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% Paper 42: The Geometry of Other Minds -- An Anti-Solipsism Theorem.
% A philosophical/psychological theorem backed by the corpus geometry (Hopf bundle). Consolidates
% and expands P04 + Addenda 373/374/375. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Pinned by verify_P373/P374/P375 (the formal claims); this paper is the prose treatment.
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\title{\textbf{The Geometry of Other Minds:\\ An Anti-Solipsism Theorem}\\[0.4em]
\large Shared reality, private experience, and synchronicity\\
on the Hopf bundle}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Solipsism --- the thesis that only one's own mind exists, or that one cannot know any reality beyond
one's own experience --- is usually treated as irrefutable but unbelievable: a position no argument
quite closes and no person quite holds. This paper argues that the difficulty is an artifact of
running two separable claims together, and that once they are separated the geometry of observation
developed in the corpus settles each. We model observers as the circle fibers of the Hopf fibration
$\pi:\Sthree\to\Stwo$, following Paper~04, and prove a single result, the \emph{Anti-Solipsism
Theorem}: (a) fibers are pairwise disjoint, so the privacy of experience is real and exactly located;
(b) every fiber is a proper subset of the one shared total space $\Sthree$, which carries
continuum-many fibers, so the thesis that only the self exists is not false but \emph{incoherent} ---
it denies the structure that constitutes the viewpoint it asserts; (c) the only channel between fibers
is the shared boundary, so intersubjective consistency is enforced; and (d) distinct fibers are
disjoint yet linked, with linking number one, so private perspectives are acausally correlated through
the bundle while the shared world changes only by action upon it. The first claim vindicates the
genuine intuition behind solipsism (the privacy of qualia); the second dissolves the metaphysical
overreach; the third explains objectivity; and the fourth locates two phenomena usually left in fog
--- synchronicity, which is the linking (correlation without contact), and agency, which is action
(the sole lever on the shared world). The diagnosis is that solipsism is a category error: the privacy
of a fiber mistaken for the privacy of the bundle. Crucially, the dissolution does not rest on the
identification of observers with Hopf fibers: parts (a)--(c) require only that a perspective be a
disjoint part of a shared whole, a structure we derive transcendentally, before any geometry, from the
partiality intrinsic to any perspective --- the horizoned, one-sided character that the solipsist's own
experience exhibits and cannot disown --- so that the strong thesis asserts a non-partial perspective,
which is no perspective at all. We are careful, however, about how much partiality alone delivers: it
forces a \emph{larger} whole but not yet a \emph{shared} one, since a solipsist may relabel the excess as
his own further possible experience (immanent idealism). Sharedness --- an exceeding \emph{independent} of
the self --- is earned on firmer corpus ground: from forced agreement (Paper~27: a lone reading constrains
nothing; reality begins where two readings are forced to match) and, decisively, from error (a reading
that can \emph{fail} requires a corrector it does not author, which a self-given stream cannot be). Error
reaches an independent reference, closing the immanent-horizon retreat; that the reference is \emph{other
minds} rather than a mute resisting world, and that the shared whole has the corpus's geometry, rests not
on proof but on evidence --- the same $(B^4,S^3)$ structure of forced agreement computes
$\alpha^{-1}=137.036$, which a contentless private stream has no reason to. Only the synchronicity of (d)
depends on the specifically Hopf geometry, the linking. We mark throughout what is forced (the
transcendental core, the independence-from-error step, and the topology) and what is corpus-conditional or
evidential (that the shared whole has the Hopf form, that the independent reference is other minds, and
that the linking is felt as synchronicity). This is a philosophical and psychological result carried by a mathematical spine, and
the prose is weighted accordingly.
\end{abstract}

\tableofcontents

\section{Introduction: the problem that will not die}
\label{sec:intro}

There is a peculiar kind of philosophical problem that survives every attempt to kill it not because
the attempts are weak but because the problem feeds on a real intuition that the attempts leave
standing. Solipsism is the clearest example. Almost no one believes it; almost no one can refute it to
their own satisfaction; and the gap between those two facts has been an embarrassment to philosophy
since Descartes sat by his fire and discovered that the one thing he could not doubt was the doubting
itself. From that single certain point --- \emph{I think, therefore I am} --- the road back out to a
world, to other people, to the conviction that the room he sat in was not a private theater staged for
an audience of one, turned out to be unexpectedly long, and at several points it looked as though it
might not connect at all.

The trouble is not that solipsism is obviously true. The trouble is that it is built out of something
that is obviously true. My experience is mine in a way that nothing else is. When I see red, taste
salt, feel the dull ache of a long day, there is something it is like to be me undergoing those
states, and that something is not available to anyone else in the way it is available to me. You may
infer it, sympathize with it, photograph my brain while I undergo it, but you cannot \emph{have} it.
The privacy of experience is not a theory; it is the most immediate datum there is. Solipsism takes
that datum and inflates it: from \emph{my experience is private} it slides to \emph{only my experience
is real}, and from there to \emph{perhaps only I am real}. The slide is so smooth that it is hard to
say exactly where the error enters --- and a problem whose error is hard to locate is a problem that
does not die.

The corpus this paper belongs to is, on its face, a theory of physics: a single geometric density on
the three-sphere from which the fine-structure constant, the structure of the gravitational coupling,
and a good deal of particle phenomenology are read off. It is an unlikely place to find an answer to
solipsism. But Paper~04 of that corpus, in the course of arguing that classical observation is not an
optional add-on to physics but a geometrically necessary layer of it, proved a handful of small
theorems about what happens when more than one observer inhabits the same world. Those theorems were
stated and then largely left alone, their philosophical reach unremarked. This paper takes them
seriously, sharpens them with two later results (Addenda~373 and~374), and shows that together they
amount to a precise and, I will argue, decisive treatment of the solipsism problem --- one that does
not refute the solipsist so much as show that the position, taken strictly, cannot be coherently
occupied, while the intuition that tempts people toward it is preserved exactly and given its proper
home.

The strategy has three moves, and they fall into two layers that the reader should keep distinct, since
they have different strengths and answer to different burdens. The first move is to insist on a
distinction the tradition has tended to blur, between the privacy of experience and the privacy of
reality. The second --- and this is the load-bearing one --- is to show, before any model is introduced,
that to be a perspective at all is to be a part of a shared whole: a transcendental argument, in the
manner of Kant and Strawson and Wittgenstein, that proceeds not from a geometry but from the partiality
intrinsic to any point of view, a feature the solipsist's own experience hands him and he cannot
coherently deny. This move dissolves strong solipsism on its own, and depends on no physics and no
bundle. The third move then introduces the corpus's specific geometric model of the shared whole --- the
circle fibers of the Hopf fibration --- not to establish that a perspective is a part, which is already
done, but to say which whole, with what further structure; and it is at this level, and only this level,
that two further things usually left in philosophical or psychological twilight --- the status of
meaningful coincidence, and the limits of what the will can do --- acquire exact locations. The first
two moves are forced and self-standing; the third is the corpus's wager, and is offered as such. The
upshot is that solipsism is a category error of a specific and nameable kind, dissolved without
reliance on the model, while the model contributes a precise account of synchronicity and agency for
those willing to grant it.

A word on register. The corpus's papers are, almost without exception, mathematics and physics carried
by a thin line of prose: the equations do the work and the words point at them. This paper inverts
that ratio deliberately. The result it reports is philosophical and psychological, and a result of that
kind is not established by exhibiting an equation but by making one see why the equation bears the
weight that is placed on it. The mathematics here is genuine and load-bearing --- the disjointness of
the fibers, the proper-subset embedding, the single shared total space, the linking number one are all
exact, and each is pinned by a machine-checked verifier --- but the mathematics is the spine, not the
body. The body is the argument about minds, and that argument is made in words.

\section{Two solipsisms, and why conflating them is the whole trouble}
\label{sec:two}

Begin with the distinction. Strip away the rhetoric and there are two quite different theses that go
under the name of solipsism, and the perennial life of the problem comes almost entirely from the
habit of treating them as one.

