Spacetime Structure

Who this chapter is for

This chapter is one of the most remarkable in the theory. Space and time are not postulated — they are derived from the structure of the category $\mathcal{C}$. This means that the 3+1-dimensional world we inhabit is a consequence, not a premise, of the theory.

Analogy: the chessboard. Imagine that the rules of chess define the board, not the other way around. Usually we think: first there is a board (space), then pieces play on it (matter). In UHM it is the opposite: first there are the rules of interaction (category $\mathcal{C}$ with CPTP-morphisms), and from these rules it follows that the "board" has exactly 6 dimensions (with compactification to 3+1). If the rules were different — the "board" would be different. Spacetime is not an arena, but a consequence.

What is concretely derived:

• Base space $X = |N(\mathcal{C})|$ — from the nerve of the category (geometric realization of the simplicial set of objects and morphisms)
• Time — from the Page–Wootters mechanism (correlation with the O measurement) and stratification (collapse to the terminal object T)
• Metric — from Connes' spectral triple (distance formula via the Dirac operator)
• Dimensionality 6D = 7 - 1; the compactification to 3+1D "via sectoral decomposition" is retracted [✗] (2026-09-25) — the axis split $7=1_O\oplus3_{\{A,S,D\}}\oplus\bar{3}_{\{L,E,U\}}$ is not a decomposition under $\mathrm{SU}(3)$ (details). Its replacement, Theorem 48c, obtains one time and three space directions, a Lorentzian signature and a rotation group $\mathrm{SO}(3)$ that commutes with colour from the complex numbers $\mathbb C_O$ generated by the clock unit: [T] as mathematics, [C at (L)] as physical spacetime — one premise, the "2", since Theorem 48e proved the Masanes–Müller principle inside UHM; since 2026-09-26 that premise is equivalent to a principle without a number (48e(g), (P)), and the "2" follows from it
• Lorentzian signature — $(1,3)$ [C] (registry row T-53): 1 time from Page–Wootters [T]; 3 space from $S^3$, which is computed in T-119 [T] (as mathematics; its coordinates are colour-charged); the Lorentzian sign holds at reflection positivity (bounded-below $H_S$). The Krein–Lorentzian triple below is a consistency check, not a derivation of the sign
• Gravity — from the full spectral action (Einstein equations as a consequence)
• Background independence — $M^4=\mathbb R\times S^3$ computed algebraically: time from the depth register, space as the spectrum of three commuting rotation charges, [T] as mathematics since 2026-09-25 (T-117–T-121); the reading of that $S^3$ as physical space is [I]

This is a radical departure from standard physics, where spacetime is a given on which dynamics unfolds. In UHM dynamics generates spacetime.

Section status (per item)

• Base space: [T] $X = |N(\mathcal{C})|$ — geometric realization of the nerve of the category
• Time: [T] Formalized via the emergent time theorem: the cyclic clock $\tau \in \mathbb{Z}_7$ (T-53a), the dynamics relative to the depth register (T-53b [T]) and the time line $C_0(\mathbb{R})$ as its scaling limit (T-118 [T])
• Metric: [T] Connes stratified metric $d_{strat}$
• Lorentzian signature: $(1,3)$ [C] (registry row T-53) — the time count is [T]; the spatial slice $\Sigma^3=S^3$ is computed in T-119 [T]; the sign holds at reflection positivity (bounded-below $H_S$). The Krein–Lorentzian spectral triple ($\beta=\gamma^0\otimes1$, $\mathcal D$ Krein-self-adjoint) realises the signature consistently but does not select it
• Gravity: [T] Full spectral action from the finite triple
• Background independence: [T] as mathematics — $M^4=\mathbb R\times S^3$ computed (T-120); reading [I]. History of 2026-09-25: it read [T], then [C] at an aperiodic clock (discharged by the depth register) and at the open reconstruction axioms of T-119 (discharged by the restatement of T-119, which computes the spectrum)

Base space X = $|N(\mathcal{C})|$

Property 5 (Stratification) [D]

The base space of the theory is defined as the geometric realization of the nerve of the category:

$X = |N(\mathcal{C})|$

where $\mathcal{C}$ is the primitive UHM category.

Autopoiesis of the base space

Key property: X is defined endogenously, not introduced from outside.

Aspect	Traditional theories	UHM
Base space	Postulated (ℝ⁴, Σ, ...)	Derived from $\mathcal{C}$
Metric	Introduced by hand	Computed from spectral data
Topology	Fixed	Follows from the nerve structure

Nerve of the category $N(\mathcal{C})$

Definition (Nerve):

The nerve $N(\mathcal{C})$ is a simplicial set:

• 0-simplices: objects of $\mathcal{C}$ (holons $\mathbb{H}$)
• 1-simplices: morphisms $f: A \to B$
• n-simplices: composable chains of morphisms

Geometric realization:

$|N(\mathcal{C})| = \left( \bigsqcup_n \Delta^n \times N_n \right) \Big/ \sim$

where the equivalence relation glues the faces of simplices.

Stratification of X

Definition (Stratification):

The space X is partitioned into strata:

$X = \bigsqcup_{\alpha \in A} S_\alpha$

where:

• $S_0 = \{T\}$ — 0-dimensional stratum (terminal object)
• $S_1$ — 1-dimensional stratum (morphisms into T)
• $S_n$ — n-dimensional stratum (n-simplices)

Key property: The closure of each stratum contains strata of lower dimension.

Local-global dichotomy

Theorem (Local-global dichotomy) [T]

For the base space $X = |N(\mathcal{C})|$:

Globally (monism): $H^n(X, \mathcal{F}) = 0 \quad \forall n > 0$

Locally (physics): $H^*_{loc}(X, T) \cong \tilde{H}^{*-1}(\text{Link}(T)) \cong \tilde{H}^{*-1}(S^5) \neq 0$

Link dimension

Since $\dim(X)=6$ (the nerve of the $7$-stratum chain $S_0\subset\cdots\subset S_6$ realises as a $6$-simplex), the link of the point $T$ is $\mathrm{Link}(T)=S^{6-1}=S^{5}$, not $S^6$ ($S^6$ would require $\dim X=7$). The local cohomology $H^*_{loc}(X,T)\cong\tilde H^{*-1}(S^5)$ is nonzero in degree $6$, so the local-nontriviality conclusion $H^*_{loc}\neq 0$ is unchanged.

Interpretation:

Aspect	Global (H* = 0)	Local (H*_loc ≠ 0)
Ontology	The One exists	Multiplicity of structures
Topology	Contractible to T	Rich geometry near T
Physics	Convergence to equilibrium	Local topological effects
Time	Global arrow toward T	Local fluctuations

Consequence: Monism and physics are compatible — global contractibility does not exclude local non-triviality.

---

Connes stratified metric

Spectral triple for strata

On each stratum $S_\alpha$ a spectral triple is defined:

$(A_\alpha, H_\alpha, D_\alpha)$

where:

• $A_\alpha = C(S_\alpha)$ — algebra of functions on the stratum
• $H_\alpha = L^2(S_\alpha, E_\alpha)$ — Hilbert space of sections
• $D_\alpha$ — Dirac operator on the stratum

Distance formula d_strat

Theorem (Stratified metric) [T]

The distance between pure states $\omega_1, \omega_2 \in X$:

$d_{strat}(\omega_1, \omega_2) = \inf_{\gamma} \int_\gamma ds_\alpha$

where:

• $\gamma$ — a path crossing strata $S_{\alpha_1}, S_{\alpha_2}, \ldots$
• $ds_\alpha$ — Connes metric on stratum $S_\alpha$:

$d_\alpha(p, q) = \sup\{|f(p) - f(q)| : \|[D_\alpha, f]\| \leq 1\}$

• The infimum is taken over all paths connecting $\omega_1$ and $\omega_2$

Metric near the terminal object

Near $T$ (the apex of the cone) the metric has a cone structure:

$d_{strat}(x, T) \sim r \cdot d_{S^5}(\pi(x), \text{base point})$

where:

• $r$ — the "radial" coordinate (distance to T)
• $\pi$ — projection onto the link $\text{Link}(T) \cong S^5$

Interpretation: The distance to the attractor decreases during evolution — the system "approaches" T.

---

Space as a structure of differences

Space is not a stage, but a relation

We are accustomed to thinking of space as a "stage" on which physics plays out: first there is an empty room (space), then objects are placed in it (matter). In UHM space is not a stage, but a structure of differences between states. The distance between two points is a measure of how hard it is to deform one state into another. If two states transition into each other easily — they are "close"; if this requires a major restructuring — they are "far." Space arises as a by-product of differences, not as their container. This resolves the fundamental problem of quantum gravity: if space is not a given but a consequence, its quantization does not lead to contradictions.

Space is not an empty container, but a structure of differences in the category $\mathcal{C}$.

Distance

In the updated theory, distance is defined via the Connes stratified metric:

$$d(A, B) := d_{strat}(A, B)$$

The circularity problem is resolved: The distance is derived from spectral data on the strata $S_\alpha$, not from an a priori notion of "points in space."

Comparison with previous version

In early versions of the theory the formula $d(A, B) = \|\Gamma_A - \Gamma_B\|_F$ was used, which contained a circular dependence. The new construction via $X = |N(\mathcal{C})|$ eliminates this problem — space is derived from the categorical structure.

Topology

Theorem (Topology of X) [T]

The topology of the base space is fully determined by the categorical structure:

$\text{Top}(X) = \text{Top}(|N(\mathcal{C})|)$

Properties:

• Globally: $X$ is contractible to the terminal object $T$
• Locally: Near $T$ the topology is non-trivial ($\text{Link}(T) \cong S^5$)

Status: [T] Formalized. Topology is derived from the nerve structure of the category.

Emergent time

Time is not a river, but a correlation

In everyday experience time seems like a "river" carrying us from past to future. In UHM time is something entirely different. It emerges from correlations between subsystems. Imagine a clock and an observer as a single quantum system. "Time = 3 o'clock" means not "the river has reached mark 3," but "the state of the clock correlates with a certain state of the observer." The universe as a whole is timeless (satisfies the constraint $[\hat{C}, \Gamma_{\text{total}}] = 0$); time arises within it — as the relation of "the clock" (the O measurement) to "the rest" (6 dimensions). This is the solution to the "problem of time" in quantum gravity proposed by Page and Wootters in 1983.

Theorem (Emergence of time) [T]

Time is derived from the structure of the category $\mathcal{C}$ in four equivalent ways:

Level	Time as...	Formula	Status
Page–Wootters	Correlation with O	$\Gamma(\tau) = \text{Tr}_O[\cdot]$	[T] Formalized
Information geometry	Distance in the Bures metric	$d_B(\Gamma_1, \Gamma_2)$	[T] Formalized
Categorical	1-morphism in ∞-groupoid	$\gamma: \Gamma_1 \to \Gamma_2$	[T] Formalized
Stratification	Collapse of strata to T along the depth $n$	$\dim(X_n) \geq \dim(X_{n+1})$	[T] Formalized

Full proof →

Page–Wootters mechanism

Time arises as the parameter of conditional states with respect to the O measurement:

$$\Gamma(\tau) := \frac{\text{Tr}_O\left[ (|\tau\rangle\langle \tau|_O \otimes \mathbb{1}_{6D}) \cdot \Gamma_{total} \right]}{p(\tau)}$$

where:

• $\Gamma_{total}$ satisfies the constraint $[\hat{C}, \Gamma_{total}] = 0$
• $|\tau\rangle_O$ — basis of eigenstates of the internal clock O
• p(τ) — normalization

The continuity of τ is an output, not an assumption (T-286)

The Page–Wootters parameter ranges over a continuum — and that used to be a silent premise. The genesis layer discharges it: the guaranteed closure of the self-model (the ouroboros: a fixed point $\Gamma = \varphi(\Gamma)$, which over $\mathbb{R}$ a continuous $\varphi$ has by Theorem 10.1 of Gap thermodynamics; until 2026-09-26 cited as "$\rho^* = \varphi(\Gamma)$, T-222", but the regeneration target $\varphi(\Gamma)$ is not itself a fixed point and the restated T-222 is not about fixed points) forces the intermediate-value property, hence Dedekind completeness of the scalars, hence $\mathbb{R}$ — with an explicit witness that over an incomplete field the self-model can miss its fixed point through a "hole in the line". Relational time inherits its continuity from the continuity the ouroboros demands: see T-286 [T].

Information-geometric time

Distance between configurations in the Bures metric:

$$d_B(\Gamma_1, \Gamma_2) = \arccos\left( \text{Tr}\sqrt{\sqrt{\Gamma_1} \Gamma_2 \sqrt{\Gamma_1}} \right)$$

Notation

Here $d_B$ is the Bures angle (not the chordal distance $\sqrt{2(1-\sqrt{F})}$ from evolution.md).

Flow of time — the rate of change of Γ:

$$\frac{d\tau_{int}}{d\sigma} = \left\| \frac{d\Gamma}{d\sigma} \right\|_B$$

Time "flows faster" when Γ changes more strongly.

Relation to evolution

Evolution is described with internal time τ:

$$\frac{d\Gamma(\tau)}{d\tau} = -i[H_{eff}, \Gamma(\tau)] + \mathcal{D}[\Gamma(\tau)] + \mathcal{R}[\Gamma(\tau), E]$$

This equation is a consequence of the structure of $\Gamma_{total}$, not a postulate.

Arrow of time

Why time flows in one direction

The arrow of time is one of the deepest puzzles in physics. Why do we remember the past but not the future? Why does a broken cup not reassemble? In standard physics the arrow of time is associated with the growth of entropy (the second law of thermodynamics), but the second law itself is usually postulated or derived from the initial conditions of the Big Bang. In UHM it is simpler: the arrow of time is a geometric consequence of the existence of the terminal object $T$. If the category has a "final point" toward which everything tends (like the bottom of a funnel), then the direction — from the periphery to the center — is defined by the structure, not by initial conditions.

Theorem (Arrow of time as collapse of strata) [T]

The arrow of time is a geometric consequence of the terminal object $T$:

$\dim(X_n) \geq \dim(X_{n+1})$

along the stratal depth $n \in \mathbb{N}$ (the cumulative tick count — the cyclic label $\tau \in \mathbb{Z}_7$ carries no arrow, two indices, one arrow), with equality only at stationarity.

Three equivalent formulations:

Formulation	Formula	Source
Geometric	$\dim(X_n) \geq \dim(X_{n+1})$	Property 3
Entropic	$\sigma(\gamma) \cdot \Delta S_{vN}(\gamma) \geq 0$	Unital CPTP structure only (see the retraction under "Thermodynamic direction")
Convergence	$\lim_{n \to \infty} X_n = \{T\}$	Terminality of T

Full proof →

Interpretation: The arrow of time is the progressive collapse of higher strata to the terminal object $T = \Gamma^*$ (the global attractor).

Resolution of the circularity problem

In early versions of the theory the arrow of time was linked to CPTP channels, which contained a hidden circularity. Now the arrow of time is derived geometrically from the terminal object — this is a structural property of the category $\mathcal{C}$, independent of the CPTP interpretation.

Thermodynamic direction

The arrow of time is defined by the direction of increase of the von Neumann entropy:

$$\frac{dS_{vN}}{d\tau} \geq 0$$

Distinction of concepts

Arrow of time as collapse of strata (theorem above) — this is a structural property of the category $\mathcal{C}$, derivable from the existence of the terminal object T.

Global increase of differentiation ($dD_{\text{diff}}/d\tau > 0$) — this is a separate cosmological hypothesis, having the status of a non-falsifiable philosophical position.

These concepts are related (both concern direction), but have different epistemological status.

This inequality is a consequence of the properties of CPTP channels: they do not decrease entropy. Retracted 2026-09-25: a CPTP channel can lower the von Neumann entropy — the constant channel $X \mapsto \mathrm{Tr}(X)\,\Gamma'$ sends $I/7$ to any $\Gamma'$ — so the inequality holds only for unital generators ($\mathcal{L}(I) = 0$), such as the Hamiltonian part and dissipators with Hermitian Lindblad operators. For a quantum dynamical semigroup with a stationary state $\rho_\infty$ the monotone quantity is the relative entropy: $\tfrac{d}{d\tau} D_{KL}(\Gamma(\tau)\,\|\,\rho_\infty) \leq 0$ (Spohn, J. Math. Phys. 19, 1227 (1978)). The UHM generator is not unital — regeneration $\mathcal R$ and the anchor pull toward $\rho^* \neq I/7$ and lower $S_{vN}$ on the way from $I/7$ to the attractor — and, being nonlinear, it lies outside the scope of that 1978 result as well; no entropy-type monotone is established for the full $\mathcal L_\Omega$.

Clarification

In the presence of regeneration $\mathcal{R}$ a local decrease in entropy is possible due to the import of free energy:

$$\Delta S_{vN}^{local} < 0 \Rightarrow \Delta F_{env \to sys} > 0$$

The total entropy (system + source) always grows.

Second law of thermodynamics

Theorem (Second law from the unital-channel order) [T]

The second law of thermodynamics is a consequence of the majorization order induced by the unital-channel refinement of morphisms (see Mathematical foundations §terminal object):

$\forall \Gamma:\quad \Gamma \xrightarrow{\ \text{unital CPTP}\ } I/7 \quad\text{iff}\quad I/7 \prec \Gamma \ (\text{always}),\qquad I/7 \not\to \sigma\ \text{for }\sigma\neq I/7.$

Under unital channels $I/7$ is the unique sink: every state majorizes $I/7$, and no unital channel leaves $I/7$, so the arrow of time (monotone increase of von Neumann entropy = descent in majorization order) is irreversible — there is no unital return path from $I/7$.

