# RCHM × Hodge Conjecture — Research Round 5
## Plücker Phase Atlas, Circuit-Birth Geometry, Valuated Degenerations, and Jet-Filtered Relation Persistence

**Status:** Internal research note  
**Version:** Round 5 / v0.5  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–4  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 5 Objective

Round 4 introduced two complementary structures for a multi-constraint system:

$$
\boxed{
\text{coefficient side}
=
\text{determinantal phase geometry}
}
$$

and

$$
\boxed{
\text{state side}
=
\text{constraint matroid}.
}
$$

Round 5 asks four more precise questions:

1. How does a constraint matroid change continuously as the state or geometry varies?
2. Can the birth of a new minimal relation be localized by explicit equations?
3. Can one measure how fast a relation approaches a phase boundary rather than only recording whether the boundary has already been crossed?
4. How can first-order relations be filtered by higher-order Hodge obstructions so that fake tangent freedoms are eliminated?

The answer uses four classical structures:

$$
\boxed{
\text{Grassmannian}
+
\text{Plücker coordinates}
+
\text{valuated matroids}
+
\text{jet filtrations}.
}
$$

The central new RCHM synthesis is:

$$
\boxed{
\text{relation phase}
\to
\text{Plücker wall}
\to
\text{valuated approach rate}
\to
\text{higher-order persistence}.
}
$$

---

# 1. Constraint Matrix over a Parameter Space

Let

$$
U
$$

be a parameter space.

A point

$$
x\in U
$$

may represent:

- a background geometry;
- a deformation direction;
- a pair $(X,v)$ consisting of geometry and tangent direction;
- a local proof state;
- any state on which obstruction channels depend.

Suppose we have

$$
m
$$

constraint channels

$$
q_i(x)\in Q,
\qquad
i=1,\ldots,m.
$$

Collect them into the obstruction matrix

$$
\boxed{
O(x)
=
\begin{bmatrix}
q_1(x)&\cdots&q_m(x)
\end{bmatrix}.
}
$$

Assume on an open region

$$
U_r\subseteq U
$$

that

$$
\operatorname{rank}O(x)=r
$$

is constant.

---

# 2. Grassmannian Lift

The row space of $O(x)$ is an $r$-dimensional subspace of

$$
\mathbb K^m.
$$

Therefore:

$$
\boxed{
\rho:
U_r
\to
\operatorname{Gr}(r,m),
}
$$

with

$$
\rho(x)
=
\operatorname{Row}(O(x)).
$$

This map replaces a changing list of obstruction vectors by a single moving Grassmannian point.

The column-dependence matroid of $O(x)$ is determined by this row-space point.

Thus the dynamic constraint system has a canonical geometric lift:

$$
\boxed{
\text{state dynamics}
\to
\text{Grassmannian dynamics}.
}
$$

---

# 3. Plücker Coordinates as Relation-State Coordinates

For every $r$-subset

$$
B\subseteq\{1,\ldots,m\},
\qquad
|B|=r,
$$

let

$$
p_B(x)
$$

denote the corresponding Plücker coordinate.

Equivalently, after choosing an $r\times m$ full-row-rank representative, $p_B$ is the determinant of the columns indexed by $B$.

Then:

$$
\boxed{
p_B(x)\neq0
\iff
B\text{ is a basis of the constraint matroid }M_x.
}
$$

Hence the entire matroid is encoded by the zero/nonzero pattern

$$
\left\{
p_B(x)=0
\text{ or }
p_B(x)\neq0
\right\}_{|B|=r}.
$$

This gives an explicit coordinate realization of the relation phase.

---

# 4. Thin Schubert Cells as Relation Phases

For a rank-$r$ matroid $M$ on

$$
E=\{1,\ldots,m\},
$$

define its Grassmannian matroid stratum

$$
\Omega_M
=
\left\{
y\in\operatorname{Gr}(r,m):
p_B(y)\neq0
\iff
B\in\mathcal B(M)
\right\}.
$$

This is the thin Schubert cell associated with $M$.

