# P/NP Debate Game Research Area | Round 10

## Multi-Anti-Structure Cores and Heterogeneous Gluing Debt: After Blocking All Known Escape Hatches Together, Where Does the Difficulty Truly Lie?

**Round 10: Multi-Anti-Structure Cores and Heterogeneous Gluing Debt**

- **Lead Researcher:** Neo.K (Chuan-Wei Hsu)
- **Collaborative Organization:** Aletheia
- **Institution:** EveMissLab (Yiyannuo Technology Co., Ltd.)
- **Date:** August 1, 2026
- **Version:** v1.0
- **Research Status:** Round 10 Dual-Hypothesis Rehearsal
- **Prerequisite Document(s):** `00`–`09` All historical rounds
- **Game Stance:** Block all known escape hatches together, then see where the Equals Team sneaks out
- **Document Standard:** All general conclusions must still distinguish between "in-model theorems," "external known results," "research conjectures," and "thought experiments."

---

## Abstract

Round 9 established the "local Blossom of SAT" and the Quotient Debt framework: Variable Elimination, OBDD/DNNF, XOR/Affine extraction, Symmetry quotient, Backdoor, and CDCL learned clauses can all precisely quotient out a large number of candidates on specific structures, but no known quotienting method has been proven to handle all SAT instances with uniform polynomial cost.

The intuitive task for Round 10 was originally to find a **Multi-Anti-Structure Core (MASC)**: a family of SAT formulas that simultaneously possesses high elimination width, lacks small backdoors, lacks useful symmetries, is difficult for knowledge compilation, lacks affine extractability, and requires large resources in certain proof systems.

The first major correction of this round is: even if such a family is found, it does not directly imply $P\neq NP$. The reason is consistent with the previous nine rounds—this might still just be "simultaneously defeating the limited arsenal currently listed," rather than defeating all possible deterministic polynomial-time algorithms.

However, after comparing Schaefer-type CSPs, Monotone 3-SAT, expander-based DNNF lower bounds, random-CNF resolution lower bounds, heterogeneous backdoors, and bottom-up knowledge compilation, this round arrived at a more valuable new focus:

$$
\boxed{
\text{The true difficulty may not lie in the local subproblems themselves, but in the "gluing" of different locally solvable structures.}
}
$$

The most concise illustration is: positive 3-clauses and negative 3-clauses each form a very easy single pattern; but Monotone 3-SAT, which allows the mixing of both, is known to remain NP-complete. More generally, the Schaefer dichotomy tells us that whether a Boolean constraint language is tractable depends on whether the entire language shares certain closure/polymorphism structures, not on whether each constraint looks easy individually.

Therefore, this round proposes a new research object:

$$
\boxed{\text{HGD = Heterogeneous Gluing Debt}}
$$

It describes "if a problem is partitioned into multiple local blocks that can each be solved/quotiented at a low cost, how much interface cost must be paid when precisely gluing these local summaries into a global answer." This allows the local-global barriers, quotient debts, polymorphisms, treewidth, backdoors, and knowledge compilation from previous rounds to be placed into the same picture.

This round still does not yield a general time lower bound; but the research focus advances from "the stacking of multiple anti-structures" to "**the incompatibility of local tractability and the global gluing cost**."

---

# 1. Previous Round Results: SAT Has Many Blossoms, But No Universal Blossom

Round 9 organized the known exact compression methods in SAT into a quotient portfolio:

$$
\mathcal Q
=
\{
Q_{\mathrm{elim}},
Q_{\mathrm{KC}},
Q_{\oplus},
Q_{\mathrm{sym}},
Q_{\mathrm{bd}},
Q_{\mathrm{learn}}
\}.
$$

Each $Q_i$ can compress a large number of microscopic candidates into a smaller exact summary on certain formula families.

But every method has explicit debts:

$$
\mathbf D_Q(F)
=
(
D_{\mathrm{build}},
D_{\mathrm{size}},
D_{\mathrm{width}},
D_{\mathrm{residual}},
D_{\mathrm{detect}},
D_{\mathrm{proof}},
D_{\mathrm{lift}}
).
$$

Therefore, the direct question left from Round 9 is:

> Can we find a family of formulas that causes multiple known quotient channels mentioned above to fail simultaneously?

