applying max, log2 and ceiling preserves
monotonicity.

Finite DRCT closure reaches \(C_\*\) after finitely many refinements, after
which the demand is constant. QED.

Thus the **measurement demand curve**

\[
\boxed{
n\mapsto B_n
}
\]

quantifies how many extra fixed-width bits are required as one demands closure
over increasingly deep future horizons.

---

# 10. Finite-horizon ideal sensors

At every depth \(n\), the same coloring construction yields an optimal abstract
observable

\[
Q_n^\dagger
\]

that distinguishes exactly the future classes visible by horizon \(n\).

Therefore an experimental program can proceed without claiming infinite
closure:

\[
\boxed{
P
\to
C_1
\to
Q_1^\dagger
\to
C_2
\to
Q_2^\dagger
\to
\cdots.
}
\]

If the target distinctions stabilize, this supplies a candidate sufficient
state. If new future distinctions continue appearing, the current descriptor
remains incomplete.

---

# 11. Edge geometry versus Shannon deficit

DRCT now has two complementary defect measures.

### Information measure

\[
D_\mu(Z)=H(C_\*\mid Z).
\]

This answers:

> On average, under distribution \(\mu\), how much predictive class information
> remains unresolved?

### Separation geometry

\[
G_Z=(\Omega,E_Z).
\]

This answers:

> Which exact pairs must some future measurement distinguish?

The information measure is distribution-sensitive.

The graph is worst-case and support-independent.

A sensor can be highly valuable in one sense and not the other. For example, a
sensor might separate a rare worst-case defect edge while contributing little
average mutual information.

Therefore DRCT should preserve both instead of collapsing observable design to a
single scalar score.

---

# 12. Residual-guided empirical approximation

In real scientific problems \(C_\*\) is not known beforehand.

A practical empirical approximation is:

1. collect states that are indistinguishable or close under the current
   descriptor \(Z\);
2. compare their later observable trajectories;
3. if two such states have reproducibly different future behavior, record an
   empirical defect pair;
4. ask candidate measurements which of these pairs they separate;
5. choose new measurements that maximize validated defect resolution;
6. repeat on genuinely unseen trajectories.

This does not prove the inferred empirical defect graph equals the true
\(G_Z\). It is a falsifiable estimator of the graph.

That distinction is critical for physical applications.

---

# 13. What v2 actually solves

For a known finite deterministic model, DRCT v2 now supplies:

\[
\boxed{
\text{current descriptor}
\to
\text{canonical predictive closure}
\to
\text{defect graph}
\to
\text{ideal missing observable}
}
\]

and, given a realizable sensor library,

\[
\boxed{
\text{defect graph}
\to
\text{minimum-cost sensor-cover problem}
\to
\begin{cases}
\text{repairing sensor set},\\
\text{uncovered-edge impossibility certificate}.
\end{cases}
}
\]

This is the requested observable-synthesis operator in the exact finite case.

---

# 14. Novelty discipline

The pieces have substantial neighboring prior art:

- graph coloring is established mathematics;
- set cover is established combinatorial optimization;
- sensor selection for observability is established control/discrete-event
  systems research;
- state distinguishability is established in automata/testing;
- mutual-information feature selection is established.

The DRCT claim should therefore **not** be "we invented graph coloring for
sensors."

The research candidate is the unified closure-defect pipeline and its use as a
representation-independent reverse-engineering calculus:

\[
\boxed{
\text{future-equivalence defect}
\to
\text{pairwise measurement specification}
\to
\text{abstract optimal observable}
\to
\text{realizable sensor cover}
\to
\text{unresolved-edge certificate}
\to
\text{recursive refinement}.
}
\]

Whether this exact synthesis is publishably novel requires a dedicated
literature search beyond the mathematical proof.