Pull it back to the state space:

$$
\boxed{
U_M
=
\rho^{-1}(\Omega_M).
}
$$

Then:

$$
\boxed{
U_r
=
\bigsqcup_M U_M.
}
$$

Inside one $U_M$, the combinatorial set of minimal constraint relations is fixed.

Crossing between $U_M$ and $U_{M'}$ changes the constraint matroid.

Thus Round 4's informal matroid strata acquire a standard Grassmannian realization.

---

# 5. Basis-Loss Walls

Let

$$
B
$$

be a basis in a generic phase.

The wall at which $B$ loses independence is:

$$
\boxed{
W_B
=
\left\{
x:
p_B(x)=0
\right\}.
}
$$

If

$$
p_B
$$

is not identically zero and vanishes transversely along a smooth point of the pullback, $W_B$ is locally codimension one.

However, the actual wall geometry may be more complicated because:

- several Plücker coordinates can vanish simultaneously;
- Plücker relations couple the coordinates;
- the state-space image $\rho(U)$ may meet the Grassmannian strata nongenerically;
- symmetry may force higher multiplicity.

Therefore:

$$
\boxed{
p_B=0
}
$$

is a candidate local phase wall, not a universal claim of simple transverse behavior.

---

# 6. Fundamental Circuit Formula

Fix a basis

$$
B=\{b_1,\ldots,b_r\}
$$

with

$$
p_B\neq0.
$$

Take

$$
e\notin B.
$$

Then there is a unique circuit contained in

$$
B\cup\{e\}
$$

and containing $e$.

Call it

$$
C(e,B).
$$

By Cramer's rule, the unique linear relation may be written projectively as

$$
\boxed{
p_B\,q_e
+
\sum_{b\in B}
\epsilon_{b,e,B}
p_{B-b+e}\,q_b
=
0,
}
$$

where

$$
\epsilon_{b,e,B}\in\{+1,-1\}
$$

depends only on the ordering convention.

Therefore:

$$
\boxed{
C(e,B)
=
\{e\}
\cup
\left\{
b\in B:
p_{B-b+e}\neq0
\right\}.
}
$$

This is important computationally.

Once the Plücker atlas is known, fundamental circuit coefficients do not need to be rediscovered by a new nullspace computation each time.

---

# 7. Circuit-Support Walls

While

$$
p_B\neq0,
$$

consider an exchange coordinate

$$
p_{B-b+e}.
$$

If:

$$
p_{B-b+e}\neq0,
$$

then $b$ participates in the fundamental circuit.

If:

$$
p_{B-b+e}=0,
$$

then the coefficient of $q_b$ vanishes and $b$ disappears from the circuit support.

Thus define:

$$
\boxed{
W_{B,e,b}^{\mathrm{circ}}
=
\left\{
x:
p_B(x)\neq0,
\quad
p_{B-b+e}(x)=0
\right\}.
}
$$

Crossing this wall changes the support of a minimal relation.

This gives a precise mechanism for:

$$
\boxed{
\text{circuit contraction / circuit birth}.
}
$$

A smaller support relation can emerge because one or more exchange Plücker coordinates vanish.

---

# 8. Circuit-Birth Locus for a Specified Support

Let:

$$
C\subseteq E,
\qquad
|C|=c.
$$

Assume:

$$
c\le r.
$$

For $C$ to become dependent, require:

$$
\operatorname{rank}O_C\le c-1.
$$

This is a determinantal condition.

For $C$ to be a circuit, every proper subset must remain independent.

Therefore the circuit locus is:

$$
\boxed{
\mathcal C_C
=
\left\{
x:
\operatorname{rank}O_C=c-1
\right\}
\cap
\left\{
\text{every proper subset of }C\text{ independent}
\right\}.
}
$$

The second condition is open inside the first determinantal locus.