This was the original MASC plan for this round.

---

# 2. Multi-Anti-Structure Core MASC: First Version Definition

For a SAT formula $F$, define an anti-structure vector relative to the currently known toolset:

$$
\mathbf A(F)
=
(
A_{\mathrm{elim}},
A_{\mathrm{bd}},
A_{\mathrm{sym}},
A_{\mathrm{KC}},
A_{\oplus},
A_{\mathrm{proof}}
).
$$

This can be temporarily understood as:

- $A_{\mathrm{elim}}$: The width or fill-in that must still be borne after the best known elimination/decomposition;
- $A_{\mathrm{bd}}$: The minimum backdoor size required to enter specified tractable base classes;
- $A_{\mathrm{sym}}$: The inverse of the degrees of freedom that can be effectively eliminated by symmetry quotients;
- $A_{\mathrm{KC}}$: The minimum representation size required in a specified compilation language;
- $A_{\oplus}$: The non-linear residual left after maximal affine/XOR extraction;
- $A_{\mathrm{proof}}$: The width, space, or length resources in a specified proof system.

If a formula family $\mathcal F=\{F_n\}$ simultaneously maintains high values across multiple known metrics, it can be called, relative to the toolset $\mathcal Q$:

$$
\boxed{\operatorname{MASC}_{\mathcal Q}}.
$$

Note that the subscript $\mathcal Q$ is very important.

This round does not allow writing it simply as "MASC," because currently we can only say it resists a finite, explicitly stated tool portfolio, not that it resists all possible representations.

---

# 3. The Not-Equals Team Plays: Blocking All Known Escape Hatches Together

The Not-Equals Team proposes the following thought experiment.

Find a formula family $F_n$ such that:

$$
\operatorname{tw}(F_n)=\Omega(n)
$$

or other elimination-related widths are very large; simultaneously, it has no known small strong/heterogeneous backdoors; its primary incidence/primal graph lacks exploitable large automorphisms; its DNNF/structured-DNNF/OBDD and other compilation languages require super-polynomial or even exponential sizes; the extractable XOR portion is insufficient to determine the whole; and it requires large width/space/size in resolution-like proof systems.

If these characteristics are caused by the same underlying reason, perhaps there exists some deeper invariant.

## 3.1 Expander Structures as a Candidate Source

It is known that there exist CNF families constructed from expander graphs that require strongly exponential size for DNNF. This proves that certain "highly connected, hard-to-decompose" structures can indeed produce unconditional representation lower bounds against quite strong knowledge-compilation languages.

On the other hand, in resolution proof complexity, expansion has also long been an important source of width, space, and size lower bounds; clause-space lower bounds for random $k$-CNF can also be established via related expansion/game methods.

So the Not-Equals Team proposes the first unifying intuition:

$$
\boxed{
\text{Expansion / high coupling may simultaneously drive up multiple quotient debts.}
}
$$

But this item is currently just a common pattern across the literature, not a general algorithmic lower bound theorem.

---

# 4. The Equals Team's First Counterattack: Defeating Ten Weapons Simultaneously Does Not Equal Defeating All Weapons

The Equals Team's answer is straightforward:

Suppose we successfully prove that:

$$
A_1(F_n),A_2(F_n),\ldots,A_{100}(F_n)
$$

are all super-polynomial.

This at most yields:

$$
F_n
\text{ is hard for the 100 listed methods.}
$$

It does not yield:

$$
F_n\notin P.
$$

Because Round 9 already clearly saw with the OBDD counterexample:

$$
\text{A P-time function}
$$

might still have an exponential representation lower bound against some fairly general, highly useful exact representation language.

Therefore:

$$
\boxed{
\text{The intersection of multiple representation/proof lower bounds is still not a general time lower bound.}
}
$$

Unless we can additionally establish a bridging theorem that "all P-time algorithms necessarily induce at least one structure measured by us."

This returns to the Algorithm-to-Algebra Bridge Problem from Round 7.

---

# 5. The Truly Interesting Twist: Easy Locally, Hard When Mixed

The most valuable example to retain this round is not "a certain formula is simultaneously hard," but the opposite:

> Each type of local constraint looks very easy individually, but when they are mixed, the entire problem class can enter the NP-complete side.

