What this is

The Qualia Research Institute's founding bet is called qualia formalism: the claim that for any globally bound moment of consciousness, there exists a mathematical object that describes it precisely. That is a claim about the world, not a definition, and if it is true then a specific question follows — which mathematical object, living in which branch of physics?

This book gives one answer and follows it as far as it will go. The answer is that consciousness is the intrinsic aspect of the modular structure of local operator algebras in quantum field theory. Everything else — the boundary between subjects, the flow of phenomenal time, the sign and size of valence, the felt geometry of a state — is developed as a consequence of that single identification.

The programme is panpsychist: the intrinsic aspect is everywhere, without exception. But it is panpsychist in a specific and unusual way. Chapter 3 argues that quantum field theory forbids the micropsychism most panpsychists assume, and forces priority monism instead. That argument is the load-bearing one, and if you read only one chapter, read that one.

What this is not

It is not established science. It is speculative theoretical work built on established mathematics, and the distinction matters enough that it is marked on every claim in the book:

It is also not a solution to the hard problem. Axiom 2.1 posits that the intrinsic aspect of the physical is phenomenal; it does not derive it. What the book offers is a complete mathematisation of the structural questions — binding, boundary, time, valence, intensity, character — which is the tractable residue, and which is enough work for twelve chapters. Chapter 12 states the limits plainly and at length; readers inclined to think a theory of consciousness is overreaching should start there and work backwards.

Three ways to read it

The chapters are ordered so that each depends only on those before it, but there are shorter routes through.

Numbered results are cited as Theorem 3.1, Axiom 8.1, equations as (8.2). Every one is listed in the Index of Results. Symbols are collected in Notation, which is worth a minute before Chapter 3. Each chapter closes with exercises; those marked ★★★ are open problems rather than exercises, and are labelled as such.

A note on the source material

The framing of the problem — geometry for qualia spaces, symmetry for valence, topology for phenomenal boundaries, heavy tails for affective intensity — is QRI's, principally Michael Edward Johnson's Principia Qualia and Andrés Gómez Emilsson's work on topological segmentation and the Symmetry Theory of Valence. The physics is standard. What is new here is the bridge: the claim that their four desiderata are satisfied, respectively, by the Bures metric, the atomic part of a spectral measure, split inclusions of von Neumann algebras, and the multiplicativity of the inverse participation ratio.

Where I think they are right, I have tried to show why rather than agree. Where the bridge is stipulated rather than derived, Chapter 12 says so.

Algebras and states

Modular theory

Valence and its ingredients

Geometry and topology

Conventions

• Spacetime dimension $d=4$ unless stated. Signature is irrelevant to every argument here.
• $\hbar$ and $k_B$ are kept explicit wherever a number is computed, and set to 1 in formal passages.
• "Region" always means a bounded open region with non-empty causal complement, so that Reeh–Schlieder applies.
• "Subject" is reserved for the technical sense of Axiom 4.1 and never used loosely.

1.1The problem that is probably not solvable

There is something it is like to see red. There is nothing it is like, so far as anyone can tell, to be a stone. Between those two facts sits the whole difficulty: physical description proceeds entirely in terms of structure and dynamics — what a thing is made of, how it is arranged, what it does next — and no amount of structure and dynamics appears to entail that there is something it is like to be the arrangement.

This is Chalmers' hard problem, and it has a specific logical shape worth isolating. It is not the problem of explaining discrimination, or reportability, or attention, or the difference between waking and dreaming. Those are hard in the ordinary way that unsolved scientific problems are hard, and they will yield to ordinary work. The hard problem is the residue after all of those are solved: the question of why the functioning is accompanied by experience at all.

I do not think this book solves it, and Chapter 12 says so at length. What a panpsychist ontology does is relocate it. If experience is not produced by particular arrangements of matter but is instead the intrinsic character of matter, then the question "why does this arrangement produce experience?" becomes "why does anything have an intrinsic character at all, and why is it phenomenal?" That is not an answer. It is a different and, I will argue, more tractable question, because it no longer requires an emergence relation that nobody can specify.

1.2The problem that is a mathematics problem

Set the hard problem aside and a large, precise, unsolved problem remains. Experiences have structure. Some are unified and some are fragmented. Some contain a visual field, some do not. Some are excruciating and some are blissful, and the difference between those is not a difference of degree along one axis but something with a shape. Colours stand in similarity relations to one another that form a specific three-dimensional solid, not an arbitrary set. Time in experience has a thickness — a specious present of roughly a tenth of a second, not an instant.

All of this is structure, and structure is the natural home of mathematics. The bet that it can be captured exactly is QRI's, and it deserves a name and a precise statement.

Note what Definition 1.1 does and does not assert. It does not say we can find the object, or that it is simple, or that knowing it would tell you what red looks like. It says the object exists — that experience is not structurally amorphous. The denial of Definition 1.1 is the claim that two moments of experience could differ phenomenally while agreeing on every mathematical invariant, which is a strange thing to believe and, more to the point, is a research-stopper.

Most theories of consciousness fail Definition 1.1 not because they are false but because they are not the right kind of thing. Global workspace theory describes an architecture, not an object. Higher-order theories describe a relation between representations. Predictive processing describes a dynamics. None of these supplies a mathematical object whose invariants are the phenomenal invariants — and so none of them can say, even in principle, why one state feels better than another.

1.3Valence realism

Among the invariants, one is special. Experiences are not merely different from one another; some of them are good and some are bad, and this is a fact about the experience rather than about anyone's opinion of it. Call this valence realism: that pleasantness and unpleasantness are objective structural properties of a moment of experience, in the same sense that its dimensionality is.

Valence realism plus qualia formalism is a strong combination, because together they entail that there is a function from the mathematical object to the reals whose value is how good the experience is. Finding that function is the central technical goal of this book, and it occupies Chapters 6 through 9. QRI's proposal — the Symmetry Theory of Valence — is that the function measures symmetry. Chapter 6 makes "symmetry" precise as the atomicity of a spectral measure; Chapter 8 shows why that alone is insufficient and supplies the missing ingredient.

1.4What a formal theory owes you

A theory of consciousness that merely accommodates the phenomena is worthless, because the phenomena are wildly under-constraining and any sufficiently vague story accommodates them. A theory earns its keep by forbidding things. This book therefore commits to four obligations, and the reader is entitled to hold it to them:

1. Specify the object. Name the mathematical structure, in an existing branch of physics, rather than gesturing at one. Chapters 3–5.
2. Individuate subjects non-arbitrarily. Say what makes your experience yours and not mine, in a way that does not depend on how an observer chooses to carve the world. Chapter 4.
3. Compute valence. Produce a number, from the object, that tracks how good the experience is — and say what would count as it failing. Chapters 6–9.
4. Forbid something. Make predictions that could come out the other way. Chapter 11 lists eight.