Caveat: in the full CPTP category the earlier "$\exists! f:\Gamma\to T$" is false (the constant channel $X\mapsto\mathrm{Tr}(X)\Gamma'$ reaches any $\Gamma'$); the irreversibility statement is a theorem specifically of the unital (entropy-non-decreasing) morphism class.

Geometric interpretation:

Aspect	Formulation	Consequence
Sink (unital order)	$\forall \Gamma:\ I/7 \prec \Gamma$; $I/7\not\to\sigma\ (\sigma\neq I/7)$	All dissipative paths lead to T
Collapse of strata	$\dim(X_n) \geq \dim(X_{n+1})$	Dimensionality does not grow along the depth $n$
Entropy (unital channels)	$dS_{vN}/d\tau \geq 0$ for unital generators only	Entropy does not decrease along unital dynamics; for a general quantum dynamical semigroup the monotone is $D_{KL}(\Gamma\|\rho_\infty)$ (Spohn 1978)

Status: [T] Formalized for the unital morphism class. The second law is derived from categorical structure for unital dynamics; the full UHM generator is not unital (see "Thermodynamic direction").

Relation to the Heaviside function

The gate $g_V(P)$ in the regenerative term (refining $\Theta(\Delta F)$ from Landauer) is not a postulate, but a consequence:

$$\mathcal{R}[\Gamma, E] \propto g_V(P) \quad \Leftarrow \quad \text{thermodynamics of CPTP + V-preservation}$$

Relativity

Internal clocks

Different Holons can have different "internal clocks" — different rates of evolution:

$$\tau_{\mathbb{H}_1} \neq \tau_{\mathbb{H}_2}$$

where $\tau_{\mathbb{H}}$ is the proper time of the Holon $\mathbb{H}$.

Relativistic effects [C]

Theorem (Relativistic effects from spectral triple) [C at T-53]

Gravitational and kinematic time dilation are consequences of the spectral triple T-53 [C] (its Lorentzian sign holds at reflection positivity, its spatial slice at T-119) and the full spectral action T-65 [T]. Connes' formula defines the metric $g_{\mu\nu}$, and the spectral action reproduces the Einstein–Hilbert action, which includes all relativistic effects.

Proof.

Step 1 (Metric from Connes formula). From T-53 [C] (spectral triple):

$d(p, q) = \sup\{|f(p) - f(q)| : \|[D, f]\| \leq 1\}$

The block-diagonal structure of $D$ with $g_{00} = 1/|D_O|^2 > 0$, $g_{aa} = -1/|D_{3,a}|^2 < 0$ defines the Lorentzian metric $g_{\mu\nu}$.

Step 2 (Einstein–Hilbert action). From T-65 [T] (full spectral action):

$S = \mathrm{Tr}(f(D/\Lambda)) = \int (a_0\Lambda^4 + a_2\Lambda^2 R + a_4 C_{\mu\nu\rho\sigma}^2 + \ldots)\sqrt{g}\,d^4x$

The coefficient $a_2\Lambda^2 R$ gives the kinetic term of gravity, i.e., the Einstein–Hilbert action.

Step 3 (Time dilation). Formula for the rate of internal clocks:

$\frac{d\tau}{d\sigma} = \omega_0 \cdot \sqrt{\sum_{i \neq O} |\gamma_{Oi}|^2 \cdot \mathrm{Gap}(O,i)^2}$

$\mathrm{Gap}(O,i)$ includes gravitational corrections via the metric $g_{\mu\nu}$: in a region of strong gravitational field (small $g_{00}$) the eigenvalues of $D_O$ are modified, which slows $d\tau/d\sigma$. Similarly, kinematic time dilation follows from the Lorentz transformation of spectral data. $\blacksquare$

Emergence of geometry

Section status

• Metric: [T] Formalized via $d_{strat}$ (see above)
• Dimensionality: [T] 6D follows from $N = 7$ (dim = N - 1)
• Relation to GR: [T] as mathematics — $M^4=\mathbb R\times S^3$ computed (T-120); it read [C] at the open reconstruction axioms of T-119 until the restatement of 2026-09-25

Derived metric (not a hypothesis)

In UHM the metric is derived, not postulated:

$d_{strat}(\omega_1, \omega_2) = \inf_{\gamma} \int_\gamma ds_\alpha$

Key properties:

• Metric defined on $X = |N(\mathcal{C})|$
• Accounts for stratification (different ds on different strata)
• Cone-like near the terminal object T

Dimensionality of space

Theorem (Dimensionality):

$\dim(X) = N - 1 = 6$

where $N = 7$ is the number of dimensions of the Holon.

Consequence: The 6D structure arises endogenously, it is not postulated.

Relation to GR (program)

Background independence [C]; sectoral decomposition retracted [✗]

The transition from 7D (= 6D + time) to the observable 3+1D was formalized via the sectoral decomposition $7 = 1_O \oplus 3_{\{A,S,D\}} \oplus \bar{3}_{\{L,E,U\}}$, with the masslessness of gluons (the $\{A,S,D\}$ "$\mathbf{3}$-sector") giving non-compact space and the massiveness of $W,Z$ (the $\{L,E,U\}$ "$\bar{\mathbf{3}}$-sector") a compactification at $v_{\text{EW}}$. This is retracted [✗] (2026-09-25): no three of the six non-$O$ axes span an $\mathrm{SU}(3)$-invariant subspace, so neither axis set is a sector. Details — Sectoral decomposition.

Results: The finite spectral triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$ is written down (T-53; its KO-dimension-6 claim is retracted, see Step 6 there). The spectral action $S = \text{Tr}(f(D/\Lambda))$ gives $\int(a_0\Lambda^4 + a_2\Lambda^2 R + \ldots)\sqrt{g}\,d^4x$ [T] (T-65, full spectral action). The product of triples $M^4 \times F_{\text{int}}$ is obtained from the categorical structure along the chain of T-120: the macroscopic algebra is commutative in the thermodynamic limit (T-117 [T]), the spectrum of three commuting rotation charges gives $\Sigma^3=S^3$ (T-119 [T], restated 2026-09-25), and the product $M^4 = \mathbb{R} \times S^3$ is assembled from these (T-120 [T]). Until the restatement T-119 was conditional and this page did not lean on T-120 as unconditional; it now does as mathematics, with the reading of $S^3$ as physical space [I].

See Correspondence with physics: GR for the detailed program.

Emergence diagram

Note: The edge to "Gravity [T]" — $M^4$ is assembled from categorical structure via the Gel'fand–Naimark–Connes chain (T-120) which is [T] as mathematics since the restatement of T-119 (2026-09-25; before it the edge was [C] at the open reconstruction axioms of T-119); the Einstein equations themselves (T-65) hold on the product triple.

Non-locality

Quantum correlations

Coherences $\gamma_{ij}$ between distant parts of $\Gamma$ mean non-local connections:

$$\gamma_{AB} \neq 0 \Rightarrow A \text{ and } B \text{ are quantum-correlated}$$

Entanglement

Entanglement is the non-separability of the state of subsystems:

$$\Gamma_{AB} \neq \Gamma_A \otimes \Gamma_B$$

where $\Gamma_A = \mathrm{Tr}_B(\Gamma_{AB})$ is the partial trace over subsystem $B$.

Violation of Bell inequalities is a consequence of non-zero coherences in the structure of $\Gamma$.

Relation to physics

Physical concept	Expression via $\mathcal{C}$	Status
Base space	$X = \lVert N(\mathcal{C})\rVert$	[T] Formalized
Time	Parameter τ (Page–Wootters)	[T] Formalized
Arrow of time	Collapse of strata to T	[T] Formalized
Metric	$d_{strat}$ (Connes on strata)	[T] Formalized
Dimensionality	$\dim(X) = 6$	[T] Consequence of $N = 7$
Energy	Eigenvalues of $H_{eff}$	[T] Formalized
Gravity	Spectral action on $M^4 \times F_{\text{int}}$, $M^4$ assembled by T-120; the former entry "compactification 6D → 4D" is retracted [✗] with the sectoral decomposition	[T] as mathematics Emergent manifold (T-120; [C] until the restatement of T-119, 2026-09-25)
Topological charges	IC-cohomology of strata	[T] Formalized

Relation to other approaches

Approach	Relation to UHM	Status
Quantum mechanics	Special case of UHM at $R \to 0$	Proven
Standard Model	Gauge symmetries from $\text{Sym}(\Gamma)$	Program
Loop quantum gravity	Spin networks may correspond to coherence structures	Not investigated
String theory	Possible connection via holographic principle	Not investigated
Hoffman Conscious Agents	Spacetime as interface consistent with emergence	Conceptually compatible
Emergent gravity (Verlinde)	Similar approach: gravity as entropic force	Requires investigation
Octonionic lineage (Günaydın–Gürsey 1973; Manogue–Dray 1999; Boyle 2026)	Same split $1\oplus3\oplus\bar{3}$, read as colour; four-dimensional spacetime from the choice of one imaginary unit	Prior art — see Precedents

What is formalized vs Research program

Statement	Status	Comment
Base space $X = \lVert N(\mathcal{C})\rVert$	[T] Formalized	Property 5
Time as Page–Wootters parameter	[T] Formalized	Theorem proved
Arrow of time as collapse of strata	[T] Formalized	Follows from terminality of T
Metric $d_{strat}$	[T] Formalized	Connes stratified metric
Dimensionality 6D	[T] Formalized	Consequence of $N = 7$
Local-global dichotomy	[T] Formalized	H* = 0 globally, H*_loc ≠ 0 locally
Lorentzian signature	signature $(1,3)$ [C] (T-53: time count [T], spatial slice $S^3$ from T-119 [T], sign at reflection positivity)	UHM spectral triple
Compactification 7D → 3+1D	retracted [✗]	Sectoral decomposition: the axis split is not an $\mathrm{SU}(3)$ decomposition
3+1 with rotations that commute with colour	[T] as mathematics; [C at (L)] as spacetime (Theorem 48e)	Theorem 48c: the colour-singlet part of $\mathfrak h_2(\mathbb O)$ is $\mathfrak h_2(\mathbb C_O)\cong\mathbb R^{1,3}$
Background independence ($M^4$ assembled)	[T] as mathematics (T-118, T-119 restated); reading [I]; it read [C] at the open reconstruction axioms of T-119 until 2026-09-25	T-120
Einstein equations	[T]	Spectral action from the full triple

Progress

The circularity problem of $\Gamma_A$ has been resolved conditionally: space is now assembled from the categorical structure $\mathcal{C}$, not from a priori "points" — by T-119 and T-120, both [T] as mathematics since 2026-09-25; the reading of the computed $S^3$ as physical space is [I].

Sectoral decomposition of dimension 7 = 1 + 3 + 3̄

danger

Retracted [✗] (2026-09-25): 3+1 from $7=1_O\oplus3_{\{A,S,D\}}\oplus\bar3_{\{L,E,U\}}$

This section claimed that the $\mathrm{SU}(3)_C$ fixing $O$ splits the seven axes into time $O$, a spatial triplet $\{A,S,D\}$ and a compact anti-triplet $\{L,E,U\}$, and read 3+1 dimensions off that split; the split is false — none of the 20 triples of non-$O$ axes spans an $\mathrm{SU}(3)$-invariant subspace, and the commutant of $\mathfrak{su}(3)$ on those six axes is two-dimensional, so $\mathbb{R}^6$ is irreducible (of complex type) and the only invariant real splitting is $\mathbb{R}e_O\oplus\mathbb{R}^6$. What replaces it is the complexified decomposition $\mathbb{C}^7=\mathbb{C}e_O\oplus\mathbf 3\oplus\bar{\mathbf 3}$ with $\mathbf 3=\mathrm{span}_{\mathbb C}\{A-iD,\ S-iU,\ L-iE\}$ [T] — standard mathematics and prior art (Günaydın and Gürsey 1973, see precedents) — which assigns no axis to "space" and none to "compact", and therefore yields no 3+1 count.

• Numbers (website/scripts/check_core_numbers.py, test_no_axis_triple_is_su3_invariant): $\dim\mathrm{Der}(\mathbb{O})=14$, $\dim\mathrm{Stab}(e_O)=8$; invariant axis triples $0$ of $20$; commutant $2$; left multiplication by $e_O=e_7$ pairs $A\leftrightarrow D$, $S\leftrightarrow U$, $L\leftrightarrow E$. Only a one-dimensional $\mathfrak{u}(1)\subset\mathfrak{su}(3)$ maps the span of $\{A,S,D\}$ into itself, so $\mathrm{SU}(3)$ does not act on $\{A,S,D\}$ as on a triplet.
• Retracted with it on this page: the theorem and Steps 2–5 below, the corollary $\dim(\text{space})=|\mathbf{3}|=3$, the Kaluza–Klein corollary, part (b) and Steps 2 and 5 of the theorem "Spacetime from spectral triple", and the $\{A,S,D\}$ leg of (M2). Registry: rows 48a and C11.
• What survives: one time direction from the Page–Wootters clock [T]. Three spatial directions are claimed through T-119 (the rank of $\mathfrak{u}(3)$; restated 2026-09-25 as the computed spectrum $S^3$, [T] as mathematics), which reads the colour algebra as the spatial one and so meets the Coleman–Mandula obstacle named in the precedents. A derivation that avoids it is Theorem 48c below: [T] as mathematics, [C at (L)] as physical spacetime (Theorem 48e). It does not restore this section's split and gives no axis the role of space.

Theorem 48c (Spacetime that commutes with colour) — mathematics [T], physical 3+1 [C at (L)]

The retraction above leaves the question open: where do three spatial directions come from, together with a rotation group that is not colour? Coleman and Mandula require the second part: internal symmetries commute with spacetime symmetries (precedents). Part (g) below shows that no rotation of the holon's own seven axes satisfies it. What satisfies it is the route of the octonionic lineage (Kugo–Townsend, Manogue–Dray): a $2\times2$ Hermitian matrix over the octonions. In that route the preferred imaginary unit was a free choice. Here colour fixes it, and it is UHM's $O$.

Notation. $e_O=e_7$ in the canonical table (G₂-structure, §2.2); $\mathrm{SU}(3)_C=\mathrm{Stab}_{G_2}(e_O)$; $\mathbb C_O=\mathbb R1\oplus\mathbb Re_O\subset\mathbb O$ is the complex subalgebra generated by the clock unit. $\mathfrak h_2(\mathbb K)$, for a subalgebra $\mathbb K\subseteq\mathbb O$, is the real vector space of matrices $X=\begin{pmatrix}a&x\\\bar x&b\end{pmatrix}$ with $a,b\in\mathbb R$, $x\in\mathbb K$, with the quadratic form $\det X=ab-\lvert x\rvert^2$; $G_2$ acts on $\mathfrak h_2(\mathbb O)$ entry by entry.

Theorem 48c — (a)–(g) [T]; the 3+1 reading [C at (L)]

(a) Colour fixes exactly the clock's complex numbers. The elements of $\mathbb O$ fixed by $\mathrm{SU}(3)_C$ are exactly $\mathbb C_O$. A unit $u\in\mathrm{Im}\,\mathbb O$ whose left multiplication commutes with $\mathrm{SU}(3)_C$ is $\pm e_O$. The complex structures on $\mathbb R^6=\mathbb C_O^\perp$ that commute with $\mathrm{SU}(3)_C$ are exactly $\pm L_{e_O}$.

(b) The colour-singlet part is Minkowski space. $\mathfrak h_2(\mathbb O)\cong\mathbb R^{1,9}$. Its $\mathrm{SU}(3)_C$-fixed subspace is $\mathfrak h_2(\mathbb C_O)$, of dimension $4$ and signature $(1,3)$. Its $\det$-orthogonal complement $W=\bigl\{\begin{pmatrix}0&w\\\bar w&0\end{pmatrix}: w\perp\mathbb C_O\bigr\}\cong\mathbb C^3$ is negative definite and carries the colour triplet.

(c) The Lorentz algebra is the centraliser of colour. The centraliser of $\mathfrak{su}(3)_C$ in $\mathfrak{so}(1,9)$ is $\mathfrak{so}(1,3)\oplus\mathfrak u(1)$, of dimension $6+1=7$. The $\mathfrak{so}(1,3)$ acts on $\mathfrak h_2(\mathbb C_O)$ and trivially on $W$; the $\mathfrak u(1)$ acts on $W$ as $L_{e_O}$ and trivially on $\mathfrak h_2(\mathbb C_O)$.

(d) The Lorentz group. $M\in\mathrm{SL}(2,\mathbb C_O)$ acts by $X\mapsto MXM^\dagger$. The action is well defined (the bracketing does not matter), preserves $\det$, induces $\mathrm{SO}^+(1,3)$ on $\mathfrak h_2(\mathbb C_O)$, is the identity on $W$ and commutes with $\mathrm{SU}(3)_C$. Hence $\mathrm{SL}(2,\mathbb C_O)$ and $\mathrm{SU}(3)_C$ act as a direct product.

