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\title{Why the Projective Consciousness Model Must Be True:\\ A Formal Demonstration from Sensorimotor Dependencies, Invariance, and Control}
\author[1]{David Rudrauf}
\author[2]{Pierre Pansu}
\affil[1]{CIAMS, Universit\'e Paris Saclay}
\affil[2]{Laboratoire de math\'ematiques d'Orsay, Universit\'e Paris Saclay}
\date{}


\begin{document}
\def\d#1{${}^{#1 D}$}


\maketitle

\begin{abstract}
Conscious experience is unified, multimodal, and perspectival: sensory contents appear in a single egocentric scene that remains coherent as the organism senses and acts. We provide a conditional necessity proof for the Projective Consciousness Model (PCM). From four minimal premises, (P1) approximately central projection of visual input, (P2) coherent integration across viewpoint changes, (P3) maximal invariance with minimal added structure, and (P4) normative control of viewpoint (e.g., information seeking under active inference or related optimal-control schemes), we derive that any internal state sufficient for coherent first-person spatial awareness and closed under viewpoint updates is, up to isomorphism, the three-dimensional real projective space $\mathbb{RP}^3$. Viewpoint changes act as collineations in $PGL(4,\mathbb{R})$, while Euclidean and affine judgments arise only through a context-dependent metric gauge layered on the projective backbone. Strictly Euclidean maps and purely topological formats either implement an equivalent projective layer to reconcile perspective with coherent updating, or violate at least one premise. This geometry yields principled accounts of perceptual constancies and systematic geometric illusions (including the Moon illusion), provides a compact control parameterization for whole-field prediction and information gain, and links naturally to known neural mechanisms for spatial updating (gain fields and predictive remapping). The same projective scaffold can be extended beyond vision to multisensory integration, valuation, social cognition, and planning. Thus, given P1-P4, projective organization is the uniquely parsimonious representational format for coherent egocentric spatial experience.
\end{abstract}


\section{Introduction}

Consciousness presents a world from a first-person point of view (PoV): sensations from multiple modalities and directions are integrated into a single, coherent spatial experience. Despite continual eye, head, and body movements, and the resulting shifts in retinal and other sensory inputs, the perceived scene remains unified and stable "from here". Any adequate theory of consciousness must explain how such a perspectivally organized field of experience is possible.


Classical theories such as Global Workspace/Global Neuronal Workspace Theory \cite{Baars1997,Dehaene2017} have emphasized global integration and broadcasting. Others, such as Integrated Information Theory (IIT) \cite{Tononi2016}, consider abstract concepts and measures of complexity/differentiation, with an emphasis on causal interactions in latest installments (?IIT~4.0?) \cite{Albantakis2023IIT4}. Other influential frameworks focus on prediction and control, including predictive processing and active inference \cite{Friston2010,Seth2014SMC,Friston2015}, on metarepresentation in Higher-Order Thought theories \cite{Rosenthal2005}, on attention and its internal model in Attention Schema Theory \cite{Graziano2022PNAS}, or on recurrent processing as a neural condition for awareness \cite{Lamme2006}. Work on global states has warned against a single graded "level" of consciousness and instead advocated a multidimensional space of capacities \cite{Bayne2016Levels}. These perspectives capture important aspects of accessibility, integration, prediction, attention, metacognition, and neural dynamics, yet they largely leave underexplained why conscious experience is specifically structured geometrically as an integrated, egocentric three-dimensional scene with a movable point of view.

Some lines of research speak more directly to this spatial, perspectival format. Merker argues, on neuroethological and clinical grounds, that brainstem and midbrain systems integrate multisensory information into an egocentric three-dimensional scene organized around a central perspective point \cite{Merker2007}. More indirectly, sensorimotor contingencies theories tie experience to lawful relations between action and sensory change \cite{o2001sensorimotor}, and predictive-processing analyses relate such contingencies to the feeling of perceptual presence \cite{Seth2014SMC}. Though not specifically focused on spatial cognition, Oizumi and colleagues have recently proposed a general "principal-bundle geometry of qualia" that aims to characterize modalities, signatures, and attributes of experience by orbit and quotient structures induced by symmetry groups acting on neural state spaces \cite{oizumi_principal_bundle_qualia}. Taken together, these contributions motivate a more explicit and formal account of the geometry of the conscious field and of its lawful update under movement.

The Projective Consciousness Model (PCM) proposes such an account. It posits that the field of consciousness (FoC) is, in essence, a three-dimensional projective space, with the organism's perspective encoded as a controllable three-dimensional virtual camera \cite{Rudrauf2017}. Euclidean and affine quantities, such as distances, sizes, and angles, are not encoded as absolute primitives but arise via contextual calibration of a gauge, for example by adopting a horizon, vertical, or scale \cite{Rudrauf2017,Williford2018}.\footnote{Throughout, a gauge is the extra, context-dependent calibration that turns the projective backbone into Euclidean and affine read-outs (sizes, angles, scale). Intuitively, fixing the gauge is like calibrating a camera (choosing a horizon and zoom): recalibration changes measured sizes and angles while leaving straightness and incidence untouched.} In this format, perceptual constancies and classic geometric illusions follow as lawful consequences of projective frame calibration and gauge choice, including the Moon illusion \cite{Rudrauf2020}. PCM has been applied to visuomotor control and social interaction \cite{Rudrauf2022JTB}, combined with virtual humans to simulate theory-of-mind assessment scenarios \cite{Rudrauf2023ACM}, and used to induce intrinsic motivation and curiosity in simulated agents exploring their environment \cite{Sergeant2025BC}. A comprehensive synthesis of PCM and its extensions is provided in \cite{Rudrauf2023BrainSci} (see also \cite{rorot2021bayesian} for discussion of PCM within Bayesian theories of consciousness).

In the present work we change methodological stance. Earlier PCM work was guided primarily by first-person phenomenology and by qualitative computational considerations. Here, instead, we show how PCM can be derived from a small set of external premises: (i) basic sensorimotor contingencies that govern how actions modulate sensory inputs \cite{o2001sensorimotor}, and (ii) general constraints on representation, inference, and control. This approach is deliberately ontologically neutral. It recovers an "internal" three-dimensional projective workspace for computational and geometrical reasons, without committing to internalism versus externalism or to representationalism versus enactivism. It parallels proposals that use predictive processing to constrain the search for neural correlates of consciousness \cite{HohwySeth2020}, but targets more specifically the format of the conscious field itself. The stance is not that projective geometry is privileged by phenomenology alone, but that, given the way embodied agents must sense and act, a projective organization is forced upon any system that satisfies the premises.

There are strong biological reasons for a geometry that preserves incidence relations under changes in viewpoint to play a central role in adaptive behavior. Incidence relations specify which entities align or intersect with which others, which surfaces occlude which others, which points lie on the same line, which lines intersect in a point, which planes intersect along an edge or contour, and how occultation is ordered. These relations are preserved by projective transformations even when metric quantities such as lengths and angles vary. Because action must succeed despite changes in viewpoint, lighting, and distance, incidence provides compact and dependable structure for perception and control. Everyday examples are intuitive: a pedestrian can anticipate collision by monitoring constant bearing to another agent, without measuring absolute speed or distance; navigating a doorway, reading a map, or threading a needle depends on alignments, intersections, and occlusion order, not on exact metric values. Socially, people infer what another agent can see by extending a straight line along the other's gaze, groups align headings for cohesion, and predators are detected when their visual axis aligns with one's own position.

Across species, depth perception and collision avoidance exploit the same constraints. Visual-cliff studies show that infants and many animals avoid the "deep" side defined by misaligned texture, indicating early sensitivity to incidence-level depth cues without prior learning \cite{Gibson1960,adolph2014fear}. Looming, that is, radial contour expansion signalling approach on a collision course, drives dedicated escape circuits \cite{de2012loom}. Interception behaviors from insects to fish and humans implement constant-bearing guidance: dragonflies minimize retinal motion of prey \cite{olberg2012visual,gonzalez2013eight,dickinson2015motor,bomphrey2016flight,plunkett2023modeling}, fish employ pursuit and evasion strategies with similar geometry \cite{peterson2021pursuit}, and engineering analyses show that constant-bearing can outperform simple pursuit under sensorimotor delay \cite{strydom2015biologically}. Human ball catching reflects the same embodied heuristics \cite{gigerenzer2021embodied}. Monocular projective cues can dominate binocular ones when the two are placed in conflict, underscoring that geometry of incidence often carries the decision \cite{yoo2023monocular}.

Perceptual organization and ecological signalling likewise rely on alignments and discontinuities. Collinearity engages automatic contour integration, and even rats show impaired detection when flankers are collinear with a target edge \cite{meier2011collinear}. Predators segment prey via discontinuities, while prey evolve disruptive coloration that introduces false edges and misleading alignments \cite{stevens2006disruptive,stevens2007predator}. Many animals behave as if agents continue behind occluders, that is, they exhibit occlusion permanence \cite{kelley2014animal}. Schooling and flocking behavior extend collinear trajectories to maintain cohesion \cite{peterson2021pursuit}, and several species, including starlings, are sensitive to the alignment of a predator's gaze \cite{carter2008subtle}. Cross-species work on size-contrast illusions, such as the Ebbinghaus illusion, reveals conserved susceptibilities modulated by ecological factors, consistent with tuning to global context and alignments \cite{santaca2025circles,kelley2014animal,feng2017animals}. In humans, shared attention and perspective taking depend on extrapolating another's line of sight, an incidence computation that is central to communication and coordinated action. Taken together, these observations suggest that alignment, intersection, and occlusion function not merely as useful cues, but as a natural representational backbone for action.

This empirical picture aligns with the formal step we take in the remainder of the paper. We derive a model in which the spatial field itself is coded in projective terms, so that changes of viewpoint correspond to lawful collineations and incidence-level facts are treated as fundamental, turning the regularities just reviewed into invariants of an internal geometry that supports perception, decision, and action. We start from four postulates (P1--P4) concerning central projection, coherent update under movement, maximal invariance with minimal gauge, and normative control of viewpoint (epistemic and pragmatic value). From these postulates we prove that an agent's internal space, understood as its FoC at time $t$ (its current framing of conscious content), must live in a three-dimensional real projective space and must update, as the agent moves or imagines moving, via projective transformations, elements of the Projective General Linear group $\mathrm{PGL}(4,\mathbb{R})$, with metric content supplied only by a context-dependent gauge. The result corresponds exactly to the Projective Consciousness Model. We then discuss how the PCM accounts for the perspectival nature of consciousness and its role in perception, action, and higher cognition, and how the PCM format embeds naturally within broader computational frameworks, including the Free Energy Principle and Active Inference \cite{Friston2010,Friston2015}. In short, given the constraints of being a situated, embodied, information-seeking agent in a three-dimensional world, the projective organization formalized by PCM emerges as the uniquely parsimonious solution.

\section{Formal Argument: Deriving PCM from First Principles}
We now present the core formal argument. First, we state a set of four minimal assumptions (Postulates) about the structural and functional requirements on a conscious agent dealing with a three-dimensional environment. These premises are intended to be simple, defensible statements grounded in well-established facts of vision, sensorimotor integration, and adaptive behavior. From these premises, we then derive a sequence of Lemmas, each addressing a facet of the problem (geometric invariances, update rules, metric calibration, etc.), and finally a theorem establishing the necessity of the projective geometric structure under normative control for the conscious field. Mathematical concepts (from projective geometry and group theory) will be explained in an intuitive manner to remain accessible to a cognitive science audience, while remaining precise.


\subsection{Postulates: Structural and Functional Premises}
\begin{description}

\item[\textbf{Postulate P1: Central Projection (Perspective Optics).}]\label{rP1}
Visual sensory input is obtained (approximately) via central projection of the 3D world onto a 2D sensor (retina). In practical terms, the imaging process is well captured by pinhole-camera geometry: 3D points in the environment project to 2D points, straight 3D lines project to straight 2D lines, and parallel lines in 3D space appear to converge toward vanishing points \cite{Hartley2003}. Central Projection preserves incidence relations (collinearity/planarity), but
Euclidean quantities (lengths, angles) are generally not preserved. In short, the raw visual data available to the brain consists of 3D to 2D perspective projections of the environment.
\item[\textbf{Postulate P2: Coherent updating across viewpoint changes.}]\label{rP2}
Organisms actively move their sense organs and bodies (eyes, head, body position) while maintaining a coherent perception of object identity and spatial layout across these viewpoint changes \cite{Duhamel1992}.
Therefore the internal state must admit coherent, whole-field updates that compose across successive movements, so that successive views are integrated into a single consistent spatial framework.

\textbf{Working reading of P1-P2.}
In the main text we only rely on two empirical regularities:
(i) a stable notion of alignment (collinearity/planarity), and
(ii) viewpoint-dependent visibility/occlusion transported coherently by admissible updates.
Together with the acceptance of the notion of ``behind'' a viewpoint and that of horizon at infinity $\Pi_\infty$, this suffices (via the Fundamental Theorem of Projective Geometry) to recover a 3D projective incidence structure as the internal state. A fully explicit axiomatization and a constructive reconstruction of the underlying 3D projective incidence structure are given in Appendix~\ref{app:proof} (Theorem~\ref{thmA:necessity}).

\item[\textbf{Postulate P3: Maximal invariance with minimal gauge.}]\label{rP3}
The minimal internal representation of the environment should be defined only up to the largest class of transformations that preserve behaviorally relevant structure, under the constraints of P1 and P2. In other words, the representation should factor out (be invariant to) as many irrelevant transformations as possible, but no more (so as not to lose information needed for behavior). \emph{Motivation for minimal geometry.} Encoding a full Euclidean metric at the backbone level is costly and unnecessary for coherent update across viewpoint changes: perspective leaves only projective relations stable, and carrying extra metric structure introduces nuisance parameters that must be continuously maintained. (As we shall see, our derivation leads to the most invariant (least committed) geometry compatible with perspective, the projective one, and adds metric only as a task-dependent gauge \cite{Hartley2003}). 
\item[\textbf{Postulate P4: Normative Control of Viewpoint (Information-Seeking Action).}]\label{rP4} The organism?s selection of viewpoints (e.g. where to look, how to position or orient itself) is guided by a principled control mechanism that aims to maximize information gain and/or achieve valued goals. For example, under the Active Inference framework, actions are chosen to minimize expected free energy (EFE), which optimally balances achieving preferred outcomes (pragmatic value) and reducing uncertainty (epistemic value) \cite{Friston2010,Friston2015}. This premise implies that the internal state variables used to evaluate such objectives should be structured in a way that makes the computation and optimization of information gain natural and tractable. While we phrase P4 in terms of active inference for concreteness, an equivalent assumption could be made using other normative control frameworks, such as stochastic optimal control or reinforcement learning with intrinsic curiosity rewards. The key point is that viewpoint control is not random or reflexive but driven by an objective function that the agent seeks to optimize. We denote this generic viewpoint-selection objective by $J$ (in active-inference conventions one typically minimizes expected free energy, so one may write $J=-\mathrm{EFE}$ when expressing the problem as a maximization.)
\end{description}

These four postulates set the stage. In summary, (P1) asserts that the geometry of sensory input (at least for vision) is inherently projective; (P2) that the agent?s internal state must globally transform in tandem with the agent?s movements to maintain a stable experience; (P3) that the internal representation should employ the largest symmetry group that retains essential structure (hinting that a projective coordinate scheme is appropriate, rather than a fully metric Euclidean one); and (P4) that the internal degrees of freedom for controlling viewpoint are aligned with a well-founded normative principle of behavior (like maximizing information gain). None of these postulates alone dictates the full PCM, but we will show that taken together, they entail that the conscious field must have a projective structure.