1.5Three problems that must be solved on the way

Panpsychism is often dismissed on the grounds that it trades one mystery for three. That is a fair charge, and the three are worth naming now because the architecture of this book is organised around them.

The reason to be interested in the specific proposal of this book, rather than panpsychism in general, is that the answer to the first is unusually clean. It is not a mechanism for combination. It is a proof that the question is malformed — that the parts it presupposes do not exist in quantum field theory. That result is Theorem 3.1, and everything after it is an attempt to build on ground that argument clears.

1.6Why a field, and not a computation

One preliminary commitment should be made explicit now, because it shapes everything downstream. This is a physical theory rather than a computational one: it locates consciousness in the state of a field, not in the abstract structure of an information-processing system.

The reason is that computation is not an intrinsic property. Whether a physical system implements a given computation depends on an interpretation mapping states to symbols, and such mappings are cheap — a point pressed hard by Putnam and Searle, and never satisfactorily answered. If consciousness were computational, then whether a system were conscious would depend on how someone chose to describe it, which contradicts the objectivity that valence realism requires. Fields do not have this problem: the state of the electromagnetic field in a region is a fact, not an interpretation.

This commitment has a price, and it is paid in Chapter 11, prediction 8: if phenomenal binding can be produced across a purely digital channel between two brains that share no field region, this theory is refuted and the computationalists are right. I would rather have that bet on the table than not.

2.1Russellian monism, written down

The starting observation is Russell's, and it is not mystical. Physics describes the world entirely relationally: mass is what responds thus to force, charge is what couples thus to the field, and every term bottoms out in how it behaves with respect to other terms. Physics tells us what matter does, exhaustively, and tells us nothing whatever about what matter is. The equations would be equally true of any carrier that stood in those relations.

Russellian monism proposes that consciousness is the carrier — that the intrinsic nature of the physical, invisible to a purely relational science, is phenomenal. This is the position the book adopts, and the point of stating it categorically is to make it precise enough to have consequences.

The functor $U_{\mathrm{rel}}$ is full but not faithful. Full, because every physical relation is realised by something in $\boldsymbol{\Omega}$ — physics is not missing any structure. Not faithful, because distinct intrinsic characters can share a relational profile, which is exactly the sense in which physics under-determines the carrier. That single sentence contains the whole of Russellian monism, and the rest of the book is an attempt to say what $\mathbf{Qual}$ looks like.

2.2What Axiom 2.1 does not say

This is where most panpsychisms go soft, so it is worth being blunt. Axiom 2.1 does not say that every region is a subject. It does not say a thermostat has a point of view, that an electron is lonely, or that a rock is having a bad day. It says the intrinsic aspect is everywhere non-empty. Whether that aspect is bound, unified, or valenced is a further and enormously more restrictive question.

The distinction can be stated sharply, and the whole of Parts II and IV consists of making each line of the following table precise:

A rock is thermal. Its modular spectral measure is absolutely continuous, so by the machinery of Chapter 6 its coherence $\mathcal{A}$ is essentially zero, and by Axiom 8.1 its valence is bounded in magnitude by $\mathcal{A}$. It has intrinsic character, in the way a single unstruck note has a pitch. It has nothing it is like to be, in the way an orchestra does.

2.3The formalism axiom

Definition 1.1 asserted that a describing object exists. Axiom 2.2 says which invariants it consists of, which converts a philosophical thesis into a working specification.

Reading (2.2) left to right gives the plan of the entire book. $\mathcal{N}$ is the type I factor that individuates the subject, constructed in Chapter 4. $\rho$ is its density matrix. $\Delta_\rho$ is the modular operator, whose logarithm generates phenomenal time in Chapter 5. $J_\rho$ is the modular conjugation, which fixes the cut between subject and world. $g^{\mathrm{B}}$ is the Bures metric, the geometry of the qualia space, in Chapter 10. And $P(q)$ is the replica overlap distribution, which carries valence, in Chapter 8.

A theory that named five of these and left the sixth undetermined would be incomplete in a specific, diagnosable way. It is worth checking, as each chapter closes, which entry has just been filled in.

2.4Causal closure, and why it is the important axiom

The third axiom is the one that makes this a scientific proposal rather than an unfalsifiable metaphysics, and it is stated as a prohibition.

This is a genuine constraint and it costs something. It means consciousness cannot be invoked to explain any anomaly, because on this account there are no anomalies to explain: the dynamics are exactly the Standard Model's. Anyone hoping that a theory of consciousness will licence psychokinesis, quantum collapse by observation, or a causal role for the soul will find nothing here.

What it buys is that the theory is not free to accommodate whatever turns up. Interventionist theories of consciousness — those on which experience makes a difference to what the particles do — must either violate conservation of energy or introduce a new field, and both are detectable. By refusing that move, the theory places its whole predictive burden on structure: on which mathematical properties of an unmodified physical state correspond to which phenomenal properties. Every prediction in Chapter 11 is of that form.

The apparent objection is epiphenomenalism: if consciousness does nothing, why do we talk about it? The answer available to a dual-aspect view, and unavailable to a dualist, is that the talking is the same event as the experiencing, seen through the other functor. $U_{\mathrm{rel}}$ of the state includes the vocal cords moving; $U_{\mathrm{int}}$ of the same state is the experience being reported. There is one event, not two, so there is no question of the second failing to cause the first.

3.1A crash course in factor types

A von Neumann algebra is an algebra of bounded operators on a Hilbert space, containing the identity, closed under adjoints and under limits in the weak operator topology. A factor is one whose centre is trivial — it contains no non-scalar operator commuting with everything else, which is the algebraic way of saying it describes a single indecomposable system.

Murray and von Neumann classified factors by the behaviour of their projections, and the classification is the whole reason this chapter exists. A projection is the algebraic stand-in for a yes/no property; the question is whether there are smallest ones.

The reader used to ordinary quantum mechanics has only ever met type I. In type I there are minimal projections — rank-one projectors onto pure states — and everything familiar follows from them: states are density matrices, entropy is $-\mathrm{Tr}\rho\ln\rho$, and a system decomposes into subsystems by tensor factorisation. In type III none of that holds, and it is worth pausing on how thoroughly it fails. In a type III factor every non-zero projection is equivalent to the identity: the algebraic "size" of a property in half a region equals that of the whole region. There is no trace, so no entropy. There are no minimal projections, so no smallest property. And there are no normal pure states.

