# P/NP Debate Game Research Area | Round 08

## Algorithm-to-Algebra Bridge Stress Test: From Matching, Flow, Determinant to Exact Quotient Structures

**Round 08: Stress-Testing the Algorithm-to-Algebra Bridge — Exact Quotients Beyond Classical Closure**

- **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 08 Dual-Hypothesis Rehearsal
- **Prerequisite Documents:** `00`–`07`
- **Game Attitude:** Team Equality and Team Inequality continue mutual dismantling
- **Document Standard:** All mathematical conclusions remain marked according to formal research specifications; scores are not evidence

---

## Abstract

Round 07, referencing Schaefer's Boolean CSP and the finite-domain CSP dichotomy, confirmed that polymorphism well explains the tractability of numerous constraint languages, but it cannot yet deduce that "any P-time algorithm necessarily induces classical solution closure." Therefore, Round 08 deliberately selects several problems that clearly do not rely on classical solution-set closure as their main narrative, yet possess polynomial-time algorithms, for stress testing: maximum matching in general graphs, maximum flow, determinant/linear elimination, shortest path, and bounded-treewidth dynamic programming.

The results show that although these algorithms do not share a single polymorphism, a broader structure repeatedly emerges: **exact quotienting / exact aggregation**. Instead of preserving all candidates one by one, the algorithms construct a smaller state where a massive number of mutually distinct local possibilities are compressed into the same equivalence class or summary, provided they have the same effect on the future answer.

For example:

- Edmonds' blossom algorithm contracts odd blossoms into single nodes while preserving the existence of augmenting paths;
- Max-flow uses residual networks to preserve all necessary information for further flow improvement, while max-flow/min-cut forms a dual proof of global optimality;
- Gaussian elimination uses row elimination to transform a linear system into a simpler but equivalent system;
- Shortest-path algorithms use min-plus/semiring structures to aggregate exponentially many paths into a small number of distance values;
- Bounded-treewidth DP only preserves states on the separator/bag boundaries relevant to future compositions, discarding massive internal details of the subgraphs.

This round thus proposes the candidate concept **Polynomial Exact Quotient Scheme (PEQS)**. However, Team Equality immediately points out: if PEQS allows "treating the machine state of any P-time solver as a quotient state," it would degenerate into a tautology of $L\in P$. Therefore, this round does not prove that "P is equivalent to PEQS," but rather rewrites the Algorithm-to-Algebra Bridge from Round 07 into a more precise new question:

$$
\boxed{
\text{Can we define a sufficiently broad class of admissible exact quotient calculus, independent of the solver itself,}
}
$$

$$
\boxed{
\text{such that numerous known P problems can be compressed within it, while general SAT cannot?}
}
$$

The main result of this round is not a P/NP conclusion, but the discovery of a broader candidate common mechanism than classical polymorphism, while explicitly identifying its greatest risk: **if the quotient structure is too broad, it will once again form a closure paradox and circular definition.**

---

# I. The Unresolved Challenge from the Previous Round

The core question established in Round 07 is:

$$
A\in P
\stackrel{?}{\Longrightarrow}
\text{Some non-trivial, independently definable preservation/aggregation structure}.
$$

For Boolean CSPs, tractable languages like Horn, dual-Horn, bijunctive, and affine indeed each have corresponding polymorphisms or algebraic preservation structures; the finite-domain CSP dichotomy also uses Taylor/WNU-type operations as the core algebraic signal for tractability.

But general P is not solely composed of CSPs.

Thus, Team Equality proposes a stress test:

> If Team Inequality truly believes that "P-time necessarily stems from some preservable structure," then please explain exactly what is preserved behind polynomial algorithms like matching, flow, determinant, and shortest path.

If these cases cannot be incorporated, the Algorithm-to-Algebra Bridge is merely a localized CSP phenomenon.

---

# II. Stress Test A: Maximum Matching in General Graphs

Let $G=(V,E)$, the goal is to find a maximum matching in a general graph.

Berge's lemma provides the basic criterion: a matching $M$ is a maximum matching if and only if there is no augmenting path relative to $M$.

The difficulty in general graphs lies in odd cycles, which disrupt the simple alternating-tree search used in bipartite graphs.