\subsection{Derivation: From Postulates to Projective Consciousness}
\label{subsec:derivation-unified}

\begin{lemma}[\textbf{Preservation of incidence relations but not metric quantities under viewpoint changes}]\label{lem:incidence-invariants}
Under P1-P2 and restricting attention to structure-less point configurations, incidence relations (point?line/plane membership;
collinearity/planarity) are preserved across
generic viewpoint changes. No non-degenerate metric quantity (lengths, angles, parallelism) is preserved without gauge fixing (see Lemma \ref{lem:metric-gauge}).
\end{lemma}

\begin{proof}[Proof sketch]
Under central projection (P1), viewpoint changes induce collineations on projective coordinates; for generic configurations of points and lines, incidence survives such transformations. Parallel 3D lines generally meet at a vanishing point in the image, so parallelism is not preserved. Likewise, the same 2D image segment can correspond to 3D segments of different physical lengths depending on their distance to the observer, so lengths and angles vary with viewpoint. Any stable metric judgment therefore requires an additional calibration (a gauge) beyond incidence.
\end{proof}

\noindent\emph{Reader's guide.}
Central (pinhole) projection (P1) preserves incidence (straightness, collinearity/planarity), but it does not preserve Euclidean quantities. In practice, the most easily accessible viewpoint-stable structure (P2) is therefore incidence-level structure. By contrast, lengths, angles, and parallelism vary with viewpoint unless a context-dependent gauge is fixed (see Lemma \ref{lem:metric-gauge}). Put differently, what survives every admissible perspective change is which points lie on which straight lines/planes, not how long or how angled they appear.


\begin{lemma}[\textbf{Local multi-view criterion for 3D collinearity}]\label{lem:two-view}
Let $A,B,C$ be three 3D points. If there exists a view in which their realizations are collinear, and this collinearity persists for all sufficiently small viewpoint changes around that view, then $A,B,C$ are aligned in 3D (i.e.\ they lie on a common 3D line of the reconstructed incidence structure).
\end{lemma}

\begin{proof}[Proof sketch]
Note that this is an empirical regularity exploited by animal vision: under small movements, accidental 2D alignments typically break, while projections of genuine 3D edges/lines remain aligned across neighboring views. The FoC therefore treats persistence of alignment under small viewpoint changes as evidence for a real 3D line, and uses this rule to build a stable incidence structure across views.
Geometrically, in an ideal pinhole camera model, a 2D line back-projects to a plane through the eye; persistence across neighboring views identifies a unique 3D line as the intersection of such planes (We use this geometrical argument because it is straightforward, not as a Euclidean premise).
\end{proof}

\begin{remark}[\emph{Alignment}]
Alignment is inferred by combining at least two nearby views (guaranteed by P2?s ongoing ego-motion), not assumed a priori in the internal space. This delivers the concept of ?straight line? without presupposing it.
\end{remark}

\noindent\emph{Reader?s guide.}
The upshot is operational: an agent can infer 3D incidence facts (which triplets are aligned, which sets are coplanar) from robust regularities observed across neighboring viewpoints, without ever measuring Euclidean lengths or angles. It is also conceptual: straightness is not assumed in the internal code; it is inferred from ordinary multi?view evidence. In the ideal pinhole model, this regularity has a clean geometric explanation (Figure~\ref{fig:pcm-fig00}), and we take its empirical counterpart to be built into the FoC's coherent updating (P2). The constraint of ?small viewpoint changes? is there to rule out degeneracies (e.g., coincident planes or special alignments) and to reflect P2: a moving organism naturally samples such view pairs while keeping feature correspondences trackable. Figure \ref{fig:pcm-fig00} illustrates the core geometrical principles underlying our rationale based on theoretical properties of the visual system (For simplicity, the illustration assumes that the external (ambient) space is approximately Euclidean).

\begin{figure}[tbp]
\centering
\includegraphics[width=0.85\linewidth]{figures/Fig_01.png}
\caption[Pipeline and gauges across $E^3$ and $\mathbb{RP}^3$.]{\textbf{From retinal projections to an internal projective space and back.}
\emph{Top (world $E^3$ with two retinal planes).} Cube and ground line $\ell$ (green), and two central projections from (black) centers onto retinal planes $\Pi_1,\Pi_2$ (each an $\mathbb{RP}^2$ sheet) (see Postulate~P1. Purple rigid frames: admissible egocentric stances; (right frame) for current internal viewing transform $\pi(t)$; (left frame) transform $\pi(t{+}1)=G\!\cdot\!\pi(t)$ updated by viewpoint change $G\in PGL(4,\mathbb{R})$ acting on internal state (Lemma \ref{lem:reconstruction-updates} and Proposition \ref{prop:PGL-action}). With two views and known projection geometry, scene points are recovered in $\mathbb{RP}^3$ by intersecting back?projection rays (Lemma \ref{lem:two-view}). Collinearity and cross?ratios are preserved: the four marked points $P,Q,R,S$ on $\ell$ satisfy $[P,Q;R,S]=1.333$ across all spaces. Dark line: horizon $\Pi_\infty$ (intersects plane at infinity).
\emph{Bottom ($\mathbb{RP}^3$; gauges and anticipated 3D views).} Fixing gauge $A$ and performing perspective division yields affine coordinates $(u,v,d)$. Left insets: (top) rendering for $\pi(t)$ and gauge $A$ (right purple frame stance); (middle) same gauge $A$ for $\pi(t{+}1)=G\!\cdot\!\pi(t)$ (left purple frame stance); (bottom) $\pi(t)$ with gauge $A'$. Changing the gauge rescales $(u,v,d)$; projective structure (collinearity and $[P,Q;R,S]$) is invariant. Main $\mathbb{RP}^3$ panel: $45^\circ$ rigid rotation artificially applied to transformed scene (after projection under rigid frame (x,y,z) and perspective division along z). Normally impossible point-of-view (different from the perspective taken to generate the scene). The internal projective space is genuinely three?dimensional, incidence is preserved, Euclidean shape is not, the cube becomes an irregular polyhedron. With viewpoint updates as $\pi \mapsto G\!\cdot\!\pi$ with $G\in PGL(4)$, the same machinery can integrate 2D inputs, anticipate new 3D perspectives and their retinal consequences (updated states projected onto $\mathbb{RP}^2$ subspaces). This closes the perception?action loop for control (see Lemma \ref{lem:control} and PCM ansatz $F_t=(\mathbb{RP}^3,\pi_t,A_t)$ in Box \ref{prelimini}). Viewpoint: carried by $(\pi,A)$ \emph{versus} a point in $\mathbb{RP}^3$. Occlusion: handled as visibility intervals along lines of sight transported coherently by $PGL(4)$ update.
}
\label{fig:pcm-fig00}
\end{figure}

\begin{lemma}[\textbf{Incidence reconstruction and real-field selection}]\label{lem:reconstruction-updates}
Under P1-P2 (as formalized in Appendix~\ref{app:proof}), the reconstructed incidence structure satisfies the axioms of a 3D projective space. Consequently, there exists a division ring $\mathbb{F}$ and a vector space $V \cong \mathbb{F}^4$ such that the structure is isomorphic to $\mathbb{FP}^3 \cong \mathbb{P}(V)$.
Under the local order postulate (see Appendix~A), one has $\mathbb{F} \cong \mathbb{R}$.
\end{lemma}

\begin{proofsketch}
From P1-P2, one verifies Veblen-Young's incidence axioms in dimension~3. See Appendix~\ref{app:proof} for the constructive proof.
\end{proofsketch}

\begin{proposition}[\textbf{Update group as collineations in $\mathrm{PGL}(4,\mathbb{R})$}]
\label{prop:PGL-action}
On the reconstructed space $\RP^{3}$, any admissible viewpoint update that preserves 3D lines is a collineation. Collineations of $\RP^{3}$ are precisely the projectivizations of invertible linear maps of $\mathbb{R}^{4}$; hence updates act via matrices in $\mathrm{PGL}(4,\mathbb{R})$, and compositions correspond to matrix multiplication.
\end{proposition}

\begin{proofsketch}
In projective geometry, for $\mathbb{P}^n$ with $n\ge 2$,
any collineation arises from a semilinear automorphism of the underlying vector space. Over $\mathbb{R}$, field automorphisms are trivial, so collineations are exactly elements of $\mathrm{PGL}(4,\mathbb{R})$ (see, e.g., \cite{Hartley2003}, Chap.~2).
\end{proofsketch}

Box \ref{prelimini} provides relevant information and notations to interpret and support the demonstration.

\medskip
\begin{infobox}
\captionof{infbox}{\textbf{PCM Preliminaries: States, Notations and Principles.}}
\label{prelimini}
Before progressing in our demonstration, it is useful at this point to introduce and discuss key formal concepts. 
We treat the conscious field (FoC) at time $t$ as
\[
F_t \;=\; \big(\mathbb{RP}^3,\;\pi_t,\;A_t\big),
\]

where $\mathbb{RP}^3$ denotes the real projective 3-space (the set of one-dimensional subspaces of $\mathbb{R}^4$), $\pi_t\!\in\!PGL(4,\mathbb{R})$ is the current internal viewing transform, and $A_t$ is a (possibly time?varying) gauge (absolute) used to recover metric read?outs (sizes, angles, scales) when required. 

We use homogeneous coordinates $[x:y:z:w]$ on $\mathbb{RP}^3$; the plane at infinity is 
$\Pi_\infty=\{w=0\}$. Fixing a gauge selects an affine chart $w\neq 0$ and defines 
affine coordinates $(u,v,d)=(x/w,y/w,z/w)$ for visualization.

Under (P1)-(P2), any admissible change of viewpoint $g$ (e.g.\ an eye/head/body rigid motion) induces a collineation $\rho(g)\in PGL(4,\mathbb{R})$ acting linearly on homogeneous coordinates, with composition $\rho(g_2g_1)=\rho(g_2)\rho(g_1)$ (see Lemma~\ref{lem:reconstruction-updates}). For a complete, self-contained reconstruction of the projective incidence space and the selection of the real field from postulates P1-P2, see Appendix~\ref{app:proof} (Theorem~\ref{thmA:necessity}).

Informally, once a gauge and an egocentric chart are fixed, this can be read as an ?egocentric field of lines of sight? or ?space of rays?? (see Remark~\ref{rem:no-internal-origin}). 

\begin{remark}[On ?space of rays?? and the egocentric origin]
\label{rem:no-internal-origin}

Crucially, there is no distinguished ?origin point? \emph{in} $\mathbb{RP}^3$ from which rays literally emanate. $\mathbb{RP}^3$ is the set of one-dimensional subspaces of $\mathbb{R}^4$ (homogeneous coordinates), i.e. the set of lines through $0\in\mathbb{R}^4$. The zero vector ($0$ itself) is excluded and there is no privileged ?origin? in $\mathbb{RP}^3$. Thus, the phrase ?space of rays? should not be read as if there were a literal point inside the represented space from which all rays emanate. \emph{Where the viewpoint lives.} The agent?s current viewpoint is carried by the collineation $\pi_t\in PGL(4,\mathbb{R})$ together with a context-dependent gauge $A_t$; changing viewpoint corresponds to multiplying by another collineation (P2). The viewpoint lives in $(\pi_t,A_t)$, it is not encoded by a point inside the space. In other words, ?moving the mind eye? is not moving a point in $\mathbb{RP}^3$ but changing the chart by which the projective state is read. \emph{What ?rays?? are in this setting.} In a chosen chart (fixed by $(\pi_t,A_t)$), each line of sight is represented by a projective line $L\subset\mathbb{RP}^3$ belonging to the pencil/congruence selected by that chart. The ``eye point'' that organizes this pencil is therefore chart-dependent rather than an intrinsic,
distinguished point of $\mathbb{RP}^3$. Finite scene points along that sightline are different points on $L$ with $w\neq 0$; the common direction of that sightline is where $L$ meets the plane at infinity $\Pi_\infty$ (the set $w=0$ fixed by the gauge $A_t$). \emph{Intuitive example (vanishing points).} Looking down a straight corridor, parallel floor tiles and wall edges seem to ?converge? at a vanishing point. In $\mathbb{RP}^3$, those parallel 3D lines are represented by families of projective lines whose images intersect at a single point on $\Pi_\infty$ (a direction). The phenomenal sense that ?everything radiates from here? comes from how $\pi_t$ and $A_t$ chart the field: it is a property of the chart and gauge on $\mathbb{RP}^3$, not evidence for a literal eye-point inside the represented space. \emph{Behind and at infinity.} The global carrier is the whole $\mathbb{RP}^3$, including directions behind the organism and the plane at infinity $\Pi_\infty$. Apparent left/right or up/down flips when crossing $\Pi_\infty$ are changes of chart/gauge in the projective atlas, not contradictions in the state. Phenomenologically two equivalent readings coexist: (i) glue a chart that renders the rear sector continuously, or (ii) reason as if a virtual rotation had been applied to $\pi_t$. Artefacts arise only if one forces a single Euclidean chart (a ``half?space?? with clipping at $\Pi_\infty$).
\end{remark}

\begin{remark}[Equivalent quotient formulation]
\label{rem:quotient-form}
For completeness, one could also encode the FoC as a quotient of triples
\[
\mathcal{T} \;=\; \{(\text{eye pose }E,\ \text{gaze direction }g,\ \text{retinal point }x)\},
\]
with the equivalence relation
\[
(E,g,x)\ \sim\ (E',g',x')\quad\Longleftrightarrow\quad
\exists\text{ a world point }X\text{ such that }x=\mathrm{proj}_{E}(X),\ x'=\mathrm{proj}_{E'}(X),
\]
i.e.\ the two triples arise as central projections of the same scene point (P1).
Up to mild genericity assumptions (P2), the quotient $\mathcal{T}/\!\sim$ is naturally isomorphic to a 3D projective incidence structure. Invoking P3 (maximal invariance, minimal gauge) collapses the redundancy of triples to the minimal sufficient state, which we identify with $\mathbb{RP}^3$ endowed with a $PGL(4)$ action. In this reading, triples are charts for reading the state; again the viewpoint itself is not a point in $\mathbb{RP}^3$ but is carried by the pair $(\pi_t,A_t)$. See also Appendix~\ref{app:proof}, definitions~\ref{defA1}-\ref{defA4}.
\end{remark}

\begin{remark}[\emph{Enriched invariants are optional, not required}]\label{rem:enriched}
Projective transformations preserve incidence relations and other projective invariants, e.g. the cross-ratio on a line. If one lets the projective group act on enriched objects (e.g., $n$-jets of curves), additional projective (differential) invariants exist. They are compatible with, but not required for, our derivation: the backbone we use later (incidence reconstruction, group action) depends only on incidence-level structure. Agents may attach such enriched features as fibers over the projective base. In practice, such enriched invariants can enhance recognition, disambiguate curvature/occlusion, and stabilize predictions under noise or sparsity, but they do not alter the base $\mathbb{RP}^3$ geometry or its $\mathrm{PGL}(4)$ updates.
\end{remark}

\begin{remark}
Physically realized moves form a subgroup $H\le \mathrm{PGL}(4,\mathbb{R})$. Postulate~P3 (maximal invariance with minimal gauge) licenses using the full $\mathrm{PGL}(4)$ as representational symmetry. Concrete actions
embed as $H\hookrightarrow \mathrm{PGL}(4)$.
\end{remark}

\begin{remark}[Occlusion as viewpoint-dependent visibility]
\label{rem:occlusion}
Occlusion can be treated as lightweight structure over the incidence backbone.
Given a projective line of sight $L\subset\mathbb{RP}^3$ (a ray in a chosen affine chart), fixing a gauge $A_t$ induces an order parameter $s$ (depth) along $L$. \emph{Opaque-case default.} For opaque scenes, visibility along a ray $L$ is a single interval $[s_-,s_+]$ (from the nearest intersection up to the first occluder). A union of intervals only arises with transparency or apertures. The transport rule under $\mathrm{PGL}(4)$ updates is unchanged. In the latter case, the visible set is a finite union of intervals $V_{\pi_t,A_t}(L)=\bigcup_k [s_k^-,s_k^+]$. Under a viewpoint update $\pi_{t+1}=G\!\cdot\!\pi_t$ with $G\in PGL(4)$, visibility transports functorially:
\[
L\mapsto G(L),\qquad V_{\pi_{t+1},A_t}(G(L)) \;=\; G\big(V_{\pi_t,A_t}(L)\big),
\]
up to the change of affine chart. Thus occlusion is a projectively coherent, viewpoint?dependent fiber over lines of sight; it does not alter the base incidence structure reconstructed in Lemma~\ref{lem:reconstruction-updates}.
\end{remark}

\begin{remark}[2D?3D bridge: collineations act internally; image homographies are special cases]\label{rem:internal_vs_image}
In computer vision, the mapping between two images is a single planar homography only when the observed scene is planar (or effectively planar). For a generic non-planar scene there is no single global 2D collineation that maps one entire image to the other: the correspondence depends on depth. Our claim is different: the internal whole-field update acts as a 3D collineation on the projective backbone (RP$^3$), and retinal predictions for any given view are obtained by projecting that updated 3D state (P1).
\end{remark}
\end{infobox}


\begin{proposition}[\textbf{Maximality of the projective backbone}]\label{rem:minimality}
Under P1-P3, the minimal sufficient backbone for coherent whole-field updating is the 3D projective space RP$^3$ together with a collineation update law (PGL$(4,\mathbb{R})$ action). Additional variables (e.g.\ velocity, efference copy, control states) may be carried as fibers for dynamics, but they are not required by the geometrical backbone nor by the update rule.
\end{proposition}

\begin{proof}[Proof sketch: empirical motivation and group-theoretic argument]
\emph{Empirical motivation.} Perceptual constancies and systematic illusions indicate that the brain does not maintain a globally veridical Euclidean model at all times; instead it relies on context-dependent recalibrations that can be non-isometric (e.g.\ horizon- and scale-dependent effects as in the Moon illusion; see Section~\ref{disc:illusions}). This supports treating metric content as a separable
overlay rather than part of the backbone. \emph{Group-theoretic closure.} The subgroup of projective transformations that preserves a chosen plane at infinity is an affine subgroup. Once the system?s repertoire includes at least one admissible incidence-preserving update that does not preserve that plane (a genuinely projective ?re-gauging?), the smallest closed update group containing the affine subgroup and that extra update is the full
collineation group PGL$(4,\mathbb{R})$. Postulate P3 then licenses using this maximal symmetry as the representational update law.
\end{proof}

