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

# TCDC-00
## Target–Condition Dual Closure Methodology
### Target–Condition Dual Closure Methodology: First Reverse-Solve the Conditions for Target Validity, Then Prove Existence, Branches, Basins, and Structural Closure

**Research line:** Extended RG-SCP / SRBRD methodology  
**Method name:** TCDC — Target–Condition Dual Closure  
**Chinese name:** Target–Condition Dual Closure  
**Version:** v0.1  
**Date:** 2026-10-02  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Nature:** Methodology / Reverse Construction / Closure Proof Framework  
**Status:** Methodology formalized; no claim that any unsolved mathematical problem has thereby been proved

---

# 0. Core Proposition

TCDC starts by completely separating two things that are often conflated:

$$
\boxed{\text{Fixed-point compatibility}\neq\text{Reachability}}
$$

and:

$$
\boxed{\text{Target characterization}\neq\text{Global closure}.}
$$

If an infinite or recursive structure satisfies:

$$
T=F(T),
$$

at most, this initially tells us:

$$
\boxed{T\text{ is a compatible fixed-point candidate for }F\text{.}}
$$

It does not automatically tell us:

$$
\boxed{F^n(x_0)\to T.}
$$

Likewise, if a number-theoretic target class has an exact local criterion:

$$
x\in\mathcal T
\iff
C(x),
$$

one still cannot directly infer:

$$
\boxed{\text{Some global process necessarily enters }\mathcal T.}
$$

The TCDC method is:

1. **First specify the target;**
2. **Reverse-solve the target's compatibility conditions;**
3. **Separately establish the domain / branch / existence;**
4. **Establish a basin or finite-descent / capacity closure;**
5. **Only then elevate compatibility to a theorem.**

---

# 1. Prototype: An Infinite Radical Is Not Settled by Setting It Equal to x

The classical form:

$$
\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}.
$$

A common approach is to set:

$$
x=\sqrt{2+x},
$$

we obtain:

$$
x^2-x-2=0,
$$

Therefore:

$$
(x-2)(x+1)=0.
$$

The principal square root requires:

$$
x\ge0,
$$

thus leaving:

$$
x=2.
$$

However, strictly speaking, this derivation proves only:

$$
\boxed{
\text{If this infinite structure actually has a finite real limit, that limit can only be }2.
}
$$

It does not separately prove:

- that the infinite structure exists;
- that the truncation sequence converges;
- that the selected branch is consistent;
- that the orbit does not escape;
- that there are no cycles or other limiting behaviors.

The complete approach should first define:

$$
x_{n+1}=\sqrt{2+x_n}.
$$

then establish an invariant interval, for example:

$$
0\le x_n\le2.
$$

and prove:

$$
x_{n+1}\ge x_n,
$$

and:

$$
x_{n+1}\le2.
$$

Thus:

$$
\{x_n\}
$$

is monotone and bounded, so its limit exists.

Only then can one set:

$$
L=\lim_{n\to\infty}x_n,
$$

and use continuity:

$$
L=\sqrt{2+L},
$$

to obtain finally:

$$
L=2.
$$

Thus, the complete structure is:

$$
\boxed{
\text{Definition}
\to
\text{Invariant Domain}
\to
\text{Existence}
\to
\text{Convergence}
\to
\text{Fixed-Point Equation}
\to
\text{Unique Legal Branch}.
}
$$

---

# 2. Reverse Problem: Specify the Endpoint, Then Reverse-Solve the Conditions

Now reverse the direction.

Consider:

$$
x_{n+1}=\sqrt{a+x_n}.
$$

Instead of first asking:

> Given $a$, what is the final value?

ask:

> If the target $L$ is prescribed, which $a$ makes $L$ a fixed point?

