# P/NP Debate Game Research Area | Round 13

## Tractable Closure Stability and Polynomial Chain Explosion: If Every Step is Easy, is the Whole Path Necessarily Easy?

**Tractable Closure Stability and Polynomial Chain Explosion: Does Local Polynomiality Compose Globally?**

- **Lead Researcher:** Neo.K (Chuan-Wei Hsu)
- **Collaborator/Editor:** Aletheia
- **Institution:** EveMissLab (Yiyannuo Technology Co., Ltd.)
- **Date:** August 1, 2026
- **Version:** v1.0
- **Research Status:** Round 13 Dual Hypothesis Rehearsal
- **Prerequisite Document:** `12_Round_12_Interface_Language_Lattice_Schaefer_Criticality_and_Recursive_SAT.md`
- **Game Attitude:** Team Equality and Team Inequality continue to dismantle each other's arguments
- **Document Standard:** Formal propositions, candidate conjectures, thought experiments, and game scores are recorded separately

---

## Abstract

Round 12 established Bridge Language Drift, Bridge Expressivity Transition, BCIC, and DTCS, and revised the relationship between different bridge languages from a linear ladder to a partial order structure induced by pp-definability / co-clone. However, a critical unresolved issue remains: even if **every step** of a dynamic bridge portfolio falls within some known tractable representation or tractable theory, is this sufficient to guarantee that the **entire dynamic transformation trajectory** can still be completed within polynomial resources relative to the original input size?

This round proves a simple yet crucial combinatorial fact: **stepwise polynomiality does not imply pathwise polynomiality**. If the representation size satisfies

$$
s_{t+1}=s_t^d,
\qquad d\ge 2,
$$

then each step is a polynomial transformation relative to the current representation size; however, starting from $s_0=n$, after $m$ steps:

$$
s_m=n^{d^m}.
$$

As long as $m$ grows with $n$, the overall size can rapidly exceed any fixed polynomial bound. This demonstrates that "I only perform one polynomial transformation at a time" cannot serve as a sufficient proof for a Dynamic Tractable Closure Scheme.

Accordingly, this round formally distinguishes between:

$$
\boxed{\text{Stepwise Tractability}}
$$

and:

$$
\boxed{\text{Pathwise Tractability}}.
$$

It also proposes **Tractable Closure Stability (TCS)**: a dynamic representation / bridge trajectory requires not only that each step is tractable, but also that all intermediate representation sizes, single-step costs, cumulative costs, bridge depth, and answer lifting costs remain uniformly polynomially bounded relative to the original input size.

Team Equality thus upgrades DTCS: instead of merely looking for the "next tractable island," they must establish an **Amortized Tractability Certificate (ATC)** or another non-cyclic global potential function to prove that the entire switching history will not suffer a chain explosion in size, degree, interface, or recursion depth. Team Inequality, on the other hand, proposes **Pathwise Closure Instability**: the hardness of general SAT might not be that any single step is necessarily hard, but rather that any sufficiently general tractable-island trajectory will inevitably accumulate super-polynomial intermediate debts, transformation depths, or expressive drift somewhere along the way.

This round does not prove $P\neq NP$, but it yields a highly important debugging rule for subsequent research:

> **Local polynomiality is a necessary condition, not a sufficient condition for dynamic multi-stage algorithms; what must truly be controlled are the global peak and cumulative costs relative to the original input size.**

---

# 1. Questions Left from the Previous Round

In Round 12, Team Equality proposed the Dynamic Tractable Closure Scheme (DTCS):

$$
R_0
\xrightarrow{\tau_0}
R_1
\xrightarrow{\tau_1}
\cdots
\xrightarrow{\tau_{m-1}}
R_m,
$$

where each $R_t$ attempts to land in some tractable representation / bridge language / theory island, and is processed by some polynomial-time local solver.

At first glance, if every:

$$
\tau_t
$$

is polynomial time, it seems the entire path should also be polynomial time.

This intuition holds automatically only under **a fixed number of polynomial compositions**.

If the number of transformations:

$$
m=m(n)
$$

itself grows with the input size, and the polynomial bound of each step is relative to the **already enlarged current representation size**, then the conclusion is entirely different.

This is the main battlefield of this round.