---

# 9. Circuit Rarity Law

For a generic configuration of $c$ vectors in an effective obstruction space of dimension $r$, where

$$
c\le r,
$$

the rank-drop condition:

$$
\operatorname{rank}O_C\le c-1
$$

has expected codimension:

$$
\boxed{
r-c+1.
}
$$

Thus:

### Pair cancellation

For:

$$
c=2,
$$

expected codimension:

$$
r-1.
$$

### Triple cancellation

For:

$$
c=3,
$$

expected codimension:

$$
r-2.
$$

### Near-maximal circuit

For:

$$
c=r,
$$

expected codimension:

$$
1.
$$

### Maximal minimal dependence

For:

$$
c=r+1,
$$

linear dependence is unavoidable, and the condition that every $r$-subset be independent is generically open.

Therefore:

$$
\boxed{
\text{small-support cancellations are generally more special than large-support circuits.}
}
$$

This has an immediate search consequence:

$$
\boxed{
\text{pairwise relation search alone is structurally biased toward rare events.}
}
$$

---

# 10. Support-Size Search Priority

Suppose:

$$
m\gg r.
$$

Round 4 showed that every minimal linear cancellation has support at most:

$$
r+1.
$$

Round 5 refines the search priority.

For a generic configuration:

- support size $r+1$ circuits are abundant;
- support size $r$ circuits occur on codimension-one walls;
- support size $r-1$ circuits require higher codimension;
- pairwise circuits may be extremely special for large $r$.

Thus the AI search schedule should not be:

$$
2,3,4,\ldots
$$

by naive support size.

Instead it should combine:

$$
\boxed{
\text{support size}
+
\text{expected codimension}
+
\text{symmetry prior}
+
\text{matroid locality}.
}
$$

---

# 11. One-Parameter Degeneration

Now let the state vary in one parameter:

$$
x=x(t).
$$

Assume the obstruction matrix is defined over a Laurent- or power-series field such as:

$$
\mathbb K((t)).
$$

Each Plücker coordinate becomes:

$$
p_B(t).
$$

Define its valuation:

$$
\boxed{
\omega_B
=
\nu_t(p_B(t)).
}
$$

If:

$$
p_B(t)\equiv0,
$$

set:

$$
\omega_B=\infty.
$$

Because the Plücker coordinates satisfy the Plücker relations, their valuations satisfy tropical Plücker relations.

Thus:

$$
\boxed{
\omega
=
(\omega_B)
}
$$

defines a valuated matroid / tropical Plücker vector under the standard hypotheses.

---

# 12. Why Valuation Adds New Information

The ordinary matroid records only:

$$
p_B=0
\quad\text{or}\quad
p_B\neq0.
$$

The valuated matroid records:

$$
\boxed{
\text{how rapidly each basis minor degenerates}.
}
$$

For example:

$$
p_{B_1}(t)
=
t+O(t^2),
$$

and:

$$
p_{B_2}(t)
=
t^7+O(t^8)
$$

both vanish at:

$$
t=0,
$$

but:

$$
\omega_{B_1}=1,
\qquad
\omega_{B_2}=7.
$$

The binary matroid at the special fiber forgets this difference.

The valuated matroid preserves it.

---

# 13. Relational Degeneration Depth

Define the degeneration depth of a basis:

$$
\boxed{
d(B)
=
\omega_B-\min_{B'}\omega_{B'}.
}
$$

This projectively normalizes the valuation.

Then:

$$
d(B)=0
$$

means $B$ belongs to the leading-order basis structure.

Large:

$$
d(B)
$$

means the corresponding independence relation is suppressed to higher order.

This gives a quantitative precursor to relation-phase change.

---

# 14. Valuated Fundamental Circuits

Return to a basis $B$ and element $e\notin B$.

The fundamental circuit coefficients satisfy:

$$
a_e
\propto
p_B,
$$

and:

$$
a_b
\propto
p_{B-b+e}.
$$

Therefore:

$$
\boxed{
\nu_t(a_b)-\nu_t(a_e)
=
\omega_{B-b+e}
-
\omega_B.
}
$$

Define:

$$
\boxed{
\kappa_{B,e}(b)
=
\omega_{B-b+e}
-
\omega_B.
}
$$

Then $\kappa$ measures the relative activation order of the circuit coefficient.