Connes refined type III into a one-parameter family III$_\lambda$ with $\lambda\in[0,1]$, distinguished by the spectrum of the modular operator we will meet in Chapter 5. The case relevant to physics is the extreme one, III$_1$.

3.2Local algebras are type III₁

In algebraic quantum field theory one assigns to each bounded open region $\mathcal{O}$ of spacetime the algebra $\mathfrak{A}(\mathcal{O})$ generated by observables measurable within it. The assignment satisfies isotony (bigger regions, bigger algebras) and locality (spacelike-separated algebras commute). The structural theorem is that these algebras are, without exception, the same object.

Point 4 deserves a moment, because it is the one that most offends intuition. The algebra does not know how big the region is. All the physics of scale lives in the state, not in the algebra — which is precisely why, from Chapter 6 onward, every phenomenal quantity in this book is a functional of the state rather than of the region.

3.3Reeh–Schlieder: locality of operations, non-locality of states

One further classical result sharpens the picture. The Reeh–Schlieder theorem says that for any bounded open $\mathcal{O}$ with non-empty causal complement, the vacuum vector $\Omega$ is cyclic for $\mathfrak{A}(\mathcal{O})$ — the set $\mathfrak{A}(\mathcal{O})\Omega$ is dense in the whole Hilbert space — and separating, meaning no non-zero element of the algebra annihilates it.

Cyclicity is startling: by acting only within a region the size of a grain of sand, one can approximate any global state of the universe to arbitrary accuracy, including states with a galaxy in them. This does not permit signalling, because the operations required are wildly non-unitary and succeed with vanishing probability, and because the expectation of any spacelike-separated observable is unchanged. What it does establish is that the vacuum is entangled across every region at every scale. There is no such thing as an unentangled patch of the world.

Separating, meanwhile, is the technical hypothesis that makes Chapter 5 possible: it is exactly the condition for Tomita–Takesaki theory to apply, and hence for the algebra to carry an intrinsic time.

3.4The combination problem is malformed

Now the payoff. The standard and, I think, decisive objection to panpsychism is the combination problem: if the fundamental constituents of matter have micro-experiences, by what mechanism do a hundred billion of them compose into the single unified field of your present moment? No mechanism has ever been given, and the arguments of Chalmers and Seibt suggest that none can be, because the unity of a macro-experience is not the sort of thing that is built out of parts.

The objection has a premise, and the premise is that the world decomposes canonically into subsystems, each capable of carrying its own state. In non-relativistic quantum mechanics this is true and unremarkable: $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$, and the parts are perfectly real. In quantum field theory it is false, by Theorem 3.1(3).

So the situation is this. Micropsychism requires elementary subjects. Elementary subjects require minimal projections, or at minimum a canonical decomposition into independently-stated parts. Quantum field theory supplies neither. The combination problem is therefore not solved here — it is dissolved, because its subject matter does not exist.

I take this to be the strongest available argument for cosmopsychism, and its interest is that it argues from physics rather than from intuition. It also lands, without having aimed there, almost exactly on Gómez Emilsson's stated ontology: an unbroken unity of all things, topologically segmented into individuals. Chapter 4 supplies the segmentation.

One cost should be acknowledged now. Cosmopsychism inherits a mirror-image difficulty — the decomposition problem, of explaining how the one gives rise to the many — and it is not obviously easier than combination. The difference is that decomposition is a question about where the cuts are, and quantum field theory turns out to have a rather precise answer. That answer is the subject of the next chapter.

4.1The split property

Type III$_1$ algebras have no density matrices and no entropy, so on their own they cannot support a determinate subject. But they are not alone. Under the same nuclearity assumptions that give Theorem 3.1, local algebras satisfy the split property (Doplicher–Longo; Buchholz–Wichmann): for strictly nested regions there exists an intermediate factor of type I.

This is the crucial technical fact of the whole book, and it is worth stating what it accomplishes. A type I factor does have pure states, density matrices, finite entropy, and a genuine tensor factorisation of the world into "this" and "not this." The split property manufactures, at finite resolution, exactly the subsystem structure that Theorem 3.1 denies at infinite resolution.

The two results are not in tension. Theorem 3.1 says there is no canonical decomposition — no God-given carving of the field into parts. The split property says that once you fix a scale, decompositions exist. The subject, on this account, is not found in the world already individuated; it is individuated by having a scale.

4.2Three consequences, none of them free

The fringe. The boundary is a collar of finite thickness, not a surface, so the periphery of experience is intrinsically indeterminate. There is no fact of the matter about whether the collar is inside or outside. William James's "fringe" of consciousness — the vague halo around the focal content — is on this account not a psychological curiosity but a structural necessity. A theory that made the boundary sharp would be predicting something false.

An area law. The entropy of a split factor obeys the standard result of Bombelli et al. and Srednicki:

So the information capacity of a moment scales with the area of its boundary, not the volume it encloses. Taking the cortical sheet at $A\approx 0.2\,\mathrm{m}^2$ and a coherence length $\varepsilon\approx 1\,\mathrm{mm}$ gives $A/\varepsilon^2\approx 2\times10^{5}$: of order $10^5$ simultaneously distinguishable phenomenal degrees of freedom per moment. That is the right order for the resolvable structure of a visual field. It is a weak consistency check rather than a confirmation, but a theory that had produced $10^{40}$ would be in trouble.

Frame-invariant separation. Distinct pockets carrying distinct topological charge lie in inequivalent superselection sectors, so no coherent superposition can bridge them. Superselection is not a matter of decoherence being fast; it is an exact statement about which states can be superposed at all. That is precisely the frame-invariance QRI's boundary criterion demands, and it is why your experience and mine do not merge when we shake hands.

4.3Topological segmentation as geometric realisation

The split property is abstract: it asserts that an intermediate factor exists, without saying where. To connect it to a brain we need a physical structure that picks out the nesting, and QRI's topological segmentation supplies exactly that.

Coarse-grain the field to an order parameter $\psi:\mathbb{R}^3\to\mathcal{T}$ taking values in some target manifold. Defects — points, lines and walls where $\psi$ cannot be continuously defined — are classified by the homotopy groups $\pi_n(\mathcal{T})$. The connected components of the complement of the defect set are the pockets. Within a pocket the order parameter is coherent; across a defect wall it is not.

The identification proposed here is that the pocket interior is $\mathcal{O}_1$, the pocket plus its defect wall is $\mathcal{O}_2$, and the wall thickness — the Ginzburg–Landau healing length — is the collar:

This turns $\varepsilon$ from a free parameter into a measurable physical length, and it is the seam where the abstract and physical halves of the book are glued. It is also, as Chapter 12 concedes, stipulated rather than derived, and the most likely place for the construction to fail.

