The Natural-Chance Red Line: Operational Determination of the Boundary Between Exact Representation and Conservative Reconstruction

This record extends the Origas Representational Jurisdiction Objection (10.5281/zenodo.22100628) by turning its qualitative fibre criterion into a quantitative and falsifiable boundary: the Natural-Chance Red Line. “Natural chance” is used operationally: it denotes unresolved task-relative ambiguity inside a declared representation P. It is not, by itself, a claim of ontic randomness or proof of a hidden physical world. Let P be a declared readout and J the external task. The central quantity is the task-relative fibre diameter: R_J(P) = sup { d(J(x), J(x′)) : P(x) = P(x′) }. R_J(P) = 0 means that J is constant on every P-fibre and therefore factors through P. R_J(P) > 0 identifies a precise jurisdictional boundary: the readout has erased distinctions that remain necessary to the task. The normalized red-line number is: Λ(J, ε; P) = R_J(P)/(2ε). When Λ(J, ε; P) > 1, no P-only procedure can uniformly guarantee the declared task tolerance ε. The record proves that deterministic closed post-processing, recursive analysis, additional compute, same-readout consensus, formal proof or AI transformations cannot shrink a distinction that is absent from the original source channel. A finite or countable family of functions of P remains informationally confined to P. A genuine crossing requires either: (1) factorization: prove J = j ∘ P on the declared domain; (2) restriction: restrict the claim to the domain for which the task-relevant fibre is zero or acceptably bounded; (3) source-sensitive refinement: add an independently acquired channel K/T and demonstrate positive separator gain: G_J(K,T | P) = R_J(P) − R_J(P,K,T) > 0. The record also introduces a task-relative order witness: χ_J(A,B;x) = d(J(ABx), J(BAx)). Therefore AB = BA may be imposed only when commutation is proved or when order is demonstrably irrelevant to the declared task. If χ_J(A,B;x) > 0 and the representation nevertheless identifies ABx and BAx, the representation contains a positive task-relevant fibre. A further exact result concerns safety. If P(x₁) = P(x₂) while Safe(x₁) ∩ Safe(x₂) = ∅, then any randomized P-only policy has worst-state failure probability ≥ 1/2; a deterministic P-only policy fails completely on at least one of the two indistinguishable states. The mathematical consequence is precise: If R_J(P) > 0 and an independently validated A+B refinement produces G_J(K,T | P) > 0, then reducing A+B back to A is non-conservative for that declared task. The framework is then tested against independent 2025–2026 scientific results rather than by examples selected only for agreement. Zhou / Connes supplies an exact non-reconstruction witness. Zeta23 supplies a formally verified method-class ceiling. GW Ori supplies a peer-reviewed empirical compatibility fibre in which additional observational channels recover information while excellent fitting does not uniquely reconstruct 3D geometry or history. 3I/ATLAS supplies a one-pass natural observer-escape benchmark: interplanetary-spacecraft geometry measurably reduces orbital uncertainty, while isotopic measurements open an independent formation-history channel; adverse frozen predictions and conventional cometary nulls are retained rather than repaired. AlphaGenome supplies a living-systems scope boundary stated by its own authors. Wang-Zahl, OpenAI/Astra and ASI-Arch are retained as controls or prospective audit targets rather than forced confirmations. The year 1981 is retained as a historical audit marker. For bibliographic precision, the CHSH quantum maximum 2√2 is the Tsirelson/Cirelson bound; the Aspect experiments supplied landmark Bell-inequality violations in agreement with quantum mechanics. The wider Origas interpretation involving the three Galilean cabins, Point 0, two differentiated physical branches +t/−t and a universal T is preserved in the record as an authorial physical interpretation and remains OPEN until independently discriminated. The paper therefore separates four statuses throughout: EXACT / EMPIRICAL / CONDITIONAL / OPEN. The record also preserves the author’s broader causal and institutional warning while separating it from the theorem layer. Its falsifiable scientific core is: identify P and J; exhibit or bound the residual fibre; preserve task-relevant AB/BA order; freeze K/T before the decisive holdout; measure separator gain; publish nulls and adverse results; and permit the framework itself to fail. This record claims priority, to the extent documented by the cited timestamped DOI chain, for the Origas formulation and cumulative synthesis of representational jurisdiction, the Natural-Chance Red Line, task-relative fibre diameter as an authority boundary, separator gain, observer-escape certificates, chronology-admissibility, Point-0 non-collapse, task-sensitive order auditing, and the Boundary Crossing Certificate architecture. THE RED LINE IS NOT CROSSED BY MORE OF THE SAME P: IT IS CROSSED ONLY BY A VALID FACTORIZATION OR BY NEW SOURCE-SENSITIVE INFORMATION. EXACTNESS IS NOT COMPLETENESS. AUTHORITY MUST NOT EXCEED INFORMATION.