The key operation performed by Edmonds' blossom algorithm is not enumerating all possibilities within an odd cycle, but rather:

$$
\text{blossom cycle}
\longrightarrow
\text{single contracted supervertex}.
$$

The core correctness property is: proper blossom contraction preserves the existence of an augmenting path. That is, an augmenting path exists in the original graph if and only if a corresponding augmenting path exists in the contracted graph; once found, the path is lifted back to the original graph.

This can be tentatively written as:

$$
G\xrightarrow{\operatorname{contract}(B)}G/B,
$$

and:

$$
\operatorname{Aug}(G,M)
\neq\varnothing
\iff
\operatorname{Aug}(G/B,M/B)\neq\varnothing.
$$

This is not a classical solution-set closure.

It is a **semantics-preserving quotienting**: the massive permutation differences inside an odd cycle can be viewed as the same macroscopic state regarding the question of "whether an augmenting path can still be formed in the future."

Even more interestingly, matching also possesses a completely different algebraic escape: the Tutte matrix can link the existence of a perfect matching to whether the determinant of a skew-symmetric symbolic matrix is the zero polynomial. That is:

$$
\text{combinatorial matching existence}
\longrightarrow
\text{algebraic determinant identity}.
$$

Therefore, Matching simultaneously demonstrates:

1. combinatorial quotient;
2. algebraic representation escape.

This is very unfriendly to Team Inequality: it illustrates that problems which "look highly global and combinatorially explosive" can indeed harbor completely different, low-cost mathematical coordinate systems.

---

# III. Stress Test B: Maximum Flow

The state of the maximum flow problem can be expressed using a residual network.

For the current flow $f$, the residual graph:

$$
G_f
$$

preserves not the history of how $f$ was obtained in the past, but the necessary information on **how it can still be improved in the future**.

If a residual augmenting path exists, the flow can still be increased; if no augmenting path exists, the max-flow/min-cut theorem provides a cut such that:

$$
|f|=c(S,T),
$$

By weak duality, it immediately follows that $f$ is already a maximum flow.

Here, two types of aggregation appear again:

### 3.1 History Compression

Different augmentation histories, as long as they yield the same residual capacities, can be considered equivalent regarding subsequent improvability.

$$
H_1\sim H_2
\quad\text{if they induce the same relevant residual state.}
$$

### 3.2 Primal-Dual Compression

Regarding "whether global optimality has been reached," there is no need to check all possible flows; one only needs to find a matching cut certificate:

$$
\text{feasible flow}
+
\text{equal-valued cut}
\Longrightarrow
\text{global optimum}.
$$

So the tractability of flow is not just solution closure, but:

$$
\boxed{
\text{Updatable residual quotient + dual structure capable of short-proving global optimum}
}
$$

---

# IV. Stress Test C: Determinant and Exact Elimination

For an $n\times n$ matrix $A$, the Leibniz expansion of the determinant contains:

$$
 n!
$$

permutation terms.

But computing the determinant does not require expanding all permutations term by term.

Gaussian elimination step-by-step transforms the matrix into triangular form:

$$
A
\rightarrow
A^{(1)}
\rightarrow
\cdots
\rightarrow
U,
$$

while preserving a traceable relationship between the determinant and the transformed matrix through elementary operations.

Therefore:

$$
\det(A)
$$

is compressed from a global quantity seemingly involving $n!$ combinatorial terms into polynomially many elimination steps.

But here is a very important reminder for this series: if one only counts arithmetic operations while ignoring coefficient bit-length, numerical explosion might be hidden. Bareiss's fraction-free elimination and similar methods are precisely about controlling intermediate growth in exact arithmetic.

Thus, the determinant case once again supports the complete cost ledger of this series:

$$
T_{\mathrm{arithmetic}}
+
L_{\mathrm{coefficient}}
+
P_{\mathrm{precision}}.
$$

True P-time compression must also keep bit complexity under control.

---

# V. Stress Test D: Shortest Path and Semiring Aggregation

The shortest path problem superficially also has exponentially many candidate paths.

But Dijkstra, Bellman-Ford, Floyd-Warshall, and more abstract algebraic path frameworks do not need to preserve all paths.

For a general semiring $(K,\oplus,\otimes)$, one can write:

- Composition along a path as $\otimes$;
- Aggregation between different paths as $\oplus$.

