# RCHM × Hodge Conjecture — Research Round 6
## Carrier Inverse Problem, Identifiability Depth, Geometric Purity, and Witness-Transition Graphs

**Status:** Internal research note  
**Version:** Round 6 / v0.6  
**Date:** 2026-09-17  
**Methodological base:** RCHM; Typed Relation & Constraint Algebra; X-Layer Mathematical Completeness; Graph-Native Mathematics  
**Previous rounds:** RCHM × Hodge Conjecture — Rounds 1–5  
**Canonical source:** UTF-8 Markdown

---

# 0. Round 6 Objective

Round 5 ended with the question:

> Can a persistent relation signature predict the algebraic witness carrier?

This is an inverse problem.

The forward direction is comparatively natural:

$$
\boxed{
\text{carrier}
\longrightarrow
\text{cohomology class / periods / Hodge locus / jets / ideals}.
}
$$

The inverse direction is harder:

$$
\boxed{
\text{Hodge signature}
\stackrel{?}{\longrightarrow}
\text{carrier}.
}
$$

Existing reconstruction results show that the inverse direction sometimes works and sometimes fails.

Movasati--Sertöz construct ideals from period data and recover equations of suitable algebraic subvarieties in projective hypersurfaces. For complete intersections, low-degree equations can be recovered under explicit conditions. Cifani--Pirola--Schlesinger obtain reconstruction results for classes of curves such as arithmetically Cohen--Macaulay curves, but also show failures such as rational quartics on quartic surfaces. Fake linear cycles provide another warning: first-order and Artinian-Gorenstein signatures may imitate those of genuine linear cycles without yielding a genuine linear support.

Therefore the correct problem is not:

$$
\text{signature}\to\text{one guessed carrier},
$$

but:

$$
\boxed{
\text{signature}
\to
\text{inverse carrier fiber}
\to
\text{geometric realizability}
\to
\text{higher-order disambiguation}.
}
$$

---

# 1. Carrier Space

Let

$$
\mathcal W
$$

denote the universe of admissible algebraic witnesses under consideration.

Decompose it by carrier type:

$$
\boxed{
\mathcal W
=
\bigsqcup_{\tau\in\mathcal T}
\mathcal W_\tau.
}
$$

Possible carrier types include:

- linear subvarieties;
- complete intersections;
- rational curves;
- ACM curves;
- Grassmannian-type varieties;
- unions or rational linear combinations of irreducible cycles;
- more general Chow/Hilbert strata.

The symbol

$$
\tau
$$

does not denote only a human-readable label.

It may include:

$$
\boxed{
\tau
=
(\dim,\deg,\text{Hilbert polynomial},\text{incidence type},\text{scheme type},\ldots).
}
$$

---

# 2. Signature Hierarchy

For a witness

$$
W\in\mathcal W,
$$

define a hierarchy of observable signatures.

A possible hierarchy is:

$$
\mathsf S^{(\le0)}(W)
=
\text{cohomological class},
$$

$$
\mathsf S^{(\le1)}(W)
=
\text{class + period/annihilator ideal + tangent Hodge data},
$$

$$
\mathsf S^{(\le2)}(W)
=
\text{previous data + second-order IVHS},
$$

and more generally:

$$
\boxed{
\mathsf S^{(\le k)}(W)
=
\text{all retained carrier-relevant invariants through level }k.
}
$$

The exact contents of level $k$ are application-dependent.

For the current Hodge program they may include:

- primitive Hodge class;
- period vector;
- associated Artinian Gorenstein ideal;
- Hilbert function;
- Hodge-locus tangent space;
- Plücker/matroid relation signature;
- valuation/degeneration profile;
- order-$j$ Hodge obstruction data;
- local scheme structure;
- field of definition.

---

# 3. Forward Signature Map

For each depth $k$, define:

$$
\boxed{
\sigma_k:
\mathcal W
\to
\mathcal S_k,
}
$$

where:

$$
\sigma_k(W)
=
\mathsf S^{(\le k)}(W).
$$

For a fixed carrier type:

$$
\sigma_{\tau,k}:
\mathcal W_\tau
\to
\mathcal S_k.
$$

The carrier inverse problem asks for the fibers of these maps.