\noindent\emph{Reader?s guide.}
Think of the internal state as a 3D space structured by an egocentric field of lines of sight. Lemma \ref{lem:two-view} showed how simple multi?view tests (e.g., two nearby views) let the agent infer which triplets of points are aligned, and which sets lie in a common plane, without assuming straightness in advance. Classical results in projective geometry then guarantee that a structure with just these incidence facts (points?lines?planes and their membership relations) is isomorphic to a 3?D projective space. In that space, a change of viewpoint is implemented by multiplying once by a single $4\times4$ homogeneous matrix; two moves simply compose as two matrix multiplications. Intuitively: ?move the eyes/head/body? $\Rightarrow$ ?apply one whole?field projective update.? 
The 2D?3D bridge is straightforward: the update acts internally in 3D projective coordinates, and retinal predictions are obtained by projecting the updated state; so we do not require the entire image warp to be one planar homography when the scene is not planar. This is why the format is both minimal and powerful: it preserves exactly what perspective preserves (incidence), and it supplies a single global rule for coherently updating every feature in the field when the viewpoint changes.


\begin{lemma}[\textbf{Metric perception is a gauge choice}]\label{lem:metric-gauge}
%\label{lem:gauge}
No non?degenerate distance/angle is invariant under $PGL(4,\mathbb{R})$ on $\mathbb{RP}^3$. Euclidean/affine metrics arise only after fixing an \emph{absolute} (e.g., plane at infinity, absolute conic), i.e., a gauge $A$ layered on the projective backbone.
\end{lemma}
\begin{proofsketch}
The full projective group is too large to preserve a non?trivial Riemannian metric. In the Cayley?Klein construction, Euclidean (or spherical/hyperbolic) metrics are recovered by specifying an absolute. Hence metric content is context?dependent gauge atop projective structure.
\end{proofsketch}
\begin{corollary}[No proper metric-preserving subgroup suffices]\label{cor:nometric}
The update group for the internal projective state cannot be confined to a proper metric-preserving subgroup of $\mathrm{PGL}(4,\mathbb{R})$ (e.g., similarities or rigid motions). Generic viewpoint changes under P1?P2 produce
collineations whose induced effects are not contained in such subgroups. Restricting to any metric?preserving subgroup either breaks P2/P3 or forces you to add gauge anyway.
\end{corollary}

\begin{remark}[Equivariant gauge (transport-compatible metric read-out)]
\label{rem:equivariant-gauge}
We call a gauge $A$ equivariant if metric read-out is compatible with backbone transport: for any collineation $G\in\mathrm{PGL}(4)$, reading out Euclidean/affine judgments after a backbone update by $G$ is equivalent to reading out before the update with the transported gauge $G\!\cdot\!A$. Intuitively, $A$ never alters incidence or the group action; it only adds metric structure where needed.
\end{remark}


\noindent\emph{Example (Euclidean gauge via an absolute).}
In homogeneous coordinates, a Euclidean metric on an affine chart can be encoded by choosing (i) a plane at infinity $\Pi_\infty$ (fixing the affine chart) and (ii) an absolute quadric (or absolute conic on $\Pi_\infty$) that determines angles. This pair is a gauge $A$ in the Cayley--Klein sense.
If the backbone is updated by a projective transformation $G\in\mathrm{PGL}(4,\mathbb{R})$, the transported gauge is $G\cdot A$ (push-forward of the absolute). Reading out Euclidean relations after updating the backbone is equivalent to reading them out before the update but with the transported gauge: this is exactly the equivariance (transport-compatibility) condition.


\noindent\emph{Reader?s guide (Frame vs.\ Gauge).} \emph{Conclusion so far:} Projective space has no built?in ruler; Euclidean/affine read?outs appear only once a context?dependent gauge $A$ (e.g., a plane at infinity or absolute conic) is fixed (Lemma~\ref{lem:metric-gauge}). Under P1 and P3, any Euclidean or affine measurements in perception (distances, sizes, angles, parallelism) therefore do not belong to the backbone?they are recovered only after gauge fixing. Intuitively, fixing the gauge is like calibrating a camera (choosing a horizon and zoom): recalibration changes measured sizes/angles while leaving straightness and incidence untouched.
Perceptual constancies and systematic illusions can be both consequences of gauge choice. When cues are rich (familiar sizes, a clear horizon or gravity axis, binocular disparity, etc.), the brain can anchor $A_t$ so that projective coordinates yield percepts (constancies) \cite{Rudrauf2017}. When cues are sparse or misleading, the same projective state $\pi_t$ interpreted under a mis?set $A_t$ produces lawful distortions or illusions. In short, metric content is a flexible overlay on a stable projective base: change the overlay and perceived lengths/angles change, while straightness and incidence do not. For near?field manipulation (tool use, grasping), local control typically fixes a robust Euclidean gauge, which explains why motor computations often look Euclidean at everyday scales \citep{Rudrauf2017,Williford2018}\footnote{On the Cayley-Klein view, Euclidean/affine/spherical/hyperbolic metrics arise by choosing an ``absolute.'' We only need the takeaway: the full projective group preserves no non-trivial metric.}. Thus many illusions are not failures of geometry but lawful outcomes of two separable mechanisms according to the PCM. Frame $F$ (perspectival structure) defines which eye-point/vanishing structure is adopted. Bistabilities as in the Necker cube are frame switches that preserve incidence \cite{Rudrauf2017}. Gauge $A$ (metric read-out) corresponds to a context-dependent calibration (horizon, vertical, scale) used when needed for metric judgments; changing $A$ warps sizes/angles but leaves straightness intact. A variety of illusions can be related to $A$ (e.g. Moon Illusion and Ames Room)\cite{Rudrauf2020}. See Section \ref{disc:illusions} for detailed discussion. Typically, real scenes engage both: $F$ fixes perspectival organization; $A$ supplies context?dependent metric.


\paragraph{Where do we stand?} Figure \ref{fig:pcm-fig00} summarizes the pipeline established by P1-P4 and Lemmas 1-4. Starting from two central projections (P1) onto retinal planes $\Pi_1,\Pi_2\subset \mathbb{RP}^3$, back?projection of corresponding image points reconstructs a 3D projective scene in which incidence relations and cross?ratios are the invariants (Lemmas 1?3). The conscious field is therefore encoded as $F_t=(\mathbb{RP}^3,\pi_t,A_t)$, where viewpoint changes compose as $\pi_{t+1}=G\!\cdot\!\pi_t$ with $G\in PGL(4,\mathbb{R})$, and metric read?outs appear only after fixing a gauge $A_t$ (Lemma 4). 

2D retinal views can be predicted a posteriori by projecting the internal $\mathbb{RP}^3$ state onto a selected projective plane $\Pi\cong\mathbb{RP}^2$ embedded in $\mathbb{RP}^3$; the retinal image is not the plane itself, but the result of a central projection onto that plane. Thus, the same $PGL(4)$ action that integrates incoming views can also anticipate counterfactual views and transports visibility/occlusion intervals coherently along lines of sight (Remark~\ref{rem:occlusion}): apply $G$ to the internal state and re?project onto any chosen retinal plane to predict what would be seen from $\pi_{t+1}$. This closes the loop from 2D views $\leftrightarrow$ internal $\mathbb{RP}^3$ $\leftrightarrow$ anticipated 2D views and sets the stage for normative viewpoint control (Lemma 5), where choosing $G$ amounts to planning perception?guiding actions. 

\begin{lemma}[\textbf{Projective control is economical for Kullback-Leibler (KL)-regularized objectives}]
\label{lem:control}
Under P4, parameterizing controllable viewpoints by a low-dimensional chart of $\mathrm{PGL}(4)$ (rotations/translations/zoom in homogeneous coordinates\footnote{Here ``zoom'' refers to the focal/scale parameter in the $\mathrm{PGL}(4)$ chart.}) makes whole-field predictions vary smoothly with parameters (concrete behavioral advantages are discussed in Section \ref{dis-control}).
We write $J(\pi,A)$ for the agent's normative viewpoint-selection objective (e.g.\ expected free energy, or an RL return with intrinsic/regularization terms),
optimized over controllable viewpoint updates $\pi$ and (when relevant) over gauge $A$. Fix time $t$ and let $G_{\mathrm{act}}\le \mathrm{PGL}(4,\mathbb{R})$ be the (finite-dimensional) Lie group of controllable viewpoint updates acting on the projective backbone $W_t\in \mathbb{RP}^{3}$
(Lemma~\ref{lem:two-view} and Proposition~\ref{prop:PGL-action}).
Let $d:=\dim G_{\mathrm{act}}$ (so for $4\times4$ projective transformation matrices $d\le 15$), and choose a smooth local chart $g:\Theta\subset\mathbb{R}^d\to G_{\mathrm{act}}$ with $g(0)=e$.\footnote{In typical embodied settings, $d$ is on the order of the physical degrees of freedom (e.g.\ pan/tilt/translation and possibly a focal/scale ``zoom'').}
Define the imagined update $\pi_{t+1}=g(\theta)\,\pi_t$.

Let $q_\theta(y)$ denote the predictive distribution over whole-field sensory outcomes $y$ (e.g.\ retinal images, proprioceptive cues) induced by applying the single collineation update $W_t\mapsto g(\theta)\!\cdot\! W_t$ and projecting through the agent's generative model (including visibility/occlusion as modeled). Assume $\theta\mapsto q_\theta$ is $C^2$ in a neighborhood of $\theta$ and has $\theta$-independent support (with $q_\theta(y)>0$ on that support).

Then the predictive KL divergence has the standard local quadratic form
\[
\mathrm{KL}\!\left(q_{\theta+\Delta\theta}\,\|\,q_\theta\right)
= \tfrac12\,\Delta\theta^\top F(\theta)\,\Delta\theta + o(\|\Delta\theta\|^2),
\]
where
\[
F(\theta):=\mathbb{E}_{y\sim q_\theta}\!\left[
\nabla_\theta \log q_\theta(y)\;\nabla_\theta \log q_\theta(y)^\top
\right]
\]
is the Fisher information matrix (the pullback of the Fisher-Rao metric on outcome distributions).

Consequently, whenever viewpoint selection is framed by a normative objective that includes KL/entropy/information terms, as in expected free energy (active inference), KL- or entropy-regularized RL
(trust-region / KL-constrained updates), and KL-control in stochastic optimal control, the local cost of a small viewpoint update is governed by the $d\times d$ matrix $F(\theta)$.
In this precise sense, a projective substrate with global whole-field updates and low-dimensional control variables provides an economy of means for minimizing $J$.
\end{lemma}

\begin{proofsketch}
The map $\theta\mapsto q_\theta$ is smooth because updates act by a single collineation on the backbone and the forward model is differentiable under the stated regularity assumptions. The displayed expansion is the standard second-order Taylor approximation of $\mathrm{KL}(q_{\theta+\Delta}\|q_\theta)$; its Hessian at $\Delta=0$ is the Fisher information matrix $F(\theta)$.
\end{proofsketch}

\begin{remark}[Beyond KL]
The same conclusion (a local quadratic penalty governed by the same Fisher-Rao pullback on the action chart) extends to other smooth $f$-divergences and to KL-based epistemic terms (e.g.\ mutual information / expected information gain), but this is beyond the scope of this report.
\end{remark}

\noindent\emph{Reader?s guide.}
Lemma~\ref{lem:control} is not a new control theory: it isolates a single structural advantage of PCM-relevant projective control. Because a candidate action is a single whole-field collineation, and because the controllable family is parameterized by a low-dimensional Lie chart, many modern normative schemes reduce locally to manipulating a small curvature matrix $F(\theta)$ rather than the full sensor space, yielding natural-gradient-like updates. This is the sense in which projective control is concise, globally coherent, and computationally economical (See Figure \ref{fig:pcm-control} for illustration).