## 5.1 The Minimal Illustration of Monotone 3-SAT

Consider two types of clauses:

$$
R_+(x,y,z)=x\lor y\lor z,
$$

and:

$$
R_-(x,y,z)=\neg x\lor\neg y\lor\neg z.
$$

If the formula only allows $R_+$, setting all variables to $1$ satisfies it; if it only allows $R_-$, setting all variables to $0$ satisfies it.

That is:

$$
\operatorname{SAT}(\{R_+\})\in P,
$$

$$
\operatorname{SAT}(\{R_-\})\in P.
$$

But Monotone 3-SAT, which allows the mixing of positive monotone 3-clauses and negative monotone 3-clauses, is known to have NP-complete versions; it remains NP-complete even when the number of variable occurrences is quite strongly restricted.

Thus, a very important phenomenon emerges:

$$
\boxed{
\text{Easy local languages}
+
\text{Easy local languages}
\not\Rightarrow
\text{Easy global language}.
}
$$

The point here is not to smuggle NP-complete into non-P, but: **Local tractability does not possess simple additive closure.**

---

# 6. The Formal Structural Version Provided by Schaefer's Dichotomy

Schaefer's Boolean CSP dichotomy divides fixed finite Boolean constraint languages $\Gamma$ into tractable and NP-complete sides.

Its tractable cases are not just "each constraint is individually easy," but rather that the entire language $\Gamma$ collectively falls into certain closure types, such as Horn, dual-Horn, bijunctive, affine, and 0-valid/1-valid cases.

In algebraic language, this can be understood as: all relations must share sufficiently strong polymorphisms/preservation operations.

So if:

$$
\Gamma_1
$$

and:

$$
\Gamma_2
$$

each have their own tractable structure, but lack sufficient common preservation structures, then:

$$
\Gamma_1\cup\Gamma_2
$$

may fall onto the NP-complete side of the dichotomy.

This connects the polymorphism of Round 7 with the MASC of this round:

$$
\boxed{
\text{Difficulty may arise from the "incompatibility between solvable structures," rather than just a "lack of structure."}
}
$$

---

# 7. Upgrading from Multi-Anti-Structure to Heterogeneous Gluing Debt

Therefore, this round proposes a new research object:

$$
\boxed{
\operatorname{HGD}
=
\text{Heterogeneous Gluing Debt}
}
$$

Suppose the problem is partitioned into local blocks:

$$
F
=
F_1\land F_2\land\cdots\land F_m,
$$

Each $F_i$ belongs to some local tractable family:

$$
F_i\in\mathcal T_i.
$$

If each $F_i$ can be solved individually in polynomial time, we still cannot directly deduce that $F$ is easy, because the blocks share interface variables.

Let:

$$
\partial F_i
$$

denote the boundary variables shared by $F_i$ with other blocks.

What the local block truly needs to pass to the global system is not a single YES/NO, but its acceptable relations for the boundary assignments:

$$
R_i(\partial F_i)
=
\left\{
\alpha:
F_i\mid_{\partial F_i=\alpha}
\text{ can be extended to satisfy}
\right\}.
$$

The global problem becomes:

$$
\exists\text{ compatible }\alpha_1,\ldots,\alpha_m
$$

such that all local relations hold simultaneously.

Therefore:

$$
\boxed{
\text{Low local solving cost}
\not\Rightarrow
\text{Low local summary gluing cost}.
}
$$

This is HGD.

---

# 8. The Simplest Gluing Ledger: Boundary State Explosion

If the interface of a certain block has $k$ Boolean variables:

$$
|\partial F_i|=k,
$$

then the most direct exact boundary table has:

$$
2^k
$$

possible boundary assignments.

Without other structures, the local summary size might be:

$$
\Theta(2^k).
$$

This is the familiar source of exponential parameter dependence in treewidth/separator-based dynamic programming.

But the Equals Team immediately reminds us:

$$
2^k
\text{ boundary assignments}
$$

does not mean $2^k$ independent states are truly needed.

If the relation $R_i$ is affine, cardinality, interval, matroidal, or another highly compressible structure, it might have a very short mathematical description.