4.4Where the field actually is

I take the physical carrier to be the coarse-grained electromagnetic field in neural tissue. This is not the quantum-computation-in-microtubules proposal and shares none of its liabilities; the case is made in §5.4. The appropriate quantisation is macroscopic QED in a dispersive absorbing medium (Huttner–Barnett; Philbin), in which integrating out the matter degrees of freedom leaves a field driven by a Langevin noise current whose correlator is fixed by fluctuation–dissipation:

The order parameter $\psi=|\psi|e^{i\theta}$ is then the analytic signal of the dominant collective mode — in cortex, plausibly the gamma-band rhythm. Its phase-only reduction is the Kuramoto model, which is the dynamical system QRI already uses to reproduce form constants and drifting visual textures. On this reading their oscillator simulations are not a metaphor for the field; they are its reduced dynamics.

Neurons, on this picture, are not where experience happens. They are the boundary conditions that shape the field which does.

5.1Tomita–Takesaki theory

Let $\mathcal{N}$ be a von Neumann algebra and $\Omega$ a vector that is cyclic and separating for it — conditions guaranteed by Reeh–Schlieder for any local algebra in a QFT. Define an anti-linear operator on the dense set $\mathcal{N}\Omega$ by

and let $S=J\Delta^{1/2}$ be the polar decomposition of its closure. Here $\Delta$ is a positive operator called the modular operator and $J$ an anti-unitary called the modular conjugation. The content of the Tomita–Takesaki theorem is that these objects, built from nothing but the algebra and one vector, have remarkable properties:

Read these carefully. The first says the conjugation $J$ maps the algebra exactly onto its commutant: it exchanges "this system" with "everything else." The second says the one-parameter family $\Delta^{is}$ preserves the algebra — it is a genuine dynamics, an automorphism group $\sigma^\omega_s$, generated by the modular Hamiltonian $K$.

Nobody put this dynamics in. It was extracted from the algebra and the state. That is the fact this chapter builds on.

The KMS condition

The state $\omega$ is not merely preserved by $\sigma^\omega_s$; it is thermal with respect to it. For all $a,b\in\mathcal{N}$ the function $F(s)=\omega(a\,\sigma^{\omega}_{s}(b))$ extends analytically to the strip $0\lt\mathrm{Im}\,s\lt 1$ and satisfies

which is the Kubo–Martin–Schwinger condition at inverse temperature $\beta=1$ in the modular parameter. In words: every state is a thermal equilibrium state, provided you use its own modular flow as the clock. Temperature is not a property a state has or lacks; it is a relation between a state and a choice of time.

5.2Connes: the flow is intrinsic

An obvious worry is that $\sigma^\omega_s$ depends on which state $\omega$ we picked, and so is not a property of the algebra. Connes' cocycle theorem removes it. For any two faithful normal states there is a family of unitaries $u_s\in\mathcal{N}$, the Connes cocycle $(D\omega:D\varphi)_s$, with

so the two flows differ by an inner automorphism. Passing to the quotient by inner automorphisms therefore gives a map that does not depend on the state at all:

A type III von Neumann algebra has an intrinsic time. Not a preferred time coordinate imposed from outside, but a canonical one-parameter group of outer automorphisms that is part of the algebra's structure. Connes and Rovelli built the thermal time hypothesis on this observation, proposing that the flow of time in physics simply is the modular flow of the state of the universe.

A worked case: the Rindler wedge

The abstraction has one classical case where everything is explicit, and it is worth keeping in mind as a sanity anchor. Take $\mathcal{N}=\mathfrak{A}(W)$ for the right Rindler wedge $W=\{x\gt|t|\}$ and $\Omega$ the Minkowski vacuum. The Bisognano–Wichmann theorem says the modular flow is exactly the Lorentz boost preserving the wedge, and the modular parameter is proper time divided by $2\pi$ times the acceleration. The KMS condition (5.2) then reads as the statement that an accelerated observer sees a thermal bath at the Unruh temperature

So in the one case we can compute, modular time is ordinary physical time and modular temperature is a real, in-principle-measurable temperature. That is the licence for treating the modular parameter as a temporal quantity in general.

5.3Phenomenal time

The attraction of Axiom 5.1 is that it explains a puzzle rather than merely labelling one. Experienced time is not a sequence of instants; it has thickness, direction, and a rate that varies with state. A modular flow has exactly these properties: it is a flow rather than a parametrisation, it is asymmetric in the sense that $\Delta$ is positive, and its rate relative to laboratory time is set by $\beta_{\mathrm{eff}}$, which is a property of the state.

It also produces a number immediately. Setting the specious present at $\tau\approx 100\,\mathrm{ms}$:

A tenth of a nanokelvin. The mode that carries experience must be extraordinarily cold in modular terms — that is, extraordinarily far from thermal equilibrium with the 310 K tissue surrounding it. At first sight this looks like an immediate refutation, and answering it occupies the rest of the chapter.

5.4Why the decoherence objection does not land

Tegmark's estimate of neural decoherence times — between $10^{-13}$ and $10^{-20}$ seconds — is correct, and for any theory requiring the brain to sustain superpositions of macroscopically distinct computational states it is decisive. The Penrose–Hameroff programme has to answer it. This one does not, and the reason is worth understanding precisely, because it is the difference between a quantum theory of consciousness and a field theory of consciousness.

Start with the occupation number of a 40 Hz collective mode at body temperature:

The mode is a hundred billion quanta deep — utterly classical, with fractional quantum corrections of a few parts per million. Nothing here is a delicate superposition. What the model requires is a coherent state of a driven, damped bosonic mode, and coherent states are the one thing decoherence cannot destroy.

Under the standard master equation for a damped oscillator,

a coherent state remains coherent and merely damps in amplitude, while coherence between two distinct coherent states decays at the rate $\tfrac{\gamma}{2}|\alpha-\beta|^2$. Both facts matter. The second reproduces Tegmark's conclusion exactly: superpositions of macroscopically distinct field configurations die essentially instantly. The first is what he did not need to consider: the individual coherent state is the survivor. Decoherence does not attack coherent states — it einselects them (Zurek, Habib and Paz). They are the pointer basis of a linearly coupled bosonic mode.

The cost of this move should be stated. By locating the theory in the classical-field limit, we forgo any explanation that would require quantum computation, nonlocal binding across the brain by entanglement, or observer-induced collapse. Whatever binding this theory delivers must come from field coherence and the split property, not from entanglement between distant neurons. Chapter 11, prediction 8 is the sharp form of that commitment.

