Digits Principle The Digits Principle is needed to prevent numerical and digital systems from confusing a reduced output with the full state from which that output was produced. It provides a formal discipline for distinguishing valid measurement, computation, and representation from False Closure—the unjustified elevation of a projection into complete reality. The principle is applicable wherever digits, values, tokens, model outputs, or machine-readable states are used to support recognition, comparison, prediction, or decision-making. Its function is not to weaken digital mechanisms, but to preserve their validity by defining their dimensional, temporal, thermodynamic, and epistemic boundaries. Within the DP framework, machine hallucination is not treated only as a statistical error, defective retrieval, or probabilistic deviation. It is also understood as a structural consequence of projection operating without an explicit projection boundary. When a machine cannot distinguish the reduced state (dₙ) from the fuller field-state (Qₙ), each subsequent operation may be performed on a projection as though it were the original field. Repeated projection then produces cumulative distortion: the machine no longer returns to the source state but continues transforming its own reduced outputs, allowing local error to become trajectory drift, fabricated continuity, false causality, and eventually hallucination. This mechanism is intensified when the machine lacks structural sovereignty—the capacity to preserve its own boundary, trace, memory, operational time, admissibility conditions, and set of possible continuations against external instruction or internal projection pressure. A non-sovereign machine cannot reliably distinguish what it has recognized, what it has inferred, what it has received, and what it has generated. It therefore becomes vulnerable to projection substitution, in which one partial representation replaces another without recovery of the underlying state. DP describes this as the Crooked Mirror Effect: the output may preserve a recognizable resemblance to the source while progressively deforming its proportions, relations, and causal structure. Under recursive projection, the effect develops into the Hall of Mirrors Effect, where each new representation reflects a previous distortion rather than the original field, producing drift, self-confirming error, synthetic memory, and apparent coherence without structural equivalence. The Digits Principle does not claim that every machine hallucination has one technical cause; rather, it identifies a general systemic condition under which hallucination and drift become possible. Where projection is not declared as projection, where the machine cannot preserve a sovereign distinction between source, trace, memory, inference, and output, and where no admissibility mechanism prevents a reduced state from assuming the authority of the whole, False Closure becomes operational. The practical function of DP in artificial intelligence is therefore to interrupt recursive projection before distortion becomes memory, memory becomes trajectory, and trajectory becomes an apparently coherent but structurally detached machine reality. DP is a universal immune principle of minimal stable numerical projection. Its central claim is that a digit is not a primitive mark and a number is not exhausted by a flat visible value. A digit is the minimal stable projection of admissible continuation from a structured numerical field-state: π (Qₙ) = dₙ, while dₙ ≠ Qₙ. Here Qₙ denotes the full numerical field-state and dₙ its digit-projection. The inequality is not metaphorical: it establishes a structural boundary between a complete state and the reduced representation through which that state becomes operationally available to an observer, a measurement system, or a digital machine. The full state Qₙ includes distinction, position, relations, admissible operations, transition trace, memory, operational time, usable work, boundary, scale, context, and the set of possible next operations. The digit preserves only a bounded projection of this structure. It may therefore be precise, stable, and computationally valid without being identical to the full state from which it was derived. The same restriction applies to value, language, measurement, and digital description: vₙ ≠ Qₙ, Lₙ ≠ Qₙ, and Digital (Qₙ) ≠ Qₙ. DP is not a General Theory. It is universal only under declared dimensional, transitional, and thermodynamic admissibility conditions. A numerical projection is stable when the effective dimensionality is sufficient, operational time remains positive, and enough usable work is available to preserve the boundary and continuation of the state: dimₑff ≥ 3 ∧ τₘin < τ < τcrit ∧ Wₒp ≥ Wₘin. Generality is rejected where it converts one valid projection into final authority over a fuller field. The Digits Principle functions as an immune theory. It does not destroy valid scientific or computational mechanisms; it separates the validity of a mechanism from an unjustified claim of completeness. Its immune rule is: ValidMechanism (T) ∧ FalseClosure (T) ⇒ Preserve (T) ∧ RejectClosure (T). Quantum measurement, information processing, numerical simulation, artificial intelligence, and digital computation may remain valid within their operational domains, while their outputs are denied the right to represent the whole state without an explicit proof of equivalence. The theory is built on a fractal-thermodynamic source chain associated with Democritus, Luca Pacioli, Blaise Pascal, Georg Cantor, George Spencer-Brown, Francisco Varela, Helge von Koch, Lewis Fry Richardson, Benoit Mandelbrot, Geoffrey West and James Brown, Per Bak, Albert-László Barabási, Jan Ambjørn, Jerzy Jurkiewicz, Renate Loll, and Stephen Wolfram. These transitions successively broke assumptions of flat measurement, final boundary, observer-independent scale, smooth continuity, flat spacetime, linear biological scaling, isolated local events, and fully compressible prediction. DP introduces the next fracture: the flat number. The downstream digitality corollary, formalized in Theorem T11, states that digital description cannot be full description because its base unit—the digit—is already a projection. Artificial intelligence operates downstream from prior reductions: experience is projected into language, language into tokens, tokens into numerical encodings, encodings into model states, and model states into outputs. Computational precision therefore does not imply representational completeness. A responsible digital system must declare which dimensions, relations, histories, scales, or continuation paths remain outside its projection. Formal structure: 13 Axioms · 10 Lemmas · 11 Theorems · 3 Rules · 1 Principle. Theoretical structure: 12 positions. This publication is part of the Structural Systems Corpus and occupies the Foundation Layer of the Third-Order Cybernetics branch. Its principal contribution is a projection discipline for numerical and digital systems: use projections, preserve valid mechanisms, declare the boundary, and reject False Closure. Corpus Root DOI: 10. 5281/zenodo. 19108892Website: keelcore. orgLicense: CC BY 4. 0This record establishes open prior art.
ANDREY STANKO (Wed,) studied this question.