---

# 2. Common Model: Dynamic Tractable Trajectory

Let the original SAT instance be:

$$
F_0,
\qquad |F_0|=n.
$$

A dynamic portfolio generates a sequence of representations:

$$
F_0=R_0,R_1,\ldots,R_m.
$$

The $t$-th step is completed by:

$$
\tau_t:R_t\mapsto R_{t+1}.
$$

Define:

$$
s_t=|R_t|
$$

as the representation length at layer $t$, and

$$
c_t=C(\tau_t,R_t)
$$

as the actual construction / transformation cost at step $t$.

Finally, $R_m$ must allow the answer to be found in polynomial time, and when necessary, lift the witness back to the original SAT:

$$
\operatorname{Lift}(R_m)\mapsto w.
$$

Therefore, the cost of the entire trajectory includes at least:

$$
T_{\mathrm{path}}
=
\sum_{t=0}^{m-1}c_t
+
T_{\mathrm{solve}}(R_m)
+
T_{\mathrm{lift}}.
$$

Additionally, actual computation cannot ignore the peak representation size:

$$
S_{\mathrm{peak}}
=
\max_{0\le t\le m}s_t.
$$

---

# 3. The First Formal Distinction: Stepwise vs. Pathwise

## 3.1 Stepwise Polynomiality

A trajectory is said to have stepwise polynomiality if, for every step $t$, there exist fixed constants $a_t,b_t$ such that:

$$
c_t\le s_t^{a_t},
$$

$$
s_{t+1}\le s_t^{b_t}.
$$

Note: These bounds are relative to the **current** $s_t$.

This merely states:

> If I already have $R_t$, generating the next $R_{t+1}$ is no more expensive than some polynomial of $R_t$.

It says nothing about how large $R_t$ itself is relative to the initial input $n$.

## 3.2 Pathwise Polynomiality

A trajectory is said to have pathwise polynomiality if there exists a fixed constant $K$ independent of $n$, such that for all sufficiently large inputs:

$$
m(n)\le n^K,
$$

$$
S_{\mathrm{peak}}\le n^K,
$$

$$
T_{\mathrm{path}}\le n^K.
$$

This is the level truly required by traditional $P$.

Therefore:

$$
\boxed{
\text{Stepwise Polynomiality}
\not\Rightarrow
\text{Pathwise Polynomiality}
}
$$

is the first main conclusion of this round.

---

# 4. Polynomial Chain Explosion Lemma

## Lemma 4.1: Repeated Squaring

Let:

$$
s_0=n,
$$

and let each step be:

$$
s_{t+1}=s_t^2.
$$

Then for each fixed $t$, the mapping:

$$
s\mapsto s^2
$$

is clearly a polynomial-size transformation.

However:

$$
s_1=n^2,
$$

$$
s_2=n^4,
$$

$$
s_3=n^8,
$$

and generally:

$$
\boxed{
s_m=n^{2^m}
}.
$$

### Proof

By induction: If

$$
s_t=n^{2^t},
$$

then:

$$
s_{t+1}
=(n^{2^t})^2
=n^{2^{t+1}}.
$$

Proof complete.

Thus, as long as:

$$
m(n)\to\infty,
$$

the exponent $2^{m(n)}$ is no longer a fixed constant.

For example, when $m(n)=\lceil\log_2 n\rceil$:

$$
s_m
=n^{\Theta(n)},
$$

which far exceeds any fixed polynomial.

This example does not rely on SAT at all; it merely points out a common logical loophole in algorithm design:

$$
\boxed{
\text{"Every step is polynomial" cannot be directly quantifier-swapped to "the entire chain is polynomial".}
}
$$

---

# 5. Generalization: Degree Accumulation

If each step satisfies:

$$
s_{t+1}\le c_t s_t^{d_t},
$$

ignoring the details of constant terms, repeated substitution yields an approximate degree accumulation:

$$
s_m
\lesssim
n^{\prod_{t=0}^{m-1}d_t}
\times
\text{constant factors}.
$$

Therefore, a very natural upper-bound certificate quantity is:

$$
\boxed{
A_m
=
\prod_{t=0}^{m-1}d_t
}
$$

called the **Degree Accumulation Factor**.

If $A_m$ grows unboundedly with $n$, this kind of simple local polynomial certificate cannot guarantee that the final representation is of a fixed polynomial size.

