Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.22844

Operational Inexpressibility at the Step-Duplicating Primitive Recursor Orientation Boundary

Moses Rahnama
View original source ↗Read source document ↗Export metadata ↓

Abstract

We identify operational inexpressibility, a structural property of term-rewriting proof systems: for a fixed input and dimension, every derivation ignores the dimension or leaves the target question unconstrained. The canonical instance is direct aggregation on the primitive recursion duplicator $F(x,y,Z)\to x$, $F(x,y,S(n))\to G(y,F(x,y,n))$, whose step argument $y$ is duplicated. A companion paper maps the non-representability frontier; we prove it is operational inexpressibility at the step-argument dimension. Sound responses split into construction methods extending the proof language and confession methods (dependency pairs, counter-projection, size-change termination, argument filtering) projecting away the unincorporable dimension under an external soundness license. Under any direct whole-term measure the recursor's mass profile coincides with that of a true circular reference: only the confession family's licensed projection separates them; non-derivability, TRS isomorphism and information equivalence are proved in Lean. Arts-Giesl soundness is a $\Pi^0_2$ principle; its subterm-criterion route is a size-change instance formalizable in $\mathrm{RCA}_0$ with an order-$\omega$ termination measure. Within the analyzed family the duplicator is the unique structurally complete member requiring confession. The confessed burden grows quadratically against linear residual proof work; a Shannon-style validator recasts the obstruction as a divergent inefficiency coefficient. An architectural necessity theorem makes the duplicator the minimal faithful record-emitter. A layer-crossing schema places the dependency-pair confession in the Feferman-Beklemishev reflection family rather than the Lawvere-Yanofsky diagonal family, matching the six-step shape of G\"odel's 1931 move. A witness-language hierarchy with minimal witness order $\kappa^*$ puts the orientation boundary at $\kappa^*(x)>0$.

Explore related subjects

cs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Moses Rahnama. 2026-04-21. Operational Inexpressibility at the Step-Duplicating Primitive Recursor Orientation Boundary. https://arxiv.org/abs/2604.22844

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Four-Valued Graph Model for Conflict Resolution: Core Framework and a Machine-Checked Formalization in Lean 4

This note consolidates the core of the Quasi-Closed World Graph Model for Conflict Resolution (QCW-GMCR), which extends the standard Graph Model for Conflict Resolution with Belnap's four-valued logic to represent option-level epistemic ambiguity, and pairs the framework with a machine-checked Lean 4 formalization. QCW-GMCR combines: (1) FOUR-valued option assignments with compositional propagation to state-level feasibility; (2) graded reachability (definite, credible, possible) based on an FDE-inspired transition-warrant semantics, with definite reachability related to FDE consequence in the Boolean fragment; (3) axiomatized deterministic reductions from four-valued assessments to binary decisions, including four canonical operators reflecting distinct risk attitudes; and (4) catastrophe-avoiding equilibrium concepts with a quasi-closed-world safety invariant. A four-valued hypergame extension captures heterogeneous subjective assessments across decision makers. We state the core definitions and results and report the parts verified in Lean 4 with mathlib, including the classical GMCR stability hierarchy, algebraic and compositional properties of FOUR-valued conjunction, properties of the canonical reductions, and the graded reachability hierarchy. The formalization also helped identify and correct earlier claims, including a knowledge-monotonicity axiom replaced by truth monotonicity. This preprint provides a stable, citable record of the framework and its current formal verification status.

cs.LO↗
arXiv2026

The Semantic Elevation Operator and the Closure of the Undecidable Class under Preservation

The undecidability of a program's static semantic properties is governed by Rice's theorem. Self-modifying systems, however, require analysing not whether a property holds now, but whether it is preserved when the system rewrites itself. We formalise this transition through a semantic elevation operator {\Lambda}{\Phi}, which turns the static question "does x satisfy P?" into the dynamic question "is P preserved after x is transformed by {\Phi}?". We prove that when {\Phi} is intensional (depending on the source code, not only on the computed function), the elevated property remains undecidable even though it breaks the extensionality that Rice's theorem requires; the proof rests on Kleene's recursion theorem, not on Rice. Consequently the class U of non-verifiable properties is closed under the elevation operator. Unbounded iteration of the operator climbs the arithmetical hierarchy -to {\Pi}02-completeness- consolidating non-verifiability as a structural fact. We further show that the supervisory regress does not terminate: no fnite tower of increasingly capable verifiers yields an unconditional certificate. A categorical reading of these results in the efective topos, in which elevation appears as an instance of Lawvere's fxed-point theorem, is left as a direction for future work.

cs.LO↗
arXiv2026

Statistical Symmetry Release for Equivariant Quantum Learning

Hard symmetry constraints reduce model complexity, but can also erase label information. Statistical symmetry release determines when finite data and quantum measurements justify relaxing such a constraint, which directions to open, and how far to move. We connect global signal detection to local, loss-dependent improvement. A two-copy twirl--swap gate estimates task information in the symmetry-breaking complement with a dimension-independent copy count under paired-state and group-unitary access; reweighting the same records resolves representation sectors. An exact duality distinguishes this Hilbert--Schmidt signal from the larger signal accessible to bounded-outcome readouts. Local improvement is governed by the release gradient and a loss-corrected double-commutator matrix. Simultaneous confidence bounds convert empirical direction selection into certified descent, using either shared Pauli measurements or scalar probes with state-independent truncation bounds. Gaussian testing lower bounds quantify the cost of searching over unknown directions in the calibrated local experiment. Independent validation controls adaptively generated models, and a fast squared-loss bound preserves the approximation--estimation rate of a nested release path. On an eight-qubit Ising model, shared measurements certify release with 6300 times fewer shots than the specified scalar estimator on the tested budget grids. Quotient quantum natural gradient then controls parameter redundancy during training. Together, these results turn symmetry relaxation into a statistically justified model-selection decision.

cs.LO↗