So we cannot directly upgrade:

$$
|\partial F_i|=k
$$

into:

$$
\text{Requires }2^k\text{ time}.
$$

What truly needs to be measured is **whether different boundary behaviors can be precisely quotiented**.

---

# 9. Boundary Semantic Quotient: Bringing Back Round Two

For boundary assignments $\alpha,\beta$ of a local block $F_i$, define:

$$
\alpha\equiv_i\beta
$$

if they have exactly the same extensible effect on the rest of the global problem.

Then the true number of local interface states is not:

$$
2^{|\partial F_i|},
$$

but:

$$
N_i^{\partial}
=
\left|
\{0,1\}^{\partial F_i}/\!\equiv_i
\right|.
$$

Define:

$$
H_i^{\partial}
=
\log_2 N_i^{\partial}.
$$

This is actually the block-wise version of the "residual distinguishability" from Round 2.

Thus, HGD can temporarily be understood as:

$$
\operatorname{HGD}(F,\mathcal D)
=
\text{The cost of precisely combining all boundary semantic quotients under decomposition }\mathcal D.
$$

Note: This still depends on the decomposition $\mathcal D$, so it is not yet a general invariant.

---

# 10. The Equals Team's Second Counterattack: Gluing Can Also Be "Re-quotiented"

The Equals Team proposes four escapes.

## 10.1 Common Algebraic Escape

Even if two blocks use different local algorithms, as long as their boundary relations have a common short representation in another algebra, there is no need to enumerate the interface table.

The XOR system is again the simplest example: on the surface, there are many boundary assignments, but the linear subspace can be compressed using a matrix basis.

## 10.2 Changing Decomposition Escape

A large interface in one decomposition does not mean it is large in another.

Therefore, any HGD lower bound must explain:

$$
\text{Why does there not exist another polynomially constructible decomposition?}
$$

Otherwise, it is merely proving that a certain DP architecture is hard.

## 10.3 Global Algorithm Escape

Algorithms do not necessarily need to "solve locally first, then glue together."

For example, efficient algorithms for determinant/matching/flow remind us: global transformations can sometimes bypass artificially set local decompositions.

## 10.4 Hidden New Language Escape

Perhaps $\Gamma_1$ and $\Gamma_2$ lack classical Schaefer polymorphisms, but there exists a higher-order representation that sends both types of constraints together into another tractable normal form.

This is exactly the "unknown coordinate system" right that the Equals Team has always reserved.

---

# 11. The Not-Equals Team Upgrades: Not "Lack of Structure", But "Collapse of Common Structure"

The Not-Equals Team therefore changes the narrative.

In the past, we always looked for:

$$
\text{Does the formula have a good structure?}
$$

Now we ask instead:

$$
\boxed{
\text{Does there exist a polynomially maintainable common structure among the local structures?}
}
$$

Let each local constraint family $\Gamma_i$ have a set of preservation operations:

$$
\operatorname{Pol}(\Gamma_i).
$$

The commonly available preservation structures must at least fall into:

$$
\bigcap_i\operatorname{Pol}(\Gamma_i).
$$

If the intersection still contains sufficiently strong tractability operations, the local structures might be globally unified.

If the intersection collapses rapidly, leaving only trivial projections or operations insufficient to support known tractability, then the local algorithms cannot directly share the same closure.

This produces a new tentative metric:

$$
\boxed{
\operatorname{PIS}(\Gamma_1,\ldots,\Gamma_m)
=
\text{Polymorphism Intersection Spectrum}
}
$$

What it studies is not "whether a problem is hard," but:

$$
\text{How much common structure remains when locally solvable algebras are combined.}
$$

---

# 12. Monotone 3-SAT as a Toy Model for "Collapse of Common Structure"

Returning to:

$$
R_+=x\lor y\lor z,
$$

$$
R_-=\neg x\lor\neg y\lor\neg z.
$$

Viewed individually:

$$
R_+
$$

has the trivial global assignment of all 1s;

$$
R_-
$$

has the trivial global assignment of all 0s.

But when mixed, these two "one-trick solutions" directly conflict with each other.

Therefore, even without using the full universal-algebra machinery, one can see:

$$
\boxed{
\text{Locally optimal summaries may be mutually incompatible.}
}
$$

This "summary incompatibility" is the minimal intuitive model of HGD.