\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{figures/Fig_02.png}
\caption[Projective control and epistemic value.]{\textbf{Projective control and epistemic value.} \emph{Panels left / right.} A vertical wall (violet) occludes a target cube (green). \emph{Expected Epistemic Value (EEV) field (insets).} In this figure, the epistemic score used for action selection is a simple proxy for illustration (Formally, active-inference epistemic value is typically defined as an expected KL / mutual information; Lemma~\ref{lem:control} states the corresponding local information geometry.) For each candidate ground cell the agent?s internal ?camera? is oriented toward the global maximum of the 3?D prior (the cube center). The prior density $Q(x)$ is then centrally projected into the field of consciousness with wall occlusion. Contributions that land on pixel $(i,j)$ are accumulated to $M_{ij}$, yielding the visible mass $V_M=\sum_{i,j} M_{ij}$ and the normalized image $p_{ij}=M_{ij}/V_M$; the Shannon entropy of this normalized image is $H(p)=-\sum_{i,j} p_{ij}\log p_{ij}$. The epistemic expected value used for action selection is the product $\mathrm{EEV}=V_M\,H(p)$, which jointly favors views that reveal more of the prior (large $V_M$) and distribute it informatively across the FoC (large $H$), in line with the projective?control rationale of P4/Lemma \ref{lem:control}. For the present figure, we computed the accumulation on the 2D image for simplicity; conceptually the same computation lives in $\,\mathbb{RP}^{3}$ (see \cite{Sergeant2025BC}). Brighter pixels indicate higher expected information gain. In the \emph{left panel}, the agent is at $\pi_{t}$ (solid orange glyph), and the dashed glyph indicates the imagined move that maximizes EEV. In the \emph{right panel}, the agent occupies that pose, i.e., $\pi_{t+1}=g\!\cdot\!\pi_{t}$ with $g\in\mathrm{PGL}(4)$; the prior is discounted on voxels that became visible, and the EEV landscape collapses around the newly revealed region. \emph{Link to the formal argument.} The figure instantiates the pipeline fixed in Figure \ref{fig:pcm-fig00}: retinal projections ($\mathbb{RP}^{2}$ sheets) are back-projected into a 3D projective FoC, and viewpoint changes act by collineations on that state ($\pi_{t}\mapsto g\!\cdot\!\pi_{t}$). This is precisely the structure established by Theorem \ref{thm:PCM} (?PCM necessity?: FoC$\cong\mathbb{RP}^{3}$ with a $\mathrm{PGL}(4)$ action and an equivariant gauge), itself derived from P1-P3 (central projection; coherent cross-viewpoint updates; maximal invariance with minimal gauge). P4 then turns this representational state space into a control landscape: parameterizing actions by projective degrees of freedom (pan / tilt / zoom / translation in $\mathrm{PGL}(4)$) yields smooth, globally consistent prediction changes and steep epistemic gradients under limited Field of View (FoV) and occlusion, as formalized in Lemma \ref{lem:control} and discussed in Section \ref{dis-control}. Together the two panels visualize how the agent uses the projective state to anticipate informative views (\emph{left panel}), apply a single whole-field projective update to guide the realization of the new stance (\emph{right panel}), and then re-evaluate uncertainty; thus closing the PCM perception-action loop}
\label{fig:pcm-control}
\end{figure}

\begin{theorem}[\textbf{PCM necessity}]
\label{thm:PCM}
Given (P1)--(P4), any internal state that is closed under viewpoint updates must (up to isomorphism) be a $3$-D real projective space $\mathbb{RP}^3$ equipped with: (i) an action of $\mathrm{PGL}(4,\mathbb{R})$ implementing viewpoint changes; and (ii) a context-dependent gauge $A$ to recover metric judgments when needed. Any alternative sufficient format either embeds this structure or violates at least one postulate.
\end{theorem}

\begin{proofsketch}
By Lemma~\ref{lem:incidence-invariants} and Lemma~\ref{lem:two-view}, alignment is a derived multi-view invariant,
yielding a well-defined incidence structure. By Lemma~\ref{lem:reconstruction-updates}, this structure is (up to
isomorphism) $\mathbb{RP}^3$ and viewpoint changes act by $\mathrm{PGL}(4,\mathbb{R})$. By P3 (maximal invariance with
minimal gauge), no superfluous metric structure is added at the backbone level; instead, metric content is supplied
only via a context-dependent gauge, as stated in Lemma~\ref{lem:metric-gauge}. Under P4, the projective parameterization
of controllable viewpoints aligns with normative objectives (Lemma~\ref{lem:control}). Hence the internal state must be a
$3$D real projective space equipped with a $\mathrm{PGL}(4)$ action and an equivariant gauge.
\end{proofsketch}

\noindent\emph{Reader?s guide.} Given Postulates P1?P4, any internal representational space that satisfies all these constraints must (up to isomorphism) be a three?dimensional projective space ($\mathbb{RP}^3$). In such an agent, the internal state must transform under viewpoint changes by the action of the projective linear group ($PGL(4,\mathbb{R})$), and an adaptive ?absolute?? gauge ($A_t$) must be maintained to recover Euclidean details when needed. In effect, the conscious field is a projective 3D model of the world (perspectival geometry) coupled to the agent?s sensory apparatus and actively controlled by the agent?s choices, i.e., the Projective Consciousness Model (PCM). No alternative type of representational format (whether a strictly Euclidean spatial map, a topological graph, or other scheme (see Section \ref{discussion}) can fulfill the premises without either embedding an equivalent projective structure or failing to account for the observed characteristics of perception and action. The preceding lemmas establish the critical pieces: From P1 and P2, the internal representation must at minimum preserve projective invariants of the environment and update via projective (collineation) transformations (Lemmas~\ref{lem:incidence-invariants} and \ref{lem:reconstruction-updates}). P3 then asserts that one should not assume more structure than necessary, which rules out building in a full Euclidean metric \emph{a priori}; instead, the representation is exactly projective up to a flexible metric gauge (Lemma~\ref{lem:metric-gauge}). This means the foundational geometry of the internal state is projective 3?space ($\mathbb{RP}^3$), with any Euclidean (or other metrical) interpretations coming from an added, context?adjustable gauge ($A_t$). The uniqueness of projective invariants and transformations for perspective vision essentially pins down this geometry of the conscious field. Finally, P4 connects this representational format to functional optimality: if the agent controls its view according to an information? or reward-seeking principle, representing that view in projective terms (like a 3D virtual camera) yields efficient identification of epistemically or pragmatically valuable actions (Lemma~\ref{lem:control}). Meanwhile, any alternative representational strategy that is not equivalent to PCM would either fail to preserve basic cross?viewpoint coherence (violating P2 or P3) or would implicitly require adding a projective?like layer on top to interpret sensory data (thus reducing to PCM anyway) (see Section \ref{sec:objections} for detailed discussion). 

\paragraph{Where this leaves the FoC.}
The lemmas together force the representational ansatz stated in Box \ref{prelimini} :
\[
F_t \;=\; \big(\mathbb{RP}^3,\;\pi_t\!\in\!PGL(4,\mathbb{R}),\;A_t\big).
\]

The projective backbone carries what is stable and globally updatable under viewpoint change; and explains lawful illusions as arising either from frame?selection ambiguities (underdetermined $F$) or from context?optimal gauge re?settings $A$ (see Section~\ref{disc:illusions}). The control landscape over $\pi_t$ provides a natural backbone for epistemic/pragmatic optimization. In short: once (P1?P4) are granted, a projectively structured FoC is not a modeling convenience but the unique parsimonious solution. In conclusion, under assumptions P1?P4, the agent?s conscious field must take the form described by PCM. Figure \ref{fig-diag} presents a summary of the main PCM algorithmic loop, recapitulating the concepts covered in theorem \ref{thm:PCM}'s demonstration. 



\begin{figure}[H]
\centering
\tikzcdset{
  math mode=false,
  cells={nodes={font=\small, align=left}},
  every label/.append style={font=\scriptsize, inner sep=1pt}
}
\begin{tikzcd}[column sep=1.0em, row sep=2.8em]
% ---------------- GRID (rows; main pipeline now in column 2) ----------------
{} & {world state $o_t$} & {} & {} & {} & {} & {} & {} \\
{} & {$Y_t^{(1,2)}$}     & {} & {} & {} & {} & {} & {} \\
{} & {$W_t\in \RP^{3}$} & {} & {} & {} & {} & {} & {} \\
{} & {select $(\pi,A)$ with $\pi\in\mathrm{PGL}(4)$ to optimize $J(\pi,A)$} & {} & {} & {} & {} & {} & {} \\
{} & {$W_t^{+} = \pi\, W_t$} & {} & {} & {} & {} & {} & {} \\
{} & {\textit{predict} $\widehat Y_t$ \ via forward proj.\ $q_{\mathrm{ret}}$} & {} & {} & {} & {} & {} & {} \\
{} & {action $u(\pi)$ (adopt stance dual to $\pi$)} & {} & {} & {} & {} & {} & {} \\
{} & {world state $o_{t+1}$}    & {} & {} & {} & {} & {} & {} \\
{PE(ret)} & {$Y_{t+1}^{(1,2)}$} & {} & {} & {} & {} & {} & {} \\
{} & {$y_{t+1}^{o}\subset \RP^{3}$} & {PE($\RP^{3}$)} & {} & {} & {} & {} & {}
%
% ---------------- CONNECTIONS (indices updated because main column is now 2) ----------------
% sensing at t
\arrow[from=1-2, to=2-2, dashed, "\scriptsize central proj.", swap]
\arrow[from=2-2, to=3-2, "\scriptsize back-proj.\ $\phi^{-1}$"]
% to selector
\arrow[from=3-2, to=4-2]
% selector ? W^+
\arrow[from=4-2, to=5-2, "\scriptsize collineation $\pi$"]
% W^+ ? predict \hat Y (implicit vertical flow)
\arrow[from=5-2, to=6-2]
% action
\arrow[from=5-2, to=7-2, out=-90, in=90]
\arrow[from=7-2, to=8-2]
% sensing at t+1
\arrow[from=8-2, to=9-2, dashed, "\scriptsize central proj.", swap]
\arrow[from=9-2, to=10-2, "\scriptsize back-proj.\ $\phi^{-1}$"]

% --- RP^3 PE: compare y_{t+1}^o with W_t^+
\arrow[from=10-2, to=10-3]                                  % y_{t+1}^o ? PE(RP^3)
\arrow[from=5-2,  to=10-3, out=0, in=140, looseness=1.0]  % W_t^+ ? PE(RP^3)

% --- Retinal PE: compare \hat Y_t with Y_{t+1}^{(1,2)}  (PE(ret) moved to row 9, col 1)
\arrow[from=6-2,  to=9-1, dashed, out=180, in=50, looseness=1.0] % \widehat Y_t ? PE(ret)
\arrow[from=9-2,  to=9-1, dashed]                                % Y_{t+1}^{(1,2)} ? PE(ret)

% --- Each PE sends its own update to the selector (no junction)
\arrow[from=9-1, to=4-2, out=180, in=180, looseness=0.51, "\scriptsize update $\nabla J$"]  % PE(ret) ? select
\arrow[from=10-3, to=4-2, out=0, in=0, looseness=0.5, "\scriptsize update $\nabla J$"'] % PE(RP^3) ? select
\end{tikzcd}

\caption{\textbf{PCM main algorithmic loop.} 
The main vertical chain shows the internal generative/recognition cycle; dashed paths are the physical sensorimotor pathway. 
(1) The current world state \(o_t\) produces one or two retinal images $Y^{(1,2)}_t$ depending on whether the agent is monocular or binocular, by central projection (dashed). 
(2) A back?projection \(\phi^{-1}\) maps \(Y_t^{(1,2)}\) to the internal projective state \(W_t\in \RP^{3}\). 
(3) The selector chooses both a collineation \(\pi\in \mathrm{PGL}(4)\) and a gauge \(A\) by optimizing \(J(\pi,A)\). The gauge $A$ supplies context-dependent metric read-out (e.g., horizon/vertical/scale) and does not affect the incidence-preserving backbone update.
(4) Applying \(\pi\) yields the predicted internal state \(W_t^{+}=\pi\,W_t\) (collineation). 
(5) From \(W_t^{+}\), the model predicts a pair (in the binocular case) of retinal images \(\widehat Y_t\) via forward projection.
We denote by $q_{\mathrm{ret}}$ the forward central-projection operator that maps a 3D projective state to one or two retinal planes (with occlusion), and by $\phi^{-1}$ (see above) its back-projection dual used for reconstruction.
(6) The action \(u(\pi)\) (adopting the stance dual to \(\pi\)) changes the world to \(o_{t+1}\). 
(7) New data are acquired: \(o_{t+1}\!\to\! Y_{t+1}^{(1,2)}\) (dashed), then \(\phi^{-1}\!: Y_{t+1}^{(1,2)}\!\mapsto\! y^{o}_{t+1}\subset \RP^{3}\). 
Two comparison nodes compute prediction errors (PE) and independently drive learning: 
PE(ret) compares \(\widehat Y_t\) to \(Y_{t+1}^{(1,2)}\) (dashed inputs), and PE(RP\(^3\)) compares \(y^{o}_{t+1}\) to \(W_t^{+}\) (solid inputs). 
Each PE returns an \(\mathrm{update}\;\nabla J\) (curved arrows) to the selector, closing the loop. 
Thus, \(\pi\) controls both (i) the internal whole?field update in \(\RP^{3}\) and (ii) the physical stance that generates the next observations; gauge \(A\) provides the context?dependent metric readout.}
\label{fig-diag}
\end{figure}

\paragraph{Summary of Argument.}
The argument says: (i) perspective leaves only projective relations stable; (ii) viewpoint moves are best modeled as a single whole?field matrix update; (iii) if you keep any extra structure beyond that, you carry needless baggage; if you throw away more, you lose what behavior needs; (iv) distances and sizes are added after the fact as a context?sensitive calibration; (v) control then becomes simple: try matrix tweaks that maximize expected information gain and/or reward. Together, this uniquely picks out a projective $3$?space with a tunable metric overlay. Each step uses only P1-P4 and standard results: Lemma~\ref{lem:incidence-invariants} (incidence?level invariants) and Lemma~\ref{lem:two-view} (two?view alignment) yield a projective incidence structure; Lemma~\ref{lem:reconstruction-updates} reconstructs $\mathbb{RP}^3$ and the $PGL(4)$ update; P3 and Lemma~\ref{lem:metric-gauge} separate metric as gauge; and Lemma~\ref{lem:control} connects the representation to normative control (P4). No conclusion is presupposed by the postulates; the chain is monotone (each lemma adds a necessary piece), and the theorem follows by conjunction. Hence the demonstration achieves the stated goal: given the premises, the PCM format is not optional but necessary.

\begin{remark}[Methodological status of the theory (relative necessity)]
The main theorem establishes a conditional or relative necessity
result: if the postulates (P1-P4) hold (central projection of inputs,
coherent cross?viewpoint integration, maximal invariance with minimal gauge,
and normative control) then  the internal representational format must be
projective (PCM). This is not a metaphysical proof that no conceivable system
could ever realize consciousness otherwise; rather, it shows that given
these widely evidenced and independently motivated premises, non?projective
formats either embed an equivalent projective layer or fail at least one
premise. In this pragmatic, scientific sense, PCM is the unique parsimonious
solution under P1-P4.
\end{remark}



\begin{table}[t]
  \centering
  \caption{Projective geometry \(\leftrightarrow\) cognition/neuroscience dictionary.}
  \label{tab:dictionary}
  \begin{tabular}{p{0.32\linewidth} p{0.62\linewidth}}
    \hline
    \textbf{Geometry} & \textbf{PCM meaning / cognitive-neural interpretation} \\
    \hline
    $\mathbb{RP}^3$ & Egocentric projective field (backbone of lines of sight; no distinguished origin in $\mathbb{RP}^3$; the current viewpoint is not a point of $\mathbb{RP}^3$ but the pair $(\pi_t,A_t)$ that fixes chart and gauge).\\
    Collineation ($PGL(4)$) & Whole?field update from eye/head/body movement (compose by matrix multiplication). \\
    Plane at infinity (absolute) & Gauge that enables Euclidean readings (parallelism, scale). \\
    Cross?ratio & Stable relational cue across viewpoint changes. \\
    Frame $F$ \& Gauge $A$ & $F$ selects perspectival structure (eye?point/vanishing set); $A$ supplies metric read?out. Illusions: (\emph{i}) underdetermined $F$ (Necker cube) and (\emph{ii}) re?gauging $A$ (Moon illusion). \\
    (Epistemic/pragmatic) value of $F$ \& $A$ & Epistemic/pragmatic gain under projective parameterization of view. KL/entropy-based regularization terms induce a Fisher-Rao metric on the projective control manifold.\\

    \hline
  \end{tabular}
\end{table}


\paragraph{Note on Local charts and reciprocity}
\label{infobox:pcm-charts}

Local?areas/topos approaches are fully compatible with the PCM.

\begin{definition}[Projective \((G,X)\)?structure and local charts]
Let \(G=\G\) act transitively on \(X=\RP^3\).
A projective \((G,X)\)?structure on \(\B\) is an atlas \(\{(U_i,\chi_i)\}\) with \(\chi_i:U_i\!\to\! X\) 
homeomorphisms onto open subsets of \(X\) and transition maps \(\chi_j\circ\chi_i^{-1}\) given by restrictions of elements of \(G\).
Operationally, covering \(\B\) by \(\{U_i\}\), we specify on each \(U_i\) a local chart with gauge
\[
(\,\RP^3,\ \pi_{t,i}\in G,\ A_{t,i}\,),
\]
patched by \(g_{ij}\in G\) on overlaps: \(\chi_j=g_{ij}\circ\chi_i,\ g_{ji}=g_{ij}^{-1},\ g_{ij}g_{jk}=g_{ik}\).
\end{definition}

Thus, a local theory of areas (multiple context?bound charts \& gauges) is not only consistent with PCM; it is the natural sheaf?of?gauges view of the same structure. Under this approach, the ?global? PCM emerges by gluing compatible locals. (Note that identifying postulates characterizing those more general concepts remains an open issue, which is beyond the scope of this article). 
See in Section~\ref{sec:objections} objection 3 for further discussion.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Discussion}
\label{discussion}

The formal result above establishes that a perspectival, projective organization of the conscious field is essentially mandatory given our premises. We now discuss how this theoretical finding sheds light on various aspects of perception and cognition, and how PCM extends beyond the simplified scenario of visual 3D geometry. Importantly, PCM is not a theory limited to vision?it is proposed as a general principle for conscious representation across modalities and domains. Below, we consider perceptual constancies and illusions, action planning and curiosity, extensions to multimodal and social cognition, and neural implementation, highlighting how each is naturally accounted for or enhanced by the projective model.