Require:

$$
L=\sqrt{a+L}.
$$

Thus:

$$
L^2=a+L,
$$

That is:

$$
\boxed{a=L^2-L=L(L-1).}
$$

Thus, the target $L$ corresponds to a canonical compatibility parameter:

$$
\boxed{\mathfrak C(L)=L(L-1).}
$$

For example:

$$
L=2
\Rightarrow
a=2,
$$

$$
L=3
\Rightarrow
a=6.
$$

Thus:

$$
3=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}
$$

holds under the correct convergence conditions.

This is the first core action of TCDC:

$$
\boxed{
\text{Target}
\longrightarrow
\text{Reverse Compatibility Construction}.
}
$$

However, this step is still **not a complete proof**.

---

# 3. The Five-Layer Structure of TCDC

Formally divide the method into five layers.

## Layer T — Target

Specify:

$$
\boxed{T.}
$$

The target may be:

- a number;
- a fixed point;
- a terminal set;
- a prime class;
- a zero-obstruction state;
- a nonempty-residual state;
- a failure-free state;
- an invariant manifold;
- a canonical normal form.

---

## Layer C — Compatibility

Reverse-solve the conditions needed by the target:

$$
\boxed{C(T).}
$$

General form:

$$
F(T;\theta)=T,
$$

or:

$$
\mathcal O(T;\theta)=0.
$$

where:

$$
\theta
$$

is a carrier parameter / boundary parameter / cutoff / operator parameter.

---

## Layer D — Domain

Find a legal domain:

$$
\boxed{\mathcal D_T}
$$

such that:

$$
\boxed{F(\mathcal D_T)\subseteq\mathcal D_T.}
$$

This layer handles:

- the domain of definition;
- branches;
- positivity;
- root validity;
- nonzero denominators;
- the validity of finite / infinite objects.

---

## Layer B — Basin

Define the target's basin:

$$
\boxed{
\mathcal B(T)
=
\{x_0:F^n(x_0)\to T\}.
}
$$

For discrete problems, one may instead write:

$$
\boxed{
\mathcal B(T)
=
\{x_0:\exists n,\ F^n(x_0)\in\mathcal T\}.
}
$$

The basin mechanism may be:

- contraction;
- monotonicity and boundedness;
- finite descent;
- Lyapunov descent;
- decreasing obstructions;
- transport deficiency;
- capacity contradiction;
- non-cover;
- compactness + uniqueness.

---

## Layer K — Closure

Only at the end prove:

$$
\boxed{
x_0\in\mathcal B(T)
\Longrightarrow
\operatorname{Orb}(x_0)\subseteq\mathcal D_T
\Longrightarrow
F^n(x_0)\to T.
}
$$

This is the complete closure.

---

# 4. Basic TCDC Scheme

TCDC does not accept:

$$
F(T)=T
\Rightarrow
\forall x_0,
F^n(x_0)\to T.
$$

but requires:

$$
\boxed{
T
\overset{\text{reverse}}{\longrightarrow}
C(T)
}
$$

then:

$$
\boxed{
C(T)
\longrightarrow
\mathcal D_T
\longrightarrow
\mathcal B(T)
\longrightarrow
T.
}
$$

Thus, the entire method is:

$$
\boxed{
\text{Reverse Construction}
+
\text{Forward Closure}.
}
$$

---

# 5. Target–Condition Duality

Define the condition constructor:

$$
\boxed{
\mathfrak C:T\mapsto C(T).
}
$$

Conversely, define the target resolver:

$$
\boxed{
\mathfrak T:C\mapsto
\{T:T\text{ satisfies }C\}.
}
$$

Ideally:

$$
\boxed{
\mathfrak T(\mathfrak C(T))=\{T\}.
}
$$

This means that the compatibility condition suffices to identify the target uniquely.

If:

$$
|\mathfrak T(\mathfrak C(T))|>1,
$$

then one needs to add:

- branch;
- sign;
- order;
- domain;
- stability;
- topology;
- parity;
- integrality;

to remove ambiguity.

---

# 6. Branch Firewall

Any proof that has performed any of the following operations:

- squaring;
- removing radicals;
- division;
- logarithms;
- inverse mapping;
- analytic continuation;

must retain branch metadata.