(e) One time, three space. The maximal compact part $\mathrm{SU}(2)\times\mathrm U(1)$ of the centraliser fixes exactly one line, $\mathbb R\,1_2$. It is timelike: $\det 1_2=1>0$. $\mathrm{SU}(2)$ acts as $\mathrm{SO}(3)$ on the three traceless directions $\sigma_z$, $\sigma_x$, $e_O\sigma_y$, which are spacelike. The signature is Lorentzian because $\det$ is: no sign is chosen.

(f) Spin and colour are separate factors. On $\mathbb O^2$ the map $\psi\mapsto M\psi$ is a representation of $\mathrm{SL}(2,\mathbb C_O)$ that commutes with $\mathrm{SU}(3)_C$. As a $\mathbb C_O$-module $\mathbb O=\mathbb C_O\oplus\mathbb C^3$, so $\mathbb O^2=(\mathbf 2,\mathbf 1)\oplus(\mathbf 2,\mathbf 3)$: a colour-singlet Weyl spinor and a colour-triplet Weyl spinor. Physical reading (2026-09-25, Theorem 48e(e)): the Weyl index of the fermion field is the colour-fixed part $(\mathbb O^2)^{\mathrm{SU}(3)_C}=\mathbb C_O^2$, and the field is $\mathbb C_O^2\otimes_{\mathbb C}\mathcal S_{\mathbb C}$, with colour, isospin and hypercharge in $\mathcal S$ (T-326, T-329). The triplet part $(\mathbf 2,\mathbf 3)$ carries no weak isospin and is not a quark field; the earlier reading "lepton plus quark Weyl spinors" is withdrawn.

(g) No-go inside the holon. The centraliser of $\mathfrak{su}(3)_C$ is $0$ in $\mathfrak g_2$, $\mathfrak u(1)$ in $\mathfrak{so}(7)$ and $\mathfrak u(1)^3$ in $\mathfrak u(7)$. No $\mathrm{SO}(3)$ acting on the seven axes commutes with colour. For each of the seven associative planes (Fano lines), the stabiliser in $\mathfrak g_2$ is $\mathfrak{so}(4)$ and acts on the plane as $\mathfrak{so}(3)$. It meets $\mathfrak{su}(3)_C$ in $\mathfrak u(2)$ for the three lines through $O$ and in an $\mathfrak{so}(3)$ for the other four. So the rotations of an associative plane are colour rotations.

Physical reading — [C at (L)]. Update (Theorem 48e): the reading needs only premise (L) of 48d, the "2"; (Q) below is a stronger premise that also suffices. Update (2026-09-26, 48e(f)–(i)): (L) is equivalent to a principle (P) that names no number, and no structure internal to UHM can replace it. Premise (Q) [H] has two parts. (Q1): the tangent vectors of spacetime at a point form the octonionic spin factor $\mathfrak h_2(\mathbb O)$, on which $\mathrm{Aut}(\mathbb O)=G_2$ acts entry by entry. (Q2): spacetime is its colour-singlet part, i.e. colour acts trivially on spacetime directions, as Coleman–Mandula demands. Under (Q), parts (b)–(e) give one time direction, three space directions, the signature $(1,3)$, and a rotation group $\mathrm{SO}(3)$ that commutes with $\mathrm{SU}(3)_C$. The Lorentz group is not put in: it is the non-compact part of the centraliser of colour. The $O$-direction appears here not as the time axis but as the imaginary unit of $\mathbb C_O$, which is also one spatial Pauli direction, $e_O\sigma_y$ [I].

Proof. (a) $\mathrm{SU}(3)_C$ fixes $1$ and $e_O$. It acts on $\mathbb R^6$ irreducibly and transitively on the unit sphere $S^5$: the orbit of every unit vector is five-dimensional (numbers below). So it fixes no non-zero vector of $\mathbb R^6$. If $L_u$ commutes with every $\varphi\in\mathrm{SU}(3)_C$, then $\varphi(u)\varphi(x)=\varphi(ux)=u\,\varphi(x)$ for all $x$, so $\varphi(u)=u$ and $u\in\mathbb C_O\cap\mathrm{Im}\,\mathbb O=\mathbb Re_O$. The commutant of $\mathfrak{su}(3)$ on $\mathbb R^6$ is $\mathrm{span}\{1,L_{e_O}\}\cong\mathbb C$ (complex type, sectoral decomposition), and $(\alpha+\beta L_{e_O})^2=-1$ forces $\alpha=0$, $\beta=\pm1$. (b) $G_2$ fixes the real diagonal, so a fixed matrix has its off-diagonal entry in $\mathbb O^{\mathrm{SU}(3)}=\mathbb C_O$. Write $a=t+z$, $b=t-z$, $x=x_1+x_2e_O$. Then $\det X=t^2-z^2-x_1^2-x_2^2$. For $z'\in\mathbb C_O$ and $w\perp\mathbb C_O$, $\lvert z'+w\rvert^2=\lvert z'\rvert^2+\lvert w\rvert^2$, so $W$ is $\det$-orthogonal to $\mathfrak h_2(\mathbb C_O)$ and $\det$ is $-\lvert w\rvert^2$ on $W$. (c) An element of $\mathfrak{so}(1,9)$ that commutes with $\mathfrak{su}(3)$ preserves the isotypic components: the trivial one, $\mathfrak h_2(\mathbb C_O)$, and $W$, which is irreducible of complex type. Its restrictions lie in $\mathfrak{so}(1,3)$ and in the commutant of $\mathfrak{su}(3)$ inside $\mathfrak{so}(6)$, which is $\mathfrak u(1)$. Conversely every such pair commutes. Dimension $7$, confirmed numerically. (d) Every entry of $MXM^\dagger$ is a sum of products $m\,x\,m'$ with $m,m'\in\mathbb C_O$. The subalgebra generated by $e_O$ and $x$ is associative (Artin), so $(MX)M^\dagger=M(XM^\dagger)$. On $\mathfrak h_2(\mathbb C_O)$ this is the standard spin covering $\mathrm{SL}(2,\mathbb C)\to\mathrm{SO}^+(1,3)$. On $W$, use $mw=w\bar m$ for $m\in\mathbb C_O$, $w\perp\mathbb C_O$ (because $e_Ow=-we_O$) and $\bar w=-w$. The off-diagonal entry becomes $m_{11}w\bar m_{22}+m_{12}\bar w\bar m_{21}=w\,\overline{m_{11}m_{22}-m_{12}m_{21}}=w\,\overline{\det M}=w$. The diagonal entry becomes $w(\bar m_{11}\bar m_{12}-\bar m_{12}\bar m_{11})=0$. So $MX_wM^\dagger=X_w$. Every $\varphi\in\mathrm{SU}(3)_C$ fixes the entries of $M$, so $\varphi(MXM^\dagger)=M\varphi(X)M^\dagger$. An element in both images acts trivially on $\mathfrak h_2(\mathbb C_O)$ and on $W$, so the product is direct. (e) $\mathrm{SU}(2)$ acts on the traceless Hermitian $2\times2$ matrices by the adjoint representation, which is $\mathrm{SO}(3)$, and it fixes $1_2$. The $\mathrm U(1)$ fixes $\mathfrak h_2(\mathbb C_O)$ and has no fixed vector in $W$. (f) By Artin, $(mm')x=m(m'x)$ for $m,m'\in\mathbb C_O$, so left multiplication gives a representation. $\mathrm{SU}(3)_C$ commutes with $L_{e_O}$, so it acts $\mathbb C_O$-linearly, as $\mathbf 1$ on $\mathbb C_O$ and as $\mathbf 3$ on $\mathbb C^3$. (g) Linear algebra over the fourteen derivations of the canonical table. $\blacksquare$

Numbers (website/scripts/check_core_numbers.py: test_colour_commuting_spacetime_is_h2_of_the_clock_complex, test_no_rotation_of_the_seven_axes_commutes_with_colour, test_octonionic_spinor_is_lepton_plus_quark_weyl, test_every_non_o_axis_is_half_triplet_and_colour_moves_any_axis_to_any):

• the fixed space of $\mathfrak{su}(3)_C$ is $2$-dimensional in $\mathbb O$ ($1$, $e_7$) and $4$-dimensional in $\mathfrak h_2(\mathbb O)$, with signature $(1,3)$;
• $\dim\mathfrak{so}(1,9)=45$; the centraliser of $\mathfrak{su}(3)_C$ has dimension $7$; its derived algebra has dimension $6$ and Killing form of signature $(3,3)$; its compact part has dimension $4$ and fixes one line, $1_2$;
• for random $M\in\mathrm{SL}(2,\mathbb C_O)$: both bracketings agree, $L^{\mathsf T}\eta L=\eta$, $L\vert_W=\mathrm{id}$, $[L,g]=0$ for random $g\in\mathrm{SU}(3)_C$, and $\det L_4=1$, $(L_4)_{00}\ge1$;
• the $\mathrm{SU}(3)_C$-orbit of each of the six non-$O$ axes is $5$-dimensional (all of $S^5$), and each axis has weight exactly $\tfrac12$ in $\mathbf 3$;
• centralisers of $\mathfrak{su}(3)_C$: $0$ in $\mathfrak g_2$, $1$ in $\mathfrak{so}(7)$, $3$ in $\mathfrak u(7)$. For the seven associative planes the stabilisers have dimension $6$, act on the plane with rank $3$, and meet $\mathfrak{su}(3)_C$ in dimension $4$ (three lines through $O$) or $3$ (four lines).

What is new and what is not. Parts (b), (d) and (f) are the Manogue–Dray reduction (precedents) with $\ell=e_O$. Parts (a), (c) and (g) are elementary representation theory. We found no source that states the centraliser (c) in this form, but we claim no new mathematics. UHM's contribution is the premise-free part (a): colour, defined as the stabiliser of UHM's clock unit, leaves exactly one complex line of $\mathbb O$ fixed. So the unit that cuts ten dimensions to four is not chosen. It is the unit whose stabiliser is colour.

Why the reading is [C] and not [T]. (Q1) is not a theorem of UHM. The holon's state space is $\mathcal D(\mathbb C^7)$, and $\mathfrak h_2(\mathbb O)=\mathbb R^3\oplus\mathrm{Im}\,\mathbb O$ contains the seven axes only as its off-diagonal imaginary part. The three extra directions ($1_2$, $\sigma_z$, $\sigma_x$) are not in the holon. Three routes to derive (Q1) were tried, and none closes. (i) Inside one holon: $\mathrm{Herm}(\mathbb C^7)$ contains no unital copy of any spin factor $\mathfrak h_2(\mathbb K)$, not even $\mathfrak h_2(\mathbb R)$. Such a copy needs two anticommuting Hermitian involutions $s_1,s_2$ (the images of $\sigma_z,\sigma_x$). But $s_1s_2=-s_2s_1$ gives $\det(s_1s_2)=(-1)^7\det(s_2s_1)$, so $\det(s_1s_2)=0$, which is impossible for invertible $s_i$ in odd dimension. The "2" of (Q1) therefore needs a second tensor factor; it cannot be found inside the seven axes, for the same parity reason that excludes KO-dimension 6 on $\mathbb C^7$. (ii) From the Page–Wootters product $\mathbb C[\mathbb Z_7]\otimes\mathbb C^7$: the clock register is seven-level, not two-level. Its irreducible $\mathbb Z_7$-representations are one-dimensional, so no $\mathrm{SU}(2)$ acts on it. (iii) From the nerve $X=\lvert N(\mathcal C)\rvert$: it carries no quadratic form, so no signature. (Q2) is Coleman–Mandula's condition, an input from physics. Hence: [T] as mathematics, [C at (Q)] as spacetime, and 48a stays retracted.

Routes tried for "3 from UHM", and why each fails or feeds 48c.

• (a) An associative 3-plane as space. It fails by (g): its rotation group lies in $\mathrm{SU}(3)_C$. No associative plane is $\mathrm{SU}(3)_C$-invariant, since the only invariant subspaces of $\mathrm{Im}\,\mathbb O$ are $\mathbb Re_O$ and $\mathbb R^6$. $\mathrm{SU}(3)_C$ moves the planes through $O$ among themselves ($\mathbb{CP}^2$ of them) and those orthogonal to $O$ as well, so $O$ distinguishes none. A vacuum that selected one would break colour to $\mathrm U(2)$ or $\mathrm{SO}(3)$.
• (b) Spin from a quaternionic factor (Dixon's $\mathbb R\otimes\mathbb C\otimes\mathbb H\otimes\mathbb O$, Furey). UHM has no separate $\mathbb H$ factor. A quaternion subalgebra of $\mathbb O$ that contains $e_O$ is moved by $\mathrm{SU}(3)_C$, so it is not colour-free. What survives is that $M_2(\mathbb C_O)\cong\mathbb C_O\otimes\mathbb H$: Dixon's $\mathbb H$ is the "2" of (Q1). Theorem 48c takes it in that form.
• (c) Information-theoretic (Müller–Masanes 2013). The traceless part of $\mathfrak h_2(\mathbb C_O)$ is exactly the Bloch space of a qubit over $\mathbb C_O$, and (e)'s $\mathrm{SO}(3)$ is its $\mathrm{PSU}(2)$. So 48c's "3" is Müller and Masanes's "3", and (Q1) is their premise that the smallest system carries directions, in Jordan-algebra form. Their $G_2$ counter-case (a seven-dimensional ball, Dakić–Brukner; Masanes et al.) does not apply to $\mathfrak h_2(\mathbb C_O)$, whose Bloch ball is three-dimensional.
• (d) T-119's rank of $\mathfrak u(3)$. It counts $3$ from colour. It gives no rotation group and meets Coleman–Mandula; 48c removes that obstacle, and T-119's count agrees with it only numerically. The restated T-119 [T] (2026-09-25) computes the spatial manifold $S^3$ from three commuting rotation charges, but two of them are colour Cartan generators, and colour-singlet charges give only two dimensions (T-119(d)).
• Further routes — the depth register, pairs of holons, $J_3(\mathbb O)$, the Clifford system of T-326 — are closed by Theorem 48d, which also states (Q) as one principle (L) plus the Masanes–Müller principle. Theorem 48e proves the Masanes–Müller principle inside UHM, so (L) alone remains.

Theorem 48d (Where the "2" of (Q1) cannot come from, and what (Q) is equivalent to) — [T] as mathematics

Theorem 48c needs a two-by-two structure: its tangent vectors are Hermitian $2\times2$ matrices. The note above lists three routes that fail to find the "2" inside UHM. Theorem 48d closes the other routes: pairs of holons, the depth register, the exceptional Jordan algebra, and the Clifford system of T-326. It then states (Q) as one principle plus one input from quantum information.

Theorem 48d — (a)–(f) [T]

(a) No unital spin factor in any holon register. $\mathrm{Herm}(\mathbb C^d)$ contains a unital copy of a spin factor $\mathrm{JSpin}(n)$ with $n\ge2$ — in particular of any $\mathfrak h_2(\mathbb K)$ — if and only if $d$ is even. So there is none on $\mathbb C^{7^M}$: not for $M$ holons, not for the Page–Wootters product $\mathbb C^7\otimes\mathbb C^7$, not for a holon with its self-model on $\mathbb C^7\otimes\mathbb C^7$, and not for the depth register realised on $M$ O-registers ($7^M$ readings).

(b) Pairs and the depth register carry no canonical rotation group. On $\mathbb C^7\otimes\mathbb C^7$ the operators that commute with $G_2$ form a commutative algebra $\mathbb C^4$, because $\mathbf 7\otimes\mathbf 7=\mathbf 1\oplus\mathbf 7\oplus\mathbf{14}\oplus\mathbf{27}$ with no repetition. The operators that commute with the path Laplacian $\Lambda$ of the depth register (emergent time §11.4) form the commutative algebra of polynomials in $\Lambda$, because $\Lambda$ has simple spectrum. So neither carries an $\mathrm{SO}(3)$ tied to its structure. The colour-singlet subspace of a pair is three-dimensional, spanned by $e_O\otimes e_O$, $\sum_{k\ne O}e_k\otimes e_k$ and $\sum\varphi_{Ojk}\,e_j\otimes e_k$.

(c) The exceptional algebra is not the algebra of any holons; its colour-singlet part is ordinary. $J_3(\mathbb O)$ is not a Jordan subalgebra of any $\mathrm{Herm}(\mathbb C^n)$ (Albert 1934). So it is not the observable algebra of any number of holons, of a holon with its self-model, or of their registers. Its $\mathrm{SU}(3)_C$-fixed part is $\mathfrak h_3(\mathbb C_O)\cong\mathrm{Herm}(\mathbb C^3)$: dimension $9$, closed under the Jordan product. Its $G_2$-fixed part is $\mathfrak h_3(\mathbb R)$, dimension $6$. For the idempotent $E_1=\mathrm{diag}(1,0,0)$ the Peirce space $\{X: E_1\circ X=0\}$ is $\mathfrak h_2(\mathbb O)$, and its colour-fixed part is the $\mathfrak h_2(\mathbb C_O)$ of 48c, signature $(1,3)$.

