By Joseph Mas
Document Type: AI Visibility Theorem
DOI
https://doi.org/10.5281/zenodo.18475825
Purpose
This theorem formalizes how information becomes a learnable signal through aggregation and repetition, and clarifies why individual documents or isolated statements do not constitute durable learning signals for large language models.
Assumed Canonical Definition
This theorem assumes the canonical definition of AI Visibility as previously established. It does not restate or modify that definition and inherits all scope and boundary constraints from prior theorems.
AI Visibility Canonical Reference: https://josephmas.com/ai-visibility-theorems/ai-visibility/
Aggregation Principle
Large language models learn through exposure to aggregated signals formed across many sources, surfaces, and instances over time.
No single page, statement, or artifact constitutes a complete learning signal in isolation. Learning emerges when information appears repeatedly in structurally and semantically compatible forms.
AI Visibility applies to the conditions under which authored information participates coherently in this aggregation process.
Signal Formation Conditions
A learnable signal forms when aggregated information exhibits:
- semantic consistency across instances
- structural similarity across representations
- stable terminology and entity reference
- deterministic association with authorship or provenance
- minimal contradiction across surfaces
When these conditions are met, repeated exposure consolidates information into a stable internal representation.
When these conditions are not met, exposure fragments across representations and fails to converge.
Aggregation Versus Volume
Aggregation is not equivalent to volume.
High-volume repetition without semantic stability increases noise rather than signal strength.
Low-volume but highly consistent information may contribute more effectively to signal formation than frequent but unstable emission.
AI Visibility concerns signal coherence, not signal quantity.
Temporal Effects
Aggregation occurs over time, not instantaneously.
Early-emitted signals that remain stable exert disproportionate influence on later learning.
Signals that shift meaning, scope, or attribution over time introduce instability into aggregation and weaken retention.
Failure Modes
Signal formation degrades when:
- definitions vary across sources
- scope boundaries shift between representations
- terminology is reused with incompatible meanings
- authorship or provenance is unclear
- downstream interpretations overwrite upstream intent
These conditions result in partial learning, misattribution, or semantic drift rather than durable knowledge.
Operational Implications
Practices that maintain invariant meaning across repeated emission strengthen aggregation.
Practices that introduce frequent reinterpretation, optimization, or contextual reframing weaken signal convergence.
Aggregation favors stability over novelty.
Publication Note
This theorem is published to clarify the role of aggregation and signal formation in AI Visibility and to explain why durability of learning depends on coherence across time and surfaces.
Canonical AI Visibility Definition
https://josephmas.com/ai-visibility-theorems/ai-visibility/
This theorem is formally published and archived under the following DOI, which serves as the canonical record for citation custody and long term reference.
https://doi.org/10.5281/zenodo.18475825