Structure: the Geometry of Quality Space

Why structure is the strongest channel

A calibration fixes constants; any theory can be fitted to a single quality. A structure — the pattern of relations among many qualities, or among the components of $\Gamma$ — can be fitted only if it has the shape the theory committed to before the data were seen. UHM commits in advance to complex projective geometry for quality, seven axes and seven Fano lines for $\Gamma$, a window and two bands for a living state. Each commitment forbids something measurable. This page lists the commitments with their statuses, the methods that test them, and the results that would count for and against. It reports no data.

What a structural test can and cannot show

• It tests geometry, not consciousness. The quality geometry of a state and the verdict $\mathrm{Cons}(S)$ are independent: the Fubini–Study distances between the eigenrays of $\Gamma$ do not depend on its spectrum, and purity does not depend on the eigenrays; the same geometry is carried by a state with $P = 0.152$ (outside the window) and one with $P = 0.312$ (inside) (measurement protocol [T]). A similarity structure shared by a person and a language model — GPT-4 matched colour-neurotypical humans on 93 colours at a matching rate of 91.4 % (Kawakita et al. 2024) — says nothing, in UHM, about whether the model is conscious.
• It supports the identity without proving it. The identity of the internal side of $\Gamma$ with experience stays [I] whatever the data say (T-214 [T]). What structure can do is give that identity many chances to fail.
• It needs calibration for its metric form. Without the map $f$ from perceived dissimilarity to $d_{\mathrm{FS}}$ (calibration, K3) every metric test below is ordinal.

The invariants

Each row gives what UHM asserts, the status of the assertion, and where it is proved or stated. "Math" is the status of the statement about $\Gamma$; "reading" is the status of its reading as a statement about experience.

#	Invariant	Math	Reading	Source
S1	Quality space is $\mathbb{CP}^{n-1}$ with the Fubini–Study distance; the enriched Yoneda embedding is an isometry, enriched isomorphism is identity	[T]	[I]	enriched Yoneda
S2	Intensity and quality are separate parameters: 6 spectral numbers against 42 eigenvector numbers	[T] (T-300)	[I]	qualia structure
S3	Content has 7 axes and 21 channels, and no more (closure T.1); $48 = 27 \oplus 14 \oplus 7$ under $G_2$	[T] (T.1, T-146, T-301)	names of channels [I]	taxonomy, decoder
S4	Irreducible quality is carried by 7 Fano holonomies $H_p$, one per line	[T] (T-301)	[I]	decoder
S5	Fano lines organise the coherences: block transparency	[T] (F-Gap-2)	—	F-Gap-2
S6	Intra-line Gap below inter-line Gap; testable as triple holonomy on 7 lines against 28 non-lines	[H] (F-Gap-1)	—	F-Gap-1
S7	A line is a parity check on its three signs	[C] (T-306)	—	what a line checks
S8	The conscious window $2/7 < P \leq 3/7$ is non-empty, and its four conditions are independent	[T] (T-124, T-124b)	criterion of consciousness: [D] core of T-153	conscious window
S9	Anything alive has $s_1 \in [1/7, 3/14]$ and $s_2 \in [1/7, 2/7]$ (F-Band)	[T] (T-321, T-323)	—	F-Band
S10	Under the canonical decay the moduli fall as $e^{-5\gamma t/21}$ and each holonomy's value stays fixed: qualities fade, they do not drift	[T] for the dynamics	[H] as a prediction about experience	how qualia die
S11	Irreducible phase and integration are opposed; at $\Phi = 1$ the typical line holonomy is $0.6387$ rad for evenly twisted content	[C] (T-307)	[I]	how much quality fits
S12	Emotional space is 30-dimensional; valence is $\mathrm{sign}(dP/d\tau)$	[T] (T-147); valence [C] (C.1)	names of emotions [I]	emotional taxonomy
S13	Number of independent slow components in fMRI lies in $[6, 12]$	[H] (F-ISF)	—	F-ISF

Dimension of quality space. The theory fixes the type of geometry, not $n = \dim\mathcal{H}_E$. In the minimal 7D formalism the $E$-sector is one-dimensional; in the 42D Page–Wootters realisation $\rho_E$ is a $7 \times 7$ block on the clock register; a genuinely many-dimensional $\mathcal{H}_E$ needs a composite substrate (the $\rho_E$ box). The dimension $n$ is therefore something to measure — and the realisability condition below gives a way to bound it from similarity data.

What the geometry forbids

A commitment is worth as much as what it excludes. From the invariants above, UHM forbids:

1. Metric violations. After calibration, perceived dissimilarities must satisfy the triangle inequality within tolerance ($\lvert d(a,b) + d(b,c) - d(a,c)\rvert < 0.15$ for violations, per prediction 4) and stay within the diameter $\pi/2$ of $\mathbb{CP}^{n-1}$.
2. Unrealisable matrices. A finite dissimilarity matrix $(d_{ij})$ is realised by rays of $\mathbb{CP}^{n-1}$ if and only if, for some phases $\theta_{ij} = -\theta_{ji}$, the Hermitian matrix $G_{ij} = \cos(d_{ij})\,e^{i\theta_{ij}}$, $G_{ii} = 1$, is positive semidefinite of rank at most $n$ (item 7 of the enriched Yoneda theorem [T]). In particular at most $n$ qualities can be pairwise at the maximal distance $\pi/2$.
3. Resolution below the probe bound. A probe set with covering radius $\delta$ has at least $\sin^{-2(n-1)}\delta$ elements; 93 probe colours cannot resolve below $\arcsin(1/\sqrt{93}) \approx 0.104$ rad even on $\mathbb{CP}^1$. A claimed resolution finer than the bound allows is an artefact.
4. A conscious state outside the bands. A system that independent evidence in $D_{\mathrm{nat}}$ calls conscious, whose reconstructed $\hat\Gamma$ lies outside the window or outside F-Band, falsifies the reconstruction; a state meeting all four criteria with $s_1$ or $s_2$ outside its band falsifies the algebra (F-Band; zero counterexamples over $20\,000$ states per band so far).
5. Drift while fading [H]. As a state decoheres under the canonical dynamics, a quality loses intensity without passing through neighbouring qualities; "there is no half-changed quale".