---

# 4. Inverse Carrier Fiber

Given observed signature:

$$
s_k\in\mathcal S_k,
$$

define:

$$
\boxed{
\mathfrak F_k(s_k)
=
\left\{
W\in\mathcal W:
\sigma_k(W)=s_k
\right\}/\sim,
}
$$

where:

$$
\sim
$$

identifies witnesses that should count as the same carrier for the current task, for example projective equivalence, cycle equivalence, or another declared equivalence.

This fiber is the correct output of carrier inference.

If:

$$
|\mathfrak F_k(s_k)|=1,
$$

the carrier is identifiable at depth $k$.

If:

$$
|\mathfrak F_k(s_k)|>1,
$$

the signature is ambiguous.

If:

$$
\mathfrak F_k(s_k)=\varnothing,
$$

the observed signature does not correspond to any admissible carrier in the current search universe.

---

# 5. Nested Inverse Fibers

Assume level $k+1$ refines level $k$.

Then there is a forgetful map:

$$
\pi_{k+1,k}:
\mathcal S_{k+1}
\to
\mathcal S_k
$$

such that:

$$
\sigma_k
=
\pi_{k+1,k}\circ\sigma_{k+1}.
$$

For compatible signatures:

$$
s_k
=
\pi_{k+1,k}(s_{k+1}),
$$

one obtains:

$$
\boxed{
\mathfrak F_{k+1}(s_{k+1})
\subseteq
\mathfrak F_k(s_k).
}
$$

Therefore genuine additional signature information can only maintain or shrink the inverse fiber.

This is the **Inverse-Fiber Monotonicity Principle**.

---

# 6. Identifiability Depth

Define the carrier identifiability depth:

$$
\boxed{
\iota(W)
=
\inf
\left\{
k:
\mathfrak F_k(\sigma_k(W))
=
\{[W]\}
\right\}.
}
$$

If no finite $k$ uniquely identifies $W$, define:

$$
\iota(W)=\infty.
$$

This creates a carrier hierarchy:

### Immediate carrier

$$
\iota(W)=0.
$$

### First-order identifiable

$$
\iota(W)=1.
$$

### Higher-order identifiable

$$
2\le\iota(W)<\infty.
$$

### Signature-nonidentifiable

$$
\iota(W)=\infty.
$$

A witness can therefore be algebraic and fully valid while still being impossible to reconstruct uniquely from the selected Hodge signature family.

---

# 7. Type Identifiability vs. Instance Identifiability

A weaker question asks only for carrier type.

Define:

$$
\mathfrak T_k(s_k)
=
\left\{
\tau:
\mathfrak F_k(s_k)\cap\mathcal W_\tau
\neq\varnothing
\right\}.
$$

If:

$$
|\mathfrak T_k(s_k)|=1,
$$

the carrier **type** is identified, even if many instances of that type remain.

Thus distinguish:

$$
\boxed{
\text{type identifiability}
}
$$

from:

$$
\boxed{
\text{instance identifiability}.
}
$$

This distinction is essential for AI search.

Often predicting the correct carrier family is enough to make a difficult equation search tractable.

---

# 8. Ambiguity Dimension

When the inverse fiber has geometric structure, define:

$$
\boxed{
A_k(s_k)
=
\dim\mathfrak F_k(s_k).
}
$$

Then, under compatible refinement:

$$
\boxed{
A_{k+1}(s_{k+1})
\le
A_k(s_k).
}
$$

This quantity measures residual carrier ambiguity.

It is preferable to calling the quantity an entropy unless a probability measure is explicitly supplied.

---

# 9. Period-Ideal Envelope

For an algebraic cycle class:

$$
\delta=[Y],
$$

Movasati--Sertöz construct an ideal:

$$
\widetilde I_\delta
$$

from periods.

For an actual carrier:

$$
Y\subset X,
$$

one has:

$$
\boxed{
I(Y)
\subseteq
\widetilde I_\delta.
}
$$

The inclusion can be strict.