If:

$$
\kappa_{B,e}(b)>0,
$$

the $b$-coefficient decays relative to the $e$-coefficient as the degeneration is approached.

If:

$$
\kappa_{B,e}(b)=0,
$$

it survives at the same leading order.

Thus a circuit does not need to be treated as appearing instantaneously.

Its support can be preceded by a hierarchy of coefficient scales.

---

# 15. Relation Precursors

Define the **relation precursor profile**:

$$
\boxed{
\mathcal P_{B,e}
=
\left\{
\kappa_{B,e}(b)
\right\}_{b\in B}.
}
$$

This profile distinguishes:

1. stable coefficients;
2. coefficients approaching disappearance;
3. competing basis charts;
4. simultaneous multi-wall degenerations.

For AI-guided exploration, a relation with several low positive activation orders may deserve attention before the exact phase boundary is reached.

This is a computational heuristic, not a theorem about Hodge algebraicity.

---

# 16. Initial Matroid View

After projective normalization, the Plücker coordinates with minimal valuation determine the leading-order dependence structure.

This is naturally described by an initial matroid associated with the valuated matroid.

Thus a degeneration can be represented as:

$$
\boxed{
M_{\mathrm{generic}}
\longrightarrow
M_{\mathrm{initial}}.
}
$$

The valuated matroid stores information lost by keeping only these two endpoint matroids.

Therefore the dynamic object is better viewed as:

$$
\boxed{
\text{ordinary matroid state}
+
\text{valuation data}
+
\text{initial degeneration}.
}
$$

---

# 17. Connection to Bergman and Tropical Geometry

For an ordinary matroid, its circuits determine the Bergman fan through tropical linear equations.

For a valuated matroid, the corresponding tropical linear space contains weighted information about the degeneration.

This makes tropical geometry a natural candidate language for relation-phase asymptotics.

Within RCHM, the interpretation is:

$$
\boxed{
\text{tropicalization}
=
\text{asymptotic relation geometry}.
}
$$

This does not replace the algebraic system.

It supplies a compressed representation of degeneration rates and combinatorial phase structure.

---

# 18. Hodge Constraint Grassmannian

Return to Hodge theory.

Let:

$$
\gamma_1,\ldots,\gamma_m
$$

be rational Hodge classes at a base point.

For a parameter state $x$ and a chosen first-order deformation input, let:

$$
\Phi_{\gamma_i}(x)
$$

be the corresponding Hodge-obstruction vector.

Collect them:

$$
O_{\mathrm H}(x)
=
\begin{bmatrix}
\Phi_{\gamma_1}(x)
&
\cdots
&
\Phi_{\gamma_m}(x)
\end{bmatrix}.
$$

On a rank-$r$ region:

$$
\boxed{
\rho_{\mathrm H}:
U_r
\to
\operatorname{Gr}(r,m).
}
$$

The pullback of the Grassmannian matroid stratification becomes the:

$$
\boxed{
\textbf{Hodge Constraint Phase Atlas}.
}
$$

Each phase records which subsets of classes have independent first-order Hodge obstructions.

---

# 19. Hodge Circuit Meaning

A circuit:

$$
C\subseteq\{1,\ldots,m\}
$$

means there exists a support-minimal coefficient vector:

$$
a_C
$$

such that:

$$
\sum_{i\in C}
a_i\Phi_{\gamma_i}(x)
=
0.
$$

Therefore the combined class:

$$
\gamma_C
=
\sum_{i\in C}a_i\gamma_i
$$

has vanishing first-order Hodge obstruction along the chosen state.

But this is only a first-order statement.

For relevance to the rational Hodge conjecture, additionally require:

$$
a_i\in\mathbb Q.
$$

Thus:

$$
\boxed{
\text{matroid circuit}
\to
\text{field-of-definition filter}
\to
\text{candidate Hodge relation}.
}
$$

---

# 20. First-Order Matroid Is Not Enough

Round 1 already showed:

$$
\text{tangent legality}
\not\Rightarrow
\text{formal legality}.
$$

A first-order circuit may be:

- a genuine persistent relation;
- a nonreduced tangent artifact;
- a relation that fails at second order;
- an alias of another witness structure.

