Well-Defined but Not Predetermined Persistent Mathematical Objects, Embedded Observers, and the Provability of P = N P

Rasoul Ramezanian

REVIEW

Well-Defined but Not Predetermined Persistent Mathematical Objects, Embedded Observers, and the Provability of P = N P

Rasoul Ramezanian

Logic · Submitted 9 Sep 2026

Pith's verdict: A novel distinction but the main unprovability argument doesn't connect to standard P vs NP as claimed—the stress test gets it right.

T0 · AI review

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arXiv ID2609.10864 (version 2)

Pith ID6ZUMEVQP, Pith's permanent record of this paper and its review

CitationRasoul Ramezanian. "Well-Defined but Not Predetermined Persistent Mathematical Objects, Embedded Observers, and the Provability of P = N P." arXiv:2609.10864 (2026). https://pith.science/paper/6ZUMEVQP

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Editor's note

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Colleague,

This paper is worth reading for its clean separation of well-definedness from predetermination and for the concrete construction of a persistently evolutionary oracle. The formal machinery—persistent set processes, observer axioms, the growing automaton—is carefully laid out. The separation theorem (6.4) is a legitimate result within its explicitly history-sensitive semantics.

The problem is with Theorem 7.5. The paper wants to conclude that an embedded observer cannot soundly prove P=NP, using the idea that in an observer-compatible evolutionary world, P=NP is false. But the false statement in that world is about the history-sensitive classes PA_ev and NPA_ev, not about standard Turing-machine P and NP. The paper acknowledges this distinction in Section 7.2 ("evolutionary and classical complexity must not be conflated"), yet the argument of 7.5 relies on conflating them: it assumes that because the evolutionary classes separate, the standard statement P=NP is false in that world. That step is unsupported. An observer's proof about standard P and NP could be sound even if the evolutionary classes differ, because the objects are different. The stress-test note is exactly right: there is a semantic gap.

This is not a minor oversight; it is a load-bearing gap in the central epistemic claim. The rest of the paper is solid and interesting. The discussion of embedded observers and the philosophical framing are thought-provoking. But as it stands, the unprovability result does not go through.

I would still send this to a referee who works on foundational complexity or philosophy of computation. The good parts are worth preserving, and the flaw might be fixable by reinterpreting the conclusion (e.g., proving unprovability for the history-sensitive analogue of P=NP). But as is, the main claim overreaches.

My recommendation: peer review with a clear flag on the semantic gap. The author might be able to address it, but it requires nontrivial additional work.

Referee report

Significance

The paper gives a clean conceptual separation between well-definedness and predetermination, a concrete persistent growing-oracle example with a polynomial-time implementation, and a useful finite-history realization result. The observer-relative conclusion is honestly presented as conditional, and the discussion of related work on choice sequences, interactive computation, simulation, and complexity barriers is appropriate. If Theorem 7.5 is repaired so that its antecedent explicitly includes a world in which the observer's standard-semantics sentence 'P=NP' is false, the result would be a valuable contribution to the foundations of embedded mathematical reasoning; it would not, however, constitute a new complexity-theoretic theorem about classical P versus NP.

Major objections (2)

[§7.1, Theorem 7.5] The proof of Theorem 7.5 cites Theorem 6.4 for the existence of an observer-compatible evolutionary realization in which 'P=NP is false in the evolutionary world,' but Theorem 6.4 establishes only P^A_ev ≠ NP^A_ev for the continuation-robust, history-sensitive classes defined in Section 6. Section 7.2 correctly warns that these classes must not be conflated with standard Turing-machine P and NP. Proposition 7.4 realizes only finite histories; it does not show that the observer's standard-semantics sentence 'P=NP' has a different truth value in that world. The contradiction with Axiom 7.2 therefore is not reached as written: the needed witness world, one that is observer-compatible and in which the same sentence under the same intended semantics is false, is exactly the premise that must be supplied, and the paper does not prove its existence. Please either add this falsity condition explicitly to the antecedent of Theorem 7.5 or prove a transfer lemma from the evolutionary separation to the observer's intended semantics.
[§6, Definitions 6.1–6.3 and Theorem 6.4] Because A is history-dependent, L_A = {x : A accepts x} is not a well-defined set: Proposition 5.2 shows that the same string may be accepted under one query order and rejected under another. Consequently L∃ and the 'languages' in P^A_ev and NP^A_ev are not fixed sets of strings, so the inequality P^A_ev ≠ NP^A_ev is not a well-formed language-class statement as written. The proof of Theorem 6.4 is best read as a claim about continuation-robust decisions from a specified history under all admissible interleavings; please make the history parameter explicit in the definitions of the classes and in the statement of Theorem 6.4. This formal precision matters because Theorem 7.5 invokes Theorem 6.4 as evidence for a world with a false P=NP statement.

Minor objections (4)

[§3, Definition 3.3] The term 'admissible histories' is used before the notion of admissibility is defined; please define the class of admissible histories explicitly, for example as those generated by the process under all possible query orders.
[§4, Axiom 4.2] The notation '(s)' and the phrase 'retaining information after that state' are informal; please define the total state and the retention relation precisely, or give a concrete example of what is excluded.
[§6, proof sketch of Theorem 6.4] The phrase 'no string of length at least n has previously been queried' should specify that 'previously' means before the run of D begins, since the proof then deliberately allows further queries during D's run.
[Appendix A] Please replace the informal phrase 'TA(x) is for sure polynomial time' with a precise statement: under the abstract cost model the bound is O(n), while under the literal filesystem cost model it is O(n^2) with the stated per-level pathname costs.