Thus:

$$
\widetilde I_\delta
$$

should not automatically be interpreted as the carrier ideal.

Instead define it as a:

$$
\boxed{
\textbf{Period-Ideal Envelope}.
}
$$

It is a signature-derived ideal known to contain geometric information about the carrier.

---

# 10. Reconstructible Regimes

For complete-intersection carriers, low-degree pieces of the period ideal can recover the geometric equations under explicit conditions.

Movasati--Sertöz show that if:

$$
Y
$$

is a complete intersection in the ambient projective space, then:

$$
\widetilde I_\delta
$$

and:

$$
I(Y)
$$

coincide through an explicit low-degree range.

Their method also reconstructs examples beyond complete intersections, including twisted cubics on quartic surfaces.

Cifani--Pirola--Schlesinger obtain reconstruction results for rational and arithmetically Cohen--Macaulay curves under numerical conditions.

Therefore there exist nontrivial subdomains:

$$
\mathcal W_{\mathrm{rec}}
\subseteq
\mathcal W
$$

where:

$$
\boxed{
\text{Hodge / period data}
\to
\text{carrier equations}
}
$$

is genuinely computable.

---

# 11. Reconstruction Is a Partial Inverse

The reconstruction operator should therefore be typed as a partial map:

$$
\boxed{
\operatorname{Rec}_k:
\mathcal S_k
\rightharpoonup
\mathcal W.
}
$$

It is defined only when sufficient reconstructibility conditions hold.

A valid algorithm should never silently upgrade:

$$
\widetilde I_\delta
$$

into:

$$
I(Y).
$$

The correct logic is:

$$
\boxed{
\widetilde I_\delta
\to
\text{candidate equations}
\to
\text{carrier realizability check}
\to
\text{symbolic verification}.
}
$$

---

# 12. Geometric Purity

A second property is required besides identifiability.

A signature-derived ideal may contain algebraic information not generated by the ideal of the intended carrier.

Movasati--Sertöz define perfection conditions for Hodge classes; Cifani--Pirola--Schlesinger show that perfection can fail.

For RCHM purposes, introduce the broader operational term:

$$
\boxed{
\textbf{Geometric Purity}
}
$$

for the condition that the portion of the signature used for carrier inference is actually generated by legitimate geometric witness data of the declared type.

This is not intended to replace the precise literature definition of a perfect class.

Rather:

$$
\text{perfectness}
$$

is one mathematically precise realization of geometric purity in the relevant period-ideal setting.

---

# 13. Identifiability and Purity Are Independent

Two signatures may be highly identifying but geometrically impure.

Conversely, a geometrically pure signature may still correspond to many carriers.

Therefore carrier inference requires two axes:

$$
\boxed{
(\text{identifiability},\text{geometric purity}).
}
$$

This gives four logical regimes.

### Regime I — identifiable and pure

Carrier reconstruction is reliable.

### Regime II — identifiable but impure

The signature strongly suggests one carrier, but part of the signal is not actually carrier-generated.

This is dangerous.

### Regime III — ambiguous but pure

The signal is geometric, but multiple carriers produce it.

Additional invariants are required.

### Regime IV — ambiguous and impure

The signature is inadequate for carrier inference.

---

# 14. Rational Quartic Warning

Let:

$$
C
$$

be a smooth rational quartic curve in a smooth quartic surface.

Cifani--Pirola--Schlesinger show that:

$$
I_{\alpha(C),2}
=
I_{C,2},
$$

so degree-$2$ data correctly recovers the quadric information of $C$.

However the class is not perfect at the next relevant level.

Thus:

$$
\boxed{
\text{partial low-degree reconstruction}
\not\Rightarrow
\text{global geometric purity of the signature ideal}.
}
$$

This is a direct warning against treating deeper truncations of an annihilator ideal as automatically more geometric.

The raw signature contains more information, but its naive carrier interpretation can become invalid.

---

# 15. Fake Linear Cycle Warning

Fake linear cycles on suitable Fermat hypersurfaces provide another failure mode.