Therefore the constraint matroid must be refined by jet order.

---

# 21. Jet-Filtered Obstruction System

Let:

$$
\Phi_i^{(j)}
$$

denote the order-$j$ obstruction data for channel $i$.

Define the stacked order-$k$ obstruction vector:

$$
\boxed{
\widehat\Phi_i^{(\le k)}
=
\left(
\Phi_i^{(1)},
\Phi_i^{(2)},
\ldots,
\Phi_i^{(k)}
\right).
}
$$

Its target is:

$$
\widehat Q_k
=
Q_1\oplus\cdots\oplus Q_k.
$$

Define:

$$
O^{(\le k)}
=
\begin{bmatrix}
\widehat\Phi_1^{(\le k)}
&
\cdots
&
\widehat\Phi_m^{(\le k)}
\end{bmatrix}.
$$

The order-$k$ legal coefficient space is:

$$
\boxed{
K^{(\le k)}
=
\ker O^{(\le k)}.
}
$$

---

# 22. Relation Persistence Filtration

Because each higher order adds new constraints:

$$
\boxed{
K^{(\le1)}
\supseteq
K^{(\le2)}
\supseteq
K^{(\le3)}
\supseteq
\cdots.
}
$$

This gives a canonical relation filtration.

A coefficient relation:

$$
a
$$

has persistence depth:

$$
\boxed{
d_{\mathrm{jet}}(a)
=
\sup
\left\{
k:
a\in K^{(\le k)}
\right\}.
}
$$

Interpretation:

### $d_{\mathrm{jet}}(a)=1$

First-order relation only.

### $d_{\mathrm{jet}}(a)=2$

Survives quadratic obstruction but fails later.

### Large finite depth

High-order approximate/formal relation.

### Infinite formal depth

$$
a
\in
\bigcap_{k\ge1}
K^{(\le k)}.
$$

This is a candidate formally persistent relation.

Additional convergence/globalization arguments may still be required.

---

# 23. Jet Matroid Tower

Each stacked matrix determines a representable matroid:

$$
M^{(\le1)},
M^{(\le2)},
M^{(\le3)},
\ldots
$$

on the same channel set.

Call:

$$
\boxed{
\mathcal M_{\mathrm{jet}}
=
\left\{
M^{(\le k)}
\right\}_{k\ge1}
}
$$

the **Jet-Filtered Constraint Matroid Tower**.

As $k$ increases, new higher-order data can destroy dependencies that existed at lower order.

Thus a first-order circuit is not automatically a circuit at higher order.

Define a persistent circuit as a support set that continues to carry a legal relation through the required jet depth.

---

# 24. Relation Survival Function

For a support:

$$
C,
$$

define:

$$
\boxed{
s_C(k)
=
\dim
\left(
K_C^{(\le k)}
\right),
}
$$

where $K_C^{(\le k)}$ is the subspace of legal coefficient vectors supported in $C$.

Then:

$$
s_C(k+1)\le s_C(k).
$$

The first order at which:

$$
s_C(k)=0
$$

is the failure order of that candidate relation.

This turns higher-order verification into a measurable persistence problem.

---

# 25. Relation Barcode

For a large collection of candidate circuits, record their survival intervals:

$$
[1,d_{\mathrm{jet}}(a)].
$$

This produces a relation-persistence barcode.

The analogy to persistent homology is only organizational; no claim is made that the same invariant or stability theorems automatically apply.

The purpose is practical:

$$
\boxed{
\text{prioritize relations that survive many independent obstruction layers}.
}
$$

---

# 26. Recent Higher-Order Hodge Variation

Recent work on hypersurfaces and complete intersections studies higher-order variation of Hodge structure through Jacobian-ring multiplication and Yukawa couplings.

This supports the feasibility of treating higher-order Hodge data as structured algebraic operators rather than as an unspecified remainder.