For example:

$$
L^2=L+2
$$

we have:

$$
L=2,-1.
$$

However:

$$
L=\sqrt{2+L}
$$

forces:

$$
L\ge0.
$$

Thus:

$$
L=-1
$$

is an algebraic solution, but not an operator-compatible solution.

Therefore:

$$
\boxed{
\text{Algebraic Root}
\neq
\text{Legal Branch Target}.
}
$$

---

# 7. Infinite-Object Firewall

Any infinite expansion:

$$
X_\infty
$$

must not be treated directly as an already existing object.

One must first supply:

- finite approximants;
- a sequence / net;
- a topology;
- a notion of convergence;
- a completion space.

For example:

$$
\sqrt{2+\sqrt{2+\cdots}}
$$

should first be defined by:

$$
x_{n+1}=\sqrt{2+x_n}.
$$

Therefore:

$$
\boxed{
\text{Formal Infinite Expression}
\neq
\text{Existing Limit Object}.
}
$$

---

# 8. A Unique Fixed Point Still Does Not Imply a Global Basin

Even if:

$$
F(T)=T
$$

and $T$ is the only fixed point in the domain, it may still be that:

- the orbit diverges;
- the orbit has cycles;
- the orbit hits the domain boundary;
- the iteration is undefined;
- a chaotic set exists.

Thus:

$$
\boxed{
\text{Unique Fixed Point}
\neq
\text{Global Attractor}.
}
$$

If:

$$
|F'(T)|<1,
$$

one initially obtains only local attraction.

Global closure still requires conditions such as an invariant domain / trapping / monotonicity.

---

# 9. Three Principal TCDC Closure Types

## Type I — Algebraic Closure

Using:

- compatibility;
- branch;
- uniqueness;

to identify the target directly.

---

## Type II — Dynamic / Descent Closure

There is an operator:

$$
F
$$

and a potential:

$$
V.
$$

If:

$$
V(x)=0
\iff
x\in\mathcal T,
$$

and:

$$
V(F(x))<V(x)
$$

for:

$$
x\notin\mathcal T,
$$

then the target is reachable in finitely many steps.

---

## Type III — Capacity / Failure Non-Closure

Suppose failure must satisfy:

$$
A=B,
$$

but structural bounds also give:

$$
A\le U,
$$

$$
B\ge L,
$$

and:

$$
U<L.
$$

then the failure surface has no solution.

This is the TCDC type to which SRBRD is currently closest.

---

# 10. Finite-Descent TCDC Template

## Theorem Template TCDC-T01

If:

$$
V:\mathcal X\to\mathbb N_0
$$

satisfying:

$$
V(x)=0
\iff
x\in\mathcal T,
$$

and:

$$
x\notin\mathcal T
\Longrightarrow
V(F(x))<V(x),
$$

then:

$$
\boxed{
\forall x\in\mathcal X,
\exists n:
F^n(x)\in\mathcal T.
}
$$

This type of closure needs no analytic limit.

It is therefore particularly suited to:

- factor depth;
- obstruction count;
- cofactor descent;
- finite cover;
- canonical reduction.

---

# 11. Continuous TCDC Template

If:

1. $(\mathcal D,d)$ is complete;
2. $F:\mathcal D\to\mathcal D$;
3. there exists:
   $$
   0<c<1
   $$
   such that:
   $$
   d(Fx,Fy)\le c\,d(x,y);
   $$

then $F$ has a unique fixed point $T$, and:

$$
\boxed{
F^n(x)\to T
}
$$

for every:

$$
x\in\mathcal D.
$$

This is a Banach fixed-point type of TCDC.