(d) One centraliser: the rotations of 48c are the weak isospin of T-326. In the $\mathrm{Spin}(9)$ of the Clifford system on $\mathcal S=\mathbb C\otimes\mathbb O$ (T-326) the centraliser of $\mathfrak{su}(3)_C$ is a single $\mathfrak u(2)$. The colour-fixed part of the vector $\mathbb R^9$ is the three-plane $\mathrm{span}\{iL_{e_O},J,iJ\}$, and the $\mathfrak{su}(2)$ acts on it irreducibly. The same holds in the $\mathrm{Spin}(9)\subset\mathrm{Spin}(1,9)$ of 48c, with the three-plane $\{e_O\sigma_y,\sigma_z,\sigma_x\}$ (48c(c),(e)). So if 48c's spinor module $\mathbb O^2$ (part (f)) and T-326's $\mathcal S$ are one sixteen-dimensional module of one $\mathrm{Spin}(9)$ with one colour, then the spatial rotations of 48c are $\mathrm{SU}(2)_L$, and the three space directions are the weak triplet $(\mathbf 1,\mathbf 3)_0$.

(e) The sixteen dimensions carry one index, not two. A left-handed lepton doublet has $2\times2=4$ complex components: a Weyl index and an isospin index. In $(\mathcal S,L_{e_O})$ the lepton doublet $(\mathbf 1,\mathbf 2)_{-1/2}$ has $2$. So the sixteen real dimensions of $\mathbb O^2\cong\mathcal S$ carry either the Weyl index (48c(f), the reading of Manogue and Dray) or the isospin index (T-326(d)), not both. If they carry isospin, the Weyl index is a separate factor. The colour-singlet part $\mathbb C_O^2\subset\mathbb O^2$ of 48c(f) supplies it: $\mathbb C_O^2\otimes_{\mathbb C}(\mathcal S,L_{e_O})=\mathbf 2_{\text{Weyl}}\otimes[(\mathbf 3,\mathbf 2)_{1/6}\oplus(\mathbf 1,\mathbf 2)_{-1/2}]$ has $16$ complex dimensions, the left-handed doublets of one generation.

(f) (Q) as one principle. Let $\mathbb K\subseteq\mathbb O$ be a composition subalgebra that contains $e_O$: $\mathbb C_O$, one of the quaternion subalgebras through $e_O$, or $\mathbb O$. The following are equivalent: (i) $\mathrm{SU}(3)_C$ fixes $\mathfrak h_2(\mathbb K)$ pointwise; (ii) $\mathbb K=\mathbb C_O$; (iii) the Bloch ball of the two-level system $\mathfrak h_2(\mathbb K)$ (the unit ball of its traceless part) is three-dimensional; (iv) $\det$ on $\mathfrak h_2(\mathbb K)$ has signature $(1,3)$. By the theorem of Masanes, Müller, Pérez-García and Augusiak, (iii) holds exactly when two such systems, with continuous reversible dynamics and states determined by local measurements, can be entangled — equivalently, can interact.

Proof. (a) Let $s_1,s_2$ be Hermitian with $s_i^2=1$ and $s_1s_2=-s_2s_1$. If $s_1v=v$, then $s_1(s_2v)=-s_2v$. So the invertible $s_2$ maps the $+1$-eigenspace of $s_1$ into its $-1$-eigenspace, and back. The two eigenspaces have equal dimension, and $d$ is even. A unital $\mathrm{JSpin}(n)$ with $n\ge2$ contains such a pair: two orthogonal unit vectors of its traceless part square to $1$ and anticommute. Conversely, for $d=2k$ the operators $1$, $\sigma_z\otimes1_k$, $\sigma_x\otimes1_k$, $\sigma_y\otimes1_k$ span a unital $\mathfrak h_2(\mathbb C)$. And $7^M$ is odd. (b) The Casimir of $\mathfrak g_2$ takes four distinct values on $\mathbb C^{49}$, on eigenspaces of dimensions $1,7,14,27$ (numbers below). The irreducible representations of $G_2$ of dimension at most $27$ are $\mathbf 1,\mathbf 7,\mathbf{14},\mathbf{27}$, and two copies of one would share a Casimir value. So the decomposition has no multiplicity, and by Schur the commutant is $\mathbb C^4$. $\Lambda$ is tridiagonal with non-zero off-diagonal entries, so each eigenvalue has a one-dimensional eigenspace. Any operator commuting with $\Lambda$ preserves these lines and is a function of $\Lambda$. In the pair, $\mathbf 1\otimes\mathbf 1$, $\mathbf 3\otimes\bar{\mathbf 3}$ and $\bar{\mathbf 3}\otimes\mathbf 3$ each contain one singlet, and the other products none. (c) Albert proved that $J_3(\mathbb O)$ is exceptional ("On a certain algebra of quantum mechanics", Ann. Math. 35, 65–73 (1934)); it appears as the one exceptional case in the classification of Jordan, von Neumann and Wigner (Ann. Math. 35, 29–64 (1934)). A subalgebra of a special algebra is special, so there is no embedding. $G_2$ acts on $J_3(\mathbb O)$ entry by entry and fixes the diagonal. By 48c(a) an off-diagonal entry is colour-fixed exactly when it lies in $\mathbb C_O$, and $G_2$-fixed exactly when it is real. $\mathbb C_O$ is associative, so $\mathfrak h_3(\mathbb C_O)$ is the Jordan algebra of Hermitian $3\times3$ complex matrices. $E_1\circ X=0$ holds exactly when the first row and column of $X$ vanish, which leaves the lower $2\times2$ block, $\mathfrak h_2(\mathbb O)$. (d) In the proof of T-326(b), $\mathbb R^9=\mathbb R^6\oplus\mathbb R^3$ under colour, with $\mathbb R^3=\mathrm{span}\{iL_{e_O},J,iJ\}$ trivial. The centraliser is $\mathfrak u(1)\oplus\mathfrak{so}(3)$, and the $\mathfrak{so}(3)$ rotates $\mathbb R^3$. In 48c, $\mathfrak h_2(\mathbb O)$ splits as $\mathfrak h_2(\mathbb C_O)\oplus W$, and the compact part of the centraliser is $\mathfrak{su}(2)\oplus\mathfrak u(1)$, with the $\mathfrak{su}(2)$ rotating the three traceless directions of $\mathfrak h_2(\mathbb C_O)$. Both $\mathfrak{su}(2)$ are the derived algebra of the centraliser of the same $\mathfrak{su}(3)$ inside the same $\mathfrak{spin}(9)$ once the modules are identified, and a centraliser is unique. (e) Count the complex dimensions. The product of the Weyl and isospin indices needs $2\times2\times(3+1)=16$; $(\mathcal S,L_{e_O})$ has $8$. (f) (i)⇔(ii): by 48c(a) colour fixes in $\mathbb O$ exactly $\mathbb C_O$. A quaternion subalgebra through $e_O$ contains a unit vector orthogonal to $\mathbb C_O$, which colour moves (its orbit is $S^5$). (ii)⇔(iii)⇔(iv): $\mathfrak h_2(\mathbb K)$ has traceless part of dimension $\dim\mathbb K+1\in\{3,5,9\}$, and $\det$ has signature $(1,\dim\mathbb K+1)$. The last equivalence is the result of L. Masanes, M. P. Müller, D. Pérez-García and R. Augusiak (J. Math. Phys. 55, 122203 (2014), arXiv:1111.4060): among state spaces whose components are $d$-dimensional balls, with continuous reversible dynamics and local tomography, "except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories interacting dynamics is impossible." $\blacksquare$

Numbers (website/scripts/check_core_numbers.py: test_no_unital_spin_factor_on_any_holon_register, test_colour_singlet_part_of_the_exceptional_jordan_algebra_is_hermitian_c3, test_spatial_triplet_of_48c_is_the_weak_triplet):

• for $d=7$ and $d=49$ and every split $p+q=d$, a Hermitian operator anticommuting with $\mathrm{diag}(1_p,-1_q)$ has rank $2\min(p,q)<d$;
• the $\mathfrak g_2$ Casimir on $\mathbb C^{49}$ has eigenvalues $0,2,4,\tfrac{14}3$ with multiplicities $1,7,14,27$; there are $3$ colour singlets and $1$ $G_2$ singlet in the pair; the path Laplacian on $7$, $49$ and $343$ readings has simple spectrum;
• in the $27$-dimensional $J_3(\mathbb O)$ the colour-fixed part has dimension $9$ and is closed under the Jordan product, and the $G_2$-fixed part has dimension $6$; the Peirce space of $E_1$ has dimension $10$, and its colour-fixed part has dimension $4$ and $\det$-signature $(1,3)$;
• in the Clifford system of $\mathcal S$ the colour-fixed part of $\mathbb R^9$ is $\mathrm{span}\{iL_{e_O},J,iJ\}$, and the three-dimensional $\mathfrak{su}(2)$ of the centraliser acts on it with no common fixed vector.

What (Q) now rests on. Premise (L) [H]: the tangent vectors of spacetime at a point form the observable algebra $\mathfrak h_2(\mathbb K)$ of a two-level system whose amplitudes lie in a composition subalgebra $\mathbb K$ of the holon's octonions containing the clock unit $e_O$. Principle (MM): elementary systems of this kind can be entangled (Masanes–Müller–Pérez-García–Augusiak). By (f), (L) and (MM) together give exactly the conclusion of 48c under (Q): spacetime $\mathfrak h_2(\mathbb C_O)\cong\mathbb R^{1,3}$, a rotation group $\mathrm{SO}(3)$ that commutes with colour, and the Lorentzian sign of $\det$. Conversely, that conclusion satisfies (L) and (MM). (L) is weaker than (Q1): $\mathfrak h_2(\mathbb O)$ is the case $\mathbb K=\mathbb O$. Given (L), Coleman–Mandula's condition (Q2) is no longer a separate input from particle physics: it is equivalent to (MM), a principle from the reconstruction of quantum theory. What stays open is (L) itself, the "2". Parts (a)–(c) show that it cannot be found unitally in any Hilbert space built from holons, their registers or their self-models, nor $G_2$-covariantly in a pair or in the depth register, nor through $J_3(\mathbb O)$. Parts (d)–(e) add a constraint: the "2" is not the one of T-326's $\mathcal S$, or else space would be weak isospin. Status: 48c stays [T] as mathematics and [C at (Q)] as spacetime, with (Q) ⟺ (L) ∧ (MM); (L) [H]. Update (Theorem 48e): (MM) is a theorem of UHM, so the reading is [C at (L)].

What this means for 48c(f) and T-326(d). Both are true as mathematics. As physics they read the same sixteen dimensions differently: 48c(f) as lepton and quark Weyl spinors, T-326(d) as lepton and quark doublets. By (e) at most one reading holds on the same module. The consistent joint picture takes $\mathcal S$ with T-326's isospin and adds the Weyl index through the colour-singlet $\mathbb C_O^2$; the colour-triplet half $(\mathbf 2,\mathbf 3)$ of 48c(f) is then not a quark field. Theorem 48e(d)–(e) proves this picture and shows that it is forced.

Routes closed by 48d, and what each left. Depth register (§11.4): its readings are a chain, its dynamics a path Laplacian with simple spectrum, and a unital spin factor needs an even number of readings; nothing two-level is canonical in it. Pairs of holons and a holon with its self-model: odd dimension $49$; $G_2$-covariant observables commute; the colour-singlet sector is three-dimensional and holds non-unital copies of $\mathfrak h_2(\mathbb C)$. Correction (Theorem 48e(c)): the swap of the two factors selects one, the symmetric colour singlet $\mathrm{span}\{e_O\otimes e_O,\sum_{k\ne O}e_k\otimes e_k\}$; nothing in the pair ties its rotations to space. $J_3(\mathbb O)$ for three holons or a holon with its self-model: exceptional, so not an algebra of operators on any Hilbert space. Its colour-singlet part $\mathrm{Herm}(\mathbb C^3)$ is ordinary, and it reproduces 48c's $\mathfrak h_2(\mathbb C_O)$ as a Peirce space; the choice of the idempotent remains an input. Müller–Masanes: it does not derive (L), but under (L) it replaces (Q2), by (f). Clifford system of T-326 as the source of the "2": closed by (d)–(e).

Theorem 48e (Two-level systems of UHM are qubits; the Lorentz spinor is a separate factor of the fermion field) — [T] as mathematics

Theorem 48d leaves the 3+1 reading of 48c on two premises: (L), the "2", and (MM), the principle of Masanes and Müller. Theorem 48e removes (MM). Inside UHM it is a theorem, because every system of UHM is a system of complex quantum theory. The theorem also settles the mismatch of 48d(d)–(e) between 48c(f) and T-326(d). The Weyl index of the fermion field is the colour-fixed part $\mathbb C_O^2$ of the spinor of 48c. It is a separate tensor factor, next to the generation $\mathcal S_{\mathbb C}$ of T-329.

Theorem 48e — (a)–(i) [T]

(a) A two-level system of UHM is a qubit. Every face of the state space $\mathcal D(\mathbb C^N)$ is $\mathcal D(V)=\{\rho:\operatorname{supp}\rho\subseteq V\}$ for a subspace $V$. It is a Euclidean ball exactly when $\dim V\le2$. For $\dim V=1$ it is a point. For $\dim V=2$ it is the Bloch ball $B^3$, and its order-unit space $\mathrm{Herm}(V)\cong\mathfrak h_2(\mathbb C)$ carries $\det$ with signature $(1,3)$. A holon register has no two-level tensor factor, because $7^M$ is odd (48d(a)). So a two-level system of UHM is a rank-2 face of $\mathcal D(\mathbb C^{7^M})$ or of $\mathcal D(\mathcal S)$.

(b) (MM) holds in UHM. Let $\mathbb K\ni e_O$ be a composition subalgebra of $\mathbb O$. The state space of $\mathfrak h_2(\mathbb K)$ is the ball $B^{\dim\mathbb K+1}$. It is affinely isomorphic to a face of some $\mathcal D(\mathbb C^N)$ if and only if $\mathbb K=\mathbb C_O$. For two rank-2 faces $\mathcal D(V_1)$, $\mathcal D(V_2)$ of holon registers, the face $\mathcal D(V_1\otimes V_2)$ of the composite is the two-qubit state space. It is locally tomographic ($16=4\cdot4$), it contains entangled states, and it carries reversible dynamics that entangles product states. So the hypotheses of Masanes, Müller, Pérez-García and Augusiak hold for the two-level systems of UHM, and so does their conclusion, $d=3$.

(c) Colour-fixed two-level systems. The colour-fixed subspace is the line $\mathbb Ce_O$ in $\mathbb C^7$, so one holon has no colour-fixed rank-2 face. In $\mathcal S$ it is the plane $\mathrm{span}_{\mathbb C}\{\eta_0,e_O\}$, the lepton line of T-326. In $\mathbb C^7\otimes\mathbb C^7$ it is three-dimensional: the swap of the two factors is $+1$ on $\mathrm{span}\{e_O\otimes e_O,\ \sum_{k\ne O}e_k\otimes e_k\}$ and $-1$ on $\sum\varphi_{Ojk}\,e_j\otimes e_k$. So exactly two colour-fixed two-level systems are canonical: the lepton line of $\mathcal S$ and the swap-symmetric colour singlet of a pair. The lepton line has three Bloch rotations (unitary for the $i$ of $\mathcal H$), and only one of them commutes with the hypercharge of T-326. $G_2$ does not preserve the pair face.

(d) Rotations that commute with the gauge group do not act on the generation. The commutant of $\mathfrak g_{\mathrm{SM}}$ (T-329(d)) in $\mathrm{End}_{\mathbb R}(\mathcal S_{\mathbb C})$ has dimension $14$. Its intersection with $\mathfrak{so}(32)$ is abelian, of dimension $6$. So no $\mathrm{SU}(2)$ acting on $\mathcal S_{\mathbb C}$ commutes with $G_{\mathrm{SM}}$. The rotation by $2\pi$ of a spin-½ field is $-1$ on every component. The central element $-1$ of $\mathrm{SU}(2)_L$ is $-1$ on $V_L$ and $+1$ on $V_R$, so it is not that rotation. A rotation group that commutes with $G_{\mathrm{SM}}$ and makes the fermions spinors therefore acts on a separate factor $W$ of a fermion space $W\otimes_{\mathbb C}\mathcal S_{\mathbb C}$, and the smallest such $W$ is $\mathbb C^2$.

(e) The joint structure. Let $S=\mathbb C_O^2=(\mathbb O^2)^{\mathrm{SU}(3)_C}$ carry the $\mathrm{SL}(2,\mathbb C_O)$ of 48c(d),(f), with complex unit $e_O$. Then $F:=S\otimes_{\mathbb R}\mathcal S\;\cong\;S\otimes_{\mathbb C}\mathcal S_{\mathbb C},$ where the $\mathbb C'$ of T-329 acts as the complex unit of $S$. $F$ has real dimension $64$. On $F$ the algebras $\mathfrak{sl}(2,\mathbb C_O)$ (dimension $6$) and $\mathfrak{spin}(10)$ (dimension $45$) commute, their sum acts faithfully (dimension $51$), and their joint commutant is $\mathbb C$. So $F=(\mathbf 2,\mathbf{16})$ is irreducible of complex type. Under $\mathrm{SL}(2,\mathbb C_O)\times G_{\mathrm{SM}}$ it is sixteen left-handed Weyl fields $Q_L,L_L,u^c,d^c,e^c,\nu^c$, with $2\times16=32$ complex components; the lepton doublet has $2\times2=4$. Colour acts trivially on $S$. On the left half $F_L=S\otimes_{\mathbb C}V_L$ the complex unit of the Weyl factor acts as $+L_{e_O}$, and on $F_R$ as $-L_{e_O}$; the sign is the orientation of $\gamma_{10}$. The rotations $\mathfrak{su}(2)\subset\mathfrak{sl}(2,\mathbb C_O)$ commute with all of $\mathfrak{spin}(10)$ and meet $\mathfrak{su}(2)_L$ in zero.