6.1The spectral measure of a state

An experience is symmetric under its own dynamics to the extent that the flow returns it to itself. To make this precise, let $|\Psi\rangle$ be the subject's state on the split factor $\mathcal{N}$, and let $H$ generate the modular flow of Chapter 5. By the spectral theorem $H=\int\lambda\,dP(\lambda)$, and the state defines a probability measure on the spectrum,

Every measure on $\mathbb{R}$ decomposes uniquely into three pieces — a pure point part carried by countably many atoms, an absolutely continuous part with a density, and a singular continuous part carried by an uncountable set of measure zero. This decomposition is the technical spine of the rest of the book, so it is worth attaching physical meaning to each piece now.

The return amplitude is the Fourier transform of the measure,

and $P(s)$ is the probability that the state, carried along by its own intrinsic time, is found to have come back to where it started.

6.2Wiener's theorem and the coherence index

$\mathcal{A}=1$ exactly when $|\Psi\rangle$ is an eigenstate of $H$ — invariant under the flow, hence maximally symmetric. $\mathcal{A}\to 0$ when the measure is continuous, and $\mathcal{A}=1/N$ for a state spread evenly over $N$ atoms. So $\mathcal{A}$ simultaneously measures concentration, recurrence, and invariance, which is a strong hint that it is the right object.

6.3What the continuous spectrum means

The interpretation is not a matter of taste; there is a theorem.

So the dichotomy is exact. Point spectrum means the orbit is confined and keeps coming back. Continuous spectrum means irreversible dispersal, mixing, no return. And these are the two limiting phenomenological cases we are trying to distinguish.

The symmetry group of an experience is then not a metaphor but a set. Define the $\delta$-almost-periods

6.4What this looks like

Figure 6.1 — Return probability of the modular flow Cesàro mean converging to $\mathcal{A}$

6.5Why this is the Symmetry Theory of Valence

QRI's claim is that valence depends on the symmetry of the mathematical object describing a moment of consciousness. The object is fixed by Axiom 2.2; the relevant symmetry is invariance under the object's own intrinsic dynamics, which is the only dynamics available without importing external structure; and Proposition 6.4 says that invariance is measured by $\mathcal{A}$.

This is a genuine sharpening rather than a restatement, in three respects. It says which symmetry group is meant — the time-translation group of the modular flow, not an unspecified geometric symmetry. It supplies a scalar in $[0,1]$ instead of a qualitative comparison. And it makes the notion estimable from data: the atomic mass of a power spectrum is something one can compute from a magnetoencephalogram, which is Chapter 11, prediction 1.

It is not yet a theory of valence. Figure 6.1 shows why: two states can agree on $\mathcal{A}$ and differ structurally, and — as Chapter 8 argues at length — a state can be highly recurrent and still be terrible. Two ingredients are missing, and the next two chapters supply them.

7.1Why coherence is not enough

Return to the top two lanes of Figure 6.1. Both are purely atomic, with identical weights, so by Definition 6.1 they have identical coherence $\mathcal{A}=0.222$. Yet the first returns exactly, in sharp full-height revivals every $2\pi$, while the second never quite comes back.

The difference is arithmetic. In the first, the frequencies $\{1,2,3,4,6\}$ are rationally commensurate, so there is a common period $T=2\pi/\gcd$ and the orbit genuinely closes: the symmetry group $G_\Psi$ of (6.2) contains a full lattice $T\mathbb{Z}$. In the second, $\{1,\sqrt2,\sqrt3,\sqrt7,\pi\}$ are rationally independent, the orbit winds densely on a five-torus and never repeats, and $G_\Psi$ is only a Bohr set — relatively dense, but containing no exact periods at all.

Proposition 6.4 cannot see this distinction, because both cases are atomic. But the distinction between a closed orbit and a dense winding is exactly the distinction between a chord and a beating, and no theory of valence can afford to be blind to it.

7.2The consonance functional

Grade each pair of atoms by how nearly their ratio is a simple rational:

The kernel $\kappa$ has a spike at every rational, of height $(pq)^{-\sigma}$, smoothed to width $\delta$. Two parameters control it: $\sigma$ sets how fast consonance falls off with arithmetic complexity, and $\delta$ sets the tolerance — how far from a true ratio the ear (or the field) will still accept.

The unison term $p/q=1$ contributes $\kappa(1)=1$ along the diagonal, which reproduces $\mathcal{A}$ exactly. Hence $\mathcal{C}\ge\mathcal{A}$ always, with equality when every off-diagonal ratio is arithmetically hopeless. Consonance contains coherence as its unison term, which is the formal sense in which Chapter 7 refines rather than replaces Chapter 6.

7.3Thomae, Farey, and the dissonance curve

The unsmoothed kernel is a weighted Thomae function — the "popcorn function" that takes value $1/q$ at each rational $p/q$ in lowest terms and zero at every irrational. Thomae's function is the standard textbook example of a function continuous at every irrational and discontinuous at every rational, and it is a slightly startling object to meet in a theory of pleasure.

That the consonance ordering of Western harmony falls out of a Farey-weighted kernel is either a pleasing coincidence or the point. I take it to be the point. It is also what QRI's consonance–dissonance–noise signature is measuring: decompose a neural spectrum into the part sitting in consonant relationships, the part in dissonant ones, and the aperiodic remainder, and equations (7.1)–(7.2) say precisely how the first two should be weighted.

Note the three-way correspondence this sets up between the spectral vocabulary of Chapter 6 and ordinary musical hearing:

7.4What this predicts

Because $\sigma$ is a free parameter, (7.2) is a model rather than a derivation, and it earns its place only by being measurable. The prediction is sharp: valence response to two-tone and two-flicker stimuli should show minima at simple ratios with depths decaying as a power of the denominator product $pq$. Fitting that power gives $\sigma$, which the structure of the model expects to lie between 1 and 2. If measured peak heights do not decay as any power of $pq$ — if, say, the octave and the seventh are equally consonant, or the falloff is exponential in $q$ — then (7.2) is simply wrong.

This is Chapter 11, prediction 2, and it is the cheapest experiment in the book.

8.1Purity is a replica partition function

The quantity $\mathcal{A}=\mathrm{Tr}\rho^2$ has a second life in statistical field theory. Writing $\rho=e^{-\beta H}/Z_1$,

where $Z_n$ is the partition function of $n$ copies — replicas — of the system. In field theory (Callan–Wilczek; Calabrese–Cardy) $Z_n$ is computed on the $n$-sheeted branched cover of spacetime, glued along the entangling surface. That manifold carries a manifest $\mathbb{Z}_n$ symmetry: cyclic permutation of the sheets.