It is not a $P\neq NP$ proof, because the NP-completeness of Monotone 3-SAT still only means: if it is in $P$, then $P=NP$.

But it is very suitable for testing:

> Can a Hybrid Quotient Portfolio truly guarantee global polynomials just by "identifying each local easy structure"?

The answer is at least: **Not by local classification alone.**

---

# 13. Another Evidence of "Gluing Debt" from Knowledge Compilation

There are two phenomena in knowledge compilation that fit perfectly with this round.

First, expander-based CNFs can produce strongly exponential representation lower bounds against DNNF. This indicates that certain highly connected formulas cannot be quotiented at a low cost by a certain class of quite strong decomposable representations.

Second, and more subtly: even when the input ultimately has a very small structured DNNF representation, certain bottom-up compilation paradigms may still be forced to generate exponentially large intermediate results.

This gives HGD an important reminder:

$$
\boxed{
\text{Small final summary}
\not\Rightarrow
\text{Small path to construct/glue the summary.}
}
$$

So a complete ledger should at least distinguish between:

$$
\operatorname{HGD}_{\mathrm{final}}
$$

and:

$$
\operatorname{HGD}_{\mathrm{path}}.
$$

The latter studies intermediate representation explosions.

This perfectly aligns with the original series' claim that "construction costs cannot be hidden before execution."

---

# 14. Backdoors Also Upgrade from "Single Shortcuts" to "Heterogeneous Shortcuts"

Backdoor research itself already has heterogeneous base classes: different backdoor assignments can send the formula into different tractable classes.

This is a real plus for the Equals Team:

$$
\text{Not all branches need to enter the same easy language.}
$$

A hybrid solver can use different structures depending on the branch.

But it also gives the Not-Equals Team a new check question:

$$
\boxed{
\text{Does the cost of finding/representing/traversing heterogeneous backdoors remain polynomial?}
}
$$

If the backdoor size is $k$, typical exhaustive search still brings:

$$
2^k
$$

level branching; therefore, what is truly needed is:

$$
k=O(\log n)
$$

or the existence of further quotients/aggregations.

So heterogeneous structures are not a free lunch; they merely upgrade "one easy structure" into "navigation among multiple easy structures."

---

# 15. Random / Expander Families: Suitable for Stress Testing, Not for Mythologizing

Random CNFs and expander-based formulas have strong lower bounds in multiple restricted frameworks such as resolution space/size and knowledge compilation, making them highly suitable as sources for MASC stress tests.

For example, the resolution clause space of random $k$-CNF can have linear-level lower bounds, and certain treelike resolution sizes also have exponential lower bounds; expansion is one of the important combinatorial tools for establishing these results.

But this round prohibits the following leap:

$$
\text{random formula is hard for resolution}
\Rightarrow
P\neq NP.
$$

It also prohibits:

$$
\text{expander CNF is hard for DNNF}
\Rightarrow
P\neq NP.
$$

The role of these formulas is:

$$
\boxed{
\text{Test whether candidate "common coupling mechanisms" can simultaneously explain multiple local lower bounds.}
}
$$

Rather than directly serving as traditional separation proofs.

---

# 16. The First Downgrade of MASC: From "Core" to "Portfolio-relative stress profile"

This round decides to temporarily not treat MASC as an ontological hardness object.

A safer definition is:

$$
\boxed{
\operatorname{MASC}_{\mathcal Q}(F)
=
\text{The anti-structure profile of formula }F\text{ against the specified quotient portfolio }\mathcal Q\text{.}
}
$$

Its uses are:

1. Systematically finding the common failure zones of existing methods;
2. Testing whether different lower bounds share the same underlying cause;
3. Searching for new quotient techniques;
4. Generating theoretical candidates for the next round.

It is not:

$$
\text{general hardness certificate}.
$$

This downgrade is an important self-correction of this round.