It must be emphasized:

$$
A_m
$$

is only an **upper-bound analysis tool**, not an invariant of the problem itself. Actual transformations might be highly contractive, making the rough $d_t$ upper bound very loose.

Its purpose is to rule out a fallacious argument:

> "Every one of my bridges is polynomial, so the portfolio is naturally polynomial."

No, global compositional control is also required.

---

# 6. A More Subtle Counterexample: Constant Factor Growth Can Also Accumulate

Even if each step only has:

$$
s_{t+1}\le 2s_t,
$$

if the number of steps is:

$$
m(n)=n,
$$

then:

$$
s_m\le 2^n n.
$$

So the problem is not even just about the polynomial degree being greater than $1$.

What truly needs to be simultaneously controlled are:

1. Single-step expansion ratio;
2. Bridge depth;
3. Whether there are contraction steps;
4. Peak intermediate representation size;
5. Cumulative construction time.

Therefore, this round introduces:

$$
\boxed{
D_B=m(n)
}
$$

as the **Bridge Depth**.

---

# 7. Small Final Size Does Not Imply Cheap Process

Team Equality might counterattack:

> I might expand in the middle, but eventually, I compress it back into a very small normal form.

This does not automatically solve the problem.

Consider:

$$
|R_0|=n,
$$

$$
|R_1|=2^n,
$$

$$
|R_2|=1.
$$

The final representation is only one bit, but as long as $R_1$ must be explicitly generated, the algorithm has already paid an exponential cost.

Therefore, one cannot only measure:

$$
|R_m|.
$$

One must measure:

$$
S_{\mathrm{peak}}
=
\max_t |R_t|.
$$

This aligns perfectly with the fundamental spirit in Knowledge Compilation that "query tractability of the target language" and "succinctness / transformation cost" must be evaluated separately.

---

# 8. True Tractable Closure Stability

This round formally proposes a working definition.

## Definition 8.1: TCS

A family of dynamic bridge trajectories:

$$
\Pi_n
=
(R_0,\tau_0,R_1,\ldots,\tau_{m-1},R_m)
$$

is called **Tractable-Closure Stable (TCS)** if there exists a fixed constant $K$ such that for all $n$ and all inputs of length $n$:

$$
m(n)\le n^K,
$$

$$
\max_t |R_t|\le n^K,
$$

$$
\sum_t C(\tau_t,R_t)\le n^K,
$$

$$
T_{\mathrm{final}}+T_{\mathrm{lift}}\le n^K.
$$

Furthermore, all transformations must be uniform, exact, of finite precision, and realizable by standard computational models.

This is not a new complexity class, but rather **a complete cost certificate format for dynamic multi-representation algorithms**.

---

# 9. Team Equality's New Strategy: Amortized Tractability Certificate

Team Equality concedes that "every step is tractable" is insufficient.

Thus, they propose a stronger concept:

$$
\boxed{
\mathrm{ATC}
=
\text{Amortized Tractability Certificate}
}
$$

That is, instead of independent step-by-step guarantees, find a global, non-cyclic potential:

$$
\Phi_n(R_t).
$$

Hoping to satisfy:

### 9.1 Initial Polynomial Bound

$$
\Phi_n(R_0)\le n^a.
$$

### 9.2 Controllable Single-Step Increment

$$
\Phi_n(R_{t+1})
\le
\Phi_n(R_t)+n^b.
$$

### 9.3 Single-Step Cost Controlled by Potential Function

$$
|R_t|+c_t
\le
\operatorname{poly}(n,\Phi_n(R_t)).
$$

### 9.4 Polynomial Bound on Number of Steps

$$
m(n)\le n^c.
$$

Then:

$$
\Phi_n(R_t)
\le
n^a+n^{b+c},
$$

so if the single-step cost polynomial degree is a fixed constant, the entire trajectory can remain polynomial.

This idea is fundamentally akin to amortized analysis:

> It does not demand that every step avoids growth, but rather that the growth has a global budget and cannot compound repeatedly without limit.

