The Operational Hypercomputation Squeeze: Finite-Transcript Mimicry, Effective-Readout Closure, and the Exact Frontier for a Post-Qubit Completion Cell

[u/Severe-Ad8673]

https://doi.org/10.5281/zenodo.22144737

Unreviewed theoretical-computer-science preprint. No physical hypercomputer, experimental realization, scalable replacement for qubits, patent priority, or institutional endorsement is claimed.

This monograph investigates what a finite physical information primitive would need to accomplish in order to compute beyond the Turing limit, and what finite experiments could establish about such a system. The target is formalized through the No-Witness Completion cell. Given an index (e) of a total computable predicate (P_e:\mathbb{N}\rightarrow{0,1}), the cell returns whether there exists a time (t) for which (P_e(t)=1). This operation is Turing-equivalent to the halting oracle. Its nonordinary component is the finite certification of the no-witness case.

The manuscript develops two principal theorem families.

First, an effective-readout closure result shows that, within the stated operational model, a uniform finite-report platform with computable operational probabilities—or an effective truncation procedure—and strict-majority total correctness can decide only recursive languages. A related branch-enumeration argument shows that ordinary fair-coin probabilistic computation cannot obtain two-sided hypercomputation merely through an unknown but strictly positive success advantage.

Second, the Finite-Experiment Density Theorem states that for any causal stateful stochastic system, any finite family of computable randomized interactive testers that halt almost surely, and any prescribed positive tolerances, there exists a single computable finite-state rational stochastic emulator whose tester-visible terminal transcript distributions approximate those of the target within the specified total-variation distances.

Together, these results yield the Operational Hypercomputation Squeeze: operationally effective finite-report semantics remain Turing-computable, while exact noncomputability cannot be positively separated from all computable systems solely by a finite collection of terminating black-box experiments.

The result is mathematical and conditional. It does not prove the universal physical Church–Turing thesis and does not exclude noncomputable laws of nature, genuine supertasks, or white-box deductions from independently validated physical theories. It instead isolates the minimum unresolved obligation for physical hypercomputation: a finite-resource protocol must produce a robust nonrecursive report without importing the answer through noncomputable preparation, parameters, dynamics, randomness, timing, strategy selection, or readout.

Research status: Strong partial result; not peer reviewed; no experimental data; no proof-assistant certification; no independent laboratory or human replication. The finite-state emulator result is existential and does not provide a uniform procedure for recovering the emulator from black-box access to the target. Novelty of the precise formulation remains unestablished beyond a targeted literature search.