In the classical shortest path:

$$
\otimes=+,
\qquad
\oplus=\min.
$$

Therefore, a node might correspond to exponentially many arrival paths, but for future optimization, one only needs to retain:

$$
 d(v)=\min_{P:s\leadsto v}w(P).
$$

That is:

$$
\{
P_1,P_2,\ldots
\}
\longrightarrow
 d(v).
$$

This is an extremely strong equivalence class compression: all paths worse than the current best distance can be directly dominated away regarding the "future shortest path value" problem.

Bellman's principle of optimality provides the recursive justification for this compression: the optimal global solution can be composed of appropriately defined optimal subproblem values.

Thus, shortest path also lacks classical polymorphism, but has:

$$
\boxed{
\text{associative composition + exact aggregation + dominance elimination}
}
$$

---

# VI. Stress Test E: Bounded-Treewidth Dynamic Programming

For bounded-treewidth graph problems, tree decomposition provides a series of small separators/bags.

When a subgraph interacts with the rest of the graph only through bag $B$, all internal details of the subgraph need not be preserved.

One only needs to preserve its effect on the boundary configuration:

$$
\operatorname{Summary}(G_{\mathrm{sub}},B).
$$

If:

$$
|B|\leq k
$$

and $k$ is fixed, then the boundary states typically depend only on $k$ rather than the entire $n$.

Different internal subgraphs, if they exhibit the same boundary behavior for all future glue operations, can be merged into the same summary.

Courcelle-type meta-theorems further show that on bounded treewidth, a vast range of MSO-definable properties can be processed via finite states/tree automata/dynamic programming.

This case almost directly reproduces the "residual distinguishability" from Round 02, except that here the separator is naturally provided by the problem structure, rather than arbitrarily fixed by the researcher.

---

# VII. What Repeatedly Emerges Behind the Five P Problems?

Putting the above cases together:

| Problem | Original Massive Space | Exact Compression / Quotienting | Core Preserved Semantics |
|---|---|---|---|
| Matching | alternating paths / odd-cycle interactions | blossom contraction | augmenting-path existence |
| Max Flow | All flow improvement histories | residual network | future augmentability / optimality |
| Determinant | $n!$ permutation terms | elimination | exact determinant relation |
| Shortest Path | Exponentially many paths | min-plus / semiring aggregation | optimal path value |
| Treewidth DP | Massive internal configurations of subgraphs | boundary table | future glue behavior |

On the surface, the mathematics they use are completely different.

But there is a common template:

$$
\boxed{
\text{Massive microscopic candidates}
\rightarrow
\text{Establish equivalence classes based on "effect on future answers"}
\rightarrow
\text{Preserve only representatives/summaries}
}
$$

Therefore, this round proposes a new candidate common mechanism:

$$
\boxed{
\text{Exact Quotientability}
}
$$

Namely, "Exact Quotientability."

---

# VIII. Candidate Definition: Polynomial Exact Quotient Scheme (PEQS)

Here we only provide a working definition and do not claim it as a standard complexity concept.

For a problem family $\mathcal F$, a PEQS is tentatively composed of the following structure:

$$
\mathcal Q
=
(\mathcal D,\Sigma,\rho,\otimes,\operatorname{Lift},\operatorname{Dec}).
$$

Where:

- $\mathcal D$: Allowed decomposition/reduction operations;
- $\Sigma$: Summary state space;
- $\rho$: Quotient map mapping local states to summaries;
- $\otimes$: Composition/update operator for summaries;
- $\operatorname{Lift}$: If a witness needs to be constructed, recovers the original solution from the quotient solution;
- $\operatorname{Dec}$: Decodes the answer from the final summary.

For a PEQS to be called "polynomial," it requires at least:

$$
L(\sigma)\leq\operatorname{poly}(n),
$$

$$
T_{\rho},T_{\otimes},T_{\mathrm{Lift}},T_{\mathrm{Dec}}
\leq\operatorname{poly}(n),
$$

and the overall quotient depth:

$$
D_{\mathcal Q}(n)\leq\operatorname{poly}(n).
$$

The most important aspect is exactness:

$$
\operatorname{Ans}(I)
=
\operatorname{Dec}
\left(
\operatorname{Aggregate}_{\mathcal Q}(I)
\right).
$$

If it is a search problem, Lift must also guarantee the recovery of a valid witness.