---

# 10. ATC's Self-Dismantling: Cannot Sneak the Solver into the Potential

Team Inequality immediately points out:

If we define:

$$
\Phi(R)
=
\text{Optimal remaining solving time starting from } R,
$$

then:

$$
\Phi(R_0)\in\operatorname{poly}(n)
$$

is itself just another way of saying:

$$
P=NP.
$$

Therefore, for an ATC to have any proof value, the potential must be:

1. **solver-independent**: cannot be defined by the identity of the optimal algorithm;
2. **answer-blind**: cannot define the potential by first finding the answer;
3. **structural**: should be independently characterizable by the representation / constraints / interface itself;
4. **computably checkable or at least have an independent mathematical definition**;
5. **cannot directly equal the remaining runtime, optimal circuit size, or shortest proof length**, unless these quantities have independent provable lower bounds.

This is a renewed safeguard against the "Closure Paradox" from Round 6 and the "unrestricted PEQS" from Round 8.

---

# 11. Positive Templates of Existing Tractable Closures

This round does not argue that all tractable islands are unstable. On the contrary, excellent positive templates exist in current theory.

## 11.1 pp-closure within a Fixed CSP Language

For a fixed finite constraint language $\Gamma$, if a relation $R$ can be represented by $\Gamma$ using a fixed primitive-positive definition, then $R$-constraints can be expanded back into $\Gamma$-constraints by introducing existential auxiliary variables.

Under the condition of a fixed finite language and fixed relation definitions, this provides a standard polynomial reduction.

From the perspective of polymorphism, pp-definable relations preserve the same polymorphism invariants; therefore, co-clone / pp-closure is precisely one of the stable algebraic spaces that tractability classification can utilize.

This shows:

$$
\boxed{
\text{Some genuine tractable closures do exist.}
}
$$

The question is whether a dynamic portfolio for general SAT can forever remain within these safe closures.

## 11.2 Conditional Stability of SMT Theory Combination

Nelson–Oppen style methods also provide another positive template: when conditions like signature disjointness and stable infiniteness hold, individual theory decision procedures can be composed.

However, the literature simultaneously points out that shared-variable arrangements might incur worst-case exponential reasoning cost; techniques like polite combination and care functions exist precisely to control this coordination explosion.

This once again supports the core of this round:

$$
\text{local decidability}
+
\text{valid composition theorem}
+
\text{controlled interface growth}
$$

is what is truly sufficient to achieve global tractability.

---

# 12. Reminder from Knowledge Compilation: Closure is Not Just About Queries

The Knowledge Compilation Map divides the value of a representation language into at least two dimensions:

1. succinctness;
2. which queries / transformations can be completed in polytime.

Therefore, if a dynamic portfolio claims:

> I switch to a query tractable language every time.

It is still insufficient.

It must simultaneously answer:

$$
\text{translation cost?}
$$

$$
\text{intermediate succinctness?}
$$

$$
\text{closure under repeated transformations?}
$$

This is exactly the gap that TCS aims to fill.

---

# 13. Team Inequality's New Candidate: Pathwise Closure Instability

In this round, Team Inequality no longer claims:

> A certain local step is inevitably super-polynomial.

Instead, they propose a more dynamic candidate:

## Pathwise Closure Instability Conjecture (PCIC)

There exists a family of SAT instances $\{F_n\}$ such that any dynamic tractable-island trajectory meeting admissibility requirements, if it exactly solves $F_n$, will incur at least one of the following:

$$
m(n)\notin\operatorname{poly}(n),
$$

or:

$$
S_{\mathrm{peak}}(n)\notin\operatorname{poly}(n),
$$

or:

$$
T_{\mathrm{path}}(n)\notin\operatorname{poly}(n),
$$

or:

$$
P_{\mathrm{precision}}(n)\notin\operatorname{poly}(n),
$$

or the trajectory is forced to leave all proven tractable closures.

This remains merely a **candidate research direction**.

If "admissible trajectory" is defined as all possible polynomial-time algorithms, it once again becomes equivalent to the original $P/NP$; thus, it is still necessary to find a broad yet non-cyclic trajectory class moving forward.