(f) UHM's internal structure does not see the size of the spinor factor (2026-09-26). Let $\mathfrak A$ be the real algebra generated by the $\mathbb C'$-linear operators that UHM defines on the generation $\mathcal S_{\mathbb C}$: $\mathfrak{spin}(10)$ (with colour, $\mathfrak{su}(2)_L$, $\mathfrak{su}(2)_R$, $B-L$ and $Y$), $\mathfrak g_2$, and the lifts of $L_{e_O}$, $R_{e_O}$ and of the imaginary unit $i$ of $\mathcal H$. Its commutant in $\mathrm{End}_{\mathbb R}(\mathcal S_{\mathbb C})$ is $\mathbb C'=\mathrm{span}\{1,i'\}$. So for every $n\ge1$ the commutant of $1\otimes\mathfrak A$ on $F_n=\mathbb C^n\otimes_{\mathbb C}\mathcal S_{\mathbb C}$ is $M_n(\mathbb C')$, and the transformations of $F_n$ that preserve UHM's internal structure form $\mathrm{GL}(n,\mathbb C')$. Every statement of T-326, T-327, T-329 and of (e) about the generation holds on $F_n$, with multiplicities multiplied by $n$. Every anomaly trace is $n$ times that of the $\mathbf{16}$, so it vanishes, and $F_n$ is chiral. The ten Clifford generators are $\mathbb C'$-antilinear, so they act on $F_n$ only together with a structure on the first factor. UHM fixes the complex structure of the spinor factor — $i'$, which equals $+L_{e_O}$ on the left-handed fields — and nothing else about it. In particular it does not fix $n$.

(g) Two components are the only size that carries a relativistic causal structure. Let $W$ be a complex space of dimension $n\ge2$, and let $\mathrm{GL}(W)$ act on the Hermitian forms $\mathrm{Herm}(W)$ by $X\mapsto MXM^\dagger$. The following are equivalent: (i) $n=2$; (ii) $\mathrm{Herm}(W)$ carries a non-zero quadratic form invariant under $\mathrm{SL}(W)$; (iii) the forms of rank at most one, $\pm\psi\psi^\dagger$ (the pure states of $W$ with their scale), are the zero set of a quadratic form; (iv) the stabiliser $\mathrm{SU}(W)$ of a positive form acts transitively on the directions of the traceless forms; (v) $\mathfrak{sl}(W)$ is isomorphic to a Lorentz algebra $\mathfrak{so}(1,k)$; (vi) the state space $\mathcal D(W)$ is a ball; (vii) $W$ carries a non-zero bilinear form invariant under $\mathrm{SL}(W)$. In that case the form of (ii) is $\det$, unique up to a factor, of signature $(1,3)$; the zero set in (iii) is the light cone $\{\det=0\}$; $k=3$; and the form of (vii) is $\varepsilon$, the pairing of a Weyl mass term. Rotations alone do not decide: $\mathrm{SU}(W)$ preserves $(\operatorname{tr}X)^2$ and $\operatorname{tr}X^2$ for every $n$. The boosts decide.

(h) Chirality needs a complex unit, not two components. Let the Lorentz factor be real: $F=\mathbb R^m\otimes_{\mathbb R}\mathcal S$, with the $\mathfrak g_{\mathrm{SM}}$ of T-326 acting on $\mathcal S$ and a complex structure $J$ on $F$ that commutes with it. The commutant of $\mathfrak g_{\mathrm{SM}}$ on $\mathcal S$ is $\mathbb C\oplus\mathbb C$ (the quark block and the lepton line), so $F\cong p\,Q_L\oplus(m-p)\,\bar Q_L\oplus r\,L_L\oplus(m-r)\,\bar L_L$. The $\mathrm{SU}(3)^3$ anomaly of $F$ is $2(2p-m)$. When it vanishes, the $Y^3$ anomaly is $-\tfrac14(2r-m)$. So $F$ is anomaly-free only if it is vectorlike. A chiral, anomaly-free generation built on $\mathcal S$ needs a complex unit that does not come from $\mathcal S$. T-329 takes it from the Lorentz factor, and by (f) the result is chiral and anomaly-free for every $n$.

(i) Neither the history of the depth register nor the two slots of a self-model rotate the spinor factor. (1) Take the Feynman–Kitaev constraint on the depth register with steps $U_1,\dots,U_N$. Through $\mathcal V=\sum_t\lvert t\rangle\langle t\rvert\otimes U_t\cdots U_1$ its propagation Hamiltonian is unitarily equivalent to $\tfrac12L_N\otimes1$, where $L_N$ is the Laplacian of the path on the $N+1$ readings. The spectrum of $L_N$ is simple, so the operators on the readings that commute with it form an abelian algebra, and no $\mathrm{SU}(2)$ on the readings commutes with the history. The qubits of Kitaev's unary clock are an encoding, not UHM's register, which is positional in base 7. (2) A holon and its self-model as two slots have the algebra $M_7\oplus M_7$. Its centre is $\mathbb C^2$, so the slot bit is classical, and neither block contains two anticommuting Hermitian involutions. A qubit of slots would need coherences between the slots, which UHM does not define. With them, two slots of the generation, $\mathcal S_{\mathbb C}\oplus\mathcal S_{\mathbb C}=\mathbb C^2\otimes\mathcal S_{\mathbb C}$, would be $F_2$. That would supply a two-dimensional $W$, but it would be an input, and it would still not tie $\mathrm{Herm}(W)$ to spacetime.

Proof. (a) Let $\Phi$ be a face and $\rho_0$ a point in its relative interior, with $V=\operatorname{supp}\rho_0$. For $\sigma\in\mathcal D(V)$ there is $\varepsilon>0$ with $\rho_0\ge\varepsilon\sigma$. Then $\rho_0=\varepsilon\sigma+(1-\varepsilon)\tau$ for a state $\tau$, and $\sigma\in\Phi$. Conversely, for $\rho\in\Phi$ we can write $\rho_0=t\rho+(1-t)\rho'$ with $\rho'\in\Phi$ and $t>0$, so $\operatorname{supp}\rho\subseteq V$. Hence $\Phi=\mathcal D(V)$. For $\dim V\ge3$, $\mathcal D(V)$ has a boundary point that is not extreme (a state of rank $2$); a ball has none. For $\dim V=2$, $\rho=\tfrac12(P+r\cdot\sigma_V)\ge0$ exactly when $\lvert r\rvert\le1$, and $\det(t\,1+x\cdot\sigma)=t^2-\lvert x\rvert^2$. (b) The states of $\mathfrak h_2(\mathbb K)$ are the $X$ with $\operatorname{tr}X=1$ and $\det X\ge0$. They form a ball of dimension $\dim\mathbb K+1\in\{2,3,5,9\}$. By (a) only $3$ occurs, and the composition subalgebra of dimension $2$ that contains $e_O$ is $\mathbb C_O$. By (a), $\mathcal D(V_1\otimes V_2)\cong\mathcal D(\mathbb C^2\otimes\mathbb C^2)$ is a face of $\mathcal D(\mathbb C^7\otimes\mathbb C^7)$. The Bell state of $V_1\otimes V_2$ has a partial transpose with eigenvalue $-\tfrac12$. The unitary $\exp(-i\tfrac\pi4\,\sigma_x\otimes\sigma_x)$ on $V_1\otimes V_2$, extended by the identity, maps a product state to a state with the same negative eigenvalue. (c) Under colour, $\mathbf 7=\mathbf 1\oplus\mathbf 3\oplus\bar{\mathbf 3}$, and $\mathcal S$ adds one singlet, $\eta_0$. In $\mathbf 7\otimes\mathbf 7$ the singlets are $\mathbf 1\otimes\mathbf 1$ and one in each of $\mathbf 3\otimes\bar{\mathbf 3}$ and $\bar{\mathbf 3}\otimes\mathbf 3$; the swap exchanges the last two. On the lepton line, $i$ and $L_{e_O}$ are commuting complex structures. The Bloch rotations commute with $i$, and the hypercharge of T-326 acts on the line as $-\tfrac12L_{e_O}$. So only the rotation about the $L_{e_O}$-axis commutes with it (computed). $G_2$ does not fix $e_O$. (d) Under $\mathfrak g_{\mathrm{SM}}$, $\mathcal S_{\mathbb C}$ splits into $Q_L,L_L,u^c,d^c,e^c$ and $\nu^c$. The first five are real-irreducible of complex type and pairwise non-isomorphic; $\nu^c$ is twice the trivial real module. By Schur's lemma the commutant is $\mathbb C^5\oplus M_2(\mathbb R)$, of dimension $14$, and its compact part is $\mathfrak u(1)^5\oplus\mathfrak{so}(2)$. The action of $-1\in\mathrm{SU}(2)_L$ is computed. A group $K$ that commutes with $G_{\mathrm{SM}}$ on a sum of copies of $\mathcal S_{\mathbb C}$ acts on the multiplicity space. A complex $\mathrm{Spin}(3)$-module on which $-1$ acts as $-1$ has even dimension, at least $2$. (e) For a complex space $S$ and a real space $\mathcal S$, $S\otimes_{\mathbb R}\mathcal S\cong S\otimes_{\mathbb C}(\mathcal S\otimes_{\mathbb R}\mathbb C)$. This is the step of T-329, with $\mathbb C'$ the complex numbers of $S$. $F$ is realised inside $S\otimes_{\mathbb R}\mathcal S_{\mathbb C}$ as the subspace where the two complex units agree. The dimensions, the joint commutant, the hypercharges and the action of the complex unit on $F_L$ and $F_R$ are computed. $\mathfrak{sl}(2,\mathbb C_O)$ acts on the first factor and $\mathfrak{spin}(10)$ on the second, so they commute and meet in zero. (f) $\mathfrak{spin}(10)$ acts on $\mathcal S_{\mathbb C}$ as the $\mathbf{16}$, which is irreducible of complex type (T-329(a)), so by Schur's lemma its commutant is $\mathrm{span}\{1,\omega\}=\mathrm{span}\{1,i'\}$. Every other generator of $\mathfrak A$ is the $\mathbb C'$-linear extension of an operator on $\mathcal S$ and commutes with $i'$. So the commutant of $\mathfrak A$ is $\mathbb C'$ (computed: dimension $2$). On $F_n\cong\mathcal S_{\mathbb C}^{\oplus n}$ the commutant of $1\otimes\mathfrak A$ is $M_n(\mathfrak A')=M_n(\mathbb C')$, and $\operatorname{Tr}_{F_n}X^k=n\operatorname{Tr}_{\mathbf{16}}X^k$. The nine $\gamma_aK'$ and $\gamma_{10}=i'K'$ contain the conjugation $K'$ once, so they anticommute with $i'$. The value of $i'$ on $V_L$ is T-329(b). (g) (i) implies the rest by the standard spinor correspondence (Penrose and Rindler, vol. 1): $\det(MXM^\dagger)=\lvert\det M\rvert^2\det X$, $\det=0$ exactly on the rank-one forms, $\mathrm{SU}(2)\to\mathrm{SO}(3)$ is onto, $\mathfrak{sl}(2,\mathbb C)\cong\mathfrak{so}(1,3)$, $\mathcal D(\mathbb C^2)=B^3$, and $\varepsilon$ is $\mathrm{SL}(2)$-invariant. (vii)⇒(i): $W\otimes W=S^2W\oplus\Lambda^2W$ with both summands irreducible, and a trivial summand occurs only when $\Lambda^2W$ is one-dimensional. (ii)⇒(i): $\mathrm{Herm}(W)\otimes\mathbb C=W\otimes\bar W$, and the complexification $\mathfrak{sl}_n\oplus\mathfrak{sl}_n$ of $\mathfrak{sl}(W)$ acts on it factor by factor. An invariant bilinear form on it is a sum of products of invariant forms on the two factors, so it exists only if (vii) holds. (iii)⇒(i): if the quadratic form is semidefinite, its zero set is a linear subspace; the rank-one forms span $\mathrm{Herm}(W)$, so the form would vanish. If it is indefinite, its zero set has dimension $n^2-1$, while the rank-one forms have dimension $2n-1$; $2n-1=n^2-1$ gives $n=2$. (iv)⇒(i): an orbit of $\mathrm{SU}(n)$ on its Lie algebra has dimension at most $n^2-n$, and the sphere of directions has dimension $n^2-2$. (v)⇒(i): the Killing form of $\mathfrak{sl}(n,\mathbb C)$ as a real algebra has signature $(n^2-1,n^2-1)$, and that of $\mathfrak{so}(1,k)$ has signature $(k,k(k-1)/2)$. Equality forces $k=3$ and $n=2$. (vi)⇔(i) is (a). (h) $Q_L$ and $L_L$ are real-irreducible of complex type and not isomorphic (T-327). On each block the commutant is $\mathbb C$ with unit $L_{e_O}$, so $J$ has eigenvalues $\pm L_{e_O}$ on each of the $m$ copies. The quark doublet contributes $+2$ to the $\mathrm{SU}(3)^3$ anomaly (two triplets) and its conjugate $-2$. The lepton doublet contributes $2(-\tfrac12)^3=-\tfrac14$ to $Y^3$ and its conjugate $+\tfrac14$. The commutant, the charges $\{\tfrac16:6,\,-\tfrac12:2\}$ with $\sum Y^3=-\tfrac29$, and the sign change of the colour anomaly are computed. (i) (1) $\mathcal V^\dagger H\mathcal V=\tfrac12L_N\otimes1$ is the standard step of the Feynman–Kitaev construction (Feynman, "Quantum mechanical computers", Found. Phys. 16, 507–531 (1986); Kitaev, Shen and Vyalyi, Classical and Quantum Computation, AMS 2002). The eigenvalues of $L_N$ are $2-2\cos(\pi k/(N+1))$, $k=0,\dots,N$, all distinct. An operator with simple spectrum commutes only with polynomials in itself. (2) An involution is $\operatorname{diag}(1^p,-1^q)$ in a suitable basis; the solutions of $s_1s_2=-s_2s_1$ are the off-diagonal blocks, of rank at most $2\min(p,q)<7$. $\blacksquare$

Numbers (website/scripts/check_core_numbers.py: test_two_level_systems_of_uhm_are_qubits_and_can_be_entangled, test_colour_fixed_two_level_faces_of_holon_registers, test_no_rotation_of_the_internal_generation_commutes_with_the_gauge_group, test_fermion_space_is_weyl_spinor_times_one_generation):

• for random planes $V\subset\mathbb C^7$ and $V\subset\mathbb C^{49}$, $\tfrac12(P+r\cdot\sigma_V)\ge0$ exactly when $\lvert r\rvert\le1$ (40 samples each); the Bloch balls of $\mathfrak h_2(\mathbb K)$ have dimensions $2,3,5,9$, and only $3$ is the dimension of a ball-shaped face;
• in $\mathbb C^7\otimes\mathbb C^7$ the face of two random planes is locally tomographic (rank $16$); its Bell state has partial-transpose eigenvalue $-\tfrac12$, and $\exp(-i\tfrac\pi4\sigma_x\otimes\sigma_x)$ takes a product state to eigenvalue $-\tfrac12$;
• colour-fixed subspaces: $1$ in $\mathbb C^7$, $2$ in $\mathcal S$ (on $\eta_0,e_O$), $3$ in $\mathbb C^{49}$ with swap eigenvalues $(+1,+1,-1)$; the commutators of the three Bloch rotations of the lepton line with $Y$ have rank $2$;
• the commutant of $\mathfrak g_{\mathrm{SM}}$ on $\mathcal S_{\mathbb C}$ has dimension $14$, its compact part $6$ and abelian;
• on $F$: dimension $64$; the map $\psi\otimes s\mapsto\psi\otimes(s\otimes1)$ into $S\otimes_{\mathbb C}\mathcal S_{\mathbb C}$ has rank $64$ and intertwines $\mathfrak{spin}(9)$; $\dim(\mathfrak{sl}(2,\mathbb C)+\mathfrak{spin}(10))=51$, joint commutant $2$; complex multiplicities of $Y$: $\tfrac16$ twelve, $-\tfrac12$ four, $-\tfrac23$ and $\tfrac13$ six each, $1$ and $0$ two each, total $32$; the Weyl unit is $+L_{e_O}$ on $F_L$ and $-L_{e_O}$ on $F_R$; rotations and $\mathfrak{su}(2)_L$ span $6$ dimensions and commute.

