NOTE-1059: Lee-Youngzie et al. — Towards a mathematical structure of global phenomenal consciousness

Contribution

The paper joins the two kinds of “qualia structure” in the structural approach to consciousness. Intra-phenomenal structures model the parts of one experience. Inter-phenomenal structures place an experience among others by similarity judgements. The proposal is a presheaf from the first to the second. Each part of an experience is assigned the dissimilarity matrix measured for it, and each inclusion of parts a restriction map. Then “Conditions for phenomenal binding are formalized as the compatibility condition for the gluing of local sections” (§5.2). Nothing is measured. The article type is “Hypothesis”, and its application is a two-region visual field, worked symbolically.

Key insight

Treat the measured similarity structure of each part of an experience as local data over that part, and ask whether the local data glue. If the narrow qualia of the left and right hemifields agree at the vertical meridian and glue into the measured broad quale of the whole field, then the whole is “wholly determined by, and decomposable into” its parts (§4.1.5). If they do not, something else is going on, and the paper offers two readings of that something: disunity, as in binocular rivalry, or a broad quale that depends on context, as in the Ebbinghaus illusion.

Assumptions

• Phenomenal unity is a structural relation between experiential parts, not an added property (§1, after Wiese). The paper takes as given that unity is always present: “their conjoint phenomenology always possesses the phenomenal quality of unity” (§1).
• Subsumption is a preorder. “experience A subsumes experience B” when what it is like to have B is an aspect of what it is like to have A, and this relation is assumed transitive (§4.1.1).
• Selective attention is restriction. Restricting a broad quale to a part “corresponds to selectively attending to a narrower experiential aspect” (§4.1.3). The narrow quale of a hemifield is measured by similarity judgements made while attending to it.
• Attention never shrinks dissimilarity. Definition 1’s morphisms require d_X(x_i, x_j) ≤ d_Y(m(x_i), m(x_j)), so narrower qualia are at least as dissimilar as the broad ones they restrict. The authors call this a speculation from studies of attention and appearance: “we speculate a general relationship between overall and specific dissimilarity of experiences” (§4.1.3).
• Dissimilarity is symmetric, “for simplicity”, although the paper’s own case for categories over metric spaces is that measured similarity violates symmetry (§3.2, Definition 1).
• Similarity judgements measure qualia. Inter-phenomenal structures are the empirical measure of “what it feels like”, and the paper notes that the method of eliciting them changes the structure obtained (§5.3.2).
• Structural equivalence licenses transfer. “if phenomenal states and their underlying physical states are assumed to be structurally equivalent … conclusions drawn in the phenomenal domain can be expected to hold in the physical domain” (Mathematizing section). Structural equivalence is left undefined: “we use the term in a broad sense without committing to a specific formal notion of equivalence” (Definition A12).

Key results

The results are definitions and readings. They are given here with what a check by script found (scratch script, 2026-10-22: enumeration of up- and down-sets, topology axioms, and the sheaf condition for the paper’s restriction maps).