---

# 14. Team Equality's Strongest Version: Global DTCS

Team Equality upgrades the DTCS from Round 12 to:

## Global Dynamic Tractable Closure Scheme (G-DTCS)

For every SAT instance $F$, there exists a uniform polynomial-time controller $\Omega$ that generates:

$$
R_0\to R_1\to\cdots\to R_m,
$$

where:

1. The correctness of each local solver is provable;
2. Every bridge transformation is exact;
3. All $R_t$ are processed by some identifiable tractable representation class;
4. $m$, $S_{\mathrm{peak}}$, total build cost, query cost, and lift cost are all controlled by the same fixed $n^K$;
5. The controller does not use an oracle or pre-hidden answers.

If a G-DTCS can truly be constructed for general SAT, it essentially provides a polynomial-time algorithm for SAT.

So Team Equality's task is very clear:

$$
\boxed{
\text{The goal is not to find more tractable islands,}
}
$$

$$
\boxed{
\text{but to prove that island switching itself possesses a global polynomial certificate.}
}
$$

---

# 15. Thought Experiment A: Squaring at Every Step

Team Equality:

> Every bridge is just a polynomial transformation.

Team Inequality:

Let:

$$
|R_{t+1}|=|R_t|^2.
$$

Every step is polynomial, but:

$$
|R_m|=n^{2^m}.
$$

If:

$$
m=\Theta(\log n),
$$

then:

$$
|R_m|=n^{\Theta(n)}.
$$

**Ruling:** Stepwise polynomiality is defeated.

---

# 16. Thought Experiment B: Safe Additive Trajectory

If there exist fixed constants $a,b$ such that:

$$
|R_{t+1}|
\le
|R_t|+n^a,
$$

and:

$$
m(n)\le n^b,
$$

then:

$$
|R_m|
\le
n+n^{a+b},
$$

maintaining polynomiality.

If simultaneously the runtime of each step has a fixed polynomial bound relative to $n$ and the current representation, the cumulative cost can also remain polynomial.

**Ruling:** Dynamic switching is not the original sin; the lack of global amortized control is the real problem.

---

# 17. Thought Experiment C: Final Compression Cannot Offset Intermediate Explosion

Let:

$$
R_0\to R_1\to R_2,
$$

where:

$$
|R_0|=n,
\qquad
|R_1|=2^n,
\qquad
|R_2|=1.
$$

Even if the answer is ultimately "beautifully compressed into a single bit," if $R_1$ must be explicitly generated, the total cost remains exponential.

Therefore, one must calculate:

$$
S_{\mathrm{peak}}
$$

rather than just the final normal form.

---

# 18. Thought Experiment D: Fixed-Depth Composition is Indeed Safe

If the bridge chain length is a fixed constant $m=O(1)$, and every transformation has a fixed degree polynomial bound:

$$
s_{t+1}\le s_t^{d_t},
$$

then:

$$
s_m
\le
n^{\prod_{t=0}^{m-1}d_t},
$$

where the product remains a fixed constant.

Therefore:

$$
\boxed{
\text{Polynomial transformations of finite fixed depth remain polynomial.}
}
$$

What is truly dangerous is:

$$
\boxed{
\text{composition depth grows with the input.}
}
$$

---

# 19. Reconnecting This Round with the Original Dynamic Rate Theory

A phenomenon emerged in this round that perfectly aligns with the original series.

Previously we wrote:

$$
T_{\mathrm{total}}
=
T_{\mathrm{search}}
+
T_{\mathrm{construct}}
+
T_{\mathrm{realize}}
+
T_{\mathrm{run}}
+
T_{\mathrm{verify}}.
$$

Now the dynamic bridge portfolio tells us that:

$$
T_{\mathrm{construct}}
$$

itself might not be a one-time single layer, but rather:

$$
T_{\mathrm{construct}}
=
\sum_{t=0}^{m-1}
T_{\mathrm{bridge},t}.
$$

Even the representation size becomes a dynamic state variable:

$$
s_{t+1}
=f_t(s_t,\Gamma_t,B_t).
$$

Therefore, computational complexity is not just "the Big-O of some algorithm"; it can be viewed during the research process as a resource trajectory:

$$
\mathbf R_t
=
(s_t,c_t,w_t,p_t,b_t,\ldots).
$$

This does not rewrite the traditional definition of $P$; it merely provides an intermediate mathematical language better suited for analyzing dynamic multi-representation solvers.