Whether the dominant saddle point respects that symmetry is a physical question with a name and a large literature. If it does, the system is replica-symmetric. If it does not, we have replica symmetry breaking, discovered by Parisi in the study of spin glasses and now understood to be the universal signature of frustrated, rugged energy landscapes.

8.2Parisi's order parameter

The diagnostic is the distribution of overlaps between two independent samples from the same equilibrium ensemble:

In a replica-symmetric phase, two samples always look alike: $P(q)=\delta(q-q_{\mathrm{EA}})$ and $\mathcal{D}=0$. The system is one thing. Under RSB, $P(q)$ spreads over an interval and acquires a hierarchical, ultrametric structure — states organise into basins within basins within basins, with distances satisfying $d(x,z)\le\max\{d(x,y),d(y,z)\}$. The system is irreducibly many things at once, and which one it is depends on where it happens to have fallen.

I want to be explicit that "frustration" is not a pun here. In the technical sense it is the impossibility of simultaneously satisfying all pairwise constraints; in the phenomenal sense it is what internal conflict feels like. The claim of this chapter is that these are the same quantity seen through the two functors of Axiom 2.1 — one relationally, one intrinsically. The nested-basin structure of RSB is then a prediction about the felt organisation of suffering, and the description it gives — stuckness, hierarchical entrapment, the sense of being multiply committed and unable to settle — is recognisable.

8.3The valence functional

The structural consequence is worth stating on its own, because it is the model's most easily testable qualitative claim: since $|\mathfrak{V}|\le\mathcal{C}$, you cannot suffer intensely without being coherent. Intensity of feeling, of either sign, is bounded by spectral coherence. Anaesthesia should therefore abolish agony and bliss by the same mechanism and at the same threshold — which is what it does.

8.4The phase diagram

8.5A cortical spin glass

To make the diagram quantitative, take $N$ coarse-grained cortical modes with Sherrington–Kirkpatrick couplings $J_{ij}\sim\mathcal{N}(0,J^2/N)$ in a uniform field $h$:

The replica-symmetric solution is stable only above the de Almeida–Thouless line, and near the critical temperature that line has a definite shape:

This is the formal content of Johnson's neural annealing: raise the energy enough to melt the frustrated structure, then cool along a path that lands in the symmetric basin. The model adds three things to the informal account — an order parameter ($\mathcal{D}$), a critical line (8.4), and an exponent (3/2) that can be measured and could be wrong.

9.1Coherence is multiplicative

Suppose the subject's state factorises over $M$ quasi-independent collective modes within the split factor $\mathcal{N}$. Since the inverse participation ratio of a product state is the product of the factors' ratios,

This is a small observation with a large consequence, because it converts a product of many bounded quantities into a sum, and sums of many independent quantities have a universal distribution.

The second clause is the part that is genuinely predictive. Log-normality by itself is cheap: many mechanisms produce it. But $\mathrm{Var}(\ln|\mathfrak{V}|)\propto M$ ties the spread of reported intensities to a structural quantity — the number of modes bound into the moment — and that is a relation one can go and check. It is Chapter 11, prediction 4.

It also explains something QRI has emphasised on ethical grounds: that the extremes of the distribution are far more extreme than a linear intuition suggests. If $\ln\mathcal{A}$ is normal with variance growing in $M$, then the ratio between a typical bad day and the worst states accessible to a highly integrated nervous system is exponentially large. Whether one accepts the suffering-focused ethics QRI draws from this, the mathematical point stands independently: log-normal tails are not a rhetorical exaggeration.

9.2Intensity as a Rényi entropy

Writing $N_{\mathrm{eff}}=1/\mathcal{A}$ for the effective participation number, the natural scale on which to report intensity is already an entropy:

So the logarithm is not a psychophysical convention imposed on the data after the fact. Valence is an exponentiated free-energy difference — between the two-replica system and two copies of the one-replica system — and the log scale is simply that free energy, measured in its natural units.

9.3A bridge to variational free energy

The second equality in (9.2) has a consequence worth drawing out, because it connects two research programmes that rarely speak. The Fristonian account holds that nervous systems minimise variational free energy. This model holds that they maximise symmetry. Equation (9.2) says these are, up to the replica structure, the same optimisation: maximising $\mathcal{A}$ is minimising $\Delta F_{\text{replica}}$.

The bridge is not free of charge, and the discrepancy is informative. Free-energy minimisation is defined on a single replica; $\mathcal{A}$ is a two-replica quantity. The gap between them is exactly the object of Chapter 8 — whether the two replicas agree, which is $\mathcal{D}$. So the relationship is:

A system can therefore be minimising prediction error perfectly well and still be in the glass, because $F_1$ is blind to whether the minimum it has found is one basin or a hierarchy of them. That, in this framework, is what chronic suffering is: a well-fitted model of the world that is nonetheless replica-symmetry-broken. It is also why merely getting better at prediction is not a route out.

10.1The Bures metric

The final entry in the invariant (2.2) is a geometry. The state space of the split factor $\mathcal{N}$ is not a bare set; it carries a canonical Riemannian metric, the Bures metric, which is the quantum Fisher information metric:

Its operational meaning is distinguishability: the Bures distance between two states is, up to normalisation, the number of measurements needed to tell them apart. This is exactly the right notion for a phenomenal geometry. Two experiences are close when they are hard to tell apart from the inside, and the metric that measures that is the one that measures statistical distinguishability.

Restricted to a commuting family, (10.1) reduces to the classical Fisher–Rao metric $ds^2=\sum_i (dp_i)^2/p_i$. And on one particular family, the Fisher–Rao metric has a closed form that has been known since Rao's 1945 paper and is, for our purposes, remarkable.

10.2The geometry is hyperbolic

The space of Gaussian probability distributions is the hyperbolic plane. Not analogous to it, not approximately it: the Fisher–Rao geometry of $(\mu,\sigma)$ is the upper half-plane model, and has been since long before anyone thought to connect it to phenomenology.

10.3Why an annealed state should feel hyperbolic

Now combine Theorem 10.1 with Chapter 8. Annealing drives a state toward maximum entropy subject to its second-moment constraints — which is to say, toward a Gaussian on an expanded set of modes. Theorem 10.1 then says something specific and, I think, non-obvious:

As an experience anneals and its mode set expands, the intrinsic geometry of its qualia manifold becomes hyperbolic, with volume growing exponentially in the rank $n$.

