Critical Projection and the Geometry of Meaning (CPM: Pure Structure, Zero Philosophy)

This work develops the Critical Projection and the Geometry of Meaning (CPM), a fully geometric and mathematically falsifiable framework for consciousness.

CPM formalizes consciousness as a 'critical refinement' of projection maps from a structureless difference domain D to a meaning manifold M, defined over a fixed physical substrate X.

A topological closure field B(x) is computed from persistent homology, while semantic-physical mismatch produces a raw energy functional. Tension τ(x) is defined as the force density derived from this energy under closure.

When the tension exceeds a critical threshold (τ > τ_c) under closure-maintained conditions (B ≥ 1), the projection atlas collapses into a strictly finer structure, producing a discrete phenomenal "moment" (Critical Projection Π').

The theory predicts that cloud-based AI systems cannot satisfy the closure condition (B ≈ 0) due to topological decoupling caused by virtualization, and therefore cannot exhibit critical projection events. The model provides concrete empirical falsification paths and a unified geometric explanation for discreteness, embodiment, and the limits of artificial systems.

This version (v2) substantially refines and expands the original manuscript.
It clarifies the theoretical scope, addresses potential misinterpretations,and strengthens the defensive exposition of the core arguments.
No change is made to the central claims or theoretical framework.

Note.
Related implementation: Eidoku (CPM-derived verification gate)
This CPM preprint motivates a verification-first view of LLM reliability. A minimal, reproducible implementation of this idea is provided in the companion work Eidoku, which formulates post-hoc reasoning verification as a constraint satisfaction problem independent of generation likelihood.

• Eidoku (arXiv): https://arxiv.org/abs/2512.20664

• Eidoku DOI: https://doi.org/10.48550/arXiv.2512.20664

Note.
Related implementation: Catelingo (constraint-based semantic validity checker)

This CPM preprint introduces a geometric and structural account of meaning, emphasizing that semantic validity is not reducible to likelihood or fluency.
A complementary, lightweight implementation of this perspective is provided in Catelingo, which operationalizes semantic validity as explicit constraint satisfiability at the sentence level, independent of probabilistic generation or reasoning chains.

Catelingo focuses on detecting semantic no-go cases—such as temporal inconsistencies, numerical impossibilities, and type clashes—by enforcing interpretable semantic constraints rather than optimizing plausibility.

• Catelingo (Zenodo): https://doi.org/10.5281/zenodo.18148498
• Source code (GitHub): https://github.com/ShinobuMiya/Catelingo