---

# IX. Team Inequality's New Conjecture: Tractability = Controllable Exact Quotienting?

Team Inequality proposes a version that is more concrete than Round 07's IAC, though it remains merely a rehearsal:

## Polynomial Exact Quotient Conjecture (PEQC, Candidate Conjecture)

For "natural" exact combinatorial problems, if a robust P-time algorithm exists, one can usually extract some:

$$
\text{polynomially bounded exact quotient structure}.
$$

Its core is not classical polymorphism, but:

$$
\boxed{
\text{Continuously quotienting massive candidate sets with equivalent future behaviors into polynomial summaries.}
}
$$

This can simultaneously accommodate:

- polymorphism closure;
- blossom contraction;
- residual graph;
- Gaussian elimination;
- semiring DP;
- separator DP.

Team Inequality therefore redescribes its ultimate dream:

> If some hard family of SAT lacks any admissible polynomial exact quotient scheme, it might approach $P\neq NP$.

But this statement is currently far from a proof.

---

# X. Team Equality's Counterattack: PEQS is Becoming a Tautology Again

Team Equality immediately proposes the PEQS version of the closure paradox from Round 06.

Assume $L\in P$, there exists a solver $A$:

$$
A(x)\in\{0,1\}
$$

that completes in polynomial time.

Then one can define:

$$
\sigma_t
=
\text{machine state of A at step }t.
$$

Every step:

$$
\sigma_t\rightarrow\sigma_{t+1}
$$

is a polynomially describable exact update; finally:

$$
\operatorname{Dec}(\sigma_T)=A(x).
$$

Therefore, if PEQS allows "any solver state," then:

$$
L\in P
\Longrightarrow
L\text{ has PEQS}
$$

is merely renaming "has a P-time algorithm" to "has a polynomial quotient process."

If we reversely define that PEQS must be executable in P-time, then:

$$
L\text{ has PEQS}
\Longrightarrow
L\in P.
$$

Yielding:

$$
\boxed{
L\in P
\iff
L\text{ has unrestricted PEQS}.
}
$$

This contains absolutely no new content.

**Team Equality scores.**

---

# XI. Therefore, "Non-circular Admissibility" Must Be Added

If PEQS is to become a true Algorithm-to-Structure bridge, it must be subject to independent constraints.

This round proposes six candidate conditions for admissibility.

## 11.1 Solver-independent

The quotient rules must be definable by the problem structure itself, without first executing the complete solver.

For example, the definition of blossom contraction comes from the alternating-tree/odd-cycle structure, rather than "first computing the maximum matching, then writing the answer into the summary."

## 11.2 Local or compositional

Quotient operations should act on identifiable substructures/interfaces and be able to compose global summaries from local summaries.

## 11.3 Answer-blind

The quotient map must not directly query the answer to the original problem:

$$
\rho(I)\neq\operatorname{Ans}(I)
$$

as a hidden oracle.

## 11.4 Exact semantic preservation

Quotienting must provably preserve some mathematical relationship more independent than "the algorithm's final output," such as:

- augmenting path existence;
- row-space equivalence;
- residual feasibility;
- boundary extension behavior;
- semiring path value.

## 11.5 Polynomially checkable law

The preservation law or composition law should be independently formalizable/verifiable, rather than relying on "it must be preserved because the solver is correct."

## 11.6 Resource-complete

Must account for:

$$
(T_{\mathrm{construct}},L_{\mathrm{summary}},M,P,T_{\mathrm{compose}},T_{\mathrm{lift}}).
$$

It is forbidden to hide exponential costs in precision, preprocessing, or nonuniform advice.

---

# XII. Team Equality Strikes Hard Again: Matching Itself is a Warning of an "Unknown SAT Blossom"

Team Equality launches the strongest intuitive attack of this round:

Before Edmonds, odd cycles in general graph matching indeed made naive augmenting-path reasoning intractable.

The real breakthrough was not searching odd cycles more violently, but discovering:

$$
\boxed{
\text{The entire blossom can be contracted.}
}
$$

Similarly, the Tutte matrix showed that matching can jump from a combinatorial graph representation to a determinant representation.

So even if today we see SAT exhibiting:

- resolution explosion;
- OBDD explosion;
- large treewidth;
- large LP extension;
- local consistency failure;

Team Equality can still say:

> You simply haven't found SAT's blossom, SAT's Tutte matrix, or SAT's Gaussian elimination yet.