\subsection{Perceptual constancies and illusions}
\label{disc:illusions}

Under PCM, illusions have two principled loci: (\emph{i}) projective?frame selection $F$ (choice of eye?point and vanishing structure), which determines the perspectival organization of the field; and (\emph{ii}) metric gauge $A$ layered on the projective backbone, which determines Euclidean (or other metric) read?outs (sizes, angles). Bistabilities (e.g., Necker) arise when $F$ is underdetermined; size/shape warps (e.g., Moon, Ames) arise when $A$ is re?set to maximize information given a fixed or slowly varying $F$.

\paragraph{Projective?frame ambiguity: the Necker cube.}
In PCM the Necker cube is encoded by projective invariants (junctions, collinearities) while the eye?point remains free. The single 2D outline supports exactly two 3D frames $F_\pm$ for back-projection (one being often favored by ecologically privileged priors such as view-from-above and gravity-aligned horizons, in a manner that affects the likelihood of a switch between the two). Switching percepts corresponds to a projective depth reflection that preserves the image?s incidence structure but reverses depth order, producing the observed bistability without positing two competing Euclidean reconstructions \cite{Rudrauf2017}.  Perspective cues that fix elements of $F$ (e.g., a visible horizon/vanishing set) collapse the posterior onto one frame and quench spontaneous flips; consistent with the model?s prediction that making the relevant frame components explicit deterministically selects one percept.

\paragraph{Metric gauge ambiguity: Moon illusion and Ames room.} Perceptual constancies (e.g., of size, shape, color) are often discussed in terms of recovering Euclidean properties of the world from changing sensory projections. In PCM, however, the conscious field (FoC) is fundamentally projective: Euclidean specifics emerge only after the system fixes a context-dependent gauge (an ?absolute?) on top of the projective layout. Many classic illusions then arise not from failures of geometry but from lawful gauge choices that optimize information integration under prevailing cues. 

\emph{The Moon illusion.}
The Moon-illusion account summarizes results obtained in \cite{Rudrauf2020} which are paraphrased here. PCM explains the Moon illusion as a consequence of how the FoC?s projective frame is calibrated to maximize usable information over the currently relevant portion of visual space \cite{Rudrauf2020}. Formally, a time-varying projective transform $T$ acts on the default frame $F_0$ to produce a calibrated frame $F = T(F_0)$; a key control point $I$ (with image $J$ at infinity) regulates the scope over which fine-grained information is integrated at the plane at infinity. When landscape/sky cues are abundant near the horizon, free-energy minimization selects a frame that narrows the scope in the Moon?s direction (effectively dilating the local angular metric at infinity), so the Moon?s apparent diameter increases. When cues are sparse at high elevation, the selected frame widens the scope (contracting the local angular metric), so the Moon appears smaller. The visual angle subtended by the physical Moon is unchanged in both cases; only the FoC?s metric at infinity is re?gauged.

PCM does not require, nor does it predict, that a particular distance inference causes the size change. Instead, the same projective calibration jointly constrains apparent size and whatever distance judgments observers may subsequently form. This resolves the ??size?distance paradox??: many observers report a larger and closer Moon at the horizon, others a larger and farther Moon, and some give no consistent distance report. On PCM, such variation is expected because distance judgments are secondary and depend on additional priors/cues layered onto the already re?gauged projective state. The primary driver of the illusion is the information?optimal gauge change at infinity, not a mandatory application of Emmert?s law. A VR experiment confirmed the predicted non?linear decline of matched area with elevation when environmental structure is present, but not when it is absent. The framework also accounts for classic modulations: enhancement by binocular disparity and convergence (more usable lateral information), persistence in pictures (frame calibration within the depicted perspective), and reduction when viewing inverted (through the legs), which disrupts the default vertical calibration and makes a large projective change costly. The account therefore explains both the typical phenomenology and inter/intra?subject variability without appealing to a fixed distance?size rule.

\emph{Ames room.}
Similarly, illusions like the Ames room (where a distorted room shape yields incorrect size perceptions of people inside) occur because the visual system assumes a normal rectangular room geometry (an incorrect gauge) and therefore mis-scales the occupants \cite{Rudrauf2020}. In all these cases, the underlying projective representation is sound; it is the additional metric assumption that is off. This viewpoint unifies understanding of why constancies usually hold but predictably break in certain situations: the projective layout of the scene is maintained in the brain, but the brain?s decision on how to interpret that layout in absolute terms can vary. In contrast, a theoretical agent with a fully Euclidean internal model would either not experience such illusions (if it had veridical scaling built-in, which would conflict with P3?s minimality), or would need an unexplained ?error? in its processing to produce them. PCM provides a principled explanation: illusions are not mere errors, but lawful outcomes of a projective strategy operating with context-dependent gauges.


\subsection{Active Vision, Epistemic Value, and Planning}
\label{dis-control}

A 3D internal projective camera makes informative actions salient: small changes in the parameters of a single collineation yield globally consistent predictions, creating steep epistemic gradients under limited Field of View (FoV) and occlusion. Because viewpoint updates are parameterized by a low-dimensional Lie chart and act by a single whole-field collineation,
the predictive outcome distribution varies smoothly with action parameters. For KL/entropy-based epistemic and regularization terms (including those in expected free energy and modern RL),
this induces a local Fisher-Rao geometry on the control manifold (Lemma~\ref{lem:control}), supporting stable natural-gradient / trust-region optimization under FoV limits and occlusion. For instance, this can guide saccades and approach movements toward uncertainty hotspots (Figure \ref{fig:pcm-control}).

A major motivation for including P4 as a premise was to connect the geometry of representation with the cybernetic purpose it serves: guiding adaptive action. One of the elegant outcomes of the PCM derivation is that it shows a convergence between having the ?right? representational format and being able to behave optimally in gathering information. Active vision researchers have long noted that vision is an active process: where we look next depends on what we want to know or do, and in turn determines what we see. PCM gives a concrete geometric structure to this process: the brain?s internal model behaves like a controllable 3D camera, and choosing a new fixation or viewpoint is akin to applying a transform to this internal camera. The expected impact of that transform on reducing uncertainty can be computed in the model. If the model is linear in homogeneous coordinates (which it is, as per Lemma~\ref{lem:reconstruction-updates}), it becomes tractable to simulate the effect of candidate actions and pick a good one.


Representing controllable degrees of freedom as projective transformations (pan, tilt, zoom of an internal 3D "camera") simplifies the evaluation of information-seeking objectives and can enhance exploratory behavior. In particular, using projective state variables leads to coherent, global changes in predicted observations when varied. If the agent?s internal state is structured as a 3D projective ?camera? looking at a world model, then the parameters of this camera (e.g. angles or directions of gaze, or a notional zoom/focus) have a global effect on the predicted sensory input. In active-inference terms, projective representations inherently encode the fact that certain orientations or movements constrain what can be seen (limited field of view and occlusion) and how well it can be seen (magnification effects of perspective) and thus where uncertainty can be reduced. Mathematically, because a single collineation updates the whole-field prediction coherently, small parameter tweaks produce smooth, globally consistent changes. Recent simulations have borne this out: agents that use a projective representation of their environment exhibit curiosity-driven behaviors like turning and moving toward objects to learn about them, whereas agents with a purely Euclidean state, and the corresponding group of transformation acting on it, do not show any approach behaviors \cite{Sergeant2025BC}. The perspectival magnification of information induced by projective transformations, made the projective agent consistently move to approach uncertain objects (maximizing epistemic value by anticipation of getting a better view). 

Thus, encoding the field of consciousness in projective terms (with a notion of ?what is seen from here and how well? built-in) naturally sets up the agent?s intrinsic rewards so that they favor exploratory, information-gathering actions. It also makes computing the expected reduction in uncertainty straightforward. This illustrates that PCM?s format is not only geometrically mandated but also functionally advantageous for active perception and control (see Figure \ref{fig:pcm-control}). Structuring the conscious field projectively may be evolution?s way of ensuring that agents have an intrinsic incentive to move and explore. It aligns the structure of perception with the computations needed for curiosity and active learning \cite{Friston2015}. This perspective highlights how PCM is not only about ?visual perception? narrowly construed, but about perception as an active, purposive process integrated with motor control and learning.

\subsection{Beyond Vision: Multimodal Integration and Social Cognition}
\label{episteval}
Although we have framed the discussion primarily in terms of visual geometry, PCM as a model applies to the integrated conscious field which includes multiple modalities and cognitive contents. The egocentric spatial frame is a natural scaffold for aligning inputs from different senses: for instance, the location of a sound can be represented as a direction and distance in the same egocentric space in which visual objects are represented. The auditory system by itself computes the direction of sound sources (using interaural time and level differences), effectively placing them on a ?celestial sphere? around the head. Integrating that with vision means mapping that direction into the visual field?s coordinates; something that is straightforward if those coordinates are projective (since both auditory and visual directions can be treated in homogeneous coordinates for combination). Tactile and proprioceptive information (about contact with surfaces, the position of one?s limbs, etc.) can similarly be related to such space through transformations that account for body posture. The posterior parietal cortex is known to house such multisensory spatial representations \cite{Andersen1997}, suggesting that the brain uses a common coordinate system for different modalities where possible. PCM implies that this common space is projective, which could be probed experimentally (e.g., are there projective distortions or invariances in auditory spatial perception analogous to visual ones? This remains a question for further research).

Furthermore, the perspectival structure might underlie aspects of affective and value-based appraisal. ?Appraisal? in emotion theory refers to how an individual evaluates the significance of a stimulus or situation for their goals and well-being. Many appraisals have a spatial component: something looming or approaching is threatening, things in the far periphery might be less immediately concerning, etc. A projective representation inherently encodes looming (as an increase in visual size) and peripheral versus foveal distinction. An interesting speculation raised in the broader PCM framework \cite{Rudrauf2023BrainSci} is that values (both epistemic and affective) are, in part, geometrically encoded in the field: for example, high uncertainty or novelty might be represented as a ?tension? driving the virtual camera toward that source of uncertainty (consistent with active inference?s information gain), whereas an emotionally aversive stimulus might drive avoidance (turning the perspective away or maintaining distance). In this way, the geometric variables of PCM could be linked to emotional drives: e.g., the feeling of something being ?too close for comfort? is literally about projected distance in the egocentric space.

Social cognition is another domain where perspectival organization is key. Understanding another person?s perspective (what they see, what they know) requires some ability to transform one?s egocentric representation into another frame. If each agent has a projective FoC, then taking someone else?s perspective can be formalized as applying a transform from one?s own $\mathbb{RP}^3$ coordinates to approximate the other?s $\mathbb{RP}^3$. In multi-agent simulations, one can nest these transformations: an agent can have an internal model not only of the external world but of how another agent might be viewing that world (a ?theory of mind? in spatial terms). Rudrauf \emph{et al.} \cite{Rudrauf2022JTB, Rudrauf2023ACM} have begun exploring this by combining PCM with robots and virtual humans. In \cite{Rudrauf2023ACM}, they simulate scenarios where an AI with PCM must infer what a human avatar (also modeled with PCM) knows or intends based on where that avatar is looking or moving to. The group-theoretic structure of projective transformations offers a framework for doing this systematically, because composing a transform for ?my perspective? with the inverse of ?your perspective? can translate objects from my frame to how they would appear in yours. Additionally, recent theoretical work has used the PCM to characterize how an agent?s self-model (its pre-reflective self-consciousness) can be understood in terms of the relations between perspectives \cite{Williford2022}. In simpler terms, the ambiguity in projective duality (the fact that points and planes exchange roles under certain transformations) might serve as a model for the relationship between the self-as-observer and self-as-observed, offering a new angle on the age-old philosophical problem of how the self can both be subject and object of experience.


\subsection{Anatomic-functional and neurophysiological basis of projective control}
\label{sec:disc_saccades_pcm}

The necessity result in Theorem~\ref{thm:PCM} is geometric/algorithmic and does not assume any particular neural mechanism. The following discussion is therefore offered
as plausible hypotheses about neurophysiological substrates consistent with the PCM structure.

\paragraph{Conceptual constraints: projective backbone and invariants.}
Philosophical considerations and psychophysics suggest that phenomenology may at times underdetermine geometry: similar perceptual data can be captured by incompatible Euclidean, hyperbolic, spherical, or hybrid models \citep{hatfield2003inverse,meadows2010geometry,rosar1985visual}. PCM treats spatial geometry as a joint constraint from experience, neural computation, and optimal control \citep{Rudrauf2017,Rudrauf2023BrainSci}. Under perspective optics, the stable structure under viewpoint change is given primarily by incidence relations (collinearity/planarity and vanishing structure) rather than metric quantities \citep{Hartley2003}. This motivates PCM's separation between (i) an incidence-preserving projective backbone (the charted field of consciousness, typically modeled on \(\mathbb{RP}^3\) with a viewpoint-dependent chart \(\pi_t\)) and (ii) a context-sensitive metric/gauge layer \(A_t\) that supplies local calibrations (e.g., depth scales, constancies, and task-dependent weights) \citep{Rudrauf2017,Rudrauf2023BrainSci}.

\paragraph{Constructing a 3D layout under viewpoint change: from retina to dorsal stream.}
Neurophysiology supports this division of labor. Early cortex preserves retinotopy while already responding to 3D surface configurations and supporting completion-like operations \citep{Bakin2000,Grossberg2015}. Mid-level and dorsal-stream areas extract disparity-defined depth and optic-flow/heading cues that constrain 3D scene structure \citep{DeAngelis2003,Lappe1996,PageDuffy2003}. Retinal circuits implement fast predictive codes that help compensate delays \citep{Johnston2019}, and broad evidence for nonretinotopic processing shows that object relations can be preserved across motion and occlusion, rather than being tied to a static ``flat camera'' map \citep{ogmen2010geometry}. Together, these findings motivate a constructed backbone that is robust under viewpoint change, with metric detail supplied flexibly and contextually: precisely the role of \((\mathbb{RP}^3,\pi_t,A_t)\) in PCM \citep{Rudrauf2017,Rudrauf2023BrainSci}.


\paragraph{Neural substrates of global coordinate transformations: gain fields and basis functions.}
\label{par:gainfields_candidate}

At the systems level, any implementation of projective control must support rapid transformations between retinocentric signals and more egocentric frames (head-/body-centered frames). In computational neuroscience, such coordinate changes could be realized by basis-function codes implemented by populations of multiplicatively modulated neurons
(gain fields) \citep{pouget1997spatial,blohm2009decoding}. Gain modulation by eye, head, and limb pose is
pervasive in posterior parietal and fronto-parietal circuits \citep{Zipser1988,Salinas2001,Andersen1997,AndersenBuneo2002}.
This empirical motif is compatible with the hypothesis that whole-field viewpoint changes are implemented as low-rank, globally coordinated transformations of a shared spatial code akin to collineations in \(PGL(4)\); an implementation candidate for the effective homogeneous-coordinate updates idealized by PCM. Mechanistically, mixed selectivity together with continuous eye-position signals may support smooth interpolation of such transformations across the peri-saccadic interval, consistent with temporally layered remapping effects \citep{neupane2016two}.