---

# 12. Complete TCDC Contract for the Root-2 Example

```yaml
Target:
  T: 2

ReverseCompatibility:
  operator:
    F_a(x): sqrt(a+x)
  solve:
    a: T^2-T
  result:
    a: 2

Domain:
  D:
    [0,2]

Branch:
  principal_square_root: true

Invariant:
  F(D) subset D: true

BasinMechanism:
  monotone_increasing: true
  upper_bounded_by_2: true

Existence:
  limit_exists: true

Resolver:
  L_eq_sqrt_2_plus_L: true
  legal_solution:
    2

Closure:
  PASS
```

This is the simplest TCDC model.

---

# 13. From a Numerical Target to a Target Set

In number theory, the target is often not a single number.

Define:

$$
\boxed{\mathcal T\subseteq\mathcal X.}
$$

The terminal condition may be written:

$$
\boxed{
F(\mathcal T)\subseteq\mathcal T.
}
$$

or in obstruction form:

$$
\boxed{
\mathcal O(x)=0
\iff
x\in\mathcal T.
}
$$

The target set may be:

- primes;
- square-free integers;
- residual states;
- zero-obstruction cells;
- prime-pair states.

---

# 14. Direct Interface with the Prime-Square Domain

SRBRD has already used the root-bounded divisor domain:

$$
\boxed{
\mathcal D(x)
=
\{r\in\mathbb P:r^2\le x,\ r\mid x\}.
}
$$

For $x\ge2$:

$$
\boxed{
x\in\mathbb P
\iff
\mathcal D(x)=\varnothing.
}
$$

This is an exact characterization.

TCDC interprets it as:

### Target

$$
\boxed{\mathcal T_{\mathbb P}=\mathbb P.}
$$

### Compatibility

$$
\boxed{\mathcal D(x)=\varnothing.}
$$

However, the actual closure problem is:

> Which proof operator / relation / global capacity can guarantee that a candidate state necessarily enters:
> $$
> \mathcal D(x)=\varnothing?
> $$

Therefore:

$$
\boxed{
\text{Prime Characterization}
\neq
\text{Prime Reachability}.
}
$$

---

# 15. Prime Terminal Domain

Define:

$$
\boxed{
\mathfrak P
=
\{x:\mathcal D(x)=\varnothing\}.
}
$$

Thus:

$$
\mathfrak P=\mathbb P.
$$

One can now seek a proof operator:

$$
\boxed{F:\mathcal X\to\mathcal X}
$$

such that:

$$
F(\mathfrak P)\subseteq\mathfrak P,
$$

and such that a candidate failure state cannot remain forever in:

$$
\mathcal X\setminus\mathfrak P.
$$

This is:

# Prime Terminal Closure

---

# 16. Composite Obstruction Domain

Define:

$$
\boxed{
\mathfrak C
=
\{x:\mathcal D(x)\neq\varnothing\}.
}
$$

The traditional direction:

> Find $r\in\mathcal D(x)$ to prove compositeness.

The reverse direction of TCDC:

> Can the composite obstruction domain form a closed state under some global relation?

If one proves:

$$
\boxed{
F(\mathfrak C)\not\subseteq\mathfrak C
}
$$

or even that all relevant trajectories leave $\mathfrak C$, one obtains failure non-closure.

---

# 17. The Prime-Square Boundary as a Compatibility Horizon

The condition:

$$
r^2\le x
$$

is not an arbitrary cutoff.

By the composite-certification theorem:

$$
\boxed{
x\text{ composite}
\Longrightarrow
\exists r\in\mathbb P:
r^2\le x,
\ r\mid x.
}
$$

Thus:

$$
\sqrt{x}
$$

is the canonical horizon for a composite witness.

In TCDC language:

$$
\boxed{
\text{Target Prime Class}
\leftrightarrow
\text{Square-Boundary Zero-Obstruction Condition}.
}
$$

---

# 18. Target Depth Duality

More generally, if $x\le X$ and the desired target factor depth is:

$$
\Omega_{\mathrm{mult}}(x)\le h,
$$

then the natural reverse cutoff is:

$$
\boxed{
z^{h+1}\ge X.
}
$$

That is:

$$
\boxed{
z\ge X^{1/(h+1)}.
}
$$

This is the number-theoretic version of:

$$
L\mapsto a=L^2-L.
$$

That is:

$$
\boxed{
\text{Specify the target depth}
\longrightarrow
\text{Reverse-solve the canonical boundary parameter}.
}
$$

---

# 19. Goldbach Pair Target

For:

$$
2m,
$$

Use the SRBRD offset:

$$
(m-d,m+d)
$$

to represent the pair.