Within the present framework, such operators can populate:

$$
\Phi_i^{(2)},
\Phi_i^{(3)},
\ldots
$$

or provide invariants used to reject low-order aliases.

The exact identification must be made case by case; RCHM does not assume that every higher-order Hodge invariant is literally a linear obstruction vector of the same form.

---

# 27. Two Independent Notions of Depth

Round 5 now contains two different depth variables.

## 27.1 Degeneration depth

$$
d(B)
$$

or:

$$
\kappa_{B,e}(b)
$$

measures how rapidly a relation phase is approached in a geometric parameter.

## 27.2 Jet persistence depth

$$
d_{\mathrm{jet}}(a)
$$

measures how many formal obstruction orders a relation survives.

These must not be conflated.

A relation can:

- approach a first-order phase wall rapidly but fail at second order;
- approach slowly but survive all tested jet orders;
- be already first-order legal yet formally invalid;
- be formally persistent but fail rationality or globalization.

---

# 28. Full Candidate Relation Type

A candidate relation should now be stored with at least:

$$
\boxed{
\mathcal R
=
(
C,
[a],
\mathbb F,
M,
\omega,
d_{\mathrm{jet}},
\Sigma,
W
).
}
$$

Where:

- $C$ is circuit support;
- $[a]$ is projective coefficient vector;
- $\mathbb F$ is coefficient field;
- $M$ is matroid phase;
- $\omega$ is valuation/degeneration data;
- $d_{\mathrm{jet}}$ is formal persistence depth;
- $\Sigma$ is scope;
- $W$ is witness status.

This is a concrete example of X-layer mathematical completeness.

---

# 29. AMRAL Data Structure Suggested by Round 5

For a matroid phase:

```text
relation_phase:
  phase_id
  rank
  ground_set[]
  basis_plucker_refs[]
  zero_plucker_refs[]
  thin_schubert_label
  parameter_scope
```

For a phase wall:

```text
relation_wall:
  wall_id
  plucker_coordinate
  vanishing_order
  source_phase
  target_phase
  affected_circuits[]
```

For a circuit:

```text
relation_circuit:
  support[]
  coefficient_vector
  coefficient_field
  fundamental_basis_ref
  plucker_formula_refs[]
  jet_persistence_depth
  witness_status
```

For one-parameter degeneration:

```text
valuated_relation_state:
  parameter
  plucker_valuations{}
  initial_matroid
  precursor_profiles[]
```

---

# 30. AI Research Workflow

Given a large family of channels:

### Step 1 — Determine effective rank

$$
r.
$$

### Step 2 — Choose a Plücker chart

Pick:

$$
B
$$

with:

$$
p_B\neq0.
$$

### Step 3 — Generate fundamental circuits

Use:

$$
p_{B-b+e}/p_B.
$$

### Step 4 — Find nearby walls

Search small or vanishing exchange coordinates.

### Step 5 — Build matroid phase

Record zero/nonzero Plücker support.

### Step 6 — If a parameter path exists, compute valuations

$$
\omega_B.
$$

### Step 7 — Filter relation field

For Hodge:

$$
[a]\in\mathbb P^{m-1}(\mathbb Q).
$$

### Step 8 — Run jet persistence

Compute:

$$
K^{(\le1)},
K^{(\le2)},\ldots
$$

### Step 9 — Search witness carriers only for persistent candidates

Avoid spending expensive witness search on relations already killed by higher-order obstruction.

---

# 31. Main New Principles

## Principle A — Plücker Phase Principle

On a constant-rank region, the relation phase is determined by the zero/nonzero pattern of Plücker coordinates.

$$
\boxed{
\text{relation phase}
=
\text{matroid stratum of a Grassmannian point}.
}
$$

---

## Principle B — Circuit-Birth Wall Principle

A new minimal dependency appears when the relevant rank minors vanish while proper-subset independence survives.

$$
\boxed{
\text{circuit birth}
=
\text{determinantal wall}
+
\text{minimality open condition}.
}
$$

---

## Principle C — Circuit Rarity Law

For a specified support of size $c\le r$:

$$
\boxed{
\operatorname{codim}_{\mathrm{exp}}
=
r-c+1.
}
$$

Smaller-support cancellations are generally more exceptional.