The first thesis is \emph{epistemic}, and we may call it the privacy thesis or, for brevity, weak
solipsism. It says: my experience is directly given only to me; the inner lives of others, if any, are
not accessible to me as my own is, but only inferred, mediated, reconstructed from behavior and
report. This is not a metaphysical claim about what exists. It is a claim about access, about the
asymmetry between the first person and the third. And it is, on its face, simply correct. The whole
apparatus of expression --- language, gesture, art, the long human effort to make the inside visible
from the outside --- is intelligible only on the assumption that the inside is not visible from the
outside to begin with. We would not need to tell each other how we feel if feeling were public.

The second thesis is \emph{metaphysical}, and we may call it the existence thesis or strong solipsism.
It says: only I exist; or, in its more cautious modal dress, nothing beyond my experience can be known
to exist, so the external world and other minds are, for all I can establish, possibly unreal. This is
a claim about what there is, or about the reach of what can be known to be, and it is the thesis that
philosophers have in mind when they call solipsism unanswerable. Schopenhauer, who took it more
seriously than most were willing to, called the strict version ``theoretical egoism'' and remarked
that it could never be refuted by proofs, yet was held by no one outside a madhouse --- a position, he
said, to be treated less as a conviction to be defeated than as a small frontier fortress that can be
left to stand, harmless because permanently garrisoned by no one.

The reason the existence thesis seems unanswerable is that the obvious replies all have the same shape
and the same flaw. They try to argue \emph{outward}: from the one thing the solipsist grants, the
present contents of his own experience, to the existence of something beyond it. Descartes argues
outward through the goodness of God, who would not deceive him about the existence of an external
world. The British empiricists argue outward through the involuntary, orderly, intrusive character of
sensation, which presents itself as caused by something not ourselves. Later philosophers argue outward
through inference to the best explanation: other minds are the best explanation of the behavior of
other bodies. Every one of these is an inference from experience to a world, and every one of them can
be met by the solipsist with the same shrug --- the inference may be natural, may be irresistible, may
be pragmatically obligatory, but it is still an inference, and the conclusion still outruns the
premise. The gap between \emph{this is how things seem} and \emph{this is how things are} is exactly
the gap the strong solipsist plants his flag in, and no argument that begins on the near side of it can
force its way across.

Here is the claim this paper will make good. The existence thesis seems unanswerable only because it is
being attacked on the wrong ground. As long as one accepts the solipsist's framing --- start inside
the private viewpoint, then try to get out --- one has already lost, because the framing builds the gap
into the starting line. The right response is not to argue outward across the gap but to deny that the
starting point is what the solipsist takes it to be. The private viewpoint is not a sealed room from
which one must somehow establish that there is an outside. It is, structurally, already a thing that
\emph{has} an outside, in the precise sense that it is a part of a larger object and could not be the
thing it is otherwise. To occupy a perspective is already to be located in something one did not make
and does not exhaust. If that can be shown --- not asserted, shown, from the structure of what a
perspective is --- then the existence thesis does not get refuted by an argument that bridges the gap;
it gets dissolved, because the gap it depends on was never there. The starting point already contains
the conclusion the solipsist thought he had to be argued into.

That is a strong promise, and one cannot keep it with words alone, because the solipsist will rightly
say that ``a perspective is already a part of a larger thing'' is precisely what is in dispute, and
that to assert it is to beg the question. What is needed is a model in which \emph{perspective} and
\emph{larger thing} are not vague philosophical counters but defined structures, in which the relation
between them is fixed by the mathematics and not by the philosopher's preference, and in which one can
therefore \emph{compute} whether the perspective presupposes the arena or not. The corpus supplies
exactly such a model, and it was not built for this purpose, which is part of why its verdict carries
weight. We turn to it now --- but first to a question that must be settled before any model is
introduced at all, on pain of the whole argument collapsing into the circle the careful reader already
senses.

\section{Why a perspective must be a part}
\label{sec:weak}

There is an objection that any argument of this shape invites, and it is fatal if it cannot be met. One
will build a model in which perspectives are defined as parts of a larger whole, and then announce, with
an air of discovery, that a perspective is part of a larger whole. That is not a transcendental argument;
it is a definition admiring itself in a mirror. The solipsist, presented with such a model, need not
struggle against it. He need only decline it: \emph{that is a pretty piece of geometry, but I am not a
part of anything; I have no reason to think my experience is a fiber of your bundle, and until you give
me one, your conclusion is a restatement of your premise.} If the argument of this paper amounted to no
more than the geometry, this objection would end it, and end it justly.

So before we introduce any geometry we must establish, from premises the solipsist already holds and
cannot abandon without abandoning his own position, that a perspective is a part of a shared whole. Not
a fiber of a particular bundle --- that stronger and more specific claim we will reach later, and will
treat as the corpus's wager rather than a necessity --- but the weaker structural fact that to be a
perspective at all is to be a partial presentation of something that exceeds the presentation, held in
common with others. We call this the \emph{weak structure}: a disjoint part of one shared whole. The
claim of this section is that the weak structure is not a model imposed on the perspective from outside
but a consequence of what a perspective is, derivable from the one thing the solipsist most insists
upon --- that he has a point of view.

The argument turns on a feature of experience so pervasive that it is easy to look past: every
perspective is \emph{partial}. To occupy a point of view is to be presented with some things and not
others, from somewhere and not from everywhere, against a horizon of what is not now given. Look at a
cup. You are presented with its near face; its far side is hidden; and yet you do not see a flat
coloured shape, you see a cup --- a thing with a back, a hollow, an inside you could turn to. The hidden
side is not absent from your experience; it is present \emph{as} hidden, co-intended, part of the sense
``cup'' even as it is withheld from view. Every perception of a thing is like this: one-sided, given
through a single profile, carrying with it a horizon of the profiles not currently shown. This is what
the phenomenological tradition, following Husserl, called adumbration --- the necessary one-sidedness of
the presentation of any transcendent object --- and the crucial point for us is that it is not a theory
\emph{about} experience but a feature \emph{of} it, there to be noticed in any honest description of how
things are given. The partiality is in the experience, not added to it by the philosopher.

From this single given the weak structure follows, and we can set out the steps so that each may be
inspected. \emph{First}, the solipsist has a perspective; he says so, and indeed he cannot state his
thesis except from one --- ``only I exist'' is uttered from a point of view, and a thesis with no point
of view is no one's. \emph{Second}, that perspective is partial in the sense just described: it presents
its objects one-sidedly, with horizons, against a background of the not-given. \emph{Third}, partiality
posits an excess. A presentation that is partial is partial \emph{of} something --- a presentation of
something that the presentation does not exhaust, for if it exhausted its object there would be nothing
one-sided about it and no horizon to it, and it would not be a perspective on the thing but a
coincidence with it. To be a partial presentation is, analytically, to point beyond oneself to a fuller
whole one does not contain. \emph{Fourth}, the solipsist's own privacy thesis tells us how these
partial presentations stand to one another: one's givens are given to oneself and not, in the same
manner, within any other point of view; perspectives do not share their interiors; they are, in that
exact sense, disjoint. Put the four together and the conclusion is forced: a perspective is a disjoint,
partial presentation of a whole that exceeds it and is held in common --- the weak structure. No model
was assumed. The premises were the solipsist's own: that he has a point of view, that it is private, and
that --- as his own experience shows the moment he consults it --- it is partial.

\begin{theorem}[The weak structure is forced by the partiality of perspective]
\label{thm:weak}
Let it be granted only that one has a perspective, that the perspective is private (its givens are not
given within another), and that it is partial (it presents its objects one-sidedly, with horizons).
Then the perspective is a disjoint, partial presentation of a shared whole that exceeds it: the weak
structure. The conclusion is entailed by the premises; no further modelling assumption is used.
\end{theorem}

This is an argument of the transcendental kind, in the lineage the critics of a purely model-theoretic
approach rightly point to: Kant's claim that the unity of self-consciousness presupposes a law-governed
world of objects, Strawson's that the ascription of experiences to oneself presupposes their
ascribability to others on public criteria, Wittgenstein's that the language in which one could so much
as formulate a private datum is already a shared, rule-governed practice. Each begins from something the
skeptic is already doing --- judging, ascribing, speaking --- and exhibits a condition of its
possibility that the skeptic was trying to deny. The present argument has the same form and begins from
something even harder for the solipsist to disown, because it is not an activity he might claim to
forgo but the very structure of his having any view at all: the partiality of his own experience. He
cannot consult his experience to check our claim without finding the horizons there. The premise is
read off the first person, which is the only place the solipsist grants anything.