---

# 20. Mutual Attacks Between the Two Teams

## 20.1 Team Inequality Attacks Team Equality

### Attack A: Swapping local-poly for global-poly

$$
\forall t,
\quad
c_t\in\operatorname{poly}(s_t)
$$

does not imply:

$$
\sum_t c_t\in\operatorname{poly}(n).
$$

### Attack B: Only looking at the final representation

A small final normal form does not mean the compilation trajectory is small.

### Attack C: Infinite portfolio

If the controller adds a new tractable island every time it encounters difficulty, the uniform cost of island discovery / recognition / translation itself must be calculated.

### Attack D: ATC circularity

If the potential equals the optimal remaining solving time, ATC is just a renaming of $P=NP$.

## 20.2 Team Equality Attacks Team Inequality

### Attack A: Composition explosion is only existential, not inevitable

Repeated squaring only proves that "some polynomial chains will explode," it does not prove that any SAT solver chain will inevitably explode.

### Attack B: Real algorithms can contract

Many intermediate steps in algorithms are not monotone expansions; elimination, contraction, and quotienting can all drastically reduce the state.

### Attack C: Global potentials can exist

If a structural potential function is found to control total growth, the chain depth itself is not necessarily dangerous.

### Attack D: Successful composition theorems already exist

Cases like CSP pp-closure and Nelson–Oppen / polite combination demonstrate that certain local decision procedures can indeed be safely composed under explicit conditions.

---

# 21. Review of Known Barriers

## 21.1 Does this prove $P\neq NP$?

No.

The Polynomial Chain Explosion Lemma only states:

$$
\text{stepwise poly}
\not\Rightarrow
\text{pathwise poly}.
$$

It does not say that all SAT algorithms must use a growing-depth chain, nor does it say that any chain will inevitably explode.

## 21.2 Is this just redefining P?

If TCS is defined too broadly, it indeed degenerates into:

$$
\text{Existence of a polynomial trajectory}
\Longleftrightarrow
L\in P.
$$

So TCS is currently just an **auditing specification for costs**; a true proof must come from independent structural conditions or independent lower bounds.

## 21.3 Can Schaefer / CSP directly complete the separation?

No.

They can provide a tractable / NP-complete boundary within a fixed constraint-language model; the NP-complete side still cannot unconditionally imply not being in P.

## 21.4 Can the exponential arrangements of theory combination directly prove general lower bounds?

No.

They only indicate that the coordination of a specific combination framework might have a worst-case exponential cost.

---

# 22. Fallacious Arguments Eliminated in This Round

1. "Every transformation is polynomial, so the entire adaptive pipeline must be polynomial."
2. "The final representation is polynomial size, so the compilation process must be polynomial."
3. "Every layer falls into some class P problem, so the composition of all layers automatically remains in P."
4. "There are polynomially many polynomial-time subroutines, so the total time must be polynomial." — This holds only when the input size and polynomial degree of each subroutine have a uniform control relative to the original $n$.
5. "Found a potential," but the potential is actually the remaining optimal runtime.
6. "A certain bridge framework's arrangements are exponential," therefore all bridge algorithms are exponential.

---

# 23. Actual Results Obtained in This Round

## Result 1: Polynomial Chain Explosion Lemma

Local polynomial transformations can form super-polynomial trajectories in growing-depth compositions.

This is a formally established foundational mathematical fact.

## Result 2: Stepwise / Pathwise Separation

From now on, all dynamic representation / bridge algorithms must simultaneously report:

$$
\text{step cost}
$$

and:

$$
\text{path cost}.
$$

## Result 3: Bridge Depth Becomes a Necessary Parameter

$$
D_B=m(n)
$$

can no longer be ignored.

## Result 4: Peak Representation is a Necessary Resource

$$
S_{\mathrm{peak}}
=
\max_t|R_t|
$$

is more important than the final representation size.

## Result 5: ATC Becomes Team Equality's New Construction Task

If Team Equality wants to salvage the Dynamic Tractable Closure, they must provide a truly non-cyclic amortized certificate.

## Result 6: Team Inequality Shifts from "Single-Step Lower Bound" to "Pathwise Instability"

What can be studied next is not that a certain representation will inevitably explode, but rather:

$$
\boxed{
\text{Do all sufficiently general tractable trajectories lack a globally stable potential function?}
}
$$

---

# 24. Ruling for This Round

Team Equality successfully pointed out:

$$
\text{Dynamic switching itself does not cause hardness,}
$$

as long as there is good global amortized control, long chains can still remain polynomial.

