Boundary Separation and Information Loss in Constrained Generative Systems: The Prime System as a Structural Test
Year: 2026
Type: preprint
Abstract: Paper C17 of the Sub-Limit Dynamics (SDL) programme. A constrained generative system (CGS) is usually observed through aggregates: counts, frequencies, averages. We prove two elementary but sharp information limits that separate what such observations can and cannot establish. Theorem T-I: an observation that is not injective on the admissible configurations cannot identify them, whatever the reconstruction rule. Theorem T-II: when a quantity is the sum of two components, a method that bounds the components independently in absolute value cannot establish a residual smaller than the sum of its bounds; reaching the residual requires information that links the components. We test both on the ordinary integers, built only from difference, limit, border and CUT, without assuming a successor operation or a list of primes. The arithmetic system decomposes exactly into poles (primes), composite deposits without repetition and self-deposits. The error of the prime count decomposes exactly as π(X) − li(X) = D_y(X) + R_y(X): two components of opposite sign and order X/log X that cancel to below √X up to 10^8. By T-II, controlling the two components separately cannot reach the scale O(√X log X) equivalent to the Riemann Hypothesis: what is missing is an arithmetic constraint on their compensation. We do not prove the Riemann Hypothesis. We locate where the open problem lies, separate it from what the definitions already contain, and state the recovery problem that remains. All numerical values are reproducible from the attached archive C17_verification.zip (Python scripts and outputs, computations up to 10^8, with an independent verification script).
Author: Lugli, Davide
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224637880