Now the decisive consequence for strong solipsism. Strong solipsism is the thesis that only this
perspective exists, with nothing beyond it. But by the argument just given, a perspective is partial,
and partiality is being-exceeded: a perspective with nothing beyond it would be a perspective that is
not partial, presenting everything from nowhere with no horizon --- which is not a perspective at all.
So strong solipsism, stated carefully, asserts a perspective that is not a perspective. It is not false
in the way a mistaken empirical belief is false; it is incoherent, and incoherent in a way the
solipsist can detect from the inside, by noticing that his own experience has a back side to every cup.
He need not be argued out across a gap from experience to world. He need only look at what his
experience already is, and see that it was never the sealed and self-sufficient thing his thesis
required.

\begin{corollary}[Strong solipsism dissolves; idealism is left standing]
\label{cor:weak}
Strong solipsism is incoherent: it asserts a non-partial perspective, contradicting the partiality
given in the solipsist's own experience. The dissolution does not, however, settle what the exceeding
whole \emph{is}. A solipsist in retreat may say: granted there is a whole my perspective is partial of,
but it is only the whole of my own further possible experience, not a mind-independent world or other
minds. This move is available, and it should be conceded, for it changes the subject. To say that the
exceeding whole is the totality of one's own possible experience is to relabel the shared whole, not to
abolish it; the structure --- one whole, exceeded by no single view, presented through disjoint partial
perspectives --- is left exactly in place. That position is a form of idealism, and idealism is not
solipsism. What the argument forecloses is the strong thesis that there is nothing beyond the present
view; what it leaves open is the further and quite different question of the nature of the beyond. The
fortress that falls here is the one that was actually manned; the realist--idealist dispute is a
different country.
\end{corollary}

It is worth being exact about how much and how little Theorem~\ref{thm:weak} delivers, because the
honesty of the whole paper depends on it. It delivers the weak structure --- a disjoint part of a shared
whole --- and with it the dissolution of strong solipsism. It does \emph{not} deliver any particular
geometry for that structure: not the dimension of the whole, not the shape of the space of
perspectives, not whether distinct perspectives are merely disjoint or also, in some richer way,
interwoven. Those are further facts, and if they hold they hold for further reasons. The premise it
leans on --- that experience is partial, horizoned, intentional --- is far weaker and far harder to
refuse than any claim about fibers and bundles, but it is still a substantive claim about the structure
of experience, and a skeptic radical enough to deny that his experience has horizons may refuse it. He
will do so, however, at the price of a description of his own experience that anyone can check against
their own and find false. That is the proper cost of the position, and it is a much heavier one than the
strong solipsist usually thinks he is paying.

\subsection{From ``larger'' to ``shared'': the gap partiality cannot close}
\label{subsec:shared}

There is a further demand the careful reviewer will press, and the honesty of the argument requires that
it be met rather than blurred past. Theorem~\ref{thm:weak} forces an exceeding whole; the word in its
statement is \emph{shared}; and one must ask whether the partiality of perspective actually delivers the
sharedness, or only the exceeding. The two are not the same, and Corollary~\ref{cor:weak} already records
why: the solipsist in retreat concedes that his perspective is partial of a whole, and then says that the
whole is nothing but the totality of his \emph{own} further possible experience --- the back of the cup is
a profile he could turn to, the horizon is a horizon of his own anticipations, and every excess his
partiality posits is an excess that remains, in principle, his. This is the immanent-horizon reading, and
under it the exceeding whole is real but private: a temporally and modally extended idealism in which the
``beyond'' is one's own not-yet. Partiality, pressed honestly, reaches \emph{larger}; it does not by itself
reach \emph{shared}. The whole it forces could be a whole of one. The word ``shared'' in
Theorem~\ref{thm:weak}, taken strictly, is at this point doing work the partiality premise has not yet
paid for, and we should say so before relying on it.

We must therefore earn sharedness on other ground, and the corpus has the ground --- but it is not the
ground of partiality. It is the ground of \emph{forced agreement} and of \emph{error}, and both turn on
something the partiality argument never used: that a reading can be made to \emph{match}, and can be found
to be \emph{wrong}. Take the two in turn.

The first is the lesson of Paper~27. A single observer satisfies the coherence identity $C\circ P=I$
trivially: any state space admits operators with $C=P^{-1}$, and no structure is forced, because there is
no second standard against which a projection could fail to be verified. Coherence for one is vacuous.
The identity becomes a \emph{constraint} --- something that forces a definite geometry, a canonical
structure both parties must use on what they share --- only when there is a \emph{second} reading that must
be reconciled with the first. Sharedness, on this account, is not given in the partiality of a single view
and could not be; it is what gets \emph{forced} the moment two readings are required to match. The
corpus's front page states the same thing without the apparatus: nothing is real alone; reality begins
where two readings are forced to match. This is exactly the move the immanent-horizon retreat cannot
parry by relabelling, because it does not concern the \emph{contents} a perspective anticipates --- which
the solipsist may always claim as his own --- but the \emph{forcing} of a match between readings, which by
its form requires a second reading that is not the first one's to author. A horizon of one's own
anticipations forces nothing; one cannot be made to agree with oneself.

The second, and the sharper wedge, is error. The solipsist, however radical, admits something he cannot
coherently retract: that he can be \emph{wrong}. He is surprised; he is corrected; he reaches for the cup
and his hand meets nothing, or meets more than he expected; the world resists him in ways he did not
author and would not have chosen. Grant him everything else and this remains: a reading that can \emph{fail}
is a reading answerable to something the reading is not. To be wrong is to be out of accord with a measure
that is not the erring view's own to set --- for if the standard of correctness were internal to the
stream, the stream could not depart from it, and ``error'' would have no purchase. A self-authored
experience cannot be the source of its own being-wrong; what corrects you cannot be what you are
producing, on pain of your producing your own correction, which is to say no correction at all. So error,
unlike partiality, escapes the self. Partiality posited an excess and the solipsist could keep the excess
inside; error posits an \emph{independent reference} and he cannot, because the very intelligibility of
his being mistaken requires that the corrector lie outside the reading it corrects.

This is what defeats the immanent-horizon retreat that partiality alone could not. ``It is all my own
experience'' is exactly the thing one's own experience cannot underwrite, because one's own experience
cannot be what convicts one's own experience of error. The retreat has one last hiding place --- ``then the
corrector is my unconscious, still me'' --- but to take it is already to lose the position: an inner
otherness that constrains the ego, that surprises and resists and overrules it, is a second reading wearing
the first one's name, the very disjoint other the privacy thesis admitted, relocated inward and renamed.
The ego that can be corrected by ``its'' unconscious is not the whole of the only thing that exists; it has
conceded an interior it does not author and must answer to. Either way --- second observer without or second
reading within --- the strict thesis that there is only this stream and its self-given contents is
abandoned the moment the stream concedes it can err.