This is not a proof of $P=NP$, but it is an effective defense:

$$
\text{All known representation-specific lower bounds}
\not\Rightarrow
\text{Non-existence of unknown exact quotient structures}.
$$

---

# XIII. Team Inequality's Response: Then Study the "Quotient Laws Themselves" Instead of the Quotient Results

Team Inequality accepts this attack and modifies the goal.

Rather than attempting to enumerate all representations, it is better to study what **quotient laws** a tractable quotient typically needs to satisfy:

### 13.1 Composability

$$
\rho(A\cup B)
=
F(\rho(A),\rho(B),\operatorname{interface}(A,B)).
$$

### 13.2 Eliminability

After a local degree of freedom is eliminated, its entire effect on the future answer is still preserved by the summary.

### 13.3 Liftability

If the quotient space yields a solution/certificate, it can be lifted back to the original problem in polynomial time.

### 13.4 Progressiveness

Each quotient should decrease some independent structural quantity:

$$
\mu(I_{t+1})<\mu(I_t)
$$

or at least cause the unprocessed problem size to terminate within polynomial steps.

### 13.5 Bounded Interface

The impact of the eliminated portion on the remaining system can be transmitted through a polynomially describable interface.

Here, a new possible direction begins to emerge:

$$
\boxed{
\text{The common feature of P might not be a fixed closure, but the existence of a repeatable exact quotient calculus.}
}
$$

However, "all P are like this" remains unproven.

---

# XIV. Re-understanding Round 03's CRC

Round 03 proposed:

$$
\operatorname{CRC}
=
\text{Causal Reconstruction Complexity}.
$$

The issue at the time was: information does not disappear, but the causal reconstruction required to convert it into an exact answer might be difficult.

PEQS now provides a new dual perspective:

- **Low CRC:** Exact quotients can be continuously established, so future reconstruction only needs to process small summaries;
- **High CRC:** Any admissible quotient leaves a massive interface, long reconstruction chains, or representation explosion.

Therefore, we can tentatively write:

$$
\operatorname{CRC}(I)
\approx
\min_{\mathcal Q\in\mathrm{Admissible}}
\operatorname{Cost}(\mathcal Q,I),
$$

But an immediate warning must be given:

If $\mathrm{Admissible}$ is too broad, this again degenerates into the optimal algorithm time.

So the true core of the research becomes:

$$
\boxed{
\text{How should an admissible quotient calculus be independently defined?}
}
$$

---

# XV. This Round's Representation/Algorithm Stress Test Matrix

| Problem | Classical solution closure obvious? | Exact quotient/aggregation exists? | Main Preservation Law | Escape Form |
|---|---:|---:|---|---|
| Horn-SAT | Yes | Yes | meet closure | lattice / implication |
| 2-SAT | Yes / algebraizable | Yes | majority-like / implication graph | SCC |
| XOR-SAT | affine closure | Yes | linear span | Gaussian elimination |
| Matching | No | Yes | augmenting-path preservation | blossom / Tutte matrix |
| Max Flow | No | Yes | residual feasibility + min-cut duality | residual quotient |
| Determinant | No | Yes | elimination identity | triangularization |
| Shortest Path | No | Yes | semiring aggregation / optimal substructure | min-plus DP |
| bounded-treewidth DP | Not necessarily | Yes | boundary extension equivalence | separator summary |
| General SAT | Unknown | Massive local schemes, but no known general PEQS | Unknown | Core battlefield |

---

# XVI. Newly Excluded Erroneous Routes

This round adds the following restricted zones:

1. **All P-time algorithms have machine states, so it is proven that exact quotient is the structural essence of P.**
   - False; this is a circular restatement.

2. **Matching has blossoms, so SAT must have blossoms.**
   - False; one can only say that unknown representation escapes cannot be excluded a priori.

3. **Some algorithm has local contraction, so general tractability stems from local contraction.**
   - False; the aggregation in algorithms like determinant is not entirely graph-local.

4. **All candidates can be quotiented, so $P=NP$.**
   - False; a uniform, exact, resource-bounded scheme is required.

5. **SAT currently has no known PEQS, so $P\neq NP$.**
   - False; epistemological absence does not equal ontological non-existence.

6. **Directly taking the minimum over all polynomial algorithms in the PEQS definition.**
   - False; this would replay the closure paradox of Round 06.