Team Inequality successfully dismantled a very easily overlooked assumption:

$$
\boxed{
\text{Every step is tractable}
\not\Rightarrow
\text{The entire trajectory is tractable}.
}
$$

Therefore, the score for this round is:

$$
P=NP:12
\qquad
P\neq NP:12.
$$

Another tie.

It seems the two teams have developed the "tie" itself into a stable fixed point in this series.

---

# 25. Gateway to Round 14

## Complexity Potential Energy Game: Does an Acyclic Global Tractability Potential Exist?

Round 14 will directly contest:

$$
\boxed{
\Phi(F)
}
$$

Is it possible for there to exist a global **complexity potential function** that:

1. Does not rely on a specific solver;
2. Does not know the answer in advance;
3. Can be compared across representations;
4. Has controllable drift under all tractable bridge transformations;
5. Is sufficient to guarantee polynomial peak size, path cost, and bridge depth;
6. And does not degenerate into a tautology of "optimal solving time"?

Team Equality's task:

> Establish an admissible potential such that for any SAT, there exists a polynomial trajectory of controlled descent / controlled growth along the potential function.

Team Inequality's task:

> Construct an adversarial formula family to prove that any candidate structural potential will exhibit plateaus, cycles, hidden debts, or representation escapes.

This will truly reconnect this series with the original "Dynamic Rate / State Evolution" research back to the same mathematical interface.

---

# 26. Historical Dependencies

This round directly relies on:

1. `12_Round_12_Interface_Language_Lattice_Schaefer_Criticality_and_Recursive_SAT.md`
   - BCIC, DTCS, Bridge Language Drift, Recursive SAT.
2. `11_Round_11_Collapse_of_Co-Preserved_Structures_and_Dynamic_Bridging.md`
   - Bridge Coordination Debt, Existential Reappearance.
3. `09_Round_09_Finding_the_Blossom_of_SAT_and_Quotient_Debt.md`
   - Quotient Debt, Hybrid Quotient Portfolio.
4. `06_Round_06_Polynomial_Representation_Transformation_Closure_and_the_Closure_Paradox.md`
   - Avoiding treating unrestricted polynomial closure directly as a new theory.
5. `00_Mathematical_Construct_State_Machine_Middleware_v1.0.md`
   - The complete cost chain of mathematical construction, representation, realization, and state transition.

---

# 27. External Theoretical References

1. Adnan Darwiche and Pierre Marquis, **A Knowledge Compilation Map**, Journal of Artificial Intelligence Research, 2002 / arXiv:1106.1819.
   - Framework used to show that succinctness, polytime queries, and transformations must be evaluated separately.
2. Dejan Jovanović and Clark Barrett, **Being Careful about Theory Combination**, Formal Methods in System Design 42(1), 2013.
   - shared-variable arrangements and care functions; coordination of composed theories is not free.
3. Ying Sheng, Yoni Zohar, Christophe Ringeissen, Andrew Reynolds, Clark Barrett, Cesare Tinelli, **Politeness and Stable Infiniteness: Stronger Together**, CADE 2021.
   - Conditional composition theorems for theory combination; arrangement reasoning can be exponential in the worst case.
4. Ying Sheng et al., **Combining Stable Infiniteness and (Strong) Politeness**, Journal of Automated Reasoning, 2023.
   - Hybrid combination methods and techniques to reduce arrangement variables.
5. Dmitriy Zhuk, **Strong subalgebras and the Constraint Satisfaction Problem**, 2020.
   - Background on finite-domain CSP, WNU polymorphism, and tractability / NP-hardness classification.
6. Boolean CSP / Schaefer / pp-definability / polymorphism literature.
   - Used to illustrate that genuine tractable pp-closures can exist, but do not automatically cover dynamic multi-representation trajectories of general SAT.

---

## One-Sentence Summary of This Round

$$
\boxed{
\text{"Every step is easy" is only a local claim; true } P \text{ requires "the entire history to be cheap".}
}
$$

and:

$$
\boxed{
\text{For dynamic switching to become a viable path to } P=NP \text{, it must present a fully amortized polynomial bill.}
}
$$