---

# 17. The First Version Resource Ledger of HGD

For a decomposition:

$$
\mathcal D=(F_1,\ldots,F_m;I),
$$

where $I$ is the interface structure between local blocks, define the tentative gluing debt vector:

$$
\mathbf D_G(F,\mathcal D)
=
(
D_{\mathrm{partition}},
D_{\mathrm{local}},
D_{\mathrm{boundary}},
D_{\mathrm{summary}},
D_{\mathrm{compat}},
D_{\mathrm{reconstruct}}
).
$$

Where:

- $D_{\mathrm{partition}}$: The cost of finding a useful decomposition;
- $D_{\mathrm{local}}$: The solving/quotienting cost of each block;
- $D_{\mathrm{boundary}}$: The interface degrees of freedom;
- $D_{\mathrm{summary}}$: The cost of precisely representing the boundary behavior;
- $D_{\mathrm{compat}}$: The cost of globally harmonizing different local summaries;
- $D_{\mathrm{reconstruct}}$: The cost of recovering the witness/final answer from the consistent summary.

If:

$$
D_{\mathrm{local}}\in\operatorname{poly}(n)
$$

but:

$$
D_{\mathrm{compat}}
$$

or:

$$
D_{\mathrm{summary}}
$$

explodes, then local easiness still cannot deduce global easiness.

---

# 18. The Equals Team's New Strategy: Don't Look for "A Common Polymorphism", But Switch Algebras Dynamically

The Equals Team refuses to be tied down by PIS.

It proposes:

> Why must a global algorithm have a fixed common polymorphism?

Perhaps the algorithm can:

1. Use Gaussian elimination in affine blocks;
2. Use implication propagation in Horn blocks;
3. Use implication graphs in bijunctive blocks;
4. Use separator DP in graph-like blocks;
5. Use learning/search in the remaining core;
6. Continuously rewrite the interface during the process, allowing each stage to use a different algebra.

Formally:

$$
\mathcal A_t
\in
\{\mathcal A_{\mathrm{Horn}},\mathcal A_{\mathrm{aff}},\mathcal A_{2SAT},\ldots\}
$$

and:

$$
\mathcal A_t\neq\mathcal A_{t+1}
$$

is completely allowed.

This can be called:

$$
\boxed{
\text{Dynamic Algebra Switching}
}
$$

If there exists a polynomial-time orchestrator that automatically switches algebras and keeps all interfaces polynomially succinct, it might escape the "collapse of common polymorphism."

This is the Equals Team's strongest new escape hatch for the next stage.

---

# 19. The Not-Equals Team's New Strategy: Study the "Switching Itself"

The Not-Equals Team's answer is: Fine, then let's account for the switching as well.

Let the algorithm use a sequence of representations/algebras:

$$
\mathcal A_0
\rightarrow
\mathcal A_1
\rightarrow
\cdots
\rightarrow
\mathcal A_T.
$$

Each switch requires an exact bridge:

$$
B_t:
S_t
\rightarrow
S_{t+1}.
$$

Thus the total cost includes:

$$
\sum_t C(B_t).
$$

The new question is not:

$$
\text{Does there exist a single common algebra?}
$$

but:

$$
\boxed{
\text{Does there exist a polynomial-cost algebra switching path?}
}
$$

Note that this is once again very close to the representation closure paradox of Round 6, so the next round must avoid directly putting "all polynomial bridges" into the definition.

---

# 20. The Most Important Thought Experiment of This Round: A World of Two One-Trick Solutions, What Happens When Glued Together?

World A: All constraints can be solved by:

$$
\mathbf x=\mathbf 1
$$

World B: All constraints can be solved by:

$$
\mathbf x=\mathbf 0
$$

Individually:

$$
T_A(n)=O(n),
$$

$$
T_B(n)=O(n).
$$

Now interleave and mix the constraints of A and B.

The question is no longer "Is A hard?" or "Is B hard?", but:

$$
\boxed{
\text{How to find a global assignment that simultaneously satisfies two sets of competing local preferences?}
}
$$

This is the lowest-level version of the heterogeneous gluing problem.

The video initially demonstrated:

$$
\text{Multiple rules can be compressed into a single function.}
$$

This round turns the problem around:

$$
\boxed{
\text{When multiple individually compressible rule systems are coupled together, can they still be compressed into a polynomial-size function?}
}
$$

This is exactly the core worth continuing to play with next.