\paragraph{Peri-saccadic remapping across the hierarchy: parietal, attentional, and cortical signatures.}
Saccades are a stringent test of any spatial framework because they require rapid, coordinated updating across the brain-body hierarchy. Classic single-unit work showed that parietal neurons update spatial sensitivity in anticipation of intended eye movements \citep{Duhamel1992}, and human fMRI reveals analogous remapping signatures in retinotopic visual cortex \citep{merriam2007remapping}. Attention studies further indicate that what is updated is not only sensory gain at the future fovea but also priority structure across the field: attention can be predictively remapped \citep{mathot2009evidence}, priority can shift across the entire field \citep{mirpour2012anticipatory}, and crowding can emerge at future retinotopic locations \citep{harrison2013crowding}, consistent with broad functional roles for remapping \citep{bisley2020remapping}. In V4, saccades can trigger predictive updating of attentional topography \citep{Marino2018}, suggesting that perceptual stability depends on coordinated updates of both sensory and attentional readout.

\paragraph{Fine-grained cortical physiology: multiple remapping modes and feature-tuning changes.}
Recent primate physiology strengthens this picture by showing that anticipatory updating reaches into early cortex and exhibits multiple, temporally layered modes. In macaque V4, activity can exhibit both forward (future-field) remapping and convergent remapping toward the saccade target (ST), with direction-dependent prevalence. For instance, toward-target saccades can show a sequence consistent with an initial future-field shift followed by drift toward the ST \citep{neupane2016two}. In V2, predictive receptive-field remapping is observed in a majority of neurons about \(\sim\)30\,ms before saccade onset, spanning cortical layers and cell types and appearing as a column-wide signal \citep{denagamage2024widespread}. This V2 update is not purely spatial: orientation selectivity can transiently increase peri-saccadically, plausibly via untuned suppression that improves effective signal-to-noise \citep{denagamage2024widespread}.

\paragraph{Peri-saccadic mislocalization: shift, compression, and extrastriate correlates.}
Around saccades, brief flashes reveal structured distortions of perceived space that depend on context. Psychophysics shows that flashes can exhibit a broadly uniform peri-saccadic displacement (``shift'') in darkness and a reference-dependent compression toward the ST when visual anchors are present \citep{Ross1997,Cai1997,hamker2011computational}. A recent macaque study combining psychophysics with MT/V4 recordings reconstructed peri-saccadic spatiotemporal sensitivity maps and linked them to the ST-centered bias in localization \citep{weng2024neural}. Critically, the bias was driven primarily by neurons with receptive fields (RFs) far from the ST, whereas ST-proximal ensembles showed little bias \citep{weng2024neural}. The bias-relevant sensitivity changes were also tied to enhanced ST representation, consistent with a trade-off in which target enhancement comes at the cost of transient spatial distortion \citep{weng2024neural}. Together, these results support a spatially extensive, temporally structured update distributed across the hierarchy.

\paragraph{PCM mapping: \(\pi_t\) and \(A_t\) dynamics.}
The distributed, predictive character of remapping is naturally described as a common transformation acting on a shared spatial frame rather than as independent local ``patch'' corrections. In this reading, forward remapping reflects predictive updating of the projective chart (a change in the mapping from world/layout to the experienced field), while convergent/target-centered dynamics reflect the coupling of this chart update to an upcoming line of sight and its associated priority structure.

Within PCM, the (approximately global) bulk of peri-saccadic mislocalization could be related to sampling during such time-dependent chart transition (an incomplete update), whereas ST-centered compression to a transient, anisotropic modulation of \(A_t\) that reweights space toward the impending gaze direction. The classic dissociation between shift in darkness and compression with references \citep{Ross1997,Cai1997,hamker2011computational} fits this picture rather naturally: references constrain the gauge and thereby determine whether the transient transformation is expressed mainly as a global shift or as target-centered compression. Weng et al.'s finding that neural components boosting ST representation are the same components linked to localization bias \citep{weng2024neural} provides a possible physiological anchor for this dual description: a global update accompanied by temporary target-weighted ``rescaling''. Ideally, PCM should quantitatively capture why peri-saccadic bias depends on RF distance from the ST \citep{weng2024neural} and how multiple remapping modes combine over time \citep{neupane2016two}. Both could perhaps be addressed by treating the peri-saccadic interval as a time-evolving chart trajectory \(\pi(t)\) coupled to a non-uniform, attention-like modulation of \(A(t)\), rather than as an instantaneous coordinate replacement.

\paragraph{Corollary discharge and saccade-related broadcast signals.}
Timing and parameterization of peri-saccadic updates plausibly rely on corollary discharge/efference copy. A pathway from superior colliculus (SC) to frontal eye fields via mediodorsal thalamus supports internal monitoring of impending saccades \citep{Sommer2002}, while the SC's multimodal connectivity and cell-type diversity position it as a broadcast hub for orienting-related signals \citep{Liu2022}. In humans, saccades modulate mnemonic single units in medial temporal lobe, consistent with a broad distribution of movement-related signals to memory systems \citep{Katz2022}. Within PCM, these pathways can synchronize and parameterize \(\pi_t\!\rightarrow\!\pi_{t+1}\) transitions, while attentional/priority mechanisms act as transient weights on readout, offering a route to target-centered anisotropies in both behavior and neural encoding \citep{hamker2011computational,Marino2018,weng2024neural}.

\paragraph{Vestibular anchoring of \(\dot{\pi}_t\) and \(A_t\).}
Vestibular pathways provide complementary control variables that couple viewpoint dynamics to a stable world reference. Semicircular canals encode angular-velocity signals that support predictive updating and ocular torsion compensation (a contribution to \(\dot{\pi}_t\)), whereas otolith-driven graviceptive signals help anchor perceived vertical and the horizon, stabilizing the gauge layer \(A_t\) \citep{Lacquaniti2015,Brandt2017,Zwergal2024}. These vestibular constraints interact with visual priors (e.g., horizon/ground-plane and disparity) and semantic expectations, providing a natural route for context-dependent tuning of \(A_t\) \citep{Rudrauf2020}. Vestibular integrity is critical for spatial orientation and navigation; vestibular loss is associated with impaired spatial memory and hippocampal structural change, and restoration/stimulation can improve spatial cognition \citep{Brandt2017,Zwergal2024,Smith2023}. Hence, vestibular pathways jointly support the projective update and the metric/gauge stabilization required for stable experience under self-motion.

\paragraph{Allocentric content and egocentric--allocentric transformations.}
PCM's projective control extends beyond peri-saccadic vision by positing that allocentric memory, imagery, and action planning share the same control variables. In hippocampal--entorhinal--retrosplenial circuits, grid/place codes support ordered experience and landmark anchoring \citep{rueckemann2021grid}. Entorhinal activity encodes maps of visual space and saccade direction during exploration \citep{Killian2012,Killian2015}. Retrosplenial cortex mediates egocentric--allocentric transformations and encodes egocentric boundaries \citep{Mitchell2018,Alexander2020,Alexander2023}. In PCM notation, medial temporal lobe maintains allocentric content \(M_t\), decoded to a 3D layout \(W(M_t)\), while dorsal parietal/RSC supply the current ``camera'' transform \(P(\pi_t)\), yielding predicted field-of-consciousness features \(y_t=P(\pi_t)W(M_t)\). In this architecture, \(\pi_t\) is the controllable viewpoint variable and \(A_t\) tunes metric/gauge aspects of the readout.

\paragraph{Variable geometry and the gauge layer.}
Psychophysically, the global metric of visual space is not fixed: Luneburg-style and related approaches, including Indow's extensive analyses, reveal systematic deviations from Euclidean structure without a single metric fit \citep{indow1991critical,indow1997hyperbolic,erkelens2015perspective,burleigh2018natural}. Neural and computational work similarly suggests flexible geometry: hippocampal spatial codes can show hyperbolic structure \citep{zhang2023hyperbolic}, models ask whether rodents can represent hyperbolic spaces \citep{urdapilleta2015hyperbolic}, and grid-cell codes adapt to environmental geometry \citep{jayakumar2019environmental}. Human path integration and memory on a sphere further supports the idea of local charts patched over curved manifolds \citep{kim2024cognitive}. In PCM, this variability is principled: Euclidean/affine specifics are supplied by the context-dependent gauge \(A_t\) layered on a projective base, consistent with gauge-dependent constancies and illusions \citep{Rudrauf2020}.

\paragraph{Perspective taking, imagery, and neuropsychological constraints.}
Evidence from perspective taking and neuropsychology further supports reuse of spatial control variables across perception and imagination. Egocentric versus allocentric judgments recruit partially separable networks \citep{Committeri2004}: right dorsal parietal and frontal regions for egocentric perspective taking; medial temporal structures for allocentric map-based judgments \cite{Committeri2004}. During mental navigation and previsualization, hippocampus/RSC engage as if retrieving a map, and premotor circuits co-activate, consistent with forward models that predict sensory consequences of contemplated viewpoint changes (internal updates of \(\pi_t\)) and with context-dependent selection of \(A_t\). Neuropsychology underscores the continuity of perception and imagery. Right parietal damage yields hemispatial neglect for both perception and imagined scenes \citep{Bisiach1978,Corbetta2011}. Hippocampal damage impairs the construction of novel scenes \citep{Hassabis2007}.

\paragraph{Neuro-computational models of peri-saccadic updating.}
At an algorithmic level, classical neuro-computational models already implement many aspects of what PCM idealizes. A Lateral Intra Parietal (LIP)-centered framework links movement-related signals and recurrent interactions to predictive remapping and saccadic suppression of displacement \citep{ziesche2014brain}. The taxonomy reviewed in \cite{hamker2011computational} relates mislocalization to retinal--extraretinal timing mismatch and to re-entry/attention-like mechanisms that yield ST-centered compression when references are available. Where these models typically diverge from PCM is geometry: most assume an (approximately) Euclidean internal space that is shifted and locally warped, whereas PCM elevates updating to a projective chart change with an explicitly flexible metric layer.

\paragraph{Convergence from machine learning: projective cameras, equivariance, and 3D-aware control.}
Finally, ``convergent engineering'' in machine learning independently motivates a similar decomposition. NeRF-style models encode scenes as 3D fields rendered from arbitrary viewpoints by explicit projective cameras \citep{kosiorek2021nerfvae,dellaert2020neural,nerfsurvey2025}. Group-equivariant architectures improve generalization by embedding symmetry actions into representations \citep{sangalli2023movingframe,liu2025se3,kang2024equigspr}. Reinforcement-learning agents increasingly couple 3D-aware latent scene representations to control policies, learning to act from single views via internally simulated views \citep{cho2025singleview}. Independently of PCM, artificial agents tasked with seeing, modelling, and acting in 3D are converging on a format in which (i) the world is encoded in an internal 3D latent space, (ii) viewpoint is an explicit transformation variable, and (iii) rendering from that latent produces predictions; a practical instantiation of PCM?s projective generative model with controllable perspective.

\paragraph{Active inference as gauge-theoretic projective control.}
In parallel, predictive-processing/free-energy accounts view the brain as minimizing (expected) free energy under a generative model \citep{Friston2010}. This program has been tied to gauge-theoretic and information-geometric formalisms \citep{sengupta2016neuronal,ramstead2018answering} and to canonical cortical computations \citep{keller2018predictive,schindler2017connectivity,dora2021deep}. PCM can be read as a concretization of these accounts in the spatial domain: the field of consciousness is a projective state space; viewpoint changes are gauge actions on that space; and optimization selects both frame \(\pi_t\) and metric gauge \(A_t\) in light of epistemic and pragmatic value \citep{Rudrauf2017,Rudrauf2023BrainSci}.

\paragraph{Synthesis and constraints for PCM.}
Across levels, evidence supports PCM's decomposition into a neurally credible incidence-preserving backbone; fast whole-field updates of \(\pi\); a context-dependent gauge \(A\); and allocentric content plus egocentric--allocentric transforms \citep{Rudrauf2017,Rudrauf2023BrainSci}. The hypothesis is that: (i) early/mid vision builds an incidence?preserving backbone \cite{Bakin2000,DeAngelis2003,Lappe1996,PageDuffy2003,Grossberg2015,ogmen2010geometry}; (ii) PPC/FEF/SC and semicircular canals implement fast, low?rank whole?field updates consistent with collineations (\(\pi_t\)) \cite{Zipser1988,Salinas2001,Andersen1997,AndersenBuneo2002,Duhamel1992,Sommer2002,Liu2022}; (iii) otoliths, horizon/ground?plane, disparity and semantic priors supply a context?dependent gauge \(A_t\) \cite{Lacquaniti2015,Brandt2017,Zwergal2024,Rudrauf2020}; (iv) RSC?MTL circuits maintain allocentric content and gate egocentric?allocentric transformations while shaping the metric/gauge layer \cite{Mitchell2018,Alexander2020,Alexander2023,rueckemann2021grid,Killian2012,Killian2015,zhang2023hyperbolic,urdapilleta2015hyperbolic,jayakumar2019environmental,kim2024cognitive}. Peri-saccadic findings then tighten constraints: predictive remapping constrains \(\pi(t)\) \citep{neupane2016two,denagamage2024widespread,mathot2009evidence,mirpour2012anticipatory,harrison2013crowding,bisley2020remapping,Marino2018}; shift and ST-centered compression constrain transient \(A(t)\) \citep{Ross1997,Cai1997,hamker2011computational}; RF-distance effects constrain non-uniform transition/readout \citep{weng2024neural}; and existing model classes anchor implementation while exposing the geometric gap that PCM makes explicit \citep{ziesche2014brain,hamker2011computational}. That the same structural solution now emerges independently in machine learning \cite{kosiorek2021nerfvae,nerfsurvey2025,sangalli2023movingframe,liu2025se3,kang2024equigspr,cho2025singleview} and dovetails with gauge?theoretic readings of active inference \cite{sengupta2016neuronal,ramstead2018answering} strengthens PCM?s claim to parsimony and explanatory scope \cite{Rudrauf2017,Rudrauf2023BrainSci}.

\subsection{Incidence relations and PCM natural selection}

Here we sketch why, given widely observed ecological and ethological pressures, something like a projective, incidence-preserving backbone is a plausible target of natural selection. 

If adaptive control across species is keyed to alignments, intersections, looming, and occlusion ordering, then a compact internal geometry should treat these relations as invariants. A projective format does exactly this: it makes incidence stable under viewpoint change while avoiding the metabolic and computational cost of maintaining a globally calibrated Euclidean metric at all times. In this sense, Postulate~P3 (maximal invariance with minimal added structure) can be read as an evolved efficiency principle: represent only what remains stable and actionable under the transformations imposed by the animal?s own movements. Metric quantities can still be recovered when needed,
but as a context-dependent gauge layered on the incidence backbone. This perspective does not prove PCM in an
evolutionary sense; rather, it makes the PCM solution appear as a natural convergent design for embodied agents.

This representational choice has three immediate consequences. First, it ensures coherence under movement: a single projective update keeps alignments, vanishing points, and occlusion ordering globally consistent, so the relations that guide collision avoidance, interception, and grouping do not need to be recomputed from scratch. Second, classical perceptual constancies and systematic geometric illusions fall out of projective frame and gauge selection: when cues are ambiguous, the system may settle on a projective frame that departs from the Euclidean ground truth (e.g., stretched receding tracks or apparent size changes), producing lawful, context-dependent distortions without invoking faulty sensing. Third, projective coding naturally integrates perception and action: many tasks reduce to choosing actions that realize desired projective transformations (centering a target, maximizing depth from perspective, aligning with a trajectory), aligning PCM with active inference/predictive processing views while adding a precise geometric substrate \cite{Williford2018}.
Finally, because PCM specifies a common projective space for multisensory and social computations, the same incidence structure can support auditory localization, tactile reaching around occluders, and social perspective-taking (computing another?s line of sight). In short, PCM formalizes ecological regularities: by construing the world in projective terms, brains encode what remains invariant from any viewpoint, enabling organisms to exploit collinearity, alignment, occultation, and related invariants for robust, goal-directed behavior \cite{Williford2018}. This closes the loop between comparative biology and computation: the invariants that evolution has leveraged for survival are the invariants built into the geometry of conscious space.