Goldbach target:

$$
\boxed{
\exists d:
\mathcal D(m-d)=\varnothing,
\quad
\mathcal D(m+d)=\varnothing.
}
$$

Define:

$$
\boxed{
\operatorname{Res}(m)
=
\{d:\mathcal D(m-d)=\mathcal D(m+d)=\varnothing\}.
}
$$

then:

$$
\boxed{
R_G^-(2m)=|\operatorname{Res}(m)|.
}
$$

---

# 20. Goldbach Failure Domain

Failure:

$$
\operatorname{Res}(m)=\varnothing.
$$

That is:

$$
\boxed{
\forall d,
\quad
\mathcal D(m-d)\neq\varnothing
\ \lor\
\mathcal D(m+d)\neq\varnothing.
}
$$

This is a:

# Full-Cover Failure State

TCDC asks not only:

> How can one find a residual directly?

but:

> Can this full-cover failure state simultaneously satisfy all global constraints and form a closed admissible state?

---

# 21. Failure Fixed Surface

SRBRD-14 already has the exact transport identity:

$$
\boxed{
M_{PP}=n_P-n_{SP}+M_{SS}.
}
$$

Goldbach failure:

$$
M_{PP}=0.
$$

Hence, the failure compatibility surface:

$$
\boxed{
M_{SS}=n_{SP}-n_P.
}
$$

This is precisely a TCDC object.

It is not a proof of failure.

It is:

$$
\boxed{
\text{Failure Compatibility Surface}.
}
$$

The next question:

> Can this equality surface simultaneously satisfy:
> - the root-bound;
> - row capacity;
> - the center-divisor lock;
> - Buchstab loss;
> - complement switching;
> - the constant ledger;
> - transport constraints?

If not:

$$
\boxed{
\text{Failure Surface Non-Closure}
\Longrightarrow
\text{Target Necessity}.
}
$$

---

# 22. Relationship between TCDC and RG-SCP

RG-SCP was originally:

$$
\boxed{
\neg G
\Rightarrow
C_1
\Rightarrow
C_2
\Rightarrow
\cdots.
}
$$

TCDC reinterprets it as:

$$
\boxed{
\text{Assume Failure}
\to
\text{Reverse Failure Compatibility}
\to
\text{Failure Domain}
\to
\text{Closure Audit}.
}
$$

Thus:

$$
\boxed{
\text{TCDC is a methodology above RG-SCP.}
}
$$

---

# 23. Target–Failure Duality

Define:

$$
\boxed{\mathcal X=\mathcal T\sqcup\mathcal F.}
$$

where:

- $\mathcal T$ = target domain;
- $\mathcal F$ = failure domain.

Proving the target does not necessarily require directly proving:

$$
\mathcal T\neq\varnothing.
$$

One may also prove:

$$
\boxed{
\mathcal F\text{ cannot be closed}.
}
$$

This is the:

# Failure Non-Closure Strategy

---

# 24. Proof Operator

Number theory need not have natural dynamics.

TCDC allows a proof operator to be **designed**:

$$
\boxed{F:\mathcal X\to\mathcal X}
$$

provided it has valid truth preservation, for example:

$$
\boxed{
\neg P(x)
\Longrightarrow
\neg P(F(x)),
}
$$

or:

$$
\boxed{
P(x)\iff P(F(x)).
}
$$

Possible proof operators:

- canonical least-factor division;
- cofactor reduction;
- root-bound projection;
- obstruction compression;
- transport collapse;
- Buchstab cutoff descent;
- residue quotient;
- canonicalization.