\begin{theorem}[Error forces an independent reference]
\label{thm:error}
Let it be granted that a perspective can be in error --- that some of its readings can fail, be corrected,
or be resisted by what they are about. Then there exists a reference independent of the perspective: a
standard of correctness not authored by the reading it measures. A self-authored stream cannot be the
source of its own being-wrong; correction that issues wholly from within the corrected stream is no
correction. Hence the admission of possible error is the admission of an exceeding that is not the self's
own further experience but is independent of it. This closes the immanent-horizon retreat left open by
Corollary~\ref{cor:weak}: the exceeding whole cannot be relabelled as private, because privacy cannot
ground the perspective's liability to error.
\end{theorem}

\begin{interpretation}[How far error reaches, and where evidence takes over]
\label{interp:bounds}
The reach of Theorem~\ref{thm:error} must be marked as carefully as its force, for it is easy to take it
for more than it gives. What error forces is an \emph{independent reference} --- something the reading must
answer to and does not author. That is already decisive against strong solipsism in its strict form: it
kills ``only my mind and its self-given contents,'' because a mind that can be wrong is a mind in contact
with what is not itself. But it does not, by logic alone, force that the independent reference be
\emph{other minds} specifically, as opposed to a mind-independent, non-mental reality that simply resists;
nor does it settle the realist--idealist question, which Corollary~\ref{cor:weak} already left open and
which error does not close. An independent corrector could be a world without minds in it. To get from
``there is an independent reference'' to ``there are other observers, and the shared whole has the form the
corpus describes,'' the corpus does not offer a further proof and should not pretend to. It offers
\emph{evidence}, and changes the burden accordingly. The structure that forced agreement --- two readings
made to match across a boundary, private bulk and shared surface and irreducible edge --- is, in the
corpus, not a metaphor but a definite geometry: the $(B^4,S^3)$ manifold whose boundary integral returns
$\alpha^{-1}=137.036$. A contentless private stream, answerable to nothing and shared with no one, has no
reason to carry that structure or to compute that number; a reality whose very fabric is the forcing of
agreement between observers has every reason to. This is offered as evidence and not as compulsion ---
flagged here exactly as the Hopf identification and the synchronicity reading are flagged in
Section~\ref{sec:hinge} --- but it is the corpus's actual answer to ``why \emph{shared} and not merely
larger, why \emph{minds} and not merely a wall'': not that idealism is refuted, which it is not, but that
the geometry which computes the constants of our world is the geometry of forced intersubjective agreement,
and a private theater is the one hypothesis under which that coincidence has no explanation at all.
\end{interpretation}

The bearing of all this on Theorem~\ref{thm:weak} is now exact, and the bookkeeping is worth stating
plainly so that nothing is claimed twice or claimed for free. Partiality dissolves \emph{present-moment}
solipsism and forces a \emph{larger} whole: this much is the transcendental core, and it is forced. Forced
agreement and error supply what partiality could not --- an exceeding that is \emph{independent} of the
self, defeating the immanent-horizon retreat to ``it is all my own''; this much, too, is forced, on a
premise (that one can err) the solipsist grants even more readily than partiality. What remains
\emph{evidential}, not forced, is the last leg: that the independent reference is populated by other minds
rather than a mute resisting world, and that the shared whole has the corpus's specific geometry. The word
``shared'' in Theorem~\ref{thm:weak}, then, is earned --- but earned in two stages of different strength,
and the paper marks the seam rather than smoothing it: independence by the necessity of
Theorem~\ref{thm:error}, other-mindedness and the specific geometry by the evidence of a world that computes
its own fine-structure constant out of the forcing of agreement.

With the weak structure secured on its own ground, and its \emph{sharedness} now grounded not on partiality
but on forced agreement and error where partiality fell short, we may introduce the corpus's specific model of
it without fear of circularity. The geometry that follows is not what makes a perspective a part --- that
was settled here, before any geometry, from the partiality of perspective. The geometry is the corpus's
account of \emph{which} shared whole, with \emph{what} further structure; and, as we will see, it adds
to the bare weak structure exactly one thing that the bare structure lacks and that turns out to matter
--- the interlinking of the perspectives. We turn to it now.

\section{The geometric setting: observers as fibers}
\label{sec:setting}

The model comes from Paper~04, where it is introduced not to address solipsism but to argue that the
classical, observed world is a necessary middle layer between a quantum bulk and a monadic edge. We
need only the part of that apparatus that concerns observers, and we can state it without the
surrounding physics.

The central object is the Hopf fibration. Take the three-sphere $\Sthree$, the set of unit vectors in
two complex dimensions, $\{(z_1,z_2)\in\mathbb{C}^2 : |z_1|^2+|z_2|^2=1\}$. There is a natural map from
it onto the two-sphere, $\pi:\Sthree\to\Stwo$, which sends a point to the complex line it spans; two
points map to the same place exactly when they differ by an overall phase, $(z_1,z_2)$ and
$e^{i\theta}(z_1,z_2)$. The set of points sharing a given image is therefore a circle --- the orbit of
the phase --- and these circles, one over each point of the base $\Stwo$, are the \emph{fibers} of the
fibration. The three-sphere is thus organized, completely and without remainder, into a family of
circles parameterized by the two-sphere: through every point of $\Sthree$ there passes exactly one
fiber, and the fibers fill the sphere.

Paper~04's interpretation, which it carries explicitly as interpretation rather than theorem, is that
this structure is the geometry of observation. The three-sphere $\Sthree$ is the shared classical
world --- ``where we all live together,'' in the paper's phrase --- the common arena in which physical
facts are settled and public. Each fiber, a circle $\Sone$, is a \emph{monad}: a single observational
perspective, a point of view threaded through the shared world. The base $\Stwo$ indexes the
perspectives, one for each of its points. An observer, on this picture, is not a thing sitting in the
world looking at it from a particular spot; an observer \emph{is} one of the circles out of which the
world is woven, a fiber of the bundle that is the world.

\begin{definition}[The observational bundle]
\label{def:bundle}
The shared world is the total space $\Sthree$. An observer (monad) is a fiber $M_q=\pi^{-1}(q)\cong\Sone$
over a base point $q\in\Stwo$. The collection of all observers is the fibration $\pi:\Sthree\to\Stwo$;
the base $\Stwo$ is the space of possible perspectives.
\end{definition}

Three features of this definition will do all the work, and each is a fact about fiber bundles, not a
choice we are free to make once the model is adopted. First, there is \emph{one} total space: the
fibers are not scattered across many three-spheres, one apiece, but are all substructures of a single
$\Sthree$. Second, the fibers are \emph{disjoint}: two distinct points of the base have fibers that do
not meet, because a point of $\Sthree$ lies over only one point of $\Stwo$. Third --- and this is the
fact Paper~04 does not use but which turns out to matter most --- any two distinct fibers are
\emph{linked}: they are not merely separate circles but interlocked ones, in the way the two rings of a
chain are interlocked, and the degree of their linking is exactly one. This last is the defining
property of the Hopf fibration, the thing that makes it nontrivial; it is why the map $\pi$ cannot be
unwound into a simple product of sphere and circle.

The relation of these three features to Section~\ref{sec:weak} should be made explicit at once, because
it governs how the rest of the paper is to be read. The first two --- one shared total space, disjoint
fibers --- are nothing other than the weak structure, the disjoint-part-of-a-shared-whole that
Theorem~\ref{thm:weak} already forced from the partiality of perspective, without any geometry. Any
realization of the weak structure has them; they are not special to Hopf. What is special to Hopf is the
third feature, the linking. A circle bundle can have disjoint fibers in one total space and yet have its
fibers \emph{unlinked} --- the trivial bundle $\Stwo\times\Sone$, the cylinder's higher analogue, is
exactly such a case: its fibers are disjoint and co-embedded, so it realizes the weak structure in full,
but two of its fibers can be drawn apart without crossing, their linking number zero. The Hopf bundle is
the weak structure \emph{plus} linking number one. This is the single thing the corpus's specific
geometry adds to what Section~\ref{sec:weak} established on general grounds, and we will find that it is
the thing, and the only thing, that the synchronicity result depends on.

This sharpens what the model is doing and, just as importantly, what it is not. It is not carrying the
dissolution of strong solipsism; that was carried in Section~\ref{sec:weak}, on premises the solipsist
grants, and it stands whether or not the shared whole is the Hopf three-sphere or any other realization
of the weak structure. What the Hopf model contributes is an account of \emph{which} whole and with
\emph{what} further structure --- and, through the linking, a precise locus for two phenomena, meaningful
correlation and the reach of the will, that the bare weak structure leaves undetermined. The model does
not assume that other minds exist; whether the fibers other than one's own are inhabited is left open,
and the argument does not need it. It does not assume experience is public; it entails experience is
private. What it adds to Section~\ref{sec:weak} is geometry, not the part--whole relation itself, which
is already secured. We will return in Section~\ref{sec:hinge} to the standing of the specifically Hopf
features --- the base $\Stwo$, the nontrivial linking --- which, unlike the weak structure, do rest on
the corpus's physical arguments rather than on the partiality of perspective. For now we adopt the model
and compute.