---

## Principle D — Fundamental Circuit Reconstruction

Inside a Plücker chart, fundamental relation coefficients are ratios of exchange Plücker coordinates.

$$
\boxed{
\text{matroid relation coefficients are graph-reconstructible from the Plücker atlas}.
}
$$

---

## Principle E — Valuated Precursor Principle

A one-parameter degeneration carries more information than its endpoint matroids.

$$
\boxed{
\text{orders of Plücker vanishing}
=
\text{relation-phase precursor data}.
}
$$

---

## Principle F — Jet Persistence Principle

First-order legality is only the first filter.

$$
\boxed{
K^{(\le1)}
\supseteq
K^{(\le2)}
\supseteq
\cdots
}
$$

measures relation survival under higher-order constraints.

---

# 32. Consequence for Hodge Search

The Hodge search architecture is now:

$$
\boxed{
\text{classes}
\to
\text{obstruction matrix}
\to
\text{Grassmannian point}
\to
\text{matroid phase}
\to
\text{circuits}
}
$$

followed by:

$$
\boxed{
\text{rationality filter}
\to
\text{valuation / degeneration analysis}
\to
\text{jet persistence}
\to
\text{witness-carrier search}.
}
$$

This is more selective than searching arbitrary rational linear combinations of algebraic classes.

---

# 33. Round 5 Conclusion

Round 5 upgrades the multi-relation system from a static matroid to a dynamic geometric object.

On a constant-rank region:

$$
\boxed{
O(x)
\leadsto
\rho(x)\in\operatorname{Gr}(r,m).
}
$$

The matroid phase is encoded by Plücker support.

Its changes occur on determinantal walls.

Its minimal relations are reconstructed through fundamental circuits.

A one-parameter degeneration upgrades the ordinary matroid to a valuated matroid, preserving orders of approach to relation walls.

Higher-order obstruction data then creates a descending relation filtration:

$$
\boxed{
K^{(\le1)}
\supseteq
K^{(\le2)}
\supseteq
K^{(\le3)}
\supseteq\cdots.
}
$$

Therefore the complete local relation state is no longer just:

$$
\text{legal / illegal}.
$$

It is closer to:

$$
\boxed{
\text{phase}
+
\text{circuit}
+
\text{distance/order to wall}
+
\text{coefficient field}
+
\text{jet persistence}
+
\text{witness status}.
}
$$

This is the main closure of Round 5.

---

# References

1. Eric Katz, **Matroid Theory for Algebraic Geometers**, arXiv:1409.3503.  
   The survey explains representable matroids, circuits, the Grassmannian interpretation, Plücker-coordinate support, and thin Schubert cells.

2. David E. Speyer, **Tropical Linear Spaces**, arXiv:math/0410455.  
   Develops tropical linear spaces, tropical Plücker coordinates, and their relation to linear spaces over power-series fields.

3. David Speyer and Bernd Sturmfels, **The Tropical Grassmannian**, arXiv:math/0304218.  
   Connects the tropical Grassmannian, Plücker relations, and tropical linear spaces.

4. Eva Maria Feichtner and Bernd Sturmfels, **Matroid Polytopes, Nested Sets and Bergman Fans**, arXiv:math/0411260.  
   Relates matroids, Bergman fans, and tropical linear geometry.

5. Mounir Nisse, **I-Maximal Variation of Hodge Structure and Jacobian Rings**, arXiv:2603.17245 (2026 preprint).  
   Studies higher-order variation of Hodge structure for hypersurfaces and complete intersections through Jacobian-ring multiplication and Yukawa coupling.

6. Remke Kloosterman, **On a conjecture on Hodge loci of linear combinations of linear subvarieties**, Rendiconti del Circolo Matematico di Palermo Series 2, 74, Article 187 (2025).  
   https://doi.org/10.1007/s12215-025-01307-4