Methods

Similarity geometry: MDS and representational similarity analysis

Collect full pairwise dissimilarity matrices within one modality. Multidimensional scaling gives a first picture of dimension and shape; representational similarity analysis compares the behavioural matrix with the matrix of $d_{\mathrm{FS}}$ between eigenrays of $\hat\Gamma$ (calibration, K3). The falsifiability page fixes the thresholds: Spearman correlation above 0.6, monotonicity violations below 10 %, triangle tolerance 0.15, MDS stress below 0.1 for $\mathbb{R}^k$.

The Tsuchiya paradigm: enriched categories and unsupervised alignment

Tsuchiya, Phillips and Saigo (2022) model a quality space as a category enriched in dissimilarities, and compare two subjects' spaces by unsupervised Gromov–Wasserstein alignment — no labels, only structure (Kawakita et al., iScience 2025, across people; Kawakita, Zeleznikow-Johnston, Tsuchiya & Oizumi, Sci. Rep. 14: 15917, 2024, between people and language models). UHM adopts the method and adds a commitment: the space should be a Fubini–Study geometry, so that enriched isomorphism is identity and a Gromov–Wasserstein distance of zero is an isometric bijection (what UHM adds). The precedent and its standing are on the theories page.

Realisability and the rank of quality space — proposed here [Pr]

For a calibrated matrix, search over phases for the smallest rank $n^*$ of a positive semidefinite $G$ realising it within measurement error. Two predictions follow [H]: (a) $n^*$ exists — the matrix is realisable at all; (b) within one modality $n^*$ stabilises as stimuli are added, rather than growing with the stimulus count. A competing geometry (real Euclidean, hyperbolic) is compared by cross-validated fit at equal parameter count; a systematically better fit of a non-projective geometry counts against S1.

Topological analysis

Persistent homology of the similarity structure reports the topology of the sampled set of qualities — the hue circle, for instance, is one loop. It does not see the ambient space: a circle embeds in $\mathbb{CP}^1$ as well as in the plane. Topology therefore tests S1 only through what the ambient geometry constrains — bounded diameter, the maximal-distance count, realisability — and is used here to choose stimulus sets that cover a manifold without holes in the probe cover (item 3 above).

Structure of $\Gamma$ itself

• Seven axes and the Fano blocks in self-report: confirmatory factor analysis of the 28-item audit, testing whether the 7-triad block structure fits better than random tripartitions (Console V0-VAL, a self-report analogue of F-Gap-2).
• Seven axes in neural data: the number of independent slow components in fMRI (F-ISF [H]), and the seven-component stress tensor (experiment IV.1).
• Fano holonomy: on live $\hat\Gamma$ with phases from EEG phase locking, compare $\lvert H_p\rvert$ on the 7 lines with the 28 non-line triangles. The corpus's own machine check found no difference on 180 constructed natal states ($+0.03 \pm 0.39$ rad): the Fano structure is what the dynamics preserves, so the signature must be sought in states that have passed through the Fano channel, not in constructions (current empirical status).
• Window and bands: $\hat P$, $\hat\Phi$, $\hat R$, $s_1$, $s_2$ across sleep–wake and anaesthesia with a frozen $\pi_{\mathrm{bio}}$ (calibration, K1–K2). On the uniform diagonal the window is exactly $\Phi \in (1, 2]$, $R \in [1/3, 1/2)$, $C \in (1/2, 2/3]$ (Step 5 [T]) — three readouts that must move together.
• Fading [Pr]: during anaesthetic induction or sleep onset, collect repeated similarity judgements on a fixed stimulus set at graded depth. S10 predicts that intensity ratings fall while the rank order of dissimilarities stays that of baseline; a systematic drift of qualities (a hue shifting before it vanishes) counts against S10 as a claim about experience.

Confirmation and refutation

Invariant	Would confirm	Would refute
S1 geometry	Calibrated matrices realisable at a stable $n^*$; projective geometry fits better than Euclidean at equal parameters; cross-subject alignment near-isometric	Metric violations beyond tolerance; more pairwise-maximal qualities than $n^*$; unrealisable matrices after calibration
S2 intensity/quality	Isospectral states differ in quality, not intensity; context changes quality at fixed $\rho_E$	Identical full invariants, distinguishable experience (refutation criterion)
S3 seven axes	7-triad structure beats random tripartitions; $N_{\mathrm{ISF}} \in [6, 12]$	Block structure no better than random; $N_{\mathrm{ISF}}$ systematically outside $[6, 12]$
S4–S6 Fano	Line holonomies and blocks separate from non-lines in live states	No separation in live states that have passed through the Fano channel
S8–S9 window and bands	Wake inside, N3 below, both exits as predicted; conscious states inside F-Band	Any of the falsifiers of calibration; a conscious state outside F-Band
S10 fading	Intensity falls, similarity order holds	Qualities drift before they vanish

A single refutation acts at the status of the claim it hits: against a [T] row it means an error in the proof or in the bridge to data (the experimental protocol's structural level), against an [H] row it removes the hypothesis, against an [I] reading it removes the support that reading had.

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