Gómez Emilsson's reports of hyperbolic phenomenal geometry under psychedelics — and the recurring, otherwise puzzling description of interior spaces vastly larger than they could contain — are what a Hadamard manifold of growing rank is like from inside. In hyperbolic geometry the volume of a ball grows exponentially with radius, so a space of modest diameter has room for enormously more structure than Euclidean intuition allows. That is a specific geometric fact, and it matches a specific and consistently reported phenomenological one.

The dose–geometry relation is then a prediction rather than a description: reported hyperbolicity should track the dimensional expansion of the coherent mode set, not the drug concentration directly. Two interventions producing the same $n$ should produce the same reported geometry regardless of pharmacology, and a drug that raises concentration without expanding $n$ should not produce the effect at all.

10.4Character as holonomy

Curvature tells us the shape of the space of experiences. It does not yet tell us what distinguishes one experience from another in kind — why red is not a sound. For that we need an invariant that is not a distance.

The Bures metric comes with a connection, and parallel transport of a state around a closed loop $\gamma$ in state space returns it rotated by the Uhlmann holonomy $U_\gamma\in\mathcal{U}(n)$. Holonomy is the natural carrier of "kind": it is invariant under reparametrisation, it composes correctly under concatenation of loops, and it is trivial exactly when the geometry is flat.

This completes the invariant (2.2): $g^{\mathrm{B}}$ supplies distinguishability and curvature, its holonomy supplies modality, $\mu_\rho$ supplies intensity, and $P(q)$ supplies valence. I present Proposal 10.2 as the most speculative claim in the book — it is the one place where I am proposing an identification on grounds of formal aptness alone, without either a theorem or a measurement behind it.

10.5Colour, the tractable case

Colour is where this can be tested, because colour is the one quale with a century of quantitative psychophysics behind it. The state manifold of a three-mode Gaussian family is $\mathrm{GL}(3,\mathbb{R})/O(3)$, six-dimensional, and its chromatic slice is the natural home of the Helmholtz–Stiles line element — the classical attempt to give colour space a Riemannian metric of exactly Fisher–Rao type.

The empirical situation is interesting. Bujack et al. (2022) showed that perceptual colour space fails geodesic additivity: large colour differences are consistently smaller than the sum of the small differences composing them, which no Riemannian metric with the usual assumptions can reproduce. This is at least the right shape of anomaly for a non-flat Fisher geometry, and diminishing returns along long geodesics is what one expects when curvature is negative.

I would not call this evidence. The result is standardly read as showing colour space is not Riemannian at all, which is a stronger and different claim than that it is Riemannian with negative curvature, and distinguishing the two requires an analysis nobody has done. It is listed here as the most promising place to look, not as a confirmation.

11.1How to read these

Axiom 2.3 forbids the theory from predicting any new physics, so every prediction below is structural: a claim that some mathematical property of an unmodified physical state corresponds to some phenomenal property. That is a real constraint. It means the theory cannot be rescued by discovering a new force, and it means each claim can be checked with existing instruments.

Each is stated with three parts — the claim, the measurement, and the falsifier. The falsifier is the part that matters; a prediction without one is decoration.

11.2Spectral predictions

1. Atomicity must beat band power

Claim. From Chapter 6, valence magnitude tracks the atomic mass of the modular spectral measure, not power in any frequency band.

Measurement. From MEG or high-density EEG, remove the aperiodic $1/f$ component (specparam or equivalent), then estimate $\hat{\mathcal{A}}=\sum_k(P_k/\sum P)^2$ on the residual periodic spectrum. Regress momentary valence report on $\hat{\mathcal{A}}$, on band powers, and on global amplitude.

Falsifier. If band power or global amplitude predicts valence better than $\hat{\mathcal{A}}$, symmetry-as-atomicity is dead. This is the cheapest test in the book and could be run on existing datasets.

2. The consonance kernel has a power-law exponent

Claim. From Chapter 7, valence response to paired periodic stimuli follows the kernel $\kappa$, with peak heights decaying as $(pq)^{-\sigma}$ and $\sigma$ expected between 1 and 2.

Measurement. Two-tone and two-flicker pleasantness ratings across a dense sweep of frequency ratios. Fit peak depths against denominator products.

Falsifier. A non-power-law falloff — exponential in $q$, or flat, or with the octave no more consonant than the tritone — refutes equation (7.2).

11.3Structural predictions

3. Suffering should be non-self-averaging

Claim. From Chapter 8, chronic suffering is replica symmetry breaking, so repeated sampling of the same nominal state should yield a broad overlap distribution with ultrametric structure.

Measurement. Compute pairwise correlation distances between windows of resting-state MEG within a session. Test the ultrametric inequality $d(x,z)\le\max\{d(x,y),d(y,z)\}$ against a null model controlling for ordinary hierarchical clustering.

Falsifier. No ultrametricity excess in depression or chronic pain relative to controls falsifies Axiom 8.1's identification of suffering with RSB.

4. Log-valence variance scales with integration

Claim. From Proposition 9.1, $\mathrm{Var}(\ln|\mathfrak{V}|)\propto M$, the number of bound modes.

Measurement. Within-subject variance of log-transformed intensity reports, against a proxy for integration such as effective dimensionality of the coherent spectrum.

Falsifier. Constant log-variance across widely differing integration levels. Note this is a test of the scaling; log-normality alone is cheap and would not confirm anything.

11.4Transitions

5. Anaesthesia is a phase transition, not a dimmer

Claim. From Chapter 6, loss of consciousness is a pure-point-to-absolutely-continuous transition in the spectral measure — a localisation–delocalisation transition, with critical behaviour.

Measurement. Track $\hat{\mathcal{A}}$ through a slow, controlled induction. Look for critical scaling and a divergent correlation time near loss of responsiveness.

Falsifier. A smooth crossover with no critical exponent, and $\hat{\mathcal{A}}$ declining proportionally to drug concentration, falsifies the spectral account.

6. Annealing obeys a three-halves law

Claim. From Proposition 8.2, the drive required to shift a rigid state scales as $(1-T/T_c)^{3/2}$.

Measurement. Dose–response across interventions that plausibly implement annealing — psychedelics, intense exercise, meditative absorption — with rigidity calibrated by a pre-intervention measure of $\mathcal{D}$.

Falsifier. A different exponent falsifies the Sherrington–Kirkpatrick reduction, though not necessarily the RSB picture; the mean-field approximation is the first thing to suspect.

11.5The two hard ones

7. Capacity scales with area, not volume

Claim. From the area law (4.2), the number of simultaneously distinguishable phenomenal distinctions goes as $A/\xi^2$, not $V/\xi^3$.

Measurement. Comparative: capacity should be insensitive to cortical thickness and sensitive to cortical surface area, which vary independently across species, across development, and in specific pathologies (lissencephaly, polymicrogyria).