\section{The Anti-Solipsism Theorem}
\label{sec:theorem}

Everything the model has to say about solipsism is contained in one theorem with four parts. The parts
were established separately --- (a) and (c) in Paper~04, (b) as a sharpening of Paper~04 in
Addendum~373, (d) in Addendum~374 --- and consolidated in Addendum~375; each is pinned by a
machine-checked verifier. We state the theorem, then spend the rest of the paper unfolding what each
part means.

\begin{theorem}[Anti-Solipsism]
\label{thm:main}
Let observers be the fibers of the Hopf fibration $\pi:\Sthree\to\Stwo$, as in
Definition~\ref{def:bundle}. Then:
\begin{enumerate}
\item[\textbf{(a)} \textbf{Privacy.}] Distinct fibers are disjoint, $M_i\cap M_j=\varnothing$ for
$i\neq j$. Experience is fiber-local: what is given within one perspective is not given within another.
\item[\textbf{(b)} \textbf{Embedding.}] Every fiber is a \emph{proper} subset of the single total space
$\Sthree$, and $\Sthree$ carries continuum-many fibers, $|\Stwo|=\mathfrak{c}$. To be a perspective is
to be a fiber \emph{of} the bundle; the perspective therefore presupposes the shared arena and the
multiplicity of other perspectives within it.
\item[\textbf{(c)} \textbf{Enforcement.}] Disjoint fibers admit no direct fiber-to-fiber path; the only
channel between them is the shared boundary $\Sthree$, $M_i\to\Sthree\to M_j$. Intersubjective
consistency is thereby enforced by the topology.
\item[\textbf{(d)} \textbf{Linking.}] Any two distinct fibers are linked with linking number one. Private
perspectives are thus globally correlated through the bundle although no path runs between them, while
the shared boundary itself is altered only by action upon it.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Parts (a) and (b) are immediate from the bundle structure. A point of $\Sthree$ lies over a unique base
point, so it belongs to exactly one fiber; hence distinct fibers are disjoint (a), and each fiber, being
the preimage of a single point, is a proper subset of $\Sthree$, which is the preimage of all of $\Stwo$
(b). The base $\Stwo$ has the cardinality of the continuum and $\pi$ is surjective, so there are
continuum-many fibers (b). Part (c) follows from (a): two disjoint circles in $\Sthree$ have no common
point through which a direct interaction could pass, so any relation between them is mediated by the
ambient space they jointly inhabit. Part (d) is the Hopf invariant: computing the Gauss linking integral
for the preimages of two distinct base points gives one; equivalently, the fibration's first Chern class
is the generator, which is the statement that the bundle is nontrivial. The disjointness of (a) and the
linking of (d) are jointly the assertion that distinct fibers are separate but interlocked --- they do
not meet, yet neither can be moved away without passing through the other. Each numerical claim
(disjointness with positive separation, linking number one) is verified to machine precision in
\texttt{verify\_P373.py} and \texttt{verify\_P374.py}, and the consolidated structure in
\texttt{verify\_P375.py}.
\end{proof}

\begin{remark}[Which parts need the Hopf bundle, and which need only the weak structure]
\label{rmk:decomp}
The four parts do not stand on the same ground, and keeping the difference in view is essential to the
honest reading of the result. Parts (a), (b), and (c) --- privacy, embedding, enforcement --- use only
that the fibers are disjoint and co-embedded in one total space; that is the weak structure of
Section~\ref{sec:weak}, and it holds in \emph{any} realization of it. The trivial bundle
$\Stwo\times\Sone$ satisfies (a), (b), and (c) in full, with linking number zero between its fibers; the
Hopf bundle satisfies them too, with linking number one. So (a), (b), (c) --- and with them the
dissolution of strong solipsism --- are independent of the choice between trivial and Hopf, and hence
independent of the specifically Hopf identification that the corpus's physics supplies. Only part (d),
the linking, distinguishes Hopf from the trivial bundle, and so only part (d) --- synchronicity, in
Section~\ref{sec:synchronicity} --- rests on the corpus-specific geometry. This is verified directly:
the Gauss linking integral is zero for two fibers of $\Stwo\times\Sone$ and one for two Hopf fibers
(\texttt{verify\_P376.py}). The reader who doubts the Hopf identification therefore loses part (d) and
keeps the dissolution; the reader who grants it gains, in addition, the account of synchronicity.
\end{remark}

The theorem is, as mathematics, modest: it says things about the Hopf fibration that are either standard
or one short step from standard. Its interest is entirely in what the parts \emph{mean} once the fibers
are read as perspectives, and that meaning is not modest at all. Part (a) says the solipsist's datum is
true. Part (b) says his conclusion is incoherent. Part (c) says the world's objectivity is enforced
rather than hoped for. Part (d) says that two things usually discussed in metaphor --- meaningful
coincidence and the efficacy of the will --- have exact structural locations, and different ones. We
take these in turn, beginning with the one that does the decisive work against strong solipsism.

\section{Why strong solipsism is incoherent, not merely false}
\label{sec:incoherent}

Section~\ref{sec:weak} has already established the incoherence of strong solipsism on general grounds,
from the partiality of perspective, without recourse to any geometry. This section shows the same
incoherence wearing its geometric face: how, inside the corpus's model, part (b) of the theorem says
exactly what Theorem~\ref{thm:weak} said, and why the two are one argument seen from two distances. The
reader who is satisfied by the transcendental version may treat what follows as illustration; the reader
who wants to see how the abstract part--whole relation looks once it is given a definite shape will find
it here.

The heart of the matter is part (b) of the theorem, and the right way to feel its force is to notice
what kind of claim it is. It is not an inference from the contents of experience to the existence of a
world. It is a claim about what a perspective \emph{is}. And it says: a perspective is a fiber, and a
fiber is, by its very definition, a fiber \emph{of} a bundle --- a proper part of a single shared total
space, alongside continuum-many others. You cannot have the fiber without the bundle, not because the
bundle is a good explanation of the fiber, nor because the fiber would be lonely without it, but
because the fiber is \emph{constituted} as the preimage of a point under a map whose domain is the
whole sphere. Remove the sphere and there is no map; remove the map and there is no fiber; there is, at
most, an abstract circle that is not anyone's perspective on anything.

This is why the argument does not have the form the solipsist is braced for. He is ready to resist an
inference from \emph{this is how things seem to me} to \emph{there is a world}; he has the gap manned.
But the theorem does not try to cross that gap. It observes, instead, that the very thing he is
standing on --- his being a perspective at all, his having a point of view from which things seem any
way whatever --- is already the property of being a fiber in the shared bundle. The ``I'' of \emph{only
I exist} is a fiber; and a fiber is a part of the one $\Sthree$; and the one $\Sthree$ is the shared
world containing all the other fibers. So when the strong solipsist says ``only I exist,'' he is saying,
unpacked, ``only this-fiber-of-the-one-shared-bundle exists'' --- which asserts the bundle in the act of
trying to deny everything but a part of it. The position is not false in the way ``the earth is flat''
is false, by getting a fact wrong. It is incoherent in the way ``I am the only word in this sentence''
is incoherent, by presupposing in its form what it denies in its content.