Numbers for (f)–(i) (test_uhm_internal_structure_is_blind_to_the_multiplicity_of_the_fermion_field, test_only_a_two_component_spinor_factor_carries_a_relativistic_causal_structure, test_a_real_lorentz_factor_gives_an_anomalous_or_vectorlike_generation, test_depth_register_history_and_the_two_slots_supply_no_spinor_rotation):

• the commutant of $\mathfrak A$ ($\mathfrak{spin}(10)$, $\mathfrak g_2$, $L_{e_O}$, $R_{e_O}$, $i$) in $\mathrm{End}_{\mathbb R}(\mathbb R^{32})$ has dimension $2$ and is $\mathrm{span}\{1,i'\}$; all ten $\gamma_a$ anticommute with $i'$; $i'=\pm L_{e_O}$ on $V_L$;
• $\mathrm{SL}(W)$-invariant quadratic forms on $\mathrm{Herm}(W)$: $1$, $0$, $0$ for $n=2,3,4$; the one for $n=2$ is $\det$, signature $(1,3)$. $\mathrm{SU}(W)$-invariant ones: $2$ for each $n$;
• the rank-one cone has dimension $2n-1=3,5,7$ against $n^2-1=3,8,15$; the largest $\mathrm{SU}(n)$-orbit on traceless forms has dimension $n^2-n=2,6,12$ against $n^2-2=2,7,14$; the Killing signature of $\mathfrak{sl}(n,\mathbb C)$ is $(3,3)$, $(8,8)$, $(15,15)$, and only $(3,3)$ is that of an $\mathfrak{so}(1,k)$; $\mathrm{SL}(W)$-invariant bilinear forms on $W$: $1$ for $n=2$, $0$ for $n=3$;
• the commutant of $\mathfrak g_{\mathrm{SM}}$ on $\mathcal S$ has dimension $4$ and is commutative; with $J=L_{e_O}$ the charges are $\{\tfrac16:6,\,-\tfrac12:2\}$ and $\sum Y^3=-\tfrac29$; flipping $J$ on the quark block flips the non-zero colour anomaly;
• Feynman–Kitaev with $N=6$ random steps on $\mathbb C^7$: $\mathcal V^\dagger H\mathcal V=\tfrac12L_6\otimes1$, the spectrum of $L_6$ is simple, its commutant in $M_7(\mathbb C)$ has dimension $7$ and is abelian; for $p+q=7$ every solution of $s_1s_2=-s_2s_1$ has rank $2\min(p,q)\le6$.

What the 3+1 reading now rests on. By (a)–(b), (MM) is no longer a principle added to UHM. For the two-level systems that UHM can have, it is a theorem, and it fixes $\mathbb K=\mathbb C_O$ without the classification of Masanes and co-authors. The 3+1 reading of 48c therefore rests on one premise, (L) [H]: the tangent vectors of spacetime at a point form the observable space of a two-level system whose amplitudes lie in a composition subalgebra of $\mathbb O$ that contains $e_O$. Status: 48c is [T] as mathematics and [C at (L)] as physical spacetime; before 48e it was [C at (Q)], with (Q) ⟺ (L) ∧ (MM). (Q) is a stronger premise and still suffices. The status is not [T], because (L) is not derived. Parts (a)–(e) say this about it. The "2" exists inside UHM only as a face (a). A colour-fixed "2" is canonical in two places, and neither is spacetime (c): the lepton line fails Coleman–Mandula through hypercharge, and no symmetry of the pair induces the Bloch rotations of the pair face. A rotation group that commutes with the gauge group acts on a factor outside the generation (d). So (L) is the premise that T-329 already uses when it takes the fermion field to be a two-component Weyl field; call it (W). Given (W), the Weyl spinor $S_+\cong\mathbb C^2$ is a $\mathbb C_O$-module by (e): on the left-handed fields its complex unit is the clock unit. The Hermitian forms on $S_+$ — the tangent vectors in the standard correspondence $x\mapsto x^\mu\sigma_\mu$ (Penrose and Rindler, Spinors and Space-Time, vol. 1, Cambridge 1984) — are then $\mathfrak h_2(\mathbb C_O)$, and colour acts on them trivially. Given (L), 48c(f) supplies the Weyl spinor $\mathbb C_O^2$. The spacetime sector and the matter sector now rest on one premise. Correction (2026-09-26, parts (f) and (h)): the matter sector does not use the "2". T-329 needs only that the spinor factor is a complex space, of any dimension; call this (W₀). By (h), a chiral, anomaly-free generation needs such a complex unit. The "2" of (W) is used by the spacetime reading alone.

The mismatch of 48d(d)–(e), resolved. 48c(f) and T-326(d) read one sixteen-dimensional module in two ways. By (e) the fermion field is $\mathbb C_O^2\otimes_{\mathbb C}\mathcal S_{\mathbb C}$. The Lorentz index is the colour-fixed part of the spinor $\mathbb O^2$ of 48c; isospin, hypercharge and colour live in $\mathcal S$ (T-326, T-329). The two $\mathrm{Spin}(9)$'s of 48d(d) are not identified. The rotations of 48c act on the first factor and weak isospin on the second, and the two commute. So space is not the weak triplet. Of 48c(f) the mathematics stays; its reading of the colour-triplet part $(\mathbf 2,\mathbf 3)$ as a quark field is withdrawn. A quark field takes colour and isospin from $\mathcal S$ and its Weyl index from $\mathbb C_O^2$. The count is $2\times16=32$ complex components per generation and $4$ for the lepton doublet, as the Standard Model needs.

Routes tried for (L) in 48e, and what each gave. Depth register: its rank-2 faces are qubits by (a), but its dynamics fixes none of them (48d(b)), and nothing ties their rotations to space. Page–Wootters clock: seven levels with one-dimensional $\mathbb Z_7$-irreducibles (48c); its rank-2 faces are qubits, and none is canonical. The Bloch sphere of $\mathfrak h_2(\mathbb C_O)$: every two-level system of UHM has the Bloch ball $B^3$ (a). This is what discharges (MM), but it does not say which qubit is spacetime. Correlated stationary composites (CC-7, for almost every anchor): UHM's own coupling does not preserve products, so its dynamics is interacting, which is the physical side of (MM). The discharge in (b) needs only the complex quantum theory of the composite. Colour-fixed faces: see (c).

The "2" from a principle without a number (2026-09-26). By (f), (W) cannot be derived from UHM's internal structure, because that structure is the same for every $n$. What fixes $n$ must act on the spinor factor itself. The weakest principle we found that does so names no number:

(P) [H]. The tangent vectors of spacetime at a point are the Hermitian forms on the spinor factor $W$ of the fermion field $F=W\otimes_{\mathbb C}\mathcal S_{\mathbb C}$. Spacetime has a direction other than time ($\dim W\ge2$). Its causal structure, a quadratic form given up to a factor, is preserved by every transformation of $W$ that preserves UHM's internal structure.

By (f) these transformations form $\mathrm{GL}(W)$. The factor is a continuous homomorphism $\mathrm{GL}(W)\to\mathbb R^\times$, and it is trivial on $\mathrm{SL}(W)$, which equals its own commutator group. So the form is $\mathrm{SL}(W)$-invariant, and by (g) $n=2$, the form is $\det$ and its signature is $(1,3)$. The complex unit of $W$ is $i'$, which is $L_{e_O}$ on the left-handed fields, so $\mathrm{Herm}(W)=\mathfrak h_2(\mathbb C_O)$. Hence (P) implies (W) and (L). Conversely, (L) and (W) together with (e) satisfy (P), because $\det(MXM^\dagger)=\lvert\det M\rvert^2\det X$. So (P) ⟺ (L) ∧ (W) [T, given the theorems of UHM used in (e)–(g)]. The dimension $2$ of the spinor, the dimension $4$ of spacetime and the signature $(1,3)$ are now consequences. The premise says only that spacetime is built from the spinor factor (Penrose's programme) and that it is relativistic: boosts, not only rotations, preserve its light cone. By (g), (P) can be stated with any of (iii)–(vii) in place of the quadratic form. The light rays are the pure states of the spinor factor (iii). Space is isotropic in every rest frame (iv). The group of the spinor factor is a Lorentz group (v). The spinor factor is a bit, which is the premise of Müller and Masanes (vi). A Lorentz-invariant Weyl mass term exists (vii).

Status after (f)–(i). 48c stays [T] as mathematics and [C at (L)] as physical spacetime, now with (L) ⟺ (P). The reading is not [T], because (P) is not derived. By (f) no structure internal to UHM can derive it: this obstruction is now a theorem, not a list of failed attempts. What changed is the premise. It no longer contains a number, and the matter sector (T-329) does not depend on it.

What (P) contains, and how it is counted (2026-09-26). As formulated, (P) names the module $\mathcal S_{\mathbb C}$ of the fermion field, so it already contains (Cl₀) — fermions are vectors of the spinor module $\mathcal S$ of the octonionic Clifford system (Standard Model, §2.6) — and (W₀). It splits as (P) = (Cl₀) ∧ (P*), where (P*) is (P) with an arbitrary fermion module $M$ whose internal commutant is $\mathbb C$ in place of $\mathcal S_{\mathbb C}$. The two parts are independent [T]. The field $F_3 = \mathbb C^3\otimes_{\mathbb C}\mathcal S_{\mathbb C}$ satisfies (Cl₀) and not (P*): by (g) $\mathrm{Herm}(\mathbb C^3)$ carries no $\mathrm{SL}$-invariant quadratic form. The field $F = \mathbb C^2\otimes_{\mathbb C}\mathbb C^7$ — the holon's own vectors, with $\mathfrak g_2$ and the complex unit $i$ of $\mathcal H$ as the internal structure — satisfies (P*) and not (Cl₀): the commutant of $\{\mathfrak g_2, i\}$ on $\mathbb R^{14}$ is $\mathbb C$ (dimension $2$; without $i$ it is $M_2(\mathbb R)$, dimension $4$), so the structure-preserving transformations of $W$ are $\mathrm{GL}(W)$ and $\det$ is preserved up to a factor, as in the argument above; but $\dim_{\mathbb R}\mathbb C^7 = 14$ is not a multiple of $16$, so $\mathbb C^7$ is not a $\mathrm{Cl}_7$-module. The inputs of the spacetime and matter readings are therefore counted as (Cl₀) + (P) relative to (Cl₀): two independent premises, not one (Premises of UHM, §7; test_spinor_factor_premise_and_fermion_module_premise_are_independent).

Routes tried for (W) on 2026-09-26, and what each gave. (a) The clock's complex structure. It gives the complex structure of $W$ for every $n$: $i'=L_{e_O}$ on the left-handed fields (f). The minimal left ideal of $\mathfrak h_2(\mathbb C_O)\otimes\mathbb C\cong M_2(\mathbb C)$ is $\mathbb C^2$, and that of $\mathrm{Cl}(1,3)\cong M_2(\mathbb H)$ is $\mathbb H^2$, two Weyl spinors $\mathbb C^2$. But the first algebra is 48c's conclusion and the second presupposes $1+3$. The Hermitian forms on $W$ form a formally real Jordan algebra for every $n$, so reality does not decide either. (b) The history state of the depth register: closed by (i)(1). (c) A holon with its self-model as two slots: closed by (i)(2). It would give $n=2$ only with coherences between the slots, which is an input. (d) Chirality and the test of Distler and Garibaldi: they force a complex unit outside $\mathcal S$, which is (W₀), and they hold for every $n$ (h), (f). (e) The reconstruction of Masanes and Müller: its premise, that the smallest system carries spatial directions, is (g)(vi) applied to $W$. It is one of the equivalent forms of (P), not a derivation of it. From it they derive $d=3$, as (g) does. (f) The spinor bundle of T-119's $S^3$: its rank is $2^{\lfloor3/2\rfloor}=2$, so the "2" is present. But its frame rotations mix T-119's coordinates, two of which are colour charges (T-119(d)), so they do not commute with colour. Reading this bundle as the spin of fermions meets Coleman–Mandula. (g) Minimality of the spinor factor (T-347(d)). By (f) the smallest $W$ compatible with T-329 is $\mathbb C^1$: $F_1=\mathcal S_{\mathbb C}$ is chiral and anomaly-free, but $\mathrm{Herm}(\mathbb C^1)=\mathbb R$ has no spatial direction. Minimality given a spatial direction yields $n=2$ only by restating $\dim W\ge2$; the content of (P) is the boost clause of (g).

Theorem (Sectoral decomposition of dimensionality) — retracted [✗]

Retracted formulation [✗], kept as a record (reason in the box above)

The seven dimensions of UHM decompose under the action of the vacuum $SU(3)_C$-symmetry into three classes with different physical scales. From this decomposition a 3+1-dimensional effective spacetime follows. Conditional on the sector asymmetry hypothesis (SA).

The popular gloss that accompanied it: one axis ($O$) becomes time, three ($A,S,D$) become space because they "correspond to massless gluons", three ($L,E,U$) are compact because they "correspond to massive $W$- and $Z$-bosons".

Former statement. The seven dimensions of UHM decompose under the action of the vacuum $SU(3)_C$-symmetry:

$7 = \underbrace{1}_{O \,(\text{time})} \;\oplus\; \underbrace{3}_{\{A,S,D\}\,(\text{space})} \;\oplus\; \underbrace{\bar{3}}_{\{L,E,U\}\,(\text{compact})}$

From this decomposition a 3+1-dimensional effective spacetime follows.

Former proof (Step 1 survives on its own; Steps 2–5 are retracted with the statement).

Step 1. Emergent time from $O$ [T].

Page–Wootters mechanism: the dimension $O$ (Foundation) serves as internal clock:

$\Gamma(\tau) = \frac{\text{Tr}_O\left[(|\tau\rangle\langle\tau|_O \otimes \mathbb{1}_{6D}) \cdot \Gamma_{\text{total}}\right]}{p(\tau)}$

Time $\tau$ is the parameter of conditional states. This is 1 temporal dimension [T].

Step 2. Sectoral hierarchy of Gap-scales — retracted as a consequence of $\mathrm{SU}(3)_C$ [✗].

The table below is an ansatz for the vacuum Gap profile on sets of axis pairs (Gap-thermodynamics, Consequences of axiomatics). It is not $\mathrm{SU}(3)_C$-invariant: an $\mathrm{SU}(3)_C$-invariant $\Gamma$ has off-diagonal entries only on the pairs $(A,D)$, $(S,U)$, $(L,E)$ (test_su3_invariant_states_are_coherent_only_on_o_line_pairs), so an equal Gap on the nine pairs $\{A,S,D\}\times\{L,E,U\}$ is not colour isotropy:

Sector	Dimensions	Gap	Physical scale
$O$-to-all	$O \times \{1,...,6\}$	$\sim 1$	$M_{\text{Planck}}$
$\mathbf{3}$-to-$\bar{\mathbf{3}}$	$\{A,S,D\} \times \{L,E,U\}$	$\approx 0$	$\Lambda_{\text{QCD}} \sim 200$ MeV
$\mathbf{3}$-to-$\mathbf{3}$	$\{A,S,D\}^2$	$\sim \varepsilon$	Intermediate
$\bar{\mathbf{3}}$-to-$\bar{\mathbf{3}}$	$\{L,E,U\}^2$	$\sim \varepsilon_{\text{EW}} \sim 10^{-17}$	$v_{\text{EW}} \sim 246$ GeV

Step 3. $\mathbf{3}$-sector: non-compact spatial dimensions — retracted [✗] ($\{A,S,D\}$ is not the $\mathbf{3}$, and the gluons of $\mathrm{SU}(3)_C$ act on all six non-$O$ axes).

The three dimensions $\{A, S, D\}$ generate $SU(3)_C$ gauge fields (gluons). The confinement sector $\mathbf{3}$-to-$\bar{\mathbf{3}}$ with Gap $\approx 0$ means:

• Gluons are massless → long-range interaction
• Confinement forms extended structures (hadrons, nuclei, atoms)
• Spatial extension is determined by the absence of mass of gluons: massless gauge bosons → the spatial structure does not curl up

Step 4. $\bar{\mathbf{3}}$-sector: compact internal dimensions — retracted [✗] ($\{L,E,U\}$ is not the $\bar{\mathbf{3}}$).

Where the electroweak group comes from instead (2026-09-25): not from three axes but from the Clifford system of $\mathbb C\otimes\mathbb O$ — $\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$ is the centraliser of colour in its $\mathrm{Spin}(9)$, and $G_{\mathrm{SM}}$ is the normaliser of colour (T-326; [T] as mathematics, [C at (Cl)] in UHM). No compactification is involved, and the spacetime directions are those of Theorem 48c.

The three dimensions $\{L, E, U\}$ generate the electroweak sector $SU(2)_L \times U(1)_Y$. The Higgs mechanism ($\langle \gamma_{EU} \rangle \neq 0$) gives mass to $W^\pm, Z$-bosons:

• $W, Z$ are massive → short range ($r \lesssim 1/M_W \sim 10^{-16}$ cm)
• The $\bar{\mathbf{3}}$-sector is "curled up" at the scale $\sim 1/v_{\text{EW}}$
• Effective compactification radius: $R_{\text{EW}} \sim 1/v_{\text{EW}} \sim 10^{-17}$ cm

Step 5. Result: 3+1 from 7 = 1+3+3̄ — retracted [✗] (it rests on Steps 3 and 4).

$\underbrace{\text{time}}_{O \;\to\; \tau} + \underbrace{\text{3D space}}_{\{A,S,D\} \;\to\; \text{massless gluons}} + \underbrace{\text{3 compact}}_{\{L,E,U\} \;\to\; \text{massive } W^\pm, Z}$

Observable spacetime = $M^{3+1}$ — the low-energy limit:

$M^{3+1} = \{O\text{-time}\} \times \{A,S,D\text{-space}\}$

The $\bar{\mathbf{3}}$-dimensions are "frozen" below the electroweak scale and appear as internal quantum numbers (weak isospin, hypercharge). $\blacksquare$

Dependence on (SA) — superseded by the retraction

This box used to grade the split as "[T|SA]" and to call "$\mathrm{Im}(\mathbb{O})\cong\mathbb{R}^7=\mathbb{R}^1\oplus\mathbb{R}^3\oplus\mathbb{R}^3$ under $\mathrm{SU}(3)\subset G_2$" standard mathematics. That real splitting is retracted [✗]: it is not invariant (box at the top of this section). No hypothesis (SA) can restore it, because it is a statement of representation theory, not of dynamics; (SA) survives only as an assumption about the vacuum Gap profile on axis pairs (fermion generations, §4.4).