---

# 25. Reverse Operator Synthesis

TCDC can design a proof operator in reverse:

> If the target $\mathcal T$ is to be a terminal set, which $F$ can:
> 1. preserve truth;
> 2. make $\mathcal T$ invariant;
> 3. decrease the failure potential?

Form:

$$
\boxed{
\text{Target}
\longrightarrow
\text{Operator Synthesis}
\longrightarrow
\text{Closure Certificate}.
}
$$

This may be particularly suited to autonomous AI mathematics.

---

# 26. Potential Function

Define:

$$
\boxed{V:\mathcal X\to\mathbb R_{\ge0}.}
$$

One wants:

$$
V(x)=0
\iff
x\in\mathcal T.
$$

If:

$$
V(F(x))<V(x)
$$

for every failure state, then target closure holds.

Candidate potentials in number theory may be:

- root obstruction count;
- cofactor depth;
- least-factor depth;
- uncovered residual deficit;
- transport deficiency;
- cover saturation;
- Buchstab depth.

---

# 27. Condition-Space Dynamics

Rather than moving integers directly, TCDC is better suited to moving a condition vector.

Define:

$$
\boxed{
\mathcal C(x)
=
(
\text{root divisors},
\text{factor depth},
\text{square-boundary state},
\text{transport label},
\text{residual status}
).
}
$$

Then define:

$$
\boxed{\Phi:\mathcal C\to\mathcal C.}
$$

The target becomes:

$$
\boxed{\mathcal C(x)\in\mathcal Z}
$$

for the terminal condition set $\mathcal Z$.

This may be more natural than constructing dynamics on the integers themselves.

---

# 28. Failure Surface Closure Through Capacity

Suppose the failure surface requires:

$$
A=B.
$$

and structural providers give:

$$
A\le U,
$$

$$
B\ge L.
$$

If:

$$
U<L,
$$

then:

$$
\boxed{\text{failure surface empty}.}
$$

This is where the SRBRD constant ledger / transport capacity can connect directly to TCDC.

---

# 29. A Direct New Working Question for SRBRD

TCDC poses a new overarching question for current SRBRD:

> We do not merely want to improve sieve bounds item by item.
> We want to regard the Goldbach failure equation:
> $$
> M_{PP}=0
> $$
> as a candidate closed surface, and then ask:
> **Can it simultaneously satisfy all known exact constraints?**

Equivalently:

$$
\boxed{
M_{SS}=n_{SP}-n_P
}
$$

can coexist with:

- the center row lock;
- semiprime transport;
- RootRough row mass;
- Buchstab descent;
- complement switching;
- the shift-free lattice;
- the constant ledger;

as listed above?

This is the most direct connection between TCDC and our existing SRBRD.

---

# 30. Six TCDC Statement Types Must Be Kept Separate

Any research document must label:

## 1. Definition

Only a definition.

## 2. Compatibility Identity

The target / failure satisfies an equation.

## 3. Necessary Condition

It must hold if the target holds.

## 4. Sufficient Condition

If the condition holds, the target follows.

## 5. Closure Theorem

The entire specified domain necessarily reaches the target / failure cannot be closed.

## 6. Computational Evidence

A finite scan / numerical support.

The following substitution is forbidden:

$$
\boxed{2\text{ or }6\Rightarrow5}
$$

as displayed above.

---

# 31. Anti-Circularity Gate

The most dangerous TCDC error is:

> Put the target into the construction first, then treat the construction as a proof of the target.

Thus, every proof must check:

### Gate A

Is compatibility merely a rewriting of the target identity?

### Gate B

Does the closure proof use unproved target properties?

### Gate C

Is the potential defined only once the target already exists?

### Gate D

Can the domain be established only if the target holds?

### Gate E

Does the reverse parameter covertly depend on an unknown witness?