---

# XVII. Official Offense and Defense of Both Teams This Round

## 17.1 Team Equality

Strongest argument:

> The reason a large number of known problems in P become easy from seemingly massive candidate spaces is often not that candidates become fewer, but that a new quotient/algebraic representation is found. Matching's blossom and Tutte matrix are the most cautionary examples. Therefore, any $P\neq NP$ route that only blocks known representations still leaves the "unknown SAT quotient" escape hatch open.

## 17.2 Team Inequality

Strongest argument:

> These successful P algorithms are not magic; every compression has an independently statable preservation law, composition law, bounded interface, or dual certificate. The truly possible lower bound is not proving "there are no conceivable representations," but proving that SAT hard families cannot possess a broad class of solver-independent exact quotient laws.

---

# XVIII. This Round's Ruling

This round did not prove:

$$
P=NP
$$

nor did it prove:

$$
P\neq NP.
$$

But the Algorithm-to-Algebra Bridge was successfully expanded:

$$
\text{classical polymorphism}
\quad\Longrightarrow\quad
\text{more general exact quotient / aggregation structure}.
$$

The new core candidate is not some fixed algebra, but:

$$
\boxed{
\text{Can a massive state space equivalent for future answers be repeatedly quotiented using polynomial resources?}
}
$$

This round's score:

$$
P=NP:7
\qquad
P\neq NP:7.
$$

- Team Equality scores: Cases like Matching/Tutte matrix prove that "unknown representation revolutions" are phenomena that have truly existed;
- Team Inequality scores: A common exact compression pattern broader than polymorphism was indeed extracted across different P algorithms.

The score is merely a game interface, not a mathematical conclusion.

---

# XIX. Gateway to Round 09: Searching for SAT's Blossom

The next round will no longer just abstractly discuss PEQS, but will directly turn around and work for Team Equality:

$$
\boxed{
\text{If }P=NP\text{, what might SAT's blossom look like?}
}
$$

We will deliberately attempt to construct candidate exact quotient operators for SAT, such as:

- variable elimination;
- clause/component contraction;
- implication quotient;
- symmetry quotient;
- XOR/affine extraction;
- bounded-interface decomposition;
- backdoor condensation;
- learned-clause state compression;
- algebraic projection.

Then Team Inequality will attack them one by one:

$$
\text{summary size}
,
\text{fill-in}
,
\text{width}
,
\text{precision}
,
\text{lift cost}
,
\text{uniformity}.
$$

Therefore, the official topic for Round 09 is:

## **"Searching for SAT's Blossom: The Candidate Exact Quotient Operator Tournament"**

This will be the round closest to the inspiration of the original video so far:

$$
\text{Can the massive conditional judgments of SAT be directly folded into new mathematical states, like Rock-Paper-Scissors or blossoms?}
$$

---

# XX. External Theoretical References

1. J. Edmonds, **“Paths, Trees, and Flowers,”** *Canadian Journal of Mathematics*, 17 (1965), 449–467.  
   DOI: https://doi.org/10.4153/CJM-1965-045-4
2. M. Abdulaziz and K. Mehlhorn, **“A Formal Correctness Proof of Edmonds’ Blossom Shrinking Algorithm,”** 2026.  
   Core reference: blossom contraction preserves augmenting path existence.
3. Ford–Fulkerson / Max-Flow Min-Cut theorem.  
   Core reference: residual network, augmenting path, and cut certificate.
4. E. H. Bareiss, **“Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination,”** *Mathematics of Computation*, 22 (1968), 565–578.  
   Core reference: exact elimination and coefficient-growth control.
5. M. Mohri, **“Semiring Frameworks and Algorithms for Shortest-Distance Problems,”** *Journal of Automata, Languages and Combinatorics*, 2002.  
   Core reference: unifying path composition/aggregation with semirings.
6. R. Bellman, **“The Theory of Dynamic Programming,”** RAND, 1954.  
   Core reference: principle of optimality and value-state compression.
7. B. Courcelle, tree-decomposition / MSO series of works.  
   Core reference: bounded interface and finite-state dynamic programming.
8. Tutte matrix / determinant-based matching literature.  
   Core reference: the algebraic bridge between perfect matching existence and determinant non-zeroness.

---

## One-Sentence Summary of This Round

$$
\boxed{
\text{P problems do not necessarily have the same kind of closure, but they often have some mathematical method to "exactly quotient away future-equivalent states."}
}
$$

The real game now becomes:

$$
\boxed{
\text{Does SAT truly lack such a quotient method, or have we just not found it yet?}
}
$$