Their associated Artinian Gorenstein ideals have the same Hilbert-function behavior as those of genuine linear cycles, and their Hodge loci have maximal first-order tangent dimension.

Yet they are not genuine linear cycles.

Second-order IVHS data detects additional structure, including nonreduced behavior.

Thus:

$$
\boxed{
\text{carrier template match at level }1
\not\Rightarrow
\text{carrier realization}.
}
$$

This is exactly an inverse-fiber ambiguity.

---

# 16. Carrier Aliasing

Define two witnesses:

$$
W_1,
W_2
$$

to be order-$k$ carrier aliases if:

$$
\boxed{
\sigma_k(W_1)=\sigma_k(W_2)
}
$$

but:

$$
[W_1]\neq[W_2].
$$

If their types differ:

$$
\tau(W_1)\neq\tau(W_2),
$$

they are **cross-type aliases**.

Fake linear cycle phenomena motivate this distinction.

---

# 17. Alias-Breaking Depth

For two aliased carriers define:

$$
\boxed{
b(W_1,W_2)
=
\inf
\left\{
k:
\sigma_k(W_1)\neq\sigma_k(W_2)
\right\}.
}
$$

If no selected signature level separates them:

$$
b(W_1,W_2)=\infty.
$$

This quantity tells AMRAL how deep a comparison must go before a carrier hypothesis can be ruled out.

---

# 18. Carrier Template Map

For each carrier type:

$$
\tau,
$$

define its depth-$k$ signature image:

$$
\boxed{
\mathcal I_{\tau,k}
=
\operatorname{Im}(\sigma_{\tau,k})
\subseteq
\mathcal S_k.
}
$$

Given an observed signature:

$$
s_k,
$$

the candidate carrier-type set is:

$$
\boxed{
\mathfrak T_k(s_k)
=
\left\{
\tau:
s_k\in\mathcal I_{\tau,k}
\right\}.
}
$$

In practice, one may use closures:

$$
s_k\in\overline{\mathcal I_{\tau,k}}
$$

when studying degenerations.

This yields a typed carrier-classification problem rather than an unrestricted equation search.

---

# 19. Realizability Filter

A template match is not enough.

Define the realizability fiber:

$$
\boxed{
\mathfrak R_{\tau,k}(s_k)
=
\left\{
W\in\mathcal W_\tau:
\sigma_{\tau,k}(W)=s_k
\right\}.
}
$$

A carrier type is only valid if:

$$
\mathfrak R_{\tau,k}(s_k)\neq\varnothing.
$$

Therefore:

$$
\boxed{
\text{template compatibility}
\not\Rightarrow
\text{geometric realizability}.
}
$$

Fake linear signatures illustrate why this filter is necessary.

---

# 20. Carrier Reconstruction Ladder

The inverse problem should be solved progressively.

## Level 0 — Class

Input:

$$
\alpha.
$$

Question:

$$
\text{which carrier classes are even compatible?}
$$

---

## Level 1 — Period / Artinian Ideal

Compute:

$$
I(\alpha)
$$

or:

$$
\widetilde I_\alpha.
$$

Use:

- Hilbert function;
- generator degrees;
- annihilator structure.

---

## Level 2 — Tangent Signature

Use:

$$
T\operatorname{NL}(\alpha).
$$

Compare with carrier deformation spaces.

---

## Level 3 — Relation Signature

Use:

- constraint matroid;
- Plücker support;
- circuit coefficients;
- coefficient field.

---

## Level 4 — Second and Higher Jets

Use:

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

to break first-order aliases.

---

## Level 5 — Geometric Realizability

Solve candidate ideal / Hilbert / Chow equations.

Check:

$$
W\subset X.
$$

---

## Level 6 — Exact Verification

Verify:

$$
cl(W)=\alpha
$$

and all required arithmetic, scheme-theoretic and global conditions.

Only after this level should the carrier be treated as a verified witness.

---

# 21. Candidate-Set Filtration

Let:

$$
\mathcal C_0
$$

be an initial carrier candidate set.