It is worth being precise about the modal structure here, because there is a tempting bad version of
this argument and we are not making it. The bad version says: other minds must exist because the other
fibers are there. That overshoots. The theorem does not establish that the other fibers are inhabited
--- that they are \emph{minds} rather than merely unoccupied perspectives is exactly the part the model
leaves open, and we will not pretend otherwise. What the theorem establishes is weaker and stronger at
once: weaker, because it does not populate the other fibers with experience; stronger, because what it
does establish, it establishes by necessity rather than by inference. It establishes that the shared
arena exists, with its continuum of perspectival places, as a precondition of there being any
perspective at all --- including the solipsist's own. The other \emph{places} are not in doubt; they are
entailed by the existence of one place, because places are positions in a single structure. Whether
anyone is home in them is a further question, and a real one, which Section~\ref{sec:hinge} addresses.
But strong solipsism does not even reach that question. It founders earlier, on the shared arena it
needs in order to be a someone with a view at all.

This is the sense in which solipsism is \emph{dissolved} rather than refuted. To refute a position is
to grant its terms and show its falsehood. To dissolve one is to show that its terms do not hold
together, that the position cannot be coherently stated once one is careful about what its words must
mean. The tradition has a name for arguments of this shape --- they are transcendental, in roughly
Kant's sense: they proceed by exhibiting the conditions under which the very thing the skeptic relies on
is possible, and showing that those conditions include what the skeptic doubts. Kant argued that the
unity of self-consciousness presupposes a law-governed world of objects; Strawson, that the ascription
of experiences to oneself presupposes their ascribability to others; Wittgenstein, that a language in
which one could so much as formulate the private contents the solipsist appeals to is already a public,
shared, rule-governed thing. The argument here is of the same family, but it has an advantage those
arguments lacked: its transcendental premise is not a contestable claim about the conditions of thought
or language, defended by philosophical reflection that the skeptic can always reinterpret. It is a fact
about a bundle, fixed by the mathematics, and the skeptic who wishes to resist it must reject not a
philosophical analysis but the geometric model of what an observer is --- which is a different and much
more exposed kind of move, and one we will look at squarely.

There is a further subtlety that the geometry makes visible and ordinary discussions of solipsism miss.
The solipsist imagines that the private viewpoint is metaphysically \emph{first}: that there is, in the
order of being, the private experience, and then, possibly, derivatively, uncertainly, a world. The
bundle reverses this. In the bundle the total space $\Sthree$ is not assembled out of antecedently
given fibers; the fibers are individuated only as parts of it, by the projection $\pi$. There is no
fiber prior to the bundle that the bundle then collects. Logically and structurally the shared whole and
the perspectival parts are simultaneous, co-given, neither prior; but if anything has priority it is the
whole, since it is the whole, through $\pi$, that says which circles count as fibers at all. The
solipsist's picture of a private interior that may or may not be joined to an exterior gets the order
exactly backwards. The interior is a cross-section of the exterior. The view is a view \emph{from
within} something, and the within is logically posterior to the something.

\section{The privacy that remains: qualia as fiber-locality}
\label{sec:privacy}

Having taken so much from the solipsist, we must now be scrupulous about what we leave him, because a
treatment of solipsism that proved too much would be as suspect as one that proved too little. The
characteristic overreach in the anti-solipsist direction is to slide from \emph{the shared world is
real and enforced} to \emph{there is nothing private after all}, and so into one or another behaviorism
on which the inner life is a grammatical illusion, exhaustively cashed out in dispositions to behave.
The geometry blocks that slide as firmly as it blocks the solipsist's. Part (a) of the theorem is not a
concession wrung from the model; it is a theorem with the same standing as part (b). The fibers are
disjoint. What is given within one is not given within another. Experience is fiber-local, and its
privacy is exact.

This matters because it shows that the corpus's answer is not a deflationary one. It does not dispel
the mystery of the first person by denying that there is a first person; it locates the first person
precisely, as the fiber, and certifies its privacy as a structural fact. When I see red, the seeing is
a state of my fiber; your fiber, disjoint from mine, does not contain that state and cannot be brought
to contain it, for the two circles share no point. The famous asymmetry between knowing-that and
knowing-what --- that one might know every physical fact about another's color experience and still not
know what it is like to have it, that there is something it is like to be a bat which no amount of bat
physiology delivers --- is, on this picture, the asymmetry between the base and the fiber. The base
$\Stwo$ is the space of perspectives \emph{described}, the third-person catalogue of points of view;
each base point can be specified, communicated, studied. The fiber over a point is the perspective
\emph{lived}, the circle itself; and it is a different object from its label. You can hold the label ---
``the perspective of the bat,'' ``the perspective of the person seeing red'' --- without holding the
fiber, exactly as you can name a point of $\Stwo$ without occupying the circle of $\Sthree$ that lies
over it. The explanatory gap is not a failure of physics. It is the difference between indexing a
perspective and being one.

So the corpus splits the difference that solipsism and its deflationary opponents fight over, and it
splits it cleanly because the split is built into the bundle. To the solipsist it says: you are right
that your experience is private --- here is the privacy, exactly, the disjointness of your fiber. You
are wrong that this privacy is the privacy of reality --- here is the shared world, exactly, the one
$\Sthree$ your fiber is a part of. To the behaviorist it says: you are right that the world is public
and shared and that minds meet only in it --- here is the public channel, exactly, the only route
between fibers. You are wrong that this leaves nothing inner --- here is the inner, exactly, the fiber
that the channel never enters. Each party has hold of one of the two facts that the geometry holds
together, and each has mistaken his one fact for the whole.

The deepest point in this neighborhood is that privacy and sharedness are not two facts but one fact
seen twice. The fibers are disjoint \emph{and} they are all in the one sphere; and these are not
independent features that happen to coexist, but the same structural situation read from two sides.
There is no version of the bundle in which the fibers are disjoint without being co-embedded, or
co-embedded without being disjoint, because to be a fiber just is to be one of the disjoint circles
that jointly constitute the single sphere. The privacy of my experience and the publicity of our world
are, geometrically, the recto and verso of one page. This is why the intuition that drives solipsism is
both honored and defused: honored, because the privacy it begins from is real and ineliminable; defused,
because that very privacy, correctly understood, is already the privacy of a part within a shared
whole, and so carries the shared whole with it. The solipsist's mistake was never to feel the privacy.
It was to think the privacy could stand alone.

\section{Synchronicity: correlation without contact}
\label{sec:synchronicity}

We come now to part (d), and to the most delicate stretch of the argument, because it touches a region
where it is easy to claim too much and where the corpus's discipline of marking theorem from
interpretation must be applied with special care. The bare mathematical fact is part (d) of
Theorem~\ref{thm:main}: any two distinct fibers are linked, with linking number one. The fibers are
disjoint --- they never meet --- and yet they are interlocked, in the literal topological sense that
neither can be drawn away to a great distance without crossing the other; they form, pairwise, a Hopf
link. Disjointness and linking are usually thought of as opposites; the Hopf fibration is precisely the
structure in which they coincide everywhere. Every two perspectives in this world are separate and
interwoven at once.

Consider what that means read on the perspectives. Disjointness, we have said, is the absence of any
direct channel: nothing passes from one fiber to another, no signal, no contact, no causal line ---
part (c) made this the ground of privacy and of the action-only character of the shared world.
Linking is something else. It is a relation that holds between the two fibers as wholes, a global fact
about how they sit together in the sphere, and it is there \emph{regardless} of whether anything passes
between them. Two linked rings are correlated in their configuration --- you cannot fully describe the
position of one without reference to the other --- though neither touches the other and neither pushes
the other. The linking is correlation without contact: a structural tie that is not a causal tie.

That phrase, correlation without contact, or more exactly correlation without a causal channel, is the
precise structural form of what Jung named synchronicity and struggled to define: an ``acausal
connecting principle,'' a meaningful correspondence between events that is not mediated by any chain of
cause and effect running between them. Jung's notion has always been hard to hold because it seemed to
require either a hidden causal mechanism, in which case it was not acausal, or sheer coincidence, in
which case it was not a principle. The geometry offers a third thing, and a definite one: a relation
that is genuinely not causal --- no path runs along it, nothing is transmitted --- yet genuinely a
connection, fixed by the structure of the shared bundle, present between any two perspectives whatever.
The fibers are correlated because of how the one sphere is woven, not because either acts on the other.
That is exactly the conceptual room Jung needed and could not find a model for: acausal, yet not
accidental; connecting, yet not transmitting.

