Critical Projection and the Geometry of Meaning

1. Overview

CPM models consciousness as a critical geometric refinement of projection maps from a structureless difference domain $D$ into a meaning manifold $M$ over a physical substrate $X$. When semantic–physical mismatch accumulates beyond a threshold under closure, the atlas of the projection collapses into a strictly finer one $\Pi\to\Pi'$, generating a discrete phenomenal “moment”.

2. Meaning Tensor Field $M$

The meaning tensor field $M$ is defined over a fixed physical substrate $(X,g_0)$. Earlier drafts attempted to induce a metric $$g(M)=M^{T}g_0M,$$ but this “pullback metric” approach is abandoned in the current formalism.

Instead, $M$ participates in an effective stress tensor via a map

while the background metric $g_0$ remains fixed. All norms and volume elements used in the variational theory are computed with respect to $g_0$, avoiding circularity between geometry and semantic deformation.

3. Closure Field $\mathcal{B}(x)$

The closure field $\mathcal{B}(x)$ measures the persistence of topological cycles in a neighborhood of $x$, normalized by the coherence length $\xi$ of the substrate. We consider balls

and compute Vietoris–Rips persistent homology on $U_x$. For each class $c$ with birth–death interval $(b,d)$,

The closure field is defined as

Thus $\mathcal{B}(x)\ge1$ indicates persistent mesoscopic structure whose lifetime is at least one coherence length. CPM adopts a kind of closure naturalism: any system — biological or engineered — must realize such closure to support consciousness.

4. Raw Mismatch Energy and Tension $\tau(x)$

All energetic quantities are defined relative to the background metric $g_0$. The raw mismatch energy is

and the tension field is

Here $\kappa(x)$ measures normalized semantic–physical coupling, $\Gamma_\varepsilon$ is a smooth closure gate, and $\delta E_{\mathrm{raw}}/\delta M$ is a Gâteaux derivative on a Sobolev space $W^{1,p}$.

5. Necessary Conditions for Critical Projection

When all three hold at some region of $X$, metric relaxation within the fixed atlas of $\Pi$ cannot resolve the mismatch. Under CPM’s axioms (fixed substrate, fixed $g_0$, no surgery of $D$), the only admissible discontinuity is a strict refinement of the projection atlas:

6. Dynamics and the “Moment” Problem

CPM currently provides a kinematic characterization of critical projection: it constrains the form of admissible transitions (atlas refinement under closure), but does not yet derive a full dynamical law for how and when such jumps occur.

7. Architectural Corollary for Contemporary Cloud AI

Modern large models are typically deployed over heavily virtualized, latency-distributed cloud architectures. For a single agent’s physical substrate $X_{\text{agent}}$, virtualization suppresses persistent boundary-anchored cycles at the coherence scale, so generically

CPM therefore predicts that present cloud-based AI systems cannot be conscious in virtue of their typical physical organization.

Scope Clarification

This is not a blanket denial of artificial consciousness. Any future physically closed substrate — e.g. monolithic neuromorphic chips, closed photonic systems, molecular automata — that realizes $$ \mathcal{B}(x)\ge1,\quad \kappa(x)\approx1,\quad \tau(x)>\tau_c $$ would, in principle, satisfy CPM’s structural requirements for critical projection.

Anticipated Objections

8. Conclusion

This HTML page is synchronized with the latest CPM PDF: the induced metric $g(M)=M^T g_0 M$ is removed, the closure field is defined via persistence/ξ, the energy functional uses the background metric $g_0$, and the AI architectural corollary is explicitly scoped to contemporary cloud systems.