Define recursively:

$$
\mathcal C_{k+1}
=
\left\{
W\in\mathcal C_k:
\sigma_{k+1}(W)
=
s_{k+1}
\right\}.
$$

Then:

$$
\boxed{
\mathcal C_0
\supseteq
\mathcal C_1
\supseteq
\mathcal C_2
\supseteq
\cdots.
}
$$

A practical AI system may stop when:

$$
|\mathcal C_k|=1,
$$

or when all remaining candidates lie in one carrier type, or when further invariants become more expensive than a direct geometric solve.

---

# 22. Cost-Aware Inverse Search

Each signature level has a computational cost:

$$
c_k.
$$

Each level also has expected ambiguity reduction:

$$
\Delta A_k.
$$

Define an information-per-cost heuristic:

$$
\boxed{
\eta_k
=
\frac{\Delta A_k}{c_k}.
}
$$

This is an engineering metric, not a mathematical invariant.

It lets AMRAL choose whether to compute:

- another Hodge jet;
- a period ideal truncation;
- a Hilbert-scheme solve;
- an arithmetic field test;
- a direct symbolic carrier verification.

---

# 23. Witness-Carrier Transmutation Revisited

Rounds 2–3 showed that a relation may change carrier type under deformation.

Therefore the inverse problem should not always output a fixed carrier type.

Let:

$$
\mathcal T
$$

be the set of carrier types.

Define a carrier-transition graph:

$$
\boxed{
\mathcal G_{\mathrm{car}}
=
(\mathcal T,E_{\mathrm{car}}).
}
$$

Add an edge:

$$
\tau_1\to\tau_2
$$

when there exists a validated flat family, degeneration, bridge construction or other admissible handoff preserving the relevant class/invariant.

Examples motivating such edges include:

$$
\text{two-plane relation}
\to
\text{complete intersection},
$$

and:

$$
\text{sum of linear cycles}
\to
\text{Grassmannian-type carrier}.
$$

---

# 24. Signature-Image Intersections

If:

$$
\overline{\mathcal I_{\tau_1,k}}
\cap
\overline{\mathcal I_{\tau_2,k}}
\neq\varnothing,
$$

then the two carrier types can become indistinguishable at depth $k$ near a common signature regime.

This does not prove a geometric transition.

But it identifies a **transition candidate locus**.

If a flat family is additionally constructed, then the candidate edge may be upgraded to a validated carrier-transition edge.

---

# 25. Carrier-Transition Prediction

Given a persistent circuit signature:

$$
s,
$$

do not ask only:

$$
\operatorname{Carrier}(s)=?
$$

Instead compute:

$$
\boxed{
\operatorname{CarrierNeighborhood}(s)
=
\left\{
\tau:
s\in
\overline{\mathcal I_{\tau,k}}
\right\}.
}
$$

Then inspect:

$$
\mathcal G_{\mathrm{car}}
$$

for validated transitions among these types.

This yields a safer prediction:

$$
\boxed{
\text{relation signature}
\to
\text{candidate carrier neighborhood}
}
$$

rather than a premature single-label classification.

---

# 26. Relation Signature Plus Carrier Signature

Round 5 produced a relation descriptor:

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

Round 6 augments it with inverse-carrier information:

$$
\boxed{
\mathcal R^+
=
(
\mathcal R,
\mathfrak T_k,
A_k,
\text{purity status},
\iota,
\text{realizability status},
\text{carrier-transition refs}
).
}
$$

This is close to the X-layer representation required for AMRAL.

---

# 27. A Proposed Carrier Confidence Lattice

Do not compress carrier status into one scalar probability.

Use a partial status lattice.

### C0 — Unclassified

No carrier family assigned.

### C1 — Template-compatible

One or more carrier templates match.

### C2 — Type-isolated

Only one carrier type remains.

### C3 — Geometrically realizable

An actual carrier of the type exists.

### C4 — Cohomologically verified

The carrier represents the target class.

### C5 — Formally/deformation verified

The required handoff or persistence conditions are verified.