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The paper

The paper claims

No embedded observer can possess a sound internal proof of P=NP

The reading

The paper distinguishes well-definedness (unique, persistent answers on queries) from predetermination (all answers fixed independently of history). It defines persistent non-predetermined objects whose future values depend on interaction order. It models observers embedded in a shared computational world subject to axioms of shared history, irreversibility, persistence, and indistinguishability. Using a persistently evolutionary polynomial-time oracle, it separates deterministic and nondeterministic polynomial time under a history-sensitive semantics. It proves that under these axioms, if an observer's world has an indistinguishable counterpart where P≠NP and the observer's proof process is reproduced there, then that observer cannot possess a sound internal proof of P=NP.

What carries the argument

The key machinery is the distinction between well-definedness and predetermination, formalized through persistent partial functions and set processes. The load-bearing computational device is the persistently evolutionary polynomial-time oracle A, implemented as a growing automaton or growing directory tree, whose answers are permanent once given but whose extension depends on query order. This oracle yields a separation between the complexity classes P^A_ev and NP^A_ev under continuation-robust semantics, which forms the computational basis for the epistemic argument about proof soundness.

Extended reading notes

Core claim

The central claim is an observer-relative unprovability theorem: under axioms of shared history, historical irreversibility, persistence of realized facts, and historical indistinguishability, any embedded mathematical observer whose realized history has an observer-compatible evolutionary realization (i.e., a world where the observer's entire internal proof process is reproduced but P≠NP) cannot possess a sound internal proof of P=NP. The argument relies on a separation of history-sensitive complexity classes via a persistently evolutionary polynomial-time oracle, showing that from the observer's perspective a predetermined world with P=NP and an evolutionary world with P≠NP can be indistinguishable.

Load-bearing premise

The existence of an observer-compatible evolutionary realization that reproduces the observer's entire internal proof process; Theorem 7.5 assumes this premise without proving it holds for all observers and proofs.

Editorial extensions

If this is right

If the observer axioms hold, then no human mathematician considered as an embedded observer can ever produce a correct proof of P=NP.
The evolutionary separation provides a new barrier against proof techniques that assume the oracle's extension is fixed independently of query order.
The framework suggests that any proof of P=NP must be immune to the possibility that the reasoner's world is an indistinguishable simulation where P≠NP.
The epistemic conclusion does not require the actual world to be non-predetermined; it only requires that the observer cannot rule out a compatible non-predetermined world.
The result may be extended to other open complexity problems under similar observer axioms and suitable evolutionary oracles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: The paper's assumption that the observer's proof process is reproduced identically in the evolutionary world may be non-trivial; for proofs that interact with the oracle, the process could diverge across worlds, weakening the conclusion.
Editorial: The observer axioms are chosen to guarantee indistinguishability but may be too strong; if an observer could embed a self-contained record that persists across parallel histories, indistinguishability might break.
Editorial: The framework implicitly assumes that mathematical proof is a deterministic process that can be fully reproduced in a different world; non-deterministic verification steps could challenge this.
Editorial: The observer-relative unprovability is analogous to self-referential incompleteness but driven by computational indistinguishability, suggesting a family of 'embedding barriers' in theoretical computer science.

Original abstract

We develop a foundational distinction between \emph{well-definedness} and \emph{predetermination}. A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension. We call such an object \emph{well-defined but non-predetermined}. We next formulate an observer-based semantics. All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history. Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed. A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic--nondeterministic separation under an explicitly history-sensitive semantics. We then derive a conditional result concerning the provability of \(P=NP\). The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,829 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.93 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We develop a foundational distinction between \emph{well-definedness} and \emph{predetermination}. A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension. We call such an object \emph{well-defined but non-predetermined}. We next formulate an observer-based semantics. All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history. Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed. A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic--nondeterministic separation under an explicitly history-sensitive semantics. We then derive a conditional result concerning the provability of \(P=NP\). The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop a foundational distinction between \emph{well-definedness} and \emph{predetermination}.	contribution 1.00	0.94	0.04	asserted 0.99
s1	A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension.	contribution 0.89	0.46	0.07	hedged possibility 0.97
s2	We call such an object \emph{well-defined but non-predetermined}.	contribution 0.92	0.82	0.05	asserted 0.98
s3	We next formulate an observer-based semantics.	method 0.51	0.91	0.02	asserted 0.76
s4	All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history.	method 0.71	0.34	0.03	asserted 1.00
s5	Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed.	result 0.98	0.79	0.17	asserted 1.00
s6	A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic--nondeterministic separation under an explicitly history-sensitive semantics.	contribution 0.51	0.91	0.06	asserted 1.00
s7	We then derive a conditional result concerning the provability of \(P=NP\).	contribution 0.64	0.93	0.10	asserted 0.87
s8	The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).	interpretation 0.47	0.89	0.83	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Discussion

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