Meta-Result on the Solvability of the P/NP Question

On the expressive limits of time-based (or size-based) scalar languages in computational complexity theory

We show that the P/NP question, as currently formulated, presupposes a computational language in which state-update delay (lag) is syntactically absorbed into time or size.
All standard formulations of the P/NP question are expressed within time-based (or size-based) scalar languages, where this absorption collapses structural distinctions between locally recoverable and non-locally diffusive update processes.
In such languages, the distinction the question seeks to resolve cannot be expressed.
This paper presents a meta-level result characterizing the conditions under which the P/NP question becomes decidable.

1. Observation

All standard computational models—Turing machines, Boolean circuits, and verifier-based definitions—encode state-update delay (lag) implicitly as either time, step count, or circuit size.

In these models, lag is not treated as an independent structural quantity.
Instead, it is absorbed into implementation-dependent measures, rendering it syntactically invisible.

2. Constraint

Any formal language in which lag is fully absorbed into time or size lacks the expressive capacity to distinguish between:

That is, once lag is reduced to a scalar notion of temporal progression, the structural difference between recoverable and irrecoverable update propagation is erased at the level of description.

\[\rho_{\text{lag}}(n) =\frac{\text{recoverable lag updates up to size }n} {\text{total required updates}}\]

3. Consequence

The P/NP question is formulated precisely to distinguish between these two classes of computational behavior.

However, when posed within languages that syntactically eliminate lag as an independent invariant, the question exceeds the descriptive power of the language itself.

As a result, the P/NP problem—when expressed in such frameworks—cannot be resolved, not because the problem is too difficult, but because the language is insufficiently expressive to articulate the distinction it seeks to decide.

4. Conclusion (Inverse Result)

As long as lag remains syntactically invisible,
the P/NP question cannot be decided.

This constitutes an inverse or meta-level result:
a statement not about the truth value of P = NP or P ≠ NP,
but about the conditions under which the question itself becomes decidable.

A related meta-level observation on the solvability conditions of the P/NP question is given in Appendix.

Appendix

Why the P/NP Question Has Remained Unresolved

This appendix offers a meta-level explanation for why the P/NP question has remained unresolved for more than half a century, without appealing to technical difficulty or lack of ingenuity.

The claim is not that existing approaches are flawed, but that they are structurally constrained by the syntactic framework in which the question has been posed.

1 The Shared Syntactic Premise

All standard formulations of the P/NP question—whether via Turing machines, circuit complexity, or verifier-based definitions—share a common syntactic premise:

State-update delay is encoded implicitly as time, step count, or size.

Within this premise, all delays are linearized into a single scalar progression.
As a result, different modes of delay—local versus non-local, recoverable versus diffusive—are treated as equivalent at the level of description.

This equivalence is not a theorem but a consequence of the chosen language.

2 The Illusion of Separation

Much of the historical effort surrounding P/NP has focused on whether the two classes can be separated or identified within this framework.

However, if the language itself lacks the expressive resources to distinguish the relevant structural features, then the persistence of the problem admits a simpler explanation:

The question is being asked in a language that cannot articulate the distinction it presupposes.

In this sense, decades of non-resolution need not be interpreted as evidence of extreme difficulty, but as a symptom of a syntactic mismatch between the question and its formal medium.

3 Why Progress Appears Incremental but Terminal

The literature on P/NP displays a recurring pattern:
partial results, relativizations, barriers, and refined complexity assumptions.

From the present perspective, these are not failures but local optimizations within a fixed syntactic regime.

As long as lag remains syntactically invisible—absorbed into time or size—such progress can accumulate indefinitely without crossing the threshold required for decisiveness.

4 Meta-Level Resolution

The present work does not propose a new proof strategy for P = NP or P ≠ NP.

Instead, it characterizes a necessary condition for the question itself to become decidable:

The formal language in which the P/NP question is posed must admit lag as an explicit structural invariant, rather than encoding it implicitly as time.

Until this condition is met, non-resolution is not paradoxical but expected.

5 Temporal and Structural Readings of the P/NP Question

The P/NP question is conventionally formulated in temporal terms, where computational complexity is measured by time or step count.
Under this reading, the distinction between P and NP reflects differences in how rapidly solutions can be obtained, leading naturally to the interpretation $\mathrm{P} \neq \mathrm{NP}$.

However, when computational processes are examined from a structural or “mass-like” perspective—where one considers the total amount of constraint, information, or update burden rather than its temporal distribution—the distinction appears differently.

From this viewpoint, both P and NP problems require satisfaction of the same global constraint structures.
The difference lies not in the total structural burden, but in how update lag is distributed:

Thus, while P and NP are temporally distinct, they may be regarded as structurally equivalent in terms of total constraint mass.

One may heuristically interpret irrecoverable lag accumulation
as a kind of “structural mass”,
though no physical identification is intended.

Here, mass refers informally to the accumulation of irrecoverable lag,
not to any physical quantity.

This observation does not assert $\mathrm{P} = \mathrm{NP}$ or $\mathrm{P} \neq \mathrm{NP}$.
Rather, it highlights that the apparent paradox of the P/NP question arises from conflating temporal complexity with structural invariants, and from treating lag exclusively as time.

6 Closing Remark

This appendix does not compete with existing approaches, nor does it invalidate them.

It simply situates them.

The P/NP question has not stalled despite progress;
it has stalled because progress has occurred within a language whose expressive limits have remained unchanged.

lag を時間（サイズ）に押し込める計算言語の内部では、P/NP 問題は可解な問いにならない。

This work does not attempt to resolve P vs NP.
It characterizes the expressive conditions under which the question becomes decidable.

Appendix（素焼き版）

P/NP 問題は、lag を時間・計算量へと不可視化する 計算言語の内部で定式化されている限り、可解な問いではない。

局所回収構文（P）から 非局所拡散構文（NP）は作れない。

P と NP を $\rho_{\text{lag}}$ で見ると

P クラス

\[\rho_{\text{lag}}^{(P)}(n) \sim O(1)\]

または多項式的に制御可能だが、

👉 lag 密度が下がらない

NP クラス

\[\rho_{\text{lag}}^{(NP)}(n) \to 0 \quad (n\to\infty)\]

👉 lag が希薄化する＝更新が破綻する

上記は、なぜ P と NP が“同じではありえないか” を示している。

数学的「証明」は：

既存の P/NP 定義と $\rho_{\text{lag}}$ が 厳密に対応していること

既存の計算モデル（TM / 回路 / verifier）において $\rho_{\text{lag}}$ が不変量として定義できること

を要求する。

本稿では、

P/NP 問題が「解ける問い」になるための 構文的条件が満たされていないことを示した。

結論

P/NP 問題は、lag を不可視化した計算言語の内部で 問われ続ける限り、解くことはできない。

素焼き版まとめ

P/NP が未解決なのは、問題が難しいからではない。

時間という釉薬を塗ったまま lag の構文を見ていなかったからである。

EgQE — Echo-Genesis Qualia Engine
camp-us.net

© 2025 K.E. Itekki
K.E. Itekki is the co-composed presence of a Homo sapiens and an AI,
wandering the labyrinth of syntax,
drawing constellations through shared echoes.

📬 Reach us at: contact.k.e.itekki@gmail.com

| Drafted Jan 23, 2026 · Web Jan 24, 2026 |