Here the discipline must be stated plainly, because the temptation to overclaim is real and the corpus
does not indulge it. What the theorem establishes is that the geometry of perspectives \emph{has the
structure of} acausal correlation --- that disjoint-but-linked is a genuine and exact relation, and
that it is the kind of relation synchronicity was always trying to name. What the theorem does
\emph{not} establish is that any particular experienced coincidence --- the dream that anticipates the
telegram, the name that surfaces in two minds at once --- \emph{is} an instance of this structural
linking rather than ordinary chance. The identification of the topological linking with felt
synchronicity is an interpretation, and it is flagged as one, exactly as the identification of fibers
with minds is flagged. The corpus licenses the \emph{possibility} and supplies the \emph{form}; it does
not certify any case, and it makes no paranormal claim. To say ``synchronicity is the linking'' is to
say that if acausal meaningful correlation between perspectives occurs, this is its geometry --- not
that one's striking coincidences are warrants of the occult. The model is, if anything, a constraint on
enthusiasm: it says that whatever synchronicity is, it is correlation and not causation, and therefore
that it carries no power, a point that becomes decisive in the next section.

It is worth observing how naturally this places synchronicity relative to the two regimes of the
theorem. The shared boundary $\Sthree$ is the domain of causation: things that happen there happen by
contact, by signal, by action, and they are public. The linking is the domain of correlation: it
relates the private fibers to one another and to the whole without any traffic across the boundary, and
it is not public in the same way --- it is not a fact one fiber transmits to another but a fact about
how they are jointly disposed. Synchronicity, on this reading, is not a spooky causation operating
where ordinary causation fails; it is not causation at all. It is the felt registration, within a
fiber, of the fiber's standing correlation with the rest of the bundle --- a correlation that was there
all along because the perspectives were never as separate as their disjointness alone suggested. They
were disjoint and linked. The privacy was real; so was the connection; and the two did not compete,
because one was about contact and the other about configuration.

\section{Action and the limits of the will}
\label{sec:action}

The same fact that locates synchronicity also draws a hard line under it, and the line is part (d)'s
second clause: the shared boundary is altered only by action upon it. This follows from part (c). The
only channel that reaches from a fiber into the shared $\Sthree$, and thence to any other fiber, is the
boundary itself --- the public route of speech, writing, motion, physical signal, the things Paper~04
lists as the means of intersubjective communication. The linking, for all that it is a real and global
correlation, supplies no such channel; it is configuration, not contact, and configuration carries no
leverage. From the fact that my fiber is linked to the whole it does not follow that I can move the
whole by adjusting my fiber. Correlation is not a handle.

This is the precise, and precisely modest, sense in which the will is limited. One can affect one's own
fiber: its phase, its trajectory, the comportment of one's own perspective is a degree of freedom one
genuinely has --- this is the affectable private layer, and Paper~08's treatment of the observer as a
trajectory through the geometry is its home. And because the fiber is linked to the whole, changes in
it are not isolated; they stand in correlation with everything. But to \emph{change the shared world}
--- to alter what is public, what other fibers will meet when they reach the boundary --- there is
exactly one way, and it is to act: to project from one's fiber onto $\Sthree$ through the public
channel, to do something that leaves a mark on the common arena. There is no shortcut through the
linking. Wishing, willing, intending, however intensely, however correlated with the whole, moves
nothing on the boundary that action does not move. ``Force the shared world without acting on it'' is
not difficult; it is geometrically impossible, blocked by the same disjointness that secures privacy.

It is striking how exactly this matches the moral structure the corpus elsewhere insists on, and worth
making the connection explicit, for it shows that the geometry of mind and the discipline of the system
built on it are the same shape. The corpus's governing constraint --- it calls it the Butlerian line
--- is that the system does not substitute intention for action, does not let a wish stand in for a
deed, does not allow the appearance of having done a thing to replace the doing. An agent in the system
executes exactly or it surfaces that it cannot; it never quietly pretends. Read against
Section~\ref{sec:action}, this is not a mere engineering policy but a recognition of the geometry: the
shared state changes only by action upon it, so a system that respects the shared state must move by
action and not by substitution. The same point appears in the corpus's refusal of a central scheduler
or broker --- its ``no central referee'' principle --- where coherence among many sub-systems is
achieved not by a coordinator issuing decisions but by each publishing its state into a shared
geometry and a composition being taken there, a midpoint on the common manifold rather than a verdict
from a throne. That is the operational image of exactly this theorem: the shared $\Sthree$ does the
coordinating, by being shared; no fiber rules another; coherence is enforced by the common arena and
realized through action into it, never by one perspective forcing another across a private channel that
does not exist.

There is a kind of consolation in this and also a kind of severity, and the paper would be dishonest to
soften either. The consolation is that one is not trapped in a private theater; one is linked, really
and structurally, to the whole, and one's own perspective is a real degree of freedom that genuinely
matters because it is genuinely correlated with everything. The severity is that this linkage grants no
magic: the world does not bend to the will, only to the deed, and the difference between wanting the
common world to be other than it is and making it so is the difference between a fiber's private state
and an action on the shared boundary --- a difference the geometry will not let collapse. To change what
we share, we must act in what we share. The will reaches exactly as far as action carries it, and not
one step further; and that is not a limitation imposed on us from outside but the very shape of there
being a shared world at all.

\section{The hinge: what is theorem and what is read}
\label{sec:hinge}

A result of this reach owes the reader a clear account of where it is load-bearing and where it leans on
interpretation, and the corpus's standing practice is to mark the join rather than paper over it. This
paper has two such joins, and it is important that they are exactly two, and that everything else is
theorem.

It is essential to be clear that the two joins are \emph{not} of equal reach, and that the dissolution
of strong solipsism does not lean on the more contentious of them. The structure of the paper was built
to make this so. The dissolution rests on Theorem~\ref{thm:weak} of Section~\ref{sec:weak}, whose only
premises are that one has a perspective, that it is private, and that it is partial --- the last being a
feature of the solipsist's own experience, not a posited model. That argument forces the weak structure,
a disjoint part of a shared whole, and on the weak structure alone strong solipsism is already incoherent
(Corollary~\ref{cor:weak}). No fiber, no bundle, no three-sphere is used. So the load-bearing premise for
the dissolution is ``a perspective is partial,'' a premise of the transcendental kind the tradition
begins from, and one the skeptic can refuse only by misdescribing his own experience.

There is, within this self-standing layer, an internal seam that Section~\ref{subsec:shared} makes
explicit and that the honest reading must carry. Partiality forces a \emph{larger} whole but not yet a
\emph{shared} one: it dissolves present-moment solipsism and leaves open the immanent-idealist retreat in
which the exceeding whole is one's own further possible experience. The step from larger to independent ---
the step that closes that retreat --- is carried not by partiality but by Theorem~\ref{thm:error}, on the
premise that one can be wrong: a reading that can fail requires a corrector it does not author, so error
escapes the self where partiality could not. That step is still forced, and on a premise even harder to
refuse than partiality; what it forces is an \emph{independent reference}. The last leg --- that the
reference is other minds rather than a mute resisting world, and that the shared whole has the corpus's
geometry --- is not forced but evidential (Interpretation~\ref{interp:bounds}): the $(B^4,S^3)$ structure
of forced agreement is the structure that computes $\alpha^{-1}=137.036$, which a private theater has no
reason to carry. So the dissolution layer is itself two-toned --- independence forced, other-mindedness
evidenced --- and the realist--idealist question proper stays open throughout.