### C6 — Global witness verified

All required global conditions are closed.

This preserves the distinction between pattern recognition and proof.

---

# 28. Application to Persistent Circuits

Suppose Round 5 finds a relation:

$$
a
$$

with large:

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

This makes it a strong candidate for geometric interpretation.

But high persistence alone does not determine the carrier.

The next pipeline is:

$$
\boxed{
a
\to
\alpha_a
\to
I(\alpha_a)
\to
\mathfrak T_k
\to
\mathfrak R_{\tau,k}
\to
W
\to
cl(W)=\alpha_a.
}
$$

If:

$$
\mathfrak T_k
$$

contains several types, compute alias-breaking signatures.

If:

$$
\mathfrak R_{\tau,k}=\varnothing,
$$

discard the false carrier template.

If a carrier exists but:

$$
cl(W)\neq\alpha_a,
$$

discard it as a geometrically relevant but incorrect witness.

---

# 29. Hodge-Specific Research Program

For an even-dimensional hypersurface and a persistent rational circuit class:

$$
\alpha
=
\sum_i a_i\gamma_i,
\qquad
a_i\in\mathbb Q,
$$

perform:

### Step 1

Construct the associated Hodge/Artinian ideal:

$$
I(\alpha).
$$

### Step 2

Extract low-degree generator pattern and Hilbert function.

### Step 3

Compare with carrier template library.

### Step 4

Check perfectness / geometric-purity conditions when available.

### Step 5

Compute second/higher IVHS signatures if aliases remain.

### Step 6

Solve candidate Hilbert/Chow realization.

### Step 7

Verify containment:

$$
W\subset X.
$$

### Step 8

Verify class:

$$
cl(W)=\alpha.
$$

### Step 9

Add validated type and carrier to:

$$
\mathcal G_{\mathrm{car}}.
$$

---

# 30. What the Literature Already Demonstrates

The current literature provides at least four calibration regimes.

## 30.1 Recoverable complete intersections

Low-degree carrier equations can be recovered from period-derived ideals under explicit hypotheses.

This calibrates:

$$
\text{identifiable + geometrically controlled}.
$$

---

## 30.2 Recoverable non-complete-intersection examples

Twisted cubics on quartic surfaces can be reconstructed in suitable regimes.

Thus reconstructibility is not restricted to complete intersections.

---

## 30.3 Rational quartic impurity

Low-degree data may partially match the geometric ideal while higher-degree perfectness fails.

This calibrates:

$$
\text{partial identification without full geometric purity}.
$$

---

## 30.4 Fake linear cycles

First-order / Artinian signatures can mimic linear-cycle signatures while the genuine linear carrier is absent.

This calibrates:

$$
\text{carrier aliasing + realizability failure + need for higher-order signatures}.
$$

---

# 31. Main New Principles

## Principle A — Carrier Inverse-Fiber Principle

The output of Hodge-to-carrier inference is an inverse fiber:

$$
\boxed{
\mathfrak F_k(s_k),
}
$$

not necessarily one carrier.

---

## Principle B — Identifiability Depth Principle

A carrier is identified only when sufficient signature depth collapses the inverse fiber to one equivalence class.

$$
\boxed{
\iota(W)
=
\inf
\{k:\mathfrak F_k=\{[W]\}\}.
}
$$

---

## Principle C — Geometric Purity Principle

A signature-derived ideal or invariant may contain non-carrier information.

$$
\boxed{
\text{more signature data}
\not\Rightarrow
\text{naively more accurate carrier ideal}.
}
$$

Geometric-origin conditions must be checked.

---

## Principle D — Realizability Principle

Template match is only a hypothesis.

$$
\boxed{
\text{template match}
\to
\text{realizability solve}
\to
\text{class verification}.
}
$$

---

## Principle E — Alias-Breaking Principle

If two carrier hypotheses share low-order signatures, increase signature depth until they separate or are proven indistinguishable under the chosen invariant family.

---

## Principle F — Transition-Neighborhood Principle

When witness-carrier transmutation is possible, infer a carrier neighborhood and transition graph rather than forcing one static carrier label.