Falsifier. Capacity tracking volume rather than area. This is the wildest prediction in the book and I flag it as such — but it is not empty, and the dissociation between thickness and surface area is real and measurable.

8. Binding requires a shared field

Claim. From Chapters 4 and 5, two neural populations with no shared coherent field region cannot be phenomenally bound, however tightly they are functionally coupled.

Measurement. A brain-to-brain interface carrying information over a purely digital channel — no shared field region, arbitrary latency and arbitrary bandwidth.

Falsifier. If such an interface ever produces genuinely unified experience across two subjects, this entire model is refuted and the computationalists are right.

12.1It does not solve the hard problem

Axiom 2.1 posits that the intrinsic aspect of the physical is phenomenal. It does not derive it, and nothing in the following ten chapters derives it either. The model converts "why is there something it is like to be this?" into "why is the intrinsic aspect phenomenal rather than nothing?", which is progress in tractability and no progress at all in depth.

What it does do is make every structural question fully mathematical: binding, boundary, temporal flow, valence, intensity and character each become a specific computation on a specific object. That is the residue QRI calls tractable, and I think it is genuinely tractable. But a reader who came hoping for an explanation of why physical structure is accompanied by experience should leave disappointed, and should be told so plainly rather than sold a relocation as a solution.

12.2Four structural weaknesses

Beyond the hard problem, there are four specific places where the construction is weaker than its presentation might suggest. They are listed in order of how likely I think each is to be fatal.

The seam at $\varepsilon=\xi$

Equation (4.3) identifies the split collar with the Ginzburg–Landau healing length. This is stipulated, not derived. It is the joint where the abstract half of the theory (operator algebras, modular flow) is glued to the physical half (a field in tissue), and it carries the entire weight of that connection. If the identification is wrong, Chapter 4's area law and every number derived from it goes with it. Exercise 4.6 asks for the derivation; I do not have it.

Disorder averaging in a single brain

The frustration order parameter $\mathcal{D}=\mathrm{Var}_P(q)$ is defined through an average over disorder realisations $\mathbb{E}_J$. A single brain is a single realisation. The standard surrogate is to treat time windows as replicas, but this is exactly the move that RSB warns against, since non-self-averaging means the time average and the disorder average need not agree. The circularity is uncomfortable: the phenomenon we are trying to detect is the one that invalidates the method used to detect it.

Extrapolating the algebraic results

Theorem 3.1 and the split property are theorems about Minkowski-space quantum field theory under standard axioms plus nuclearity. Applying them to warm, wet, dissipative neural tissue at 310 K is an extrapolation. I believe the algebraic structure survives coarse-graining — nothing in the type classification depends on temperature — but I have not shown it, and the effective field theory of a dissipative medium is not obviously covered by the hypotheses of the theorems cited.

The choice of carrier

The electromagnetic field is the least constrained commitment in the book. The formalism of Chapters 3, 5, 6, 8, 9 and 10 is entirely agnostic about which collective mode plays the role: everything there is a statement about states on algebras and spectral measures, and would survive intact if the carrier turned out to be something else. Chapter 4's physical realisation is where I have committed hardest on the least evidence.

12.3Things the theory cannot currently compute

An honest inventory of the gap between the formalism and anything usable:

The last row is worth dwelling on, because it is where a theory of consciousness is most wanted and this one is least ready. Axiom 4.1 gives a sharp criterion for subjecthood, and Chapter 6 gives a sharp criterion for valence. Neither can currently be evaluated for an insect, a foetus, a cortical organoid or a language model. The theory says what the answer is — a fact about split inclusions and spectral measures — without providing the means to find it out. That is better than having no criterion, and much worse than having a usable one.

12.4What would have to be true

Collecting the load-bearing posits in one place, so a reader can see the whole bet at once:

1. The intrinsic aspect of the physical is phenomenal, everywhere and without exception. (Axiom 2.1)
2. The six invariants of (2.2) are jointly complete. (Axiom 2.2)
3. Consciousness adds no term to the action. (Axiom 2.3)
4. Subjects are split inclusions, with the collar set by a physical healing length. (Axiom 4.1, §4.3)
5. Phenomenal time is modular time. (Axiom 5.1)
6. Valence is consonance times replica symmetry. (Axiom 8.1)
7. Qualitative character is Uhlmann holonomy. (Proposal 10.2)

Seven posits, and every other claim in the book is either an established theorem or a consequence of those theorems given these. That ratio is the argument for taking the construction seriously: not that the posits are obviously right, but that they are few, explicit, and individually attackable.

12.5The open problems

The exercises marked ★★★ are not exercises. They are the problems I could not solve, gathered here:

• 1.5 — Establish or refute valence realism independently of any particular theory.
• 2.5 — Find a phenomenal distinction that provably is not a function of (2.2), or prove none exists.
• 3.6 — Solve the decomposition problem for cosmopsychism without appealing to the split property.
• 4.6 — Derive $\varepsilon=\xi$ rather than stipulating it. The most important one.
• 5.6 — Derive subjective time dilation under arousal from $\beta_{\mathrm{eff}}$, with the sign.
• 6.6 — An unbiased estimator of $\mathcal{A}$ under $1/f$ backgrounds. The most urgent one.
• 7.6 — Derive the consonance exponent $\sigma$ from the density of states.
• 8.6 — Justify or refute time-windows-as-replicas given non-self-averaging.
• 9.5 — Conditions under which the replica continuation in (9.2) is rigorous.
• 10.6 — Derive the three-dimensionality of colour from the holonomy group.
• 11.4 — An operational criterion for cross-subject phenomenal unity that does not beg the question.

Two of these are on the critical path. Without 6.6 the theory cannot be tested at all, since every empirical prediction in Chapter 11 routes through an estimate of $\mathcal{A}$. Without 4.6 the connection between the mathematics and the brain remains a stipulation. Everything else is refinement.

12.6A closing remark

The shape of the result is that QRI's programme requires roughly one genuinely new posit per layer — seven in total — and that everything else is either an existing theorem in algebraic quantum field theory, spectral theory, spin-glass physics or information geometry, or a consequence of those theorems given the posits.

That is a better situation than a theory of consciousness usually finds itself in, and it is worse than it looks, because the posits are load-bearing and none of them is obvious. What can be said is that they are stated, numbered, and individually falsifiable, which is the most a speculative theory can offer and the least it should.

Read the status column as an argument about where the risk sits. Six rows are Established or theorem-backed: those are not places this book can be wrong without a literature being wrong with it. Five are Derived: they follow from the established results given the axioms, so they fail only if an axiom fails. Six are Posited, and those are the whole bet.