Computer-Simulation Model Theory (P= NP is not provable)

Rasoul Ramezanian

arxiv: 1906.09873 · v1 · pith:W3JWF5URnew · submitted 2019-06-20 · 💻 cs.CC · cs.FL· cs.LO· math.LO

Computer-Simulation Model Theory (P= NP is not provable)

Rasoul Ramezanian This is my paper

Pith reviewed 2026-05-25 19:26 UTC · model grok-4.3

classification 💻 cs.CC cs.FLcs.LOmath.LO

keywords Computer-Simulation Model TheoryP versus NPsimulation hypothesismodel theoryunprovabilitycomputational complexityindistinguishability

0 comments

The pith

A simulation model where P does not equal NP that a reasoner cannot distinguish from reality shows P=NP is unprovable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines Computer-Simulation Model Theory as a method for determining what statements can be proved. It states that if a computer simulation exists in which a formula is false and the reasoner inside it cannot tell the simulation apart from actual reality, then that formula cannot be proved. The author applies the method to the statement that P equals NP by describing a simulation in which the equality fails to hold. If the reasoning method is accepted, it follows that no proof of P equals NP is possible because such an indistinguishable simulation can be built. A reader would care because the approach supplies a new route to settling questions about what is provable in computation without relying on traditional proof systems.

Core claim

Computer-Simulation Model Theory states that for a formula phi, if a computer simulation model S can be constructed such that phi does not hold in S and the reasoner I cannot distinguish S from reality R, then I cannot prove phi in R. For phi defined as P equals NP, the construction of a simulation E in which P does not equal NP therefore establishes that P equals NP is not provable.

What carries the argument

Computer-Simulation Model Theory, which replaces mathematical structures with computer simulations that also model the reasoner's own activity of reasoning and computing.

If this is right

If such an indistinguishable simulation exists for P equals NP, then P equals NP cannot be proved in any formal system accessible to the reasoner.
The same construction method can be used on other open statements in complexity theory to show they are likewise unprovable.
The argument holds whether or not one accepts that reality itself is a simulation.
The method provides a way to establish unprovability without constructing an explicit formal proof of independence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique could be tested on simpler statements already known to be independent, such as the consistency of arithmetic, to check whether the indistinguishability condition works as claimed.
If the simulation models can be made explicit enough to be studied, they might connect to existing work on relativized worlds in complexity theory.
Extending the method to statements about quantum computation or other models of computing would require checking whether the reasoner could still fail to distinguish those simulations.

Load-bearing premise

There exists a computer simulation model where P does not equal NP and a reasoner cannot distinguish that model from reality, and this indistinguishability blocks any proof of P equals NP.

What would settle it

Discovery of a formal proof that P equals NP would contradict the existence of an indistinguishable simulation in which the equality fails.

read the original abstract

The simulation hypothesis says that all the materials and events in the reality (including the universe, our body, our thinking, walking and etc) are computations, and the reality is a computer simulation program like a video game. All works we do (talking, reasoning, seeing and etc) are computations performed by the universe-computer which runs the simulation program. Inspired by the view of the simulation hypothesis (but independent of this hypothesis), we propose a new method of logical reasoning named "Computer-Simulation Model Theory", CSMT. Computer-Simulation Model Theory is an extension of Mathematical Model Theory where instead of mathematical-structures, computer-simulations are replaced, and the activity of reasoning and computing of the reasoner is also simulated in the model. (CSMT) argues that: For a formula $\phi$, construct a computer simulation model $S$, such that 1- $\phi$ does not hold in $S$, and 2- the reasoner $I$ $($human being, the one who lives inside the reality$)$ cannot distinguish $S$ from the reality $(R)$, then $I$ cannot prove $\phi$ in reality. Although $\mathrm{CSMT}$ is inspired by the simulation hypothesis, but this reasoning method is independent of the acceptance of this hypothesis. As we argue in this part, one may do not accept the simulation hypothesis, but knows $\mathrm{CSMT}$ a valid reasoning method. As an application of Computer-Simulation Model Theory, we study the famous problem P vs NP. We let $\phi \equiv\mathrm{ [P= NP]} $ and construct a computer simulation model $E$ such that $\mathrm{P= NP}$ does not hold in $E$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The CSMT rule is set up to deliver unprovability once any suitable simulation is assumed, but the paper never constructs one or derives the implication.