\subsection{Objections, Alternatives, and Counterarguments}
\label{sec:objections}

We anticipate several questions or objections regarding the claim that consciousness must be projectively structured. Below we recast alternative modeling options as explicit objections and address them alongside broader conceptual concerns. The goal is to show where alternatives either secretly rely on PCM?s projective layer or fail one of the postulates, while keeping compatibilities in view.

\paragraph{Objection 1: Why not a Euclidean internal model augmented with perspective computation?}
One could argue that the brain might maintain a full Euclidean 3D model of the environment (with absolute distances, etc.), and simply compute perspective views from it on demand. Wouldn't that also explain coherence across viewpoints? \textbf{Counterargument:} Such a proposal effectively splits the problem into two parts: (i) maintaining an accurate Euclidean map, and (ii) rendering perspective images for conscious awareness. But step (ii) is itself a projection operation, meaning the brain would still be implementing a perspective transform to get what we actually see. In other words, this model doesn?t avoid PCM; it just hides it. The ?conscious field? in that scenario corresponds to the rendered scene, which necessarily lies in a projective format. Meanwhile, maintaining a full Euclidean map continuously is costly and brittle: as P2 and P3 articulate, trying to enforce metric accuracy at all times is unnecessary and impractical; the visual system is content to let certain dimensions (absolute scale, exact shape) remain uncalibrated as long as it preserves projective consistency. Empirically, a strictly Euclidean-internal-model theory would predict that we could never be systematically fooled by perspective (because the internal map would ?know? the reality), yet illusions like the Ames room demonstrate that we are fooled, implying the brain does not always maintain a veridical Euclidean model and instead relies on context to gauge metrics. Thus, a Euclidean-first approach either collapses into PCM (with an extra rendering step) or conflicts with actual perceptual behavior (by predicting no illusions). PCM by contrast captures why the brain usually doesn?t encode more than necessary (projective structure suffices) and why it can be misled in specific, lawful ways when context is lacking.

\paragraph{Objection 2: Why not a purely topological or graph-based representation of space?}
Perhaps the brain only cares about connectivity and qualitative spatial relations (e.g., ?object A is left of object B, behind object C?) rather than precise geometry. Would such a qualitative map violate any premises? \textbf{Counterargument:} A purely topological representation would violate P1 and P2 in practice. Under P1, the sensory data arriving are inherently geometrical (perspective projections), and a topological model would need some way to relate those data to its format. If the internal model doesn?t encode straightness or angles at all, how would it predict that as you move, objects should deform continuously and certain lines should align? It could not; unless one augments it with additional geometric rules, which is precisely what PCM provides. Under P2, an agent with only a graph of connections would have no straightforward way to update spatial relationships after a movement. Imagine storing ?A is left of B? as a purely qualitative fact; if you turn around, that relation might swap. How would a graph handle this systematically? It would have to recompute from scratch or store multiple context-dependent graphs. In essence, a graph model either fails to maintain coherence across transformations or must implicitly include coordinate-like information to do so. By contrast, the PCM geometrical model carries the full continuous relationships that can simply be transformed by a matrix. The fact that we experience a stable, continuously transforming world when we move suggests the brain?s representation is metric enough to allow interpolation and isomorphisms (as projective geometry is) rather than a discrete graph. In short, a topological model undergenerates the phenomena (loses structure present in experience) and lacks a principled update mechanism, making it incompatible with P2.

\paragraph{Objection 3: Isn?t a ``local areas''/topos or patchwork approach an alternative to PCM?}\label{obj:local-areas}

One might propose that perception operates by stitching together local charts or areas, each with its own context-specific calibration, rather than a single global projective backbone. \textbf{Counterargument (compatibility via gauge patching):} While our formal result singles out a projective backbone for the FoC, in practice perception could operate by stitching together local projective charts with context?specific gauges. Different tasks, viewpoints, or cue regimes select different $A_{t,i}$ on patches $U_i$, with overlap transition maps $g_{ij}\!\in\!PGL(4,\mathbb{R})$ enforcing compatibility. On this sheaf?like reading, classic constancies reflect stable gauge alignment across patches, whereas some illusions arise when locally optimal gauges (given the cues) fail to agree globally. Thus, the local areas perspective does not compete with PCM; it is the operational way projective structure could be implemented and cohered in real agents. This patchwise formulation is the operational reading of the local chart note at the end of
Section \ref{infobox:pcm-charts}.


\paragraph{Objection 4: Aren?t predictive processing / Bayesian brain frameworks sufficient without PCM?}
Since the brain is a prediction machine, perhaps predictive coding alone (or with generic latent states) can maintain perceptual coherence across movement, without any explicit geometric model. \textbf{Counterargument:} Predictive processing provides a normative lens (minimize prediction errors / free energy), but not the representational substrate. PCM is compatible with predictive processing and can be viewed as specifying the form of the internal generative model for spatial cognition: a projective simulation of the world, with the current state $\pi_t$ rendering whole-field predictions. Under this view, minimizing free energy proceeds by adjusting $\pi_t$ (itself controlling saccades/attention) and updating scene content. This also suggests concrete implementations (e.g., prediction-error neurons compare observed flow/disparity to that predicted by changes in $\pi_t$) and explains expectation-driven misframings as temporary ?wrong? projective settings under strong priors.

\paragraph{Objection 5: Could active inference alone (without an explicit geometric model) account for coherence?}
Maybe an agent can learn to keep perception coherent across movements simply by updating internal states to reduce surprise with no geometry needed. \textbf{Counterargument:} Without the right state-space, active inference could converge to a very suboptimal solution (e.g., relearn the world after each move). Human/animal behavior suggests a structured model that can be manipulated quickly when views change. Our formal results show that a projective structure makes exploration and planning tractable: actions have intelligible, global effects on predictions through a single collineation, yielding smooth information geometry and clearer epistemic gradients. With no global group action, one either recomputes over superfluous state components or loses discriminability, hindering optimization.

\paragraph{Objection 6: Is PCM falsifiable? Could there be a conscious system that doesn?t use projective geometry?} \textbf{Counterargument:} PCM makes specific, testable predictions. Failures would undermine PCM, for example, robust violations of projective invariants in controlled conditions, or neural dynamics during imagined viewpoint change that cannot be captured by parameter shifts within a persistent coordinate frame. Cross-species or artificial systems with demonstrably non-projective conscious spatial representations would also be challenging. To date, evidence has aligned with PCM?s expectations, but the claims are empirical and open to test.

\paragraph{Objection 7: The PCM does not address the ?hard problem?? of consciousness.} \textbf{Counterargument:} PCM is not intended to solve the ?hard problem? (the question of why subjective experience exists and how it causally integrates with material embodiment). It tackles what could be called the ?real problem? or the ?how-it-works problem? of consciousness \citep{Nagel1974}; explaining the functional, structural, and dynamical properties of conscious experience. By doing so, it significantly narrows the hard problem: whatever the ultimate explanation of subjectivity may be, it will have to apply to a system whose architecture is projective. In other words, PCM tells us what kind of physical/functional system manifests the thing we call spatial consciousness. This is a necessary step for any eventual solution to the hard problem, because an explanation of why certain brain activity is accompanied by experience must surely take into account the nature of that activity. If PCM is correct, then any satisfactory theory of why we have first-person experiences will likely reference the projective organization of our perception (for instance, one can speculate that a projective global workspace is what gives rise to a unified point-of-view ?self? \cite{Rudrauf2023BrainSci,Williford2018,Williford2022}). These are speculative bridges, but the point is that PCM doesn?t leave the hard problem harder: it circumscribes it. We now know that the conscious mind uses projective geometry; the enigma is why projective geometry implemented in neural tissue gives rise to a feeling of being in the world. That reformulated hard problem is more concrete than before, and thus progress on the easy problems (like PCM) is widely seen as a necessary precursor to tackling the hard problem. In summary, PCM doesn?t solve the hard problem, but it?s a major step toward a complete scientific theory of consciousness by solving many ?harder-than-the-easy? problems that lie in between.

\paragraph{Wrapping up.}
PCM is not at odds with other frameworks; it often provides the structural backbone they lack. A Euclidean world map must render via a projective transform to match experience; a purely topological graph undergenerates lawful whole?field updates and lacks a principled update rule; predictive processing/active inference benefit from a geometrically appropriate state-space that makes information- and reward- seeking control tractable. Thus, given the premises (P1?P4), the perspectival, projective organization of the FoC is not an optional modeling convenience but the unique parsimonious solution.



\section{Conclusion}
We have presented a formal argument that the peculiar perspectival structure of conscious experience is no accident of evolution or quirk of our senses, but a necessary consequence of basic principles of embodiment, perception, and action. By assuming only that our sensory input is perspectivally projected, that we maintain coherence across movements, that we favor the most invariant (parsimonious) representation, and that our actions are guided by an optimality principle, we derived that the brain?s internal representation of the world must take the form of a 3D projective space with appropriate transformations: essentially the Projective Consciousness Model. Along the way, we saw that this framework naturally accounts for how the world appears stable to us despite constant eye and body motion, how we achieve perceptual constancies and why we sometimes experience systematic illusions, how the brain?s known mechanisms can implement such transformations, and how this representation dovetails with strategies for information-seeking and planning. 

PCM provides a unifying geometrical structure underlying conscious perception, one that transcends individual modalities and extends into cognitive domains like social perspective-taking and self-representation. An important aspect of this model is that it can operate within various theoretical paradigms of brain function: it complements the Free Energy Principle/Active Inference by specifying the form of the internal generative model (perspectival and projective), but it could equally be used in other control schemes where an internal model is needed. The distinctive claim of PCM is that, whatever else the brain is doing, it encodes the world in a way analogous to a projective simulation, and this is the only viable solution if we want an integrated, first-person viewpoint that is efficient and behaviorally useful.

If PCM is true, it opens several avenues for future research: empirically, one can look for further evidence of projective encoding in neural activity or behavior (e.g., are there signatures of projective invariants in neural population responses? Can we design new illusions by manipulating projective frame and gauge cues as PCM suggests?). In artificial intelligence and robotics, implementing PCM-like representations might improve active vision systems or curiosity-driven learning agents, as initial studies indicate \cite{Sergeant2025BC}. Philosophically, PCM offers a fresh lens on the age-old problem of how the subjective perspective is constituted, potentially connecting to formal definitions of the self. 

Ultimately, by demonstrating that the projective structure is not just a convenient descriptive model but a mandatory one under broad premises, we hope to shift the discussion from ?why might the brain use projective geometry?? to ?given it must use projective geometry, what else can we understand about the mind and brain from this fact??. PCM, in our view, is not merely a theoretical possibility: it is an inevitability waiting to be fully recognized in the science of consciousness.

\appendix
\section{Appendix A. Constructive proof of the projective structure and the $\mathrm{PGL}(4)$ update law}
\label{app:proof}



\noindent\textbf{Source and structure.} This appendix follows a constructive scheme (P1a-P1c, P2a-P2f; Lemmas~\ref{lemA:two-points-one-line}--\ref{lemA:two-planes}) to rebuild a 3D projective incidence space from 2D views and to select the real field. It also yields the $\mathrm{PGL}(4)$ update law for viewpoint changes.

\subsection{Postulates (P1, P2) with explicit subclauses}
\begin{definitionA}[Points and views]\label{defA1}
We posit a family of 2D views. A \emph{point}$^{\!3D}$ is specified by the set of all its
realizations $(\text{view},\ \text{point}^{\!2D})$ across views. In each view, through two distinct points$^{\!2D}$
passes a unique line$^{\!2D}$ (\textbf{P1a}); parallel lines meet at points$^{\!2D}$ at infinity so that any two
lines$^{\!2D}$ intersect (\textbf{P1b}). Nondegeneracy: there exist at least two distinct lines$^{\!2D}$ in each view (\textbf{P1c}); every line$^{\!2D}$ in a view contains at least 3 points$^{\!2D}$ (\textbf{P1d}).
\end{definitionA}

\begin{definitionA}[Alignment and lines$^{\!3D}$]\label{defA2}
We postulate a coherence rule: if three points$^{\!3D}$ $A,B,C$ have collinear realizations in one view and this
persists in neighbouring views, then the realizations are collinear in \emph{all} views (\textbf{P2a}). Define
\emph{alignment}$^{\!3D}$ by: $A,B,C$ are aligned$^{\!3D}$ iff their realizations are collinear in every view;
the line$^{\!3D}$ $(AB)$ is the set of points$^{\!3D}$ aligned$^{\!3D}$ with $A,B$. The realization of a line$^{\!3D}$ in
a view is a full line$^{\!2D}$, if not just a point (\textbf{P2b}).
\end{definitionA}

\begin{definitionA}[Planes$^{\!3D}$ and lines of view]\label{defA3}
A plane$^{\!3D}$ is the union of the lines$^{\!3D}$ intersecting two distinct intersecting lines$^{\!3D}$. Fix a view and
a pixel $p$; the locus of points$^{\!3D}$ whose realization is $p$ is a line$^{\!3D}$ with the viewpoint removed
(\textbf{P2c}) (viewpoint is shorthand for selecting a view/camera centre). 
\end{definitionA}

\begin{definitionA}[Continuity and local order]\label{defA4}
Geometric constructions are continuous, (\textbf{P2d}). 
Points\d3 locally divide neighborhoods on lines\d3, (\textbf{P2e}). 
There exists a line$^{\!3D}$ and a point\d3 on it whose complement is separable, locally compact and not totally disconnected, (\textbf{P2f}). 
\end{definitionA}

\begin{remarkA}[\textbf{Comments on (P2b), (P2e), (P2f): vanishing points, front/behind, and mild topological regularity}]\label{remA:P2bP2eP2f-comments}
\textbf{(P2b) Full projective lines in a view.}
In the perspective of painters, two parallel lines converge to the same vanishing point, giving the impression that most lines are seen as half-lines in 2D views. What does the opposite half-line correspond to?
To the invisible part of the line that lies behind the observer. Assumption \textbf{(P2b)} means that the 2D
view is treated as a projective completion: it includes the vanishing point itself and the continuation
corresponding to the behind-the-observer branch of the same 3D line. (This is an idealization of the incidence
structure, not a claim that the biological retina literally ``sees behind''.) \textbf{(P2e) Local order along the line of sight.} Assumption \textbf{(P2e)} expresses that subjects can distinguish, along their line of sight, what is right in front of them within reach and what is behind them, also within reach. Importantly, this is a local front/behind order along a line of view; it does not commit us to a single global order across the whole projective line past the plane at infinity (compare the discussion of chart changes and visibility in Remark~\ref{rem:no-internal-origin} and Remark~\ref{rem:occlusion}). \textbf{(P2f) Why ``separable'', ``locally compact'', ``not totally disconnected''.} Assumption \textbf{(P2f)} invokes abstract topological concepts. In the present context, ``separable'' and ``locally compact'' are mild regularity/finite-complexity assumptions (Euclidean manifolds satisfy both). By contrast, in totally disconnected spaces one cannot move at all: continuous curves are constant. Thus \textbf{(P2f)} assumes that nontrivial motion is possible along a line$^{\!3D}$.
\end{remarkA}