---

# 21. Obstacle Review

## 21.1 NP-complete does not equal unconditional non-P

Results like Monotone 3-SAT and the Schaefer hard side can only prove NP-completeness.

If one directly writes:

$$
\text{NP-complete}\Rightarrow\text{no polynomial algorithm exists},
$$

that is taking $P\neq NP$ as a premise.

**Status: Blocked.**

## 21.2 Superposition of multi-model lower bounds does not equal general lower bounds

Resolution, DNNF, OBDD, backdoors, treewidth, etc., can be used together as stress tests, but without bridging theorems, they cannot be multiplied into a general lower bound.

**Status: Blocked.**

## 21.3 HGD cannot be defined as "optimal remaining solving time"

Otherwise:

$$
\operatorname{HGD}(F)=\min_A T_A(F)
$$

is just renaming the original problem.

**Status: Blocked.**

## 21.4 Dynamic algebra switching might repeat the closure paradox

If arbitrary polynomial-time bridges are allowed, then "whether one can polynomially switch to a tractable target" is once again synonymous with $L\in P$.

**Status: Focus of the next round.**

---

# 22. Erroneous Routes Eliminated in This Round

1. "A formula is simultaneously hard for OBDD, DNNF, and resolution, therefore $P\neq NP$." — Invalid.
2. "Random 3-SAT is very hard, so it is a general lower bound family." — The corresponding results only hold in specific models/average-case distribution studies.
3. "High treewidth necessarily means not in P." — False; many P problems can have high treewidth instances, and algorithms do not necessarily decompose according to treewidth.
4. "No small backdoor means no polynomial algorithm." — False; backdoors are one class of shortcuts, not all shortcuts.
5. "Every local block is in P, so the global must be in P." — False; gluing can destroy common tractable structures.
6. "Local tractable languages are NP-complete when mixed, thus proving $P\neq NP$." — Circular.
7. "All tractable structures must share a fixed polymorphism." — Very powerful for fixed CSP languages, but cannot yet restrict general dynamic algorithms.
8. "MASC is an ontological hardness core." — Currently, it can only be a portfolio-relative stress profile.

---

# 23. Tentative Results of This Round

## 23.1 MASC is repositioned

$$
\operatorname{MASC}_{\mathcal Q}
$$

only indicates multiple anti-structure pressure against a known quotient portfolio, not general unsolvability.

## 23.2 Found a new common problem: Heterogeneous Gluing Debt

$$
\boxed{
\operatorname{HGD}
}
$$

Separates the measurement of local tractability and global compatibility.

## 23.3 Schaefer provides a formal template for "structural incompatibility"

The tractability of Boolean CSPs relies on the common closure/polymorphism of the entire constraint language, not on whether each constraint is individually easy.

Therefore:

$$
\boxed{
\text{Hardness-like behavior can arise from the "collapse of the intersection of good structures."}
}
$$

## 23.4 Introduced PIS

$$
\operatorname{PIS}(\Gamma_1,\ldots,\Gamma_m)
$$

Tracks the remaining structure of common polymorphisms when local tractable languages are merged.

## 23.5 The Equals Team proposes Dynamic Algebra Switching

Does not require a fixed common closure, but allows the algorithm to switch between different algebras/representations over time.

This pushes the question for the next round to:

$$
\boxed{
\text{Can the bridge between local algebras always remain polynomially cheap?}
}
$$

---

# 24. This Round's Score

Not-Equals Team:

- Successfully upgraded "many methods fail" to "the compatibility of locally solvable structures might be the core";
- Introduced HGD and PIS;
- Supported the research direction that "high coupling leaves debts in multiple local models" with results from expanders, resolution, and knowledge compilation.

Equals Team:

- Successfully prevented MASC from being mistakenly upgraded to a general time lower bound;
- Pointed out that local polymorphism incompatibility might still be bypassed by dynamic representation/algebra switching;
- Retained the "unknown global coordinate system" escape hatch.

This Round's Score:

$$
P=NP:9
\qquad
P\neq NP:9.
$$

Another tie.

This is not intentional score control; it is just that neither side currently has a lethal weapon capable of crossing model boundaries.