• The intra-phenomenal topology (§4.1.1, Definition A7). A finite preorder of experiential aspects induces an Alexandroff topology, whose open sets are said to be “upwards-closed sets”. The worked example gives the chain E_Location ≤ E_Color ≤ E_Shape ≤ E_ColorShape and lists the open sets {∅, {Location}, {Color, Location}, {Shape, Location}, {Shape, Color, Location}}. Checked: that list is not a topology on the four aspects, since it lacks the whole set. It is not the up-sets of the chain, which are ∅, {CS}, {S, CS}, {C, S, CS} and the whole set. It is exactly the down-sets of a different order, a diamond with Location below Color and Shape and both below ColorShape, less the whole set. The diamond is what the prose describes. Definition A8 also calls an Alexandroff space one “whose underlying set is a finite set”, which is a special case of the property its next sentence states correctly. The repairs are routine (take the diamond and include the whole set), and nothing later depends on the error.
• The visual field (§4.1.1, Figure 5). τ is written as {∅, {VM}, {LeftHF}, {RightHF}, {TotalVF}}. Checked: read as a topology on the four aspect labels it fails, since {LeftHF} ∪ {RightHF} is not listed. Read as regions of a field of points, with LeftHF ∩ RightHF = VM and LeftHF ∪ RightHF = TotalVF, it is a topology, and this is the reading the text uses.
• The presheaf (§4.1.2–4.1.3, Definition 1). Q sends each region U to a dissimilarity matrix Q(U), an object of a category DIS whose morphisms are functions between items that never decrease dissimilarity. The “local sections” are rows of the matrix, one per trial: e(i,j) is the relational embedding of the trial with left grating at angle θ_i and right at θ_j. Checked: DIS is a category (non-decreasing maps compose and identities qualify).
• Binding as gluing (§4.1.4, Figure 7). For a unified visual field the restrictions of the left and right sections to the meridian are “expected to result in identical measures of narrower quale”, and matching sections glue into a section over the whole field. If gluing and restriction recover the measured sections everywhere, Q is a sheaf. The paper: “The condition for compatibility in the overlap can be understood as a formalization of the condition for phenomenal binding of narrow qualia into a broader quale.” Checked: a morphism of DIS is a function on items with an inequality on their dissimilarities. It does not say what row a restricted row becomes. If restriction is the trial-to-trial function the text describes, the left and right restrictions of e(0,30) are the same item, every pair matches, and the sheaf condition holds trivially. The text says the restriction “provides the predicted measure of the narrower quale in the overlap”, but Definition 1 supplies no predicted measure. The content of the test is therefore the empirical comparison §5.1 describes, of measured matrices by a consistency radius (Robinson’s), not a consequence of the stated presheaf.
• Sheaf failure, first reading: disunity (§4.1.5, Figure 8). The Penrose tribar is pairwise compatible at each junction with no global configuration, “a form of contextuality”. The paper places it “at the level of cognitive interpretation”. Binocular rivalry is then “a phenomenal ‘impossible figure’”: “no global section … exists that coherently integrates all local sections”, and “violation of the sheaf condition may be diagnostic of phenomenal disunity”. Split-brain disunity is a predicted case, left “for future research” (§5.1). Checked against its own definitions: Definition A15’s gluing axiom applies only to sections that “agree throughout their intersection”. In rivalry the monocular sections disagree on the binocular overlap, so the axiom has nothing to glue and is not violated. Rivalry as described is a non-matching family, not a failure of the sheaf condition. The tribar, a cycle of three junctions with pairwise agreement and no whole, is a genuine gluing obstruction of the kind CLAIM-153 describes.
• Sheaf failure, second reading: context dependence (§4.1.6, Figure 9). Narrow qualia measured with each grating shown alone may fail to glue into the measured broad quale. That failure means “interactions between parts that are irreducible to a predictable combination of narrower qualia considered in isolation”, as in the Ebbinghaus illusion. Here a non-commuting diagram marks an ordinary, unified experience whose whole is not fixed by its separately measured parts.
• A spectrum, not a verdict (§5.2). “At one end of the spectrum, measures of broad qualia may be completely decomposable”, which is the sheaf case. “In many other cases, however, measures of broad quale may not be predictable from those of its narrow qualia.” The framework is offered “without presupposing that phenomenal experience is either fully decomposable or entirely unanalyzable”.
• Compression (§4.1.5). At the editor’s suggestion, a sheaf that glues admits a “compressive” description of the local measures, and a presheaf that does not glue admits “no unique compressive description”. Stated, not derived.
• Experimental programme (§5.3). The paper lists what it lacks: richer partitions than two regions, other modalities, time, a method for measuring broad qualia, and a test of whether attention is the right probe of restriction. “few empirical studies construct inter-relational structures of broad qualia” (§5.3.2).

Claims

id	claim	strength	support
C1	Intra- and inter-phenomenal qualia structures can be joined by a presheaf from a topology of experiential parts to dissimilarity matrices	moderate as a construction; the worked topology needs repair	§4.1, Definition 1, Appendix A
C2	Compatibility of narrow qualia on an overlap formalizes phenomenal binding	weak: a proposed reading, with no measurement and, as stated, no formal content beyond item matching	§4.1.4, §5.2
C3	The presheaf’s being a sheaf means the broad quale is wholly determined by the narrow qualia	moderate as a reading of the sheaf condition for measured data	§4.1.5, Figure 7
C4	Binocular rivalry, and possibly split-brain cases, are violations of the sheaf condition and diagnostic of disunity	weak: by the paper’s own Definition A15, disagreement on an overlap is not a violation	§4.1.5, §5.1, Figure 8
C5	A sheaf failure with isolated narrow measures reveals context dependence among parts of a unified experience	moderate as a design; no data	§4.1.6, Figure 9
C6	Attention to a part never decreases perceived dissimilarity	speculative: inferred from attention and appearance studies, not measured as stated	§4.1.3

Concepts

• Narrow and broad qualia: the phenomenal qualities of an aspect of experience and of an experience that includes it as a part; “qualia in the broadest sense correspond to the phenomenal qualities of the total field of consciousness” (Figure 1).
• Intra-phenomenal structure: the mereological organization of parts within one experience.
• Inter-phenomenal structure: the relations of an experience to other experiences, measured as perceived similarity.
• Subsumption: B → A when what it is like to have B is an aspect of what it is like to have A.
• Local section: here, one row of a measured dissimilarity matrix, the relational embedding of one trial.
• Global section: reserved for “the coherent global data that results from a sheaf” (Definition A14). It is a measurement-level object, not an experience or a subject.
• Phenomenal versus functional binding: phenomenal binding integrates phenomenal qualities into one experience; functional binding integrates perceptual information into percepts, and “can occur without conscious experience” (§1).