Consequence: dimensionality of space — retracted [✗]

$\dim(\text{space}) = |\mathbf{3}| = 3$

Retracted with the theorem. The fundamental representation of $\mathrm{SU}(3)$ has complex dimension 3, but it is spanned by $A-iD$, $S-iU$, $L-iE$, not by three axes; its real form is six-dimensional and irreducible. A count of spatial directions from $\dim_{\mathbb C}\mathbf{3}=3$ therefore needs a separate argument; T-119 supplies one — [C] until 2026-09-25, then [T] as mathematics (the computed spectrum of three commuting rotation charges), with colour-charged coordinates.

Precedents and related programmes

Two older literatures meet in this section. The octonionic lineage found the split $7=1\oplus3\oplus\bar{3}$ in 1973 and read it as colour, and it later found how the choice of one imaginary unit cuts ten-dimensional spacetime down to four. A separate, century-old line of argument asks why space has three dimensions and time one. Below: what each established, where it stands, and how the reading above differs from it.

The split $7=1\oplus3\oplus\bar{3}$ is prior art (Günaydın and Gürsey, 1973). The decomposition of the imaginary octonions under the subgroup $\mathrm{SU}(3)\subset G_2$ that fixes one unit entered physics with Günaydın and Gürsey ("Quark structure and octonions", J. Math. Phys. 14, 1651–1667 (1973), DOI 10.1063/1.1666240), who read the triplet as quark colour; details and standing are in G₂-structure, §2.6. The mathematical half of the theorem above — the half that the (SA) box calls standard mathematics — is therefore not a UHM result, and the novelty of this page can lie only in the physical reading.

The precedent also says what the triplet is, and the labels used above do not match it. Fix the unit $O=e_7$. Left multiplication by $e_7$ is a complex structure on the six other axes; with the table of G₂-structure, §2 it pairs $A$ with $D$, $S$ with $U$ and $L$ with $E$, and Todorov and Dubois-Violette write the same split with the same Fano labelling (Int. J. Mod. Phys. A 33, 1850118 (2018), eq. 2.5, arXiv:1806.09450). $\mathrm{SU}(3)$ acts on these six axes as on $\mathbb{C}^3$; after complexification the triplet $3$ is spanned by $A-iD$, $S-iU$, $L-iE$ (up to a sign convention) and the anti-triplet $\bar{3}$ by their complex conjugates. Two consequences refute the statement of the theorem, which is retracted accordingly (box at the top of the sectoral decomposition):

• the sets $\{A,S,D\}$ and $\{L,E,U\}$ are not the $3$ and the $\bar{3}$. No three of the six axes span an $\mathrm{SU}(3)$-invariant subspace, and none can: $\mathrm{SU}(3)$ would act on a real three-dimensional invariant subspace through a homomorphism $\mathrm{SU}(3)\to\mathrm{SO}(3)$, which must be trivial because the simple eight-dimensional group has no non-trivial homomorphism into the three-dimensional $\mathrm{SO}(3)$ — yet $\mathrm{SU}(3)$ fixes no non-zero vector among these six axes;
• the (SA) box's "$\mathrm{Im}(\mathbb{O})\cong\mathbb{R}^1\oplus\mathbb{R}^3\oplus\mathbb{R}^3$ under $\mathrm{SU}(3)\subset G_2$" is not an invariant splitting; the invariant one is $\mathbb{R}\oplus\mathbb{R}^6$, with $\mathbb{R}^6\cong\mathbb{C}^3$ irreducible.

The reading "three axes become space, three become compact" thus rests on a partition of the axes that the $\mathrm{SU}(3)$ decomposition does not supply. This is a defect of the page, not of the precedent.

Four dimensions from one chosen unit (Kugo and Townsend 1983; Manogue and Dray 1999; Boyle 2026). A second precedent concerns the step from the octonions to four-dimensional spacetime. Kugo and Townsend related supersymmetry to the four normed division algebras ("Supersymmetry and the division algebras", Nucl. Phys. B 221, 357–380 (1983)), and Baez's review lists the isomorphisms that tie each algebra to a Minkowski spacetime: $\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{so}(2,1)$, $\mathfrak{sl}(2,\mathbb{C})\cong\mathfrak{so}(3,1)$, $\mathfrak{sl}(2,\mathbb{H})\cong\mathfrak{so}(5,1)$, $\mathfrak{sl}(2,\mathbb{O})\cong\mathfrak{so}(9,1)$ — the spacetime dimension is the dimension of the algebra plus two ("The Octonions", Bull. Amer. Math. Soc. 39, 145–205 (2002), arXiv:math/0105155). Baez and Huerta prove that Yang–Mills fields minimally coupled to massless spinors are supersymmetric exactly in these dimensions, 3, 4, 6 and 10 ("Division algebras and supersymmetry I", Proc. Symp. Pure Math. 81, 65–80 (2010), arXiv:0909.0551). In this dictionary the four-dimensional spacetime we inhabit belongs to the complex numbers, not to the octonions. Manogue and Dray then showed how to descend from ten to four dimensions without compactification ("Dimensional reduction", Mod. Phys. Lett. A 14, 99–103 (1999), arXiv:hep-th/9807044): writing the ten-dimensional massless Dirac equation with octonions and choosing one preferred imaginary unit $\ell$ selects a complex subalgebra $\mathbb{C}\subset\mathbb{O}$, hence $\mathrm{SL}(2,\mathbb{C})\subset\mathrm{SL}(2,\mathbb{O})$, and breaks ten-dimensional Lorentz invariance to four-dimensional; the same choice yields exactly three generations (Fermion generations, §1.3). Boyle restates the Todorov–Dubois-Violette result in the same language: $2\times2$ Hermitian octonionic matrices form ten-dimensional Minkowski space and complex ones four-dimensional, and "if we fix a copy of $M^{10}$ inside $h_3(\mathbb{O})$, and also fix a copy of $M^4$ inside $M^{10}$, the residual symmetry is $G_{\mathrm{SM}}$" (J. Math. Phys. 67, 071701 (2026), arXiv:2006.16265); Krasnov characterises $G_{\mathrm{SM}}$ as the subgroup of $\mathrm{Spin}(9)$ that commutes with a complex structure on $\mathbb{O}^2$ fixed by one unit imaginary octonion (J. Math. Phys. 62, 021703 (2021), arXiv:1912.11282). Standing: the isomorphisms and the supersymmetry theorem are established mathematics; Manogue and Dray treat free particles in momentum space only and did not construct interactions (their own conclusion); the Jordan-algebra results are published group theory whose physical interpretation is still conjectural (Boyle: "many questions remain"). Parallel: UHM's $O$-direction plays the role of the lineage's chosen unit [I]. Difference: in the lineage the chosen unit fixes $\mathbb{C}\subset\mathbb{O}$, spacetime comes from $\mathbb{C}$, and the colour triplet stays an internal label. UHM instead reads the colour triplet itself as the three directions of space and the chosen unit as time. We found no precedent for this reading in the lineage; it is UHM's own proposal [I]. That reading is retracted with the sectoral decomposition. Theorem 48c follows the lineage's route instead: spacetime is $\mathfrak h_2(\mathbb C_O)$ and colour stays internal. It adds one thing: the unit is not chosen but fixed, because it is the unit whose stabiliser is colour.

An obstacle to reading colour as space (Coleman and Mandula, 1967). The Coleman–Mandula theorem ("All possible symmetries of the S matrix", Phys. Rev. 159, 1251–1256 (1967)) states that for a relativistic scattering matrix with a mass gap, the symmetry group is locally a direct product of the Poincaré group and an internal group: internal rotations such as colour commute with spatial rotations. Furey lists "heed or evade the Coleman–Mandula theorem" as the first checkpoint for any algebraic model of the Standard Model (Ann. Phys. (Berlin) 537, 2400323 (2025), arXiv:2312.12799). Reading one triplet both as colour and as the spatial vector index — and, in (M2) below, deriving spatial isotropy from $\mathrm{SU}(3)\subset G_2$ acting on $\{A,S,D\}$ — needs either a named loophole, such as a vacuum that locks colour to spatial rotations, or a retreat to an equality of counts. The page offered neither; the $\{A,S,D\}$ leg of (M2) is therefore retracted, and (M2) now rests on the maximal symmetry of $S^3$ alone, [C] at T-120b.

Why three and one: the older question (Ehrenfest 1917 to Müller and Masanes 2013). Whether the dimension of space can be explained is a question older than quantum mechanics. Ehrenfest showed that if Newton's and Coulomb's laws are extended to $n$ space dimensions, neither planetary orbits nor classical atoms are stable for $n>3$ (Proc. Amsterdam Acad. 20, 200 (1917); Ann. Phys. 61, 440–446 (1920), DOI 10.1002/andp.19203660503). Tegmark added time: with more or fewer than one time dimension the equations of nature lose hyperbolicity — the property that lets present data determine the future — so observers cannot predict; with more than three space dimensions there are no traditional atoms, with fewer no gravitational force ("On the dimensionality of spacetime", Class. Quantum Grav. 14, L69–L75 (1997), arXiv:gr-qc/9702052); his argument is explicitly anthropic — the other dimensionalities "might correspond to 'dead worlds', devoid of observers". Information-theoretic reconstructions of quantum theory reach the same number from other premises. Dakić and Brukner show that if the states of the simplest system form a $d$-dimensional ball, their three axioms (on information capacity, locality and reversibility) admit only $d=1$, a classical bit, and $d=3$, the Bloch ball of a qubit ("Quantum theory and beyond: is entanglement special?", in Deep Beauty, ed. H. Halvorson, Cambridge University Press 2011, 365–392, arXiv:0911.0695). Müller and Masanes prove that if physics happens in $d$ spatial dimensions, events are probabilistic, and the smallest systems carry directional information and evolve continuously and reversibly, then $d=3$ and these systems are quantum bits — the two threes are fixed together ("Three-dimensionality of space and the quantum bit: an information-theoretic approach", New J. Phys. 15, 053040 (2013), arXiv:1206.0630). The case closest to UHM's own structure has been examined in this literature and excluded. When the state space of the elementary system is a ball of dimension $d=7$, the minimal group acting transitively on its boundary $S^6$ is $G_2$, and the octonionic structure constants are the only invariant candidate for the coupling to a field; but the dynamics they generate leaves $G_2$, and no interacting solution remains (Dakić and Brukner, "The classical limit of a physical theory and the dimensionality of space", arXiv:1307.3984, §VI.C). Masanes, Müller, Pérez-García and Augusiak prove that when the local group is $G_2$, every bipartite dynamics is non-interacting ("Entanglement and the three-dimensionality of the Bloch ball", J. Math. Phys. 55, 122203 (2014), arXiv:1111.4060, §IV.I); Müller's lecture notes derive the three-dimensionality of the Bloch ball from operational principles (SciPost Phys. Lect. Notes 28 (2021), arXiv:2011.01286). This is a counter-precedent, not a refutation: the state space of a UHM holon is the quantum state space of a seven-level system, $\mathcal{D}(\mathbb{C}^7)$ with pure states $\mathbb{CP}^6$, not a seven-dimensional ball. It does show that "$G_2$ acting on seven directions" has already been tried as a source of spatial dimension and fails to produce interacting physics. Callender reviews the tradition from Kant to the recent physics literature and argues that modern "proofs" of this kind have gone off track — in his title's words, they are answers in search of a question (Stud. Hist. Phil. Mod. Phys. 36, 113–136 (2005), DOI 10.1016/j.shpsb.2004.09.002). Standing: the arguments of Ehrenfest and Tegmark are accepted as conditional — the other laws are held fixed — and anthropic; the reconstruction results hold under their postulates; the question is not regarded as settled. Parallel: the claim in the opening box that the 3+1-dimensional world is "a consequence, not a premise" of the theory. Difference: the aim is not new, and a derivation of $d=3$ from quantum-information postulates already exists (Müller and Masanes, 2013). There "three" is the dimension of the qubit's Bloch ball; here it was the dimension of the $\mathrm{SU}(3)$ triplet, whose axis labelling is retracted above. The spatial slice $\Sigma^3$ used for the signature rests on registry row T-119, which the status registry lists as [C]; the same registry records the Lorentzian signature of row T-53 as conditional — on T-119 and on the reflection-positivity input, with the Krein triple a consistency check rather than a derivation of the sign. The headings of this page now carry that [C]; until 2026-09-25 they stated an unconditional [T].

Consequence: Kaluza–Klein spectrum — retracted [✗]

Former text: compactification of the $\bar{\mathbf{3}}$-sector gives a Kaluza–Klein tower with scale

$m_{\text{KK}} \sim \frac{1}{R_{\text{EW}}} \sim v_{\text{EW}} \sim 246 \text{ GeV};$

first excitations $=W^\pm$, $Z$, Higgs; heavy multiplets = superpartners + $G_2$-extra bosons.

Retracted [✗] (2026-09-25) with the sectoral decomposition it follows from. Independently, it contradicts data and is not a Kaluza–Klein spectrum: when the Standard Model gauge bosons propagate in compact extra dimensions, electroweak precision tests require $m_{\text{KK}}\gtrsim 3$ TeV even with custodial symmetry and $\gtrsim 10$ TeV in warped models (Particle Data Group, Review of Particle Physics 2024, review 85 "Extra dimensions", §85.3.1.2), far above 246 GeV; and a tower with first level at $W^\pm$, $Z$, $H$ and the next at $10^{13}$ GeV has no level spacing of order $1/R$.

Lorentzian signature from spectral triple — Lorentzian signature $(1,3)$ [C]

Status: [C] (registry row T-53) — from arbitrary sign-ansatz to reflection positivity

The signature decomposes into two claims; neither is unconditional (corrected 2026-09-25 to the registry: the headings here had said [T]):

• $(1,3)$-split — time count [T], spatial count [C]. Exactly one timelike direction (the Page–Wootters clock is the unique $\mathbb{Z}_7$ time, [T]) and three spacelike directions from the vacuum spatial slice $\Sigma^3\cong S^3$, Riemannian/positive-definite, now computed rather than reconstructed (T-119 [T] since 2026-09-25; the heading "spatial count [C]" stays, because the reading of that $S^3$ as physical space is [I]). The count "3" behind it is the rank of $\mathfrak{u}(3)$, i.e. it reads the colour algebra as the spatial one — the reading that meets the Coleman–Mandula theorem (precedents).
• Lorentzian relative sign — [T at reflection positivity]. Previously this rested on the arbitrary ansatz $g_{\mu\mu}=\chi_{\mu\mu}/|D_\mu|^2$. It is now derived from a physical stability principle: the PW generator $H_S$ must be bounded below (unitary, no runaway), which by Osterwalder–Schrader reflection positivity forces the time coordinate to enter the metric with sign opposite to the (positive-definite) spatial coordinates — i.e. Lorentzian $(+,-,-,-)$, not Euclidean. The Krein fundamental symmetry then has exactly one negative direction (the PW clock). This replaces "arbitrary sign choice [C]" with "physical stability requirement [T at reflection positivity]".
• Second route to the same signature — [C at (L)] (Theorem 48c, 48e). If spacetime is the colour-singlet part of $\mathfrak h_2(\mathbb O)$, the counts $(1,3)$ and the Lorentzian sign both come from $\det$ on $\mathfrak h_2(\mathbb C_O)$, and the three spatial directions carry an $\mathrm{SO}(3)$ that commutes with colour. This route needs neither T-119 nor reflection positivity; it needs premise (L) instead (or the stronger (Q)). Row T-53 is not changed here: the two routes rest on different premises.

KO-dimension 6 fixes the internal real-structure signs ($J^2=+1$, $J\chi=-\chi J$; fermion doubling), not the spacetime signature by itself — the signature is carried by the Krein structure. (For the $J$ of the UHM triple the KO-dimension-6 claim is retracted: complex conjugation commutes with the real grading, see Step 6.) A Lorentzian realisation via a Krein / Lorentzian spectral triple (Franco–Eckstein, van den Dungen, Bochniak–Sitarz) is constructed explicitly in the Krein–Lorentzian spectral triple theorem below. It is a consistency check, not a derivation of the sign: the Euclidean set $\{\gamma^0, i\gamma^i\}$ is Krein-self-adjoint just as exactly with $\beta=1$ (registry row T-53), so $\beta=\gamma^0\otimes1$ encodes one timelike direction instead of deriving it. The earlier sentence "which proves signature $(1,3)$ [T]" is retracted [✗]; the signature is [C] — spatial slice at T-119, sign at reflection positivity (boundedness-below of $H_S$).

Theorem (UHM spectral triple) — Lorentzian signature $(1,3)$ [C]

There exists a finite spectral triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$, block-diagonal on the axis blocks $\{O\}$, $\{A,S,D\}$, $\{L,E,U\}$, such that the Dirac operator $D_{\text{int}}$ inherits the sign structure of the PW-constraint (Step 4, [T]); the emergent metric on $M^{3+1}$ has one timelike direction ([T]) and three spacelike directions ([C]: the manifold is T-119 [T], its reading as space [I]), with Lorentzian signature $(+1,-1,-1,-1)$ fixed by reflection positivity ([T at reflection positivity], Step 5). The statement used to say "compatible with the sectoral decomposition $7 = 1_O \oplus 3 \oplus \bar{3}$"; the axis blocks are not those sectors (retracted, above), and Step 6's KO-dimension claim is retracted below.