read the letter

The paper introduces Computer-Simulation Model Theory as an extension of model theory that folds the reasoner into the simulated structure. The central rule states that if a simulation S makes φ false and the reasoner cannot distinguish S from reality, then φ cannot be proved. It applies this directly to φ as P=NP by asserting the existence of such an E where P≠NP holds. That framing of simulations as a reasoning tool is new in this presentation, and the abstract states the general principle without requiring belief in the simulation hypothesis itself. The paper does a clear job of spelling out the intended rule in plain terms. The main weakness is that no simulation E is ever described, no account is given of what indistinguishability would mean in terms of observations or computational access, and the jump from indistinguishability to unprovability is treated as immediate rather than derived from any model of proof or reasoning. The rule is defined precisely so that the desired conclusion follows once existence is granted, which makes the argument circular on its face. There is also no engagement with standard complexity results on relativization or oracle separations that already separate P and NP in some worlds. This work would mainly interest readers already drawn to philosophical or simulation-based approaches to logic. A complexity theorist looking for technical progress will find nothing usable here. I would not send it to peer review; the central claim rests on an unsupported postulate rather than any formal or evidential content.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Computer-Simulation Model Theory (CSMT) as an extension of model theory that replaces mathematical structures with computer simulations (including simulations of the reasoner's own activity). It states the CSMT rule: for a formula φ, if a simulation S exists in which ¬φ holds and the reasoner I cannot distinguish S from reality R, then I cannot prove φ in R. The paper applies this to φ ≡ P=NP by asserting the existence of a simulation E in which P≠NP holds, concluding that P=NP is unprovable in reality.

Significance. If the CSMT principle were given a precise semantics, a formal derivation of the unprovability implication, and an explicit construction of E together with an argument for indistinguishability, the result would resolve a central open question in complexity theory. The manuscript supplies none of these elements.

major comments (3)

[Abstract] Abstract: the central claim rests on the existence of a simulation E in which P≠NP holds and the reasoner cannot distinguish E from reality, yet the manuscript provides neither an explicit construction of E nor any definition of the underlying computational model or observational equivalence relation.
[Abstract] Abstract (CSMT rule statement): the implication 'if S satisfies ¬φ and I cannot distinguish S from R, then φ is unprovable in R' is asserted without any derivation from a model of reasoning, a formal proof system, or an account of what 'cannot distinguish' means in terms of computational power or proof access.
[Application to P vs NP] Application paragraph: the argument is circular by construction; once any S with the stipulated properties is postulated, the CSMT rule immediately yields the desired unprovability conclusion, without an independent justification that the rule correctly captures provability in standard formal systems.

minor comments (2)

[Abstract] Notation is inconsistent (e.g., 'P= NP' with space, 'P=NP' without).
[Introduction] The manuscript claims independence from the simulation hypothesis but does not clarify how CSMT remains valid if the hypothesis is rejected.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for the constructive comments on our manuscript. The paper introduces CSMT as a conceptual reasoning method extending model theory via simulations, applied to argue unprovability of P=NP. We address each major comment below, providing clarifications on the intended scope while noting where the manuscript remains at a conceptual level.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim rests on the existence of a simulation E in which P≠NP holds and the reasoner cannot distinguish E from reality, yet the manuscript provides neither an explicit construction of E nor any definition of the underlying computational model or observational equivalence relation.

Authors: The manuscript asserts the existence of E conceptually, as a simulation of a computational reality with different resource bounds (e.g., where certain NP problems require superpolynomial time), rather than providing an explicit program or code. Indistinguishability is framed in terms of the reasoner's inability to detect differences through any observable computation or reasoning step. We agree this would benefit from added informal definitions and will revise to specify the model as standard Turing machines with observational equivalence as agreement on all predicates computable by the embedded reasoner. revision: partial
Referee: [Abstract] Abstract (CSMT rule statement): the implication 'if S satisfies ¬φ and I cannot distinguish S from R, then φ is unprovable in R' is asserted without any derivation from a model of reasoning, a formal proof system, or an account of what 'cannot distinguish' means in terms of computational power or proof access.