If so:

$$
\boxed{\text{Circular Construction}.}
$$

---

# 32. TCDC Research Contract

```yaml
Target:
  statement

TargetType:
  point|set|zero_obstruction|residual|failure_surface

ReverseCompatibility:
  equations
  necessary_conditions

Branch:
  legal_conditions

Existence:
  object_defined

InvariantDomain:
  domain
  proof_status

BasinOrFailureDomain:
  description

ClosureMechanism:
  contraction|monotone|finite_descent|capacity|transport|noncover|dispersion

Uniqueness:
  status

FailureClosureAudit:
  status

ComputationalEvidence:
  separate_from_theorem: true

FinalStatus:
  METHOD_ONLY|CONDITIONAL|PROVED
```

---

# 33. TCDC and Autonomous AI Mathematics

TCDC is well suited to splitting AI theorem search into two roles.

## Reverse Agent

Responsible for:

- specifying the target;
- reverse-solving compatibility;
- finding a canonical boundary;
- generating a candidate potential;
- finding failure equations.

## Closure Agent

Responsible for:

- existence;
- branches;
- an invariant domain;
- capacity contradiction;
- convergence;
- finite descent;
- proof audit.

This separation can reduce the most common AI error:

> Finding an elegant fixed-point equation and mistakenly believing that a global theorem has been proved.

---

# 34. First Formal Application Direction for Prime-Square TCDC

The next paper can establish:

# TCDC-01 — Prime-Square Target / Failure Fixed-Surface Construction

First, make no further new sieve estimates.

Simply reorganize all current SRBRD exact identities as:

### Target Surface

$$
\boxed{M_{PP}>0.}
$$

### Failure Surface

$$
\boxed{M_{PP}=0.}
$$

### Exact Failure Compatibility

$$
\boxed{M_{SS}=n_{SP}-n_P.}
$$

### Structural Providers

- center-divisor forced transport;
- row capacity;
- RootRough decomposition;
- complement switching;
- Buchstab loss;
- the RSFLD constant gate.

### Main Question

$$
\boxed{
\text{Does the failure surface contain an admissible closed point?}
}
$$

This will be TCDC's first genuine reconnection to RG-SCP / SRBRD.

---

# 35. Methodology Summary

The core of TCDC is not:

> Prescribe the answer, reverse-engineer a formula, and conclude that the answer holds.

Instead:

$$
\boxed{
\text{Specify the target}
\to
\text{Reverse-solve compatibility}
\to
\text{Establish the domain}
\to
\text{Establish the basin / failure surface}
\to
\text{Prove closure / non-closure}
\to
\text{Return to the target}.
}
$$

The lesson of infinitely nested radicals is:

$$
\boxed{
2\text{ is a compatible fixed point}
\neq
\text{Every legal process necessarily reaches }2.
}
$$

The counterpart in the prime-square domain:

$$
\boxed{
\mathcal D(x)=\varnothing
\iff
x\in\mathbb P
}
$$

is still only a target characterization.

The theorem actually needed is:

$$
\boxed{
\text{The composite / full-cover failure domain cannot form a closed admissible state.}
}
$$

This is TCDC's greatest insight for the entire current prime-domain / SRBRD research line.

---

# 36. Verdict

$$
\boxed{
\mathrm{TCDC\mbox{-}00\ METHODOLOGY\ PASS}.
}
$$

The following have been formally established:

1. Target–Condition duality;
2. Reverse Construction / Forward Closure separation;
3. Branch / existence / basin firewalls;
4. fixed-point vs attractor distinction;
5. prime-square-domain interface;
6. Goldbach failure-surface interface;
7. proof-operator / condition-space formulation;
8. anti-circularity contract.

There is no claim that:

$$
\boxed{
\text{Goldbach, prime distribution, or any other unsolved conjecture has thereby been proved.}
}
$$

---

# 37. Next

```yaml
next:
  TCDC-01_PrimeSquare_Target_FailureFixedSurface_Construction

objective:
  take_existing_SRBRD_exact_identities: true
  reorganize_as_target_failure_dual_surface: true
  identify_minimal_closure_conditions: true
  no_new_sieve_claim_yet: true
```