The first hinge proper, then, is not the dissolution's but the \emph{geometry's}: the identification of
the shared whole with the specific Hopf bundle, observers with its fibers --- the premise adopted in
Section~\ref{sec:setting}, motivated on physical grounds in Paper~04, and carried there as
interpretation. This identification does real work, but a bounded and locatable amount of it. By the
decomposition of Remark~\ref{rmk:decomp}, parts (a), (b), (c) of Theorem~\ref{thm:main} --- privacy,
embedding, enforcement --- hold on \emph{any} realization of the weak structure, the trivial bundle no
less than the Hopf, and so do not depend on the Hopf identification at all; they are the geometric face
of what Section~\ref{sec:weak} secured without geometry. Only part (d), the linking, distinguishes the
Hopf bundle from the trivial one, and so only the synchronicity of Section~\ref{sec:synchronicity}, and
the cosmological connections that travel with it, actually rest on the Hopf identification. That layer is
honestly a \emph{consequence theorem}: given the specific geometry Paper~04 motivates, here is a
consequence that had not been noticed; it compels no skeptic into the geometry, and a reader who declines
the identification declines synchronicity with it and keeps everything else. The Hopf wager is thus the
hinge for synchronicity and for the corpus-specific shape of the shared world; it is not the hinge for
the dissolution of solipsism, which stands on the partiality of perspective whether the shared whole
turns out to be the Hopf three-sphere or anything else answering to the weak structure.

The second hinge is narrower and confined to Section~\ref{sec:synchronicity}: the reading of the
topological linking as experienced synchronicity. The linking is a theorem; that disjoint perspectives
are nonetheless correlated through the bundle is fixed by the Hopf invariant. That any particular felt
coincidence is an instance of this correlation rather than ordinary chance is not a theorem and is not
claimed. The corpus supplies the structure of acausal correlation and licenses synchronicity as its
possible name; it certifies no case and makes no paranormal assertion. A reader who rejects synchronicity
entirely loses nothing in the rest of the argument; parts (a) through (c) and the action-only result of
part (d)'s second clause stand without it. The linking is doing real work even for the skeptic, since it
is what shows that privacy does not entail isolation; the further step to felt synchronicity is offered,
flagged, and detachable.

Two clarifications forestall predictable misreadings. First, the paper does not claim that the existence
of other \emph{minds} is a theorem. The existence of the other perspectival \emph{places} --- the other
fibers --- is a theorem, entailed by the existence of one's own; whether those places are occupied by
experience is precisely what the first hinge leaves open, and the dissolution of strong solipsism does
not depend on settling it, because strong solipsism founders already on the shared arena, before the
question of other minds is even reached. Second, the paper does not claim novelty for the transcendental
\emph{strategy}; Kant, Strawson, and Wittgenstein are its ancestors and are acknowledged as such. The
claim is that the strategy, which in its classical forms rests on contestable analyses of thought or
language, is here carried by an explicit geometric model in which the crucial premise --- that the
perspective is a proper part of a shared whole --- is not argued philosophically but computed, and can
be checked.

\section{Conclusion: solipsism as a category error}
\label{sec:conclusion}

We can now say, compactly, what the corpus's answer to solipsism is. Solipsism is a category error: it
mistakes the privacy of a fiber for the privacy of the bundle. There is a real privacy --- the
disjointness of one's perspective, the fact that one's experience is lived only from within one's own
fiber and is given to no other --- and the solipsist is right to feel it and right to insist on it
against every deflationary attempt to explain it away. His single error, from which all the rest
follows, is to take that privacy as though it could be the privacy of \emph{reality}, as though being a
sealed perspective were compatible with being all there is. But a perspective is a fiber, and a fiber is
a part, and a part of a shared whole that it neither makes nor exhausts and without which it would not be
a perspective at all. The privacy is real; it is just not the privacy of the whole. It is the privacy of
a part \emph{in} the whole --- which is to say, a privacy that carries the whole with it wherever it
goes.

Set out as a list, the corpus's positions are these. The privacy of experience: true, and exactly
located, as the disjointness of the fiber. The existence of the shared world: not inferred but
presupposed, entailed by the existence of any single perspective, since a perspective is constituted as
a part of it. Strong solipsism, the thesis that only the self exists: not refuted but dissolved, shown
incoherent, since it asserts the shared bundle in the very act of denying all but a fiber of it.
Objectivity, the agreement of observers on a common world: not a fortunate coincidence but a topological
enforcement, since the only channel between perspectives runs through the one shared boundary.
Synchronicity, meaningful correlation without a causal link: given an exact structural form, the linking
of disjoint fibers, correlation without contact --- offered as the geometry such a thing would have, not
asserted of any case. And the will: limited with precision, able to affect its own fiber and thereby to
stand in correlation with the whole, but able to change the shared world only by acting upon it, never by
wishing across a private channel that the geometry does not provide.

What emerges is a picture of mind that is neither the lonely interior of the solipsist nor the
transparent public object of the behaviorist, but something the two had between them obscured: a
perspective that is genuinely private and genuinely embedded, separate from every other and interlocked
with every other, sealed in its experience and bound into one world, free in its own motion and bound to
act if it would move anything beyond itself. The solipsist was not wrong about the seal. He was wrong to
think the seal was the whole story, when it was always only one face of a single fact whose other face
is that we share, unavoidably and from the start, one world. That we cannot get out of our own
perspective and that we were never out of the common world are, in the end, the same truth, read on the
two sides of one page. The page is the bundle. We are its fibers: private, linked, and here together,
whether we argue the point or not.

\section*{Status}
This paper consolidates and expands the anti-solipsism result distributed across Paper~04 (the
multiple-observer theorems and the shared-boundary corollary), Addendum~373 (the topological enforcement
of shared reality), Addendum~374 (the linked-but-disjoint structure and the two regimes of efficacy),
Addendum~375 (the consolidated theorem), and Addendum~376 (the transcendental re-grounding and the
weak-structure decomposition, developed in response to review). It is structured in two layers, of
deliberately unequal strength. The \emph{dissolution} of strong solipsism rests on
Theorem~\ref{thm:weak} of Section~\ref{sec:weak}, a transcendental argument from the partiality of
perspective whose premises are the solipsist's own and which uses no geometry; it forces only the weak
structure --- a disjoint part of a shared whole --- on which strong solipsism is already incoherent
while idealism survives. Within that layer a seam is marked
(Section~\ref{subsec:shared}): partiality forces a \emph{larger} whole, but the \emph{sharedness} ---
an exceeding independent of the self, defeating the immanent-idealist retreat to ``it is all my own
experience'' --- is grounded not on partiality but on forced agreement (Paper~27) and on error
(Theorem~\ref{thm:error}: a reading that can fail requires a corrector it does not author). Error forces an
\emph{independent reference} only; that the reference is other minds, and that the shared whole has the
corpus's geometry, rests on evidence rather than proof --- the $(B^4,S^3)$ structure of forced agreement is
the one that computes $\alpha^{-1}=137.036$ --- and is flagged as such (Interpretation~\ref{interp:bounds}).
The realist--idealist question stays open. The \emph{geometric} layer --- the specific Hopf realization, and with it the
synchronicity result and the cosmological connections --- is a consequence theorem, conditional on the
Hopf identification motivated in Paper~04, and is the only layer that requires it; by
Remark~\ref{rmk:decomp}, privacy, embedding, and enforcement hold on the weak structure alone (the
trivial bundle, linking zero) and only synchronicity singles out Hopf (linking one). The formal content
--- disjointness, proper-subset embedding, continuum cardinality, the sole shared channel, linking
numbers zero and one for the trivial and Hopf bundles --- is theorem-grade and machine-verified in
\texttt{verify\_P373.py}, \texttt{verify\_P374.py}, \texttt{verify\_P375.py}, and \texttt{verify\_P376.py};
the transcendental core's structure is recorded in \texttt{verify\_P376.py}. The two interpretive hinges
are flagged in Section~\ref{sec:hinge}, and neither bears on the dissolution: the identification of the
shared whole with the Hopf bundle (carried from Paper~04), on which only synchronicity and the
corpus-specific geometry depend, and the reading of the linking as experienced synchronicity, which is
offered and detachable. The one substantive premise of the dissolution itself --- that experience is
partial, horizoned, intentional --- is weaker than any geometric claim and is read off the first person.
No published number changes. The paper deliberately inverts the corpus's usual prose-to-mathematics
ratio, the result being a philosophical theorem whose spine, not whose body, is mathematical.

\end{document}