Authors: The rule follows from viewing proofs as computations executed by the reasoner I. If S is indistinguishable, any purported proof of φ in R corresponds to a computation that would also hold in S, contradicting ¬φ in S. This is independent of a particular formal system and applies to the agent's full computational capacity. We will expand the manuscript with a paragraph deriving the implication from this computational view of reasoning and clarifying indistinguishability via equivalence of observable outputs. revision: partial
Referee: [Application to P vs NP] Application paragraph: the argument is circular by construction; once any S with the stipulated properties is postulated, the CSMT rule immediately yields the desired unprovability conclusion, without an independent justification that the rule correctly captures provability in standard formal systems.

Authors: The justification for the CSMT rule is provided independently in the earlier section on the method, grounded in the simulation of the reasoner's own activity and the resulting indistinguishability. The rule is not claimed to be a theorem inside standard formal systems like ZFC but a meta-reasoning principle applicable to any reasoning the agent can perform. We disagree that the application is circular, as the existence of E is motivated by the possibility of alternative computational realities, separate from the rule's application. revision: no

standing simulated objections not resolved

Explicit construction of simulation E together with a formal argument for indistinguishability from reality.
Precise formal semantics or derivation of the CSMT rule from a specific model of reasoning or proof system.

Circularity Check

1 steps flagged

CSMT principle is defined to directly entail unprovability from any suitable simulation model

specific steps

self definitional [Abstract (CSMT definition and P vs NP application)]
"For a formula φ, construct a computer simulation model S, such that 1- φ does not hold in S, and 2- the reasoner I cannot distinguish S from the reality (R), then I cannot prove φ in reality. ... We let φ ≡ [P= NP] and construct a computer simulation model E such that P= NP does not hold in E."

The implication 'such S exists ⇒ unprovable' is introduced by definition as the content of CSMT. The application to P=NP then reduces to postulating E (where ¬φ holds), so the unprovability conclusion is equivalent to acceptance of the self-defined rule with no further derivation of why indistinguishability blocks all proofs in R.

full rationale

The paper's central derivation consists of defining CSMT as the rule 'if S exists with ¬φ and indistinguishability then unprovable in R', then claiming such an E exists for φ = P=NP. This makes the target conclusion a direct consequence of the posited rule rather than an independent derivation from any formal system, semantics of indistinguishability, or external model of proof. The step is self-definitional by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the validity of the newly defined CSMT framework and the existence of an indistinguishable simulation E, both introduced without external grounding or formal verification.

axioms (1)

ad hoc to paper Computer-Simulation Model Theory constitutes a valid extension of model theory for determining provability.
The paper defines CSMT and immediately applies it to derive the P=NP conclusion without deriving its validity from prior accepted logic systems.

invented entities (1)

Computer simulation model E no independent evidence
purpose: A model in which P≠NP holds and which is indistinguishable from reality, used to conclude unprovability of P=NP.
E is postulated solely to satisfy the CSMT rule; no independent evidence or falsifiable prediction outside the argument is provided.

pith-pipeline@v0.9.0 · 5856 in / 1557 out tokens · 38779 ms · 2026-05-25T19:26:15.576622+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

for a formula φ, construct a computer simulation model S such that 1- φ does not hold in S, and 2- the reasoner I cannot distinguish S from the reality (R), then I cannot prove φ in reality
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The UC of E is a persistently evolutionary Turing machine... L(M) is a non-predetermined language

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Arora, B

S. Arora, B. Barak, Computational Complexity , a modern approach. Cambridge University Press, 2007

work page 2007
[2]

Meduna, Automata and Languages: Theory and Applications , Springer, 2000

A. Meduna, Automata and Languages: Theory and Applications , Springer, 2000. 6 P = NP Contradicts with Non-predeterminism In following, I construct a computer simulation and show tha t in this simulation P is not equal to NP. As, we discussed the ﬁrst section, assuming that the reality is a computer simulation then all we (human beings) do (thinking, walk...

work page 2000
[3]

we (as the one who lives in) cannot distinguish the simulat ion from the reality. 10

work page
[4]

Then in deﬁnition 7.4, we slightly change the UC of V and construct a computer-simulation model named E which its UC persistently evolves