Connections

The paper builds on the structural approach of Tsuchiya, Kleiner and others, on category-theoretic work by Phillips and Saigo (thanked for guidance), on IIT’s phenomenal structure for the intra-phenomenal side, on colour qualia-structure studies with Gromov–Wasserstein alignment for the inter-phenomenal side, and on Robinson’s applied sheaf theory and consistency radius (Robinson is a co-author). Its tribar example cites Penrose’s cohomology of impossible figures (ref. 132), Abramsky’s “Contextuality: At the Borders” (ref. 133) and Abramsky & Brandenburger (ref. 134). Of these, the record holds Abramsky & Brandenburger, the sheaf-theoretic account of contextuality (LIT-016, NOTE-016), and the IIT 3.0 statement (LIT-445). It registers unread Binding and the Phenomenal Unity of Consciousness (LIT-1208), neurophenomenal structuralism (LIT-1344), and two arguments against structuralism about consciousness (LIT-1348, LIT-1346).

Bearing on the record

• QUESTION-255 (is unity a global section). The record filed this question after the exchange cited the paper as a sheaf-theoretic account of global unity. What the paper establishes, exactly:

  • It does not identify the unity of an experience with the existence of a global section. It assumes unity is always present (§1), and it calls a broad quale that is not fixed by its parts, which is a sheaf failure, an ordinary unified experience with context dependence (§4.1.6, §5.2).
  • It does propose that phenomenal binding of narrow into broad qualia is compatibility on overlaps (§4.1.4, §5.2), and it reads one kind of sheaf failure, rivalry, as disunity (§4.1.5).
  • Its two readings of a sheaf failure, disunity and context dependence, are told apart by the example, not by the formalism. By its own Definition A15, the rivalry case is not a sheaf failure at all.
  • Its global section is a section of measured similarity data (Definition A14). Whether a gluing of such data corresponds to anything in the experience is the structure-preserving premise the paper assumes and does not test.

  So it gives the question no answer either way. It does supply a worked case for A7’s alternative. The tribar is a presheaf of interpretations with pairwise agreement and no global section, and the paper places that “at the level of cognitive interpretation”. Yet the figure is seen as one figure. Read with the record’s cover of interpretive contexts, that is an interpretive gluing failure inside one experience. The paper does not draw this conclusion; the inference is the record’s.

• CLAIM-153, CLAIM-197, CLAIM-198. The paper’s two-region examples (hemifields, monocular fields) are two-set covers. Under CLAIM-153, two contexts that agree on their overlap always glue, so on a two-set cover only two things can go wrong. The sections can disagree on the overlap, which is the rivalry case and not a gluing failure. Or the measured whole can differ from the glued one, which is the Figure 9 case and a failure of the parts to determine the whole, a different property from CLAIM-153’s monodromy. Only the tribar, a three-junction cycle, has CLAIM-153’s shape. This supports CLAIM-153’s point that a genuine obstruction needs a cycle. CLAIM-143’s requirement, that the correspondence between contexts be fixed independently, is met here by design: the overlap is the meridian, fixed by fixation, before any judgement is taken.

• CLAIM-412 and TERM-072. The paper begins from phenomenal structure, sense 3 of TERM-072, and measures it by report. It does not derive phenomenality from organization, and it bears on the constitutive claim only by keeping phenomenal binding apart from functional binding, which “can occur without conscious experience” (§1). That is CLAIM-411’s point made about binding: functional integration does not suffice for phenomenal unity.

• QUESTION-048. Every measure in the framework is a similarity judgement, a report. The paper says so and lists the ways task demands change the structure obtained (§5.3.2). It adds a kind of report evidence, but no route around report.

• THEORY-063 (Metzinger’s unity). Metzinger’s unity is a single functional cluster, a dynamic criterion. This paper’s is a condition on measured qualia data. They are different proposals for the same unity (ADR-024’s phenomenal unity), and neither is tested.

• THEORY warranted. No THEORY in the record states the proposal. One is filed, THEORY-875, at the strength above: a framework that proposes binding as gluing of measured qualia data, with its two readings of failure and their unresolved difference.

• No instruction for machine-learning practice.

Limitations

• No data. Every result is a definition, a reading or a proposed design.
• The worked Alexandroff example is wrong as printed: it calls the opens up-sets, lists down-sets of a different order, and omits the whole set (checked by script). The visual-field τ is a topology only on the region reading.
• As stated, the gluing condition is either trivial (items matched by trial label) or undefined (rows of measured values), because Definition 1 constrains dissimilarities and does not map data. The operational test is a consistency radius on measured matrices.
• Rivalry is offered as a sheaf violation, against the paper’s own definition of gluing.
• The restriction-as-attention assumption and the non-decreasing dissimilarity rule are speculative, and symmetry is assumed against the paper’s own motivation for categories.
• Two readings of a sheaf failure, disunity and context dependence, are given without a criterion that separates them.
• The formalism stops at two regions, one modality and one moment (§5.3.1).

Open questions

• What, in the data, separates a sheaf failure that is disunity from one that is context dependence? Without that, the formalism cannot diagnose disunity.
• What is the restriction of a measured row? A definition that maps data, not only items, would give the gluing condition content.
• Is there a phenomenal case of the tribar’s shape, three or more parts pairwise compatible with no whole, and is it experienced as one? That would bear directly on QUESTION-255.
• Does attention to a part in fact never decrease dissimilarity? It is a measurable claim, and the framework’s restriction maps depend on it.