\subsection{Basic lemmas}

\begin{lemmaA}[Uniqueness of a line through two points]
\label{lemA:two-points-one-line}
Through two distinct points$^{\!3D}$ $A,B$ passes a unique line$^{\!3D}$, denoted by $(AB)$.
\end{lemmaA}

\begin{proof}
Let $A,B$ be distinct points\d3 and let $\ell$ be a line\d3 which passes through $A$ and $B$. Pick a view. Let $\pi$ denote the map which sends a point\d3 to its realization in the view. By \textbf{P2a}, for every $C\in\ell$, $\pi(A),\pi(B),\pi(C)$ are aligned\d2. By definition, there exist points\d3 $C$ and $D$ such that $\ell=(CD)$. By \textbf{P1a}, $\pi(A),\pi(B),\pi(C),\pi(D)$ all lie on the same line\d2, which therefore contains $\pi((AB))$ and $\pi(\ell)$. Since this happens in every view, by definition of alignment\d3, it follows that every point\d3 in $(AB)$ is aligned\d3 with $C$ and $D$, and so $(AB)\subset (CD)=\ell$. Also, every point\d3 in $\ell$ is aligned\d3 with $A$ and $B$, and so $\ell\subset (AB)$. We conclude that $\ell=(AB)$, i.e. $(AB)$ is the only line\d3 containing $A$ and $B$.
\end{proof}

\begin{lemmaA}[Lines in a plane meet (\emph{Pasch}-type)]
\label{lemA:pasch}
If $A,B,C,D$ are distinct points$^{\!3D}$ and the lines$^{\!3D}$ $(AB)$ and $(CD)$ meet, then also the lines$^{\!3D}$ $(AD)$ and $(BC)$ meet.
\end{lemmaA}

\begin{figure}[!ht]
\begin{center}

\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{xdxdff}{rgb}{0.49019607843137253,0.49019607843137253,1.}
\definecolor{uuuuuu}{rgb}{0.26666666666666666,0.26666666666666666,0.26666666666666666}
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-0.6618564548399316,-0.5449321698322642) rectangle (13.620282153037591,3.5189721628084194);
\draw [domain=-0.6618564548399316:13.620282153037591] plot(\x,{(-0.--3.*\x)/2.});
\draw [domain=-0.6618564548399316:13.620282153037591] plot(\x,{(--38.84351265370121-3.*\x)/10.947837551233738});
\draw [domain=-0.6618564548399316:13.620282153037591] plot(\x,{(--13.-2.*\x)/3.});
\draw [domain=-0.6618564548399316:13.620282153037591] plot(\x,{(-0.--1.*\x)/5.});
\draw [domain=-0.6618564548399316:13.620282153037591] plot(\x,{(-0.-0.*\x)/12.947837551233738});
\begin{scriptsize}
\draw [fill=qqqqff] (0.,0.) circle (2.5pt);
\draw[color=qqqqff] (-0.36948923666434264,0.5514448983261936) node {$A$};
\draw [fill=qqqqff] (12.947837551233738,0.) circle (2.5pt);
\draw[color=qqqqff] (13.152494603956649,0.4344980110559581) node {$B$};
\draw [fill=qqqqff] (2.,3.) circle (2.5pt);
\draw[color=qqqqff] (2.4372360578213117,3.270460027359169) node {$C$};
\draw [fill=qqqqff] (5.,1.) circle (2.5pt);
\draw[color=qqqqff] (5.097777743219171,1.267744582856386) node {$D$};
\draw [fill=uuuuuu] (6.5,0.) circle (2.5pt);
\draw[color=uuuuuu] (6.647323999549793,0.37602456742084034) node {$E$};
\draw [fill=ffqqqq] (7.484922891359462,1.4969845782718922) circle (2.5pt);
\draw[color=ffqqqq] (7.582899097711678,1.764768853754887) node {$F$};
\end{scriptsize}
\end{tikzpicture}

\caption{Pasch configuration}
\label{Pasch}
\end{center}
\end{figure}

\begin{proof}

Let $E$ be the common point\d3 of $(AB)$ and $(CD)$. Let us choose $D$ as a point of view. Let $\pi$ denote the map which sends a point\d3 to its realization in the view, a point\d2. By \textbf{P2c}, the aligned\d3 points\d3 $D,C,E$ have the same realization in the view, $\pi(C)=\pi(E)$. By \textbf{P2a}, since $A,B,E$ are aligned\d3, $\pi(A),\pi(B),\pi(E)$ are aligned\d2, on the line\d2 which contains $\pi(C)$ and $\pi(B)$. By \textbf{P1a}, this line\d2 is contained $\pi((BC))$. So $\pi(A)\in\pi((BC))$. So there exists a point\d3 $F\in(BC)$ such that $\pi(F)=\pi(A)$. Therefore both $F$ and $A$ belong to the set of points\d3 whose realization in the view is $\pi(A)$. According to \textbf{P2c}, this is the line\d3 passing through $A$ and $D$. Thus $F\in(BC)$ and $F\in(AD)$, so the lines$^{\!3D}$ $(AD)$ and $(BC)$ intersect (at $F$).
\end{proof}

\begin{corollaryA}[Well-defined plane through three non-aligned points]
\label{corA:plane-well-defined}
The plane$^{\!3D}$ defined by $(AB)$ and $(AC)$ coincides with that defined by $(CA)$ and $(CB)$.
\end{corollaryA}

\begin{proof}
Let $M$ be a point\d3 belonging to the plane\d3 defined by $(CA)$ and $(CB)$. By definition, there is a line\d3 $\ell$ containing $M$ and intersecting $(CA)$ in $E$ and $(CB)$ in $D$. Lemma \ref{lemA:pasch} shows that $\ell$ intersects $(AB)$ as well. So $M$ belongs to plane\d3$((AB),(AC))$. This shows that the plane\d3 defined by $(CA)$ and $(CB)$ is contained in the plane\d3 defined by $(AB)$ and $(AC)$. The same argument proves the reverse inclusion, hence the two planes coincide.
\end{proof}

\begin{lemmaA}[Every line meets every plane]
\label{lemA:line-plane-intersect}
Each line$^{\!3D}$ intersects each plane$^{\!3D}$.
\end{lemmaA}

\begin{proof}
Let $\ell$ be a line\d3, let $O,Q$ be distinct points\d3 on $\ell$. Let us pick $O$ as a point of view, $\ell$ as a line of view. Let $\pi$ denote the map which sends a point\d3 to its realization in the view. Let $P$ be a plane\d3, defined by three points $A,B,C$ which are not aligned\d3. 

Let us assume first that the realizations $\pi(A),\pi(B),\pi(C)$ are not aligned\d2. Let $\ell'$ be a line\d2 through $\pi(Q)$. By \textbf{P1b}, $\ell'$ intersects the line\d2 through $\pi(B)$ and $\pi(C)$ in a point $d$. By \textbf{P2b}, there exists a point\d3 $D$ of $(BC)$ whose realization is $d$. Similarly, there exists a point\d3 $E$ of $(AC)$ whose realization is the intersection of $\ell'$ and of the line\d2 through $\pi(A)$ and $\pi(C)$. By \textbf{P2b}, the realization of $(DE)$ is $\ell'$, hence it contains $\pi(Q)$. Thus there exists a point\d3 $G$ of $(DE)$ with realization $\pi(Q)$. By \textbf{P2c}, $G$ belongs to $\ell$. By construction, $G$ belongs to $P$, so $\ell$ and $P$ intersect.

Otherwise, the realizations $\pi(A),\pi(B),\pi(C)$ are aligned\d2. This implies that $\pi(P)$ is contained in a line\d2 $\ell'$. By \textbf{P2c}, the points\d2 are not all the same point. Say that $\pi(B)$ and $\pi(C)$ are not equal. Then $\ell'$ is the line\d2 through $\pi(B)$ and $\pi(C)$. By \textbf{P2b}, $\pi((BC))$ is the whole line\d2 $\ell'$, so there exist a point\d3 $A'$ on the line\d3 $(BC)$ such that $\pi(A')=\pi(A)$. By \textbf{P2c}, $A$ and $A'$ lie on a line\d3 $\ell''$ passing through the point of view $O$. Since $A$ does not belong to $(BC)$, $A'$ is not equal to $A$, so the line\d3 $\ell''=(AA')\subset P$. This shows that $O$ belongs to $P$. Therefore $\ell$ and $P$ intersect in $O$.
\end{proof}


\begin{lemmaA}[At least two distinct planes]
\label{lemA:two-planes}
There exist at least two distinct planes$^{\!3D}$.
\end{lemmaA}


\begin{proof}
Pick a 2D view and two distinct lines\d2 in this view (\textbf{P1c}). Since they intersect by \textbf{P1b}, let us denote them by $(ab)$ and $(ac)$. By \textbf{P2c}, there exist distinct points\d3 $A$, $A'$, $B$, $C$ with realizations $a,a,b,c$. For every point\d3 $D$ of $(A'B)$ and $E$ of $(A'A)$, by \textbf{P2a}, the realizations of $D$, of $E$ and of every point\d3 of the line\d3 $(DE)$ belong to the line\d2 $(ab)$. So the whole plane\d3 defined by the lines\d3 $(A'B)$ and $(A'A)$ maps to $(ab)$, and onto $(ab)$ by \textbf{P2b}. Similarly, the plane\d3 defined by the lines\d3 $(A'C)$ and $(A'A)$ maps to $(ac)$, and all of $(ac)$. Therefore these are distinct planes\d3.
\end{proof}

\subsection{From incidence to $\mathbb{FP}^{3}$ and to the real field}

\begin{theoremA}[Necessity of the projective structure; field selection]
\label{thmA:necessity}
Under P1a-P1d and P2a-P2d, the incidence structure is a 3D projective space; thus it is isomorphic to
$\mathbb{P}(V)$ for some $4$-dimensional vector space $V$ over some division ring $\mathbb{F}$. Under P2e (continuity/local order on lines of view), one recovers $\mathbb{F}\cong\mathbb{R}$.

\end{theoremA}

The proof of Theorem \ref{thmA:necessity} relies on the axiomatic of projective geometry. This theory has developed during the 19th century, see for example \cite{Enriques1898}. Here, we choose to separate axioms which lead to projective geometry over some division ring, from the axioms characterizing the field of real numbers. This point of view was inaugurated  by O. Veblen and J.W. Young \cite{VeblenYoung1908}. 

The clauses P1a - P1d turn 2D views into generalized projective planes, but do not suffice to characterize projective planes over division rings, so the existence of a third dimension plays a crucial role.

\subsection{From incidence to $\mathbb{FP}^{3}$}

We shall follow the presentation by A. Seidenberg, \cite{Seidenberg1962}.

This author singles out a list of 7 assumptions on a set equipped with an incidence relation, which together characterize 3-dimensional projective spaces over division rings. Our first goal is to check that each of these assumptions is satisfied in our setting.

\begin{itemize}
  \item[A0.] \emph{There exists at least one point and at least one line}. This follows from (P1c) and (P1d).
  \item[A1.] \emph{There exists at least one line through two distinct points}. This follows from  Definition A.2 of lines\d3.
 \item[A2.]  \emph{There exists at most one line through two distinct points}. This is precisely the statement of Lemma \ref{lemA:two-points-one-line}.
  \item[A3.] Pasch's Axiom. \emph{If $A,B,C,D$ are distinct points and the lines $(AB)$ and $(CD)$ intersect, then also the lines $(AD)$ and $(BC)$ intersect.} This is precisely the statement of Lemma \ref{lemA:pasch}. 
  \item[A4.]  \emph{Every line contains at least 3 distinct points.} This is (P1d).
  \item[A5.]  \emph{Not all points are on the same line}. This follows from (P1c).
  \item[A6.] \emph{Not all points are in the same plane}. This is Lemma \ref{lemA:two-planes}.\end{itemize}

Once this is done, one classically constructs a division ring $\mathbb{F}$ and one identifies the incidence relation with that of a projective space over $\mathbb{F}$ of dimension at least $3$. Lemma \ref{lemA:line-plane-intersect} implies that dimension is at most $3$.

We briefly describe how this is done, following Chapter 5 of \cite{Seidenberg1962}. The first step, the construction of the field $\mathbb{F}$, starts in the plane. One picks a plane\d3 $P$, one picks four points\d3 $A,B,C,D$ on this plane such that no 3 of them are aligned\d3, and one defines, by geometric constructions using only incidence relations, an addition and a multiplication operation on the set $\mathbb{F}=(AB)$ with $B$ deleted. The idea goes back to K. von Staudt, \cite{vonStaudt1857}. One shows that these operations obey the laws of division rings (associativity of both laws, commutativity of addition, neutral elements, existence of opposites and inverses, distributivity). Note that associativity relies on the fact that the plane $P$ is contained in a higher dimensional space.

To achieve an isomorphism between the internal state (the set of points\d3) and projective 3-space $\mathbb{FP}^3$, one picks five points\d3 $A,B,C,D,E$ such that no 4 of them lie on the same plane\d3 and one constructs a bijection, denoted by $\Phi$, between the projective 3-space $\mathbb{FP}^3$ and the internal state, mapping
\begin{align*}
[1:0:0:0]&\to A,\\
[0:1:0:0]&\to B,\\
[0:0:1:0]&\to C,\\
[0:0:0:1]&\to D,\\
[1:1:1:1]&\to E.
\end{align*}
with the property that $\Phi$ maps projective lines of $\mathbb{FP}^3$ to lines\d3. This is what we mean by isomorphism of sets equipped with incidence relations. The map $\Phi$ is uniquely determined by these requirements. 

It follows that each 2D view is isomorphic to a projective plane over the division ring $\mathbb{F}$.

\subsection{Update law as collineations}

Linear bijections of the vector space $\mathbb{F}^4$ factor and induce incidence-preserving bijections of $\mathbb{FP}^{3}$. This is still true for \emph{semi-linear} bijections, i.e.\ bijective additive maps $A:\mathbb{F}^4\to \mathbb{F}^4$ such that there exists an automorphism $\sigma$ of the division ring $\mathbb{F}$ with, for all $v\in \mathbb{F}^4$ and all $\lambda\in\mathbb{F}$,
\[
A(\lambda v)=\sigma(\lambda)\,A(v).
\]
The collection of all semi-linear bijections of $\mathbb{F}^4$ is denoted by $\overline{GL}(4,\mathbb{F})$.
The collection of the corresponding bijections of $\mathbb{FP}^3$ is also a group, denoted here by $\overline{\mathrm{PGL}}(4,\mathbb{F})$.

The converse is true.

\begin{propositionA}[Collineations and $\mathrm{PGL}(4)$ action]
\label{propA:PGL}
Any whole-field update that preserves lines$^{\!3D}$ is a collineation of $\mathbb{FP}^{3}$ and hence the projectivization of some $T\in \overline{GL}(4,\mathbb{F})$; composition corresponds to matrix multiplication mod scalars. Thus the internal update group embeds into $\overline{\mathrm{PGL}}(4,\mathbb{F})$;
see \cite{Hartley2003}.
\end{propositionA}

\subsection{To the real field}

The remaining task is to identify $\mathbb{F}$ with the field of real numbers.

One equips $\mathbb{F}$ with the topology induced by the map $\Phi$. Since geometric constructions are continuous (P2d), this topology does not depend on the choice of frame $A,B,C,D,E$ and turns $\mathbb{F}$ into a topological division ring. By construction, $\Phi$ is continuous, hence $\mathbb{F}$ is a locally compact and connected topological division ring (P2f). Theorem II of \cite{Taussky1935}, elaborating on \cite{Pontrjagin1932}, asserts that $\mathbb{F}$ is isomorphic either to the reals $\mathbb{R}$, the complex numbers $\mathbb{C}$ or the quaternions $\mathbb{H}$. The local cutpoint condition (P2e) leaves only the reals $\mathbb{R}$ as a possible field.

Since the field $\mathbb{R}$ has no nontrivial automorphisms, the group of collineations of $\mathbb{RP}^{3}$ is $\mathrm{PGL}(4,\mathbb{R})$.


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