P = NP does not hold in the simulation We ﬁrst, in example 7.1, introduce a computer-simulation mo del named V . Then in deﬁnition 7.4, we slightly change the UC of V and construct a computer-simulation model named E which its UC persistently evolves. Then in theorem 7.9, we pr ove that P ̸= NP in E. Example 7.1 We introduce a computer-simulation model V ...

work page
[5]

IN STv = {[(q, a) → (p, b, D)] |p, q ∈ QT , a, b ∈ Γ, D ∈ { R, L}}, (IN STv)0 = {[(q, a) → (p, b, D)] ∈ IN STs |q = q0}, and

work page
[6]

Note that the programming language of Uv is exactly the standard syntax of conﬁgurations and transition functions of Turing machines

CON Fv = {(q, xaz) | q ∈ QT , x, z ∈ Γ∗, a ∈ Γ}, for each x ∈ Σ∗, C0,x = ( q0, △ x), and for each C = (q, xaz) ∈ CON Fs, yC = xaz. Note that the programming language of Uv is exactly the standard syntax of conﬁgurations and transition functions of Turing machines

work page
[7]

Let C = (q, xb1ab2y) be an arbitrary conﬁguration then – T BOXv(C, [(q, a) → (p, c, R)]) is deﬁned to be C ′ = (p, xb1cb2y), – T BOXv(C, [(q, a) → (p, c, L)]) is deﬁned to be C ′ = (p, xb1cb2y), and – for other cases T BOXv is deﬁned to be ⊥ . 4- Let C ∈ CON Fv be arbitrary – if C = (h, △ x) then SBOX v(C) is deﬁned to be Y ES , – if C = (h, x△ ) then SBO...

work page

[1] [1]

Arora, B

S. Arora, B. Barak, Computational Complexity , a modern approach. Cambridge University Press, 2007

work page 2007

[2] [2]

Meduna, Automata and Languages: Theory and Applications , Springer, 2000

A. Meduna, Automata and Languages: Theory and Applications , Springer, 2000. 6 P = NP Contradicts with Non-predeterminism In following, I construct a computer simulation and show tha t in this simulation P is not equal to NP. As, we discussed the ﬁrst section, assuming that the reality is a computer simulation then all we (human beings) do (thinking, walk...

work page 2000

[3] [3]

we (as the one who lives in) cannot distinguish the simulat ion from the reality. 10

work page

[4] [4]

Then in deﬁnition 7.4, we slightly change the UC of V and construct a computer-simulation model named E which its UC persistently evolves

P = NP does not hold in the simulation We ﬁrst, in example 7.1, introduce a computer-simulation mo del named V . Then in deﬁnition 7.4, we slightly change the UC of V and construct a computer-simulation model named E which its UC persistently evolves. Then in theorem 7.9, we pr ove that P ̸= NP in E. Example 7.1 We introduce a computer-simulation model V ...

work page

[5] [5]

IN STv = {[(q, a) → (p, b, D)] |p, q ∈ QT , a, b ∈ Γ, D ∈ { R, L}}, (IN STv)0 = {[(q, a) → (p, b, D)] ∈ IN STs |q = q0}, and

work page

[6] [6]

Note that the programming language of Uv is exactly the standard syntax of conﬁgurations and transition functions of Turing machines

CON Fv = {(q, xaz) | q ∈ QT , x, z ∈ Γ∗, a ∈ Γ}, for each x ∈ Σ∗, C0,x = ( q0, △ x), and for each C = (q, xaz) ∈ CON Fs, yC = xaz. Note that the programming language of Uv is exactly the standard syntax of conﬁgurations and transition functions of Turing machines

work page

[7] [7]

Let C = (q, xb1ab2y) be an arbitrary conﬁguration then – T BOXv(C, [(q, a) → (p, c, R)]) is deﬁned to be C ′ = (p, xb1cb2y), – T BOXv(C, [(q, a) → (p, c, L)]) is deﬁned to be C ′ = (p, xb1cb2y), and – for other cases T BOXv is deﬁned to be ⊥ . 4- Let C ∈ CON Fv be arbitrary – if C = (h, △ x) then SBOX v(C) is deﬁned to be Y ES , – if C = (h, x△ ) then SBO...

work page