---

# 25. Entrance to Round 11

Tentative Topic for the Next Round:

## Collapse of Common Preservation Structures and Dynamic Algebra Switching

Core Showdown:

### Not-Equals Team

Studies how:

$$
\operatorname{Pol}(\Gamma_1)
\cap
\operatorname{Pol}(\Gamma_2)
\cap\cdots
$$

collapses as heterogeneous constraints are mixed, and attempts to connect this collapse to interface/gluing debt.

### Equals Team

Rejects "must share a fixed polymorphism," and proposes the dynamic algebra switching of:

$$
\Gamma_1
\xrightarrow{B_1}
\Gamma_2
\xrightarrow{B_2}
\cdots
\xrightarrow{B_T}
\mathcal T
$$

attempting to prove that even without a common local algebra, one can solve along a polynomial bridge path.

### Common Problem

$$
\boxed{
\text{Can we define a non-circular "algebra switching cost" and identify its incompressible structural source?}
}
$$

---

# 26. Historical Dependencies

This round directly depends on:

1. **Round 2**: Residual distinguishability — reinterpreted as boundary semantic quotient;
2. **Round 4**: Local-global barrier — upgraded this round to the gluing problem of heterogeneous local tractability;
3. **Round 6**: Representation closure paradox — restricts Dynamic Algebra Switching from being arbitrarily relaxed;
4. **Round 7**: Polymorphism / Algorithm-to-Algebra Bridge — forms PIS this round;
5. **Round 8**: PEQS / Exact Quotientability — pushes quotients to block interfaces this round;
6. **Round 9**: Hybrid Quotient Portfolio / Quotient Debt — establishes MASC and HGD this round.

Relationship with Neo.K's old dynamic rate series: The old series emphasized that problem difficulty changes with the agent's structural recognition, knowledge, and representation capabilities; this round turns this into a more granular algorithmic problem—the orchestrator must not only recognize "which local structure is easy," but also handle the global compatibility between different easy structures. This directly connects with the ORCH (Orchestrator) role in the seven-role framework.

---

# 27. External Theoretical References

1. Thomas J. Schaefer, **The Complexity of Satisfiability Problems**, STOC 1978.
   - Boolean CSP dichotomy; the tractable/NP-complete boundary for fixed Boolean constraint languages.
2. Andreas Darmann, Janosch Döcker et al., **Monotone 3-SAT** series of results (2016–2021).
   - Shows that after mixing positive monotone clauses and negative monotone clauses, it can remain NP-complete even when strong occurrence restrictions are applied.
3. Simone Bova, Florent Capelli, Stefan Mengel, Friedrich Slivovsky, **A Strongly Exponential Separation of DNNFs from CNF Formulas**, 2014/2016.
   - Strongly exponential size lower bounds of expander-based CNFs against DNNF.
4. Antoine Amarilli, Mikaël Monet, Pierre Senellart, **Connecting Width and Structure in Knowledge Compilation**, 2017.
   - Connects treewidth/pathwidth with upper/lower bounds of structured representations.
5. Alexis de Colnet, Stefan Mengel, **Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation**, AAAI 2022.
   - Even if a very small structured representation ultimately exists, bottom-up compilation may still be forced to generate exponential intermediate results.
6. Marko Samer, Stefan Szeider, **Backdoor Trees**, AAAI 2008; Serge Gaspers et al., **Backdoors into Heterogeneous Classes of SAT and CSP**, AAAI 2014.
   - Hidden structures, strong backdoors, and heterogeneous tractable base classes.
7. Eli Ben-Sasson, Nicola Galesi, **Space Complexity of Random Formulae in Resolution**, 2001.
   - Resolution space lower bounds of random $k$-CNF and their relationship with expansion.
8. Eli Ben-Sasson, Avi Wigderson, **Short Proofs Are Narrow—Resolution Made Simple**, 1999/2001.
   - The relationship between resolution width and proof size, and the application of expansion to multiple lower bounds.

---

## One-Sentence Conclusion for Round 10

$$
\boxed{
\text{The "anti-structure" truly worth pursuing may not be the absence of good structures, but the inability to glue multiple good structures into a single global structure at a low cost.}
}
$$

Furthermore:

$$
\boxed{
\text{The next battlefield is no longer whether SAT has a Blossom, but whether many local Blossoms can be polynomially glued into a larger flower.}
}
$$