---

# 32. AMRAL Representation

Suggested node:

```text
carrier_hypothesis:
  hypothesis_id
  target_relation_ref
  carrier_type
  signature_depth
  template_match
  realizability_status
  geometric_purity_status
  inverse_fiber_dimension
  identifiability_depth
  alias_refs[]
  transition_refs[]
  witness_ref
  verification_status
```

Suggested signature object:

```text
carrier_signature:
  class_ref
  period_ideal_ref
  hilbert_function
  tangent_signature_ref
  matroid_signature_ref
  plucker_signature_ref
  valuation_signature_ref
  jet_signature_refs[]
  coefficient_field
```

Suggested transition edge:

```text
carrier_transition:
  source_type
  target_type
  invariant_preserved
  family_ref
  transition_kind
  proof_status
```

---

# 33. Consequence for X-Layer Mathematical Completeness

A final theorem statement such as:

$$
\alpha
\text{ is algebraic}
$$

can hide all of the following:

- how the carrier was inferred;
- whether its type was unique;
- whether period data were perfect;
- which aliases were rejected;
- which jet order separated them;
- whether carrier type changed under deformation;
- whether the witness was directly constructed or reached by bridge object.

Therefore a graph-native mathematical record should preserve:

$$
\boxed{
\text{class}
+
\text{signature hierarchy}
+
\text{inverse fiber}
+
\text{purity}
+
\text{realizability}
+
\text{carrier transition}
+
\text{verification}.
}
$$

This is a concrete example of why theorem-only storage is insufficient for AI-native mathematics.

---

# 34. Round 6 Conclusion

Round 6 answers the question:

> Can persistent circuits predict witness carriers?

The answer is:

$$
\boxed{
\text{sometimes, but only as a typed inverse problem with ambiguity and realizability controls}.
}
$$

The correct architecture is:

$$
\boxed{
\text{persistent relation}
\to
\text{Hodge signature}
\to
\text{inverse carrier fiber}
\to
\text{geometric-purity test}
\to
\text{higher-order alias breaking}
\to
\text{realizability}
\to
\text{verified witness}.
}
$$

The principal new concept is:

$$
\boxed{
\textbf{Carrier Identifiability Depth}.
}
$$

The principal warning is:

$$
\boxed{
\text{signature resemblance is not carrier existence}.
}
$$

And the principal algorithmic lesson is:

$$
\boxed{
\text{do not ask an AI to guess one carrier prematurely;}
\text{maintain and shrink a typed inverse fiber}.
}
$$

This is the main closure of Round 6.

---

# References

1. Hossein Movasati and Emre Can Sertöz, **On reconstructing subvarieties from their periods**, Rendiconti del Circolo Matematico di Palermo Series 2, 70 (2021), 1441--1457.  
   arXiv: https://arxiv.org/abs/1908.03221  
   DOI: https://doi.org/10.1007/s12215-020-00562-x

2. Maria Gioia Cifani, Gian Pietro Pirola, and Enrico Schlesinger, **Reconstructing curves from their Hodge classes**, Rendiconti del Circolo Matematico di Palermo Series 2, 72 (2023), 945--958.  
   arXiv: https://arxiv.org/abs/2104.05576  
   DOI: https://doi.org/10.1007/s12215-021-00704-9

3. Jorge Duque Franco and Roberto Villaflor Loyola, **On fake linear cycles inside Fermat varieties**, Algebra & Number Theory 17 (2023), 1847--1865.  
   arXiv: https://arxiv.org/abs/2112.14818  
   DOI: https://doi.org/10.2140/ant.2023.17.1847

4. 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).  
   DOI: https://doi.org/10.1007/s12215-025-01307-4

5. Remke Kloosterman, **Variational Hodge conjecture for complete intersections on hypersurfaces in projective space**, Rendiconti del Seminario Matematico della Università di Padova 148 (2022), 185--201.  
   arXiv: https://arxiv.org/abs/2104.14845  
   DOI: https://doi.org/10.4171/RSMUP/110