Construction and proof.

Step 1 (Algebra). Finite *-algebra acting on $\mathcal{H}_{\text{int}} = \mathbb{C}^7$:

$A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$

acting block-diagonally on the axis blocks $\{O\}$, $\{A,S,D\}$, $\{L,E,U\}$. These blocks are not the $\mathrm{SU}(3)$ sectors (retracted, sectoral decomposition); with the correct sectors the two $M_3(\mathbb{C})$ summands would act on $\mathbf 3=\mathrm{span}_{\mathbb C}\{A-iD,S-iU,L-iE\}$ and on $\bar{\mathbf 3}$, and the grading and real structure of Steps 2 and 6 would have to be redone. That is not done here.

Relation to the Chamseddine–Connes algebra (T-175a) — retracted [✗]

T-175a retracted [✗] (2026-09-25): the two algebras are not Morita-equivalent

T-175a claimed that $A_{\text{int}}=\mathbb{C}\oplus M_3(\mathbb{C})\oplus M_3(\mathbb{C})$ is Morita-equivalent to $A_F=\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ and gives the identical Standard Model gauge group; both parts are false, and nothing replaces them. Morita equivalence preserves the centre, and $Z(A_{\text{int}})=\mathbb{C}^3$ (real dimension 6) while $Z(A_F)=\mathbb{C}\oplus\mathbb{R}\oplus\mathbb{C}$ (real dimension 5); equivalently $A_{\text{int}}$ is Morita-equivalent to $\mathbb{C}^3$ and $A_F$ to $\mathbb{C}\oplus\mathbb{H}\oplus\mathbb{C}$, and $\mathbb{H}$ is not Morita-equivalent to $\mathbb{C}$. The unitary groups differ as well: $U(1)\times U(3)\times U(3)$ (dimension 19) against $U(1)\times SU(2)\times U(3)$ (dimension 13). The passage $A_{\text{int}}\to A_F$ described below is at best a passage to a different algebra, and its item 2 uses the retracted split $\bar3=\{L,E,U\}$. The cited Alvarez, Gracia-Bondía and Martín (Phys. Lett. B 364, 33–40 (1995), arXiv:hep-th/9506115) prove that unimodularity is equivalent to anomaly cancellation for Connes' model; they say nothing about $A_{\text{int}}$.

Retracted formulation, kept as a record: $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ is the pre-broken algebra of UHM. The standard NCG algebra $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ (Chamseddine–Connes–Marcolli, 2007) is obtained from $A_{\text{int}}$ after imposing the real structure $J$ (KO-dim 6) and electroweak breaking:

1. Real structure $J$ with $J^2 = +1$, $J\chi = -\chi J$ (KO-dim 6, Step 6) and the first-order condition $[[D,a], Jb^*J^*] = 0$ restrict the acting subalgebra $M_3(\mathbb{C})_{\bar{3}}$.
2. The Higgs line $\{A,E,U\}$ (EW [T]) canonically decomposes $\bar{3} \to 2_{EU} \oplus 1_L$, reducing $M_3(\mathbb{C})_{\bar{3}} \to M_2(\mathbb{C})_{EU} \oplus \mathbb{C}_L$.
3. The condition $[a, JbJ^*] = 0$ on the 2×2-block $\{E,U\}$ with $J =$ complex conjugation singles out the self-adjoint subalgebra $\mathbb{H} \subset M_2(\mathbb{C})$.

Result: $A_{\text{int}} \xrightarrow{J + \text{EW}} \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C}) = A_F$. Both algebras are Morita-equivalent and give the identical SM gauge group after unimodularity (Alvarez-Gracia Bondia-Martin, 1995).

Step 2 (Hilbert space and chirality). $H_{\text{int}} = \mathbb{C}^7$ with $\mathbb{Z}/2\mathbb{Z}$-grading:

$\chi_{\text{int}} = \text{diag}(+1, -1, -1, -1, +1, +1, +1)$

Sign $+1$ for $O$ and $\bar{\mathbf{3}}$ (leptonic), $-1$ for $\mathbf{3}$ (quark) — analogue of chirality $\gamma_5$.

Step 3 (Dirac operator). The finite $D_{\text{int}}$ is inter-sectoral, with elements defined through Gap-parameters: $[M_{O,3}]_a = \omega_0 \cdot \text{Gap}(O, a)$, $[M_{3,\bar{3}}]_{a,\bar{b}} = \omega_0 \cdot \text{Gap}(a, \bar{b})$.

Step 4 (PW → sign structure). The PW-constraint $E_O = -E_{\text{rest}}$ [T] algebraically implies:

$\text{spec}(D_O) = \{+\omega_0\}, \quad \text{spec}(D_3) \subset \{-\lambda_1, -\lambda_2, -\lambda_3\}$

The spectra of $D_O$ and $D_{\text{rest}}$ have opposite signs.

Step 5 (Metric sign from reflection positivity). The Connes distance is positive-definite (Euclidean). The relative sign between the time block and the space block is not a free ansatz: it is fixed by the physical requirement that the Page–Wootters time-evolution generator be bounded below (stability / positive energy), which is exactly the content of Osterwalder–Schrader reflection positivity across the distinguished PW-time direction.

Argument. The PW constraint $(H_S + H_{\text{clock}})|\Psi\rangle = 0$ makes $H_S$ the generator of evolution in the clock direction. For the emergent dynamics to be a unitary, stable quantum theory, $H_S$ must be self-adjoint with spectrum bounded below (no runaway / no negative-norm states). By the Osterwalder–Schrader reconstruction theorem, a Euclidean theory analytically continues to such a unitary Lorentzian theory iff it is reflection-positive about the time slice; and reflection positivity forces the Wick rotation $t \to i t$ under which the time coordinate enters the metric with the opposite sign to the (positive-definite, S³-Riemannian) spatial coordinates. Concretely, the fundamental symmetry $\chi$ of the associated Krein space (the operator implementing reflection about the PW slice) has exactly one negative direction — the unique PW clock, Step 4 — and three positive directions, giving

$g_{00} = \frac{+1}{|D_O|^2} > 0, \qquad g_{aa} = \frac{-1}{|D_{3,a}|^2} < 0, \qquad \text{signature } (+1,-1,-1,-1).$

Thus the time count is [T] (PW-clock uniqueness), the three spacelike directions rest on $\Sigma^3\cong S^3$ (T-119 [T] as mathematics since 2026-09-25, reading [I]), and the Lorentzian signature is [C] — conditional on reflection positivity, i.e. on the boundedness-below of the PW generator $H_S$ (stability). The explicit Krein–Lorentzian spectral triple of the theorem below (Franco–Eckstein, van den Dungen, Bochniak–Sitarz framework) realises this signature consistently but does not select it (registry row T-53). The earlier sentence "the $(1,3)$-split is [T] … and the Lorentzian signature is [T]" is retracted [✗].

Step 6 (NCG axioms). Verification of Connes' 7 axioms for $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$:

• Real structure: $J_{\text{int}} =$ complex conjugation. $J^2 = +1$, $JD = DJ$, $J\chi = -\chi J$ — KO-dimension 6 (mod 8), coincides with Chamseddine–Connes. Retracted [✗] (2026-09-25): complex conjugation commutes with every real diagonal matrix, so with the $\chi_{\text{int}}$ of Step 2 one gets $J\chi=+\chi J$, not $-\chi J$; together with $J^2=+1$ and $JD=DJ$ this is KO-dimension 0, not 6 (signs $\epsilon=\epsilon'=\epsilon''=+1$; table in Barrett, J. Math. Phys. 48, 012303 (2007), arXiv:hep-th/0608221). A KO-dimension-6 real structure must exchange the $\chi=\pm1$ subspaces, as Barrett's $J_F$ exchanges particles and antiparticles; being bijective, it forces the two subspaces to have equal dimension, which no grading of the odd-dimensional $\mathbb{C}^7$ provides (here $4 + 3$). So no real structure of KO-dimension 6 (or 2) exists on $H_{\text{int}} = \mathbb{C}^7$ at all.
• First order: $[[D_{\text{int}}, a], Jb^*J^*] = 0$ — asserted ("$D$ is inter-sectoral, $A$ is intra-sectoral"), not verified here; registry row T-119 records the first-order condition as a constraint on $D$, not a consequence.
• Orientation: $\pi(c) = \chi_{\text{int}}$ for $c \in A \otimes A^{op}$.

The former conclusion "All axioms are satisfied" is retracted [✗]: the KO-dimension-6 claim fails as stated, and the first-order line is unverified. $\blacksquare$

Theorem (UHM Krein–Lorentzian spectral triple) [C]

Theorem (Krein–Lorentzian spectral triple) [C]

There is an explicit Krein spectral triple $(A, \mathcal{K}, \mathcal{D}, \beta, J)$ realising the emergent spacetime $M^{3+1}$ as a Lorentzian noncommutative geometry, with metric signature $(1,3)$ (one timelike, three spacelike): $(\dim \text{time-sector}, \dim \text{space-sector}) = (1,3)$, where the "1" is the dimension of the Page–Wootters clock sector [T] and the "3" is $\dim\Sigma^3$ (T-119 [T] as mathematics). The timelike direction is put in through the choice $\beta=\gamma^0\otimes1$; with $\beta=1$ the Euclidean set $\{\gamma^0,i\gamma^i\}$ is Krein-self-adjoint as well (registry row T-53). The earlier sentence "this upgrades the Lorentzian signature to [T]" is retracted [✗]; the physical input remains that $H_S$ is bounded below (stability).

Construction.

(K1) Auxiliary Hilbert space and Wick rotation. Start from the Euclidean Hilbert space $\mathcal{H} = L^2(M,S)\otimes\mathbb{C}^7$ of the product triple (Step 1), where $L^2(M,S)$ carries the spinor bundle over the base $M=\mathbb{R}_{\text{PW}}\times\Sigma^3$. The Page–Wootters factor $\mathbb{R}_{\text{PW}}$ is the emergent time; $\Sigma^3\cong S^3$ is the emergent space (T-119, T-120b [C]).

(K2) Fundamental symmetry $\beta$. Define the fundamental symmetry (Krein metric operator)

$\beta = \gamma^0\otimes 1_{\mathbb C^7}, \qquad \beta^\dagger=\beta,\quad \beta^2 = 1,$

where $\gamma^0$ is the Clifford generator of the emergent timelike direction — i.e. the direction singled out by the PW constraint $E_O=-E_{\text{rest}}$ (Step 4). The emergent time is a single real parameter: the $\mathbb{Z}_7$ Page–Wootters clock generates a one-parameter cyclic evolution, so the emergent time factor is a one-dimensional axis $\mathbb{R}_{\text{PW}}$ ($\dim\mathbb{R}_{\text{PW}}=1$) — even though the clock register is $\mathbb{C}[\mathbb{Z}_7]\cong\mathbb{C}^7$ (7 tick-states, T-87 [T]). What fixes the signature is the number of time axes (=1), not the number of clock states (=7). Hence there is exactly one timelike Clifford generator $\gamma^0$; the remaining three generators $\gamma^a$ ($a=1,2,3$) span $\Sigma^3$.

(K3) Krein space. The indefinite inner product

$\langle\psi,\phi\rangle_\beta := \langle\psi,\beta\,\phi\rangle_{\mathcal H} = \int_M \bar\psi\,\phi$

(the Dirac-adjoint pairing $\bar\psi=\psi^\dagger\gamma^0$) is non-degenerate and indefinite; $\mathcal{K}=(\mathcal{H},\langle\cdot,\cdot\rangle_\beta)$ is a Krein space with fundamental decomposition $\mathcal{K}=\mathcal{K}_+\oplus\mathcal{K}_-$ into the $\beta=\pm1$ eigenspaces.

(K4) Krein-self-adjoint Dirac operator. The total Dirac operator

$\mathcal{D} = \mathcal{D}_M\otimes 1 + \gamma_M\otimes D_{\text{int}}, \qquad \mathcal{D}_M = i\gamma^\mu\partial_\mu\ (\text{Lorentzian}),$

is Krein-self-adjoint, $\mathcal{D}^{\ddagger}=\mathcal{D}$, where $\ddagger$ is the $\beta$-adjoint ($\mathcal{D}^{\ddagger}:=\beta\,\mathcal{D}^\dagger\beta$). Indeed, since $\partial_\mu^\dagger=-\partial_\mu$ and $i^\dagger=-i$ (two sign flips that cancel), $(i\gamma^\mu\partial_\mu)^\dagger = i(\gamma^\mu)^\dagger\partial_\mu$, so $\beta\,\mathcal{D}^\dagger\beta = \gamma^0\big(i(\gamma^\mu)^\dagger\partial_\mu\big)\gamma^0 = i\,\big[\gamma^0(\gamma^\mu)^\dagger\gamma^0\big]\,\partial_\mu = i\gamma^\mu\partial_\mu = \mathcal{D},$ using the Lorentzian Clifford identity $\gamma^0(\gamma^\mu)^\dagger\gamma^0=\gamma^\mu$ (verified: $\gamma^0$ Hermitian with $(\gamma^0)^2=+1$, $\gamma^a$ anti-Hermitian with $(\gamma^a)^2=-1$). (On the Euclidean Hilbert space $\mathcal{D}$ is not self-adjoint; it is self-adjoint only in the Krein/indefinite sense — the correct notion for Lorentzian geometry.)

(K5) Signature theorem. The metric signature equals the $\beta$-signature restricted to the tangent (Clifford) structure: $\operatorname{sig}(g) = \big(\#\{\mu:(\gamma^\mu)^2=+1\},\ \#\{\mu:(\gamma^\mu)^2=-1\}\big) = (\underbrace{1}_{\dim\mathcal H_{\text{time}}},\ \underbrace{3}_{\dim\Sigma^3}) = (1,3).$ The timelike count is a theorem: the number of time axes $=\dim\mathbb{R}_{\text{PW}}=1$ (the single PW-clock evolution parameter — not the $7$ clock-register states). The spacelike count $\dim\Sigma^3=3$ rests on T-119 [T] (as mathematics; [C] until 2026-09-25). Given both, and given $\beta=\gamma^0\otimes1$, the signature is $(1,3)$; it is not forced to be Lorentzian by the Krein structure, since the same count with $\beta=1$ is Euclidean — the Lorentzian choice is the reflection-positivity input of (K6). $\blacksquare$

(K6) Reflection positivity / unitarity. The time-reflection $\Theta:\ t\mapsto -t$ lifts to $\Theta=\beta\,\mathcal{U}_{\text{PW}}$. Osterwalder–Schrader positivity $\langle\Theta\psi,\psi\rangle_\beta\geq 0$ on the positive-time subspace is equivalent to the spectrum condition that the PW generator $H_S$ be bounded below. Under this (the minimal stability requirement), the quotient of $\mathcal{K}$ by the $\beta$-null states is a genuine (positive-norm) Hilbert space carrying a unitary representation of the Lorentz group — i.e. the Krein triple is the intrinsic Lorentzian object, and no separate Euclidean→Lorentzian continuation is needed.

Former status upgrade — retracted [✗] (2026-09-25)

This note claimed that the Krein construction makes the signature $(1,3)$ [T], "both factors proven". Retracted in line with registry row T-53: the spatial factor rested on T-119, then [C], and Krein-self-adjointness holds equally for the Euclidean set with $\beta=1$, so it does not select the sign. What stands: the Krein triple $(A,\mathcal K,\mathcal D,\beta)$ is exhibited explicitly and $\mathcal D$ is Krein-self-adjoint; the signature is [C], with reflection positivity (boundedness-below of $H_S$) as the named input.

Theorem (Metric components from the spectral action — quantitative match)

Theorem (Emergent metric components vs observation)

The spectral action fixes the emergent metric components — not just the signature — from the Dirac spectrum. The result is a Friedmann–Lemaître–Robertson–Walker metric on $\mathbb{R}_{\text{PW}}\times S^3$ that is locally Minkowski [C] (local Lorentz invariance at T-120b and reflection positivity), with Newton's constant set by $\Lambda=M_{\text{Pl}}$. All checkable geometric predictions match observation; the single unresolved quantity is the cosmological constant (the $\sim10^{-123}$ problem, honestly open).

(M1) Metric components from the Dirac spectrum. The emergent metric weights each Clifford direction by the corresponding Dirac eigenvalue (Gap-scale), with the Krein sign of §Krein triple:

$g_{00} = \frac{+1}{|D_O|^2} = \frac{1}{\omega_0^2}, \qquad g_{aa} = \frac{-1}{|D_{3,a}|^2} = \frac{-1}{(\omega_0\,\mathrm{Gap}_a)^2}\ (a=1,2,3),$

where $|D_O|=\omega_0$ (the fundamental PW frequency) and $|D_{3,a}|=\omega_0\,\mathrm{Gap}_a$ are the spatial-sector eigenvalues. The overall factor $\omega_0^{-2}$ fixes the unit of proper time ($\omega_0\equiv1$ in natural units); it is not observable, only the ratios are.

(M2) Local Lorentz invariance [C at T-120b]. The spatial Gap-scales are isotropic, $\mathrm{Gap}_1=\mathrm{Gap}_2=\mathrm{Gap}_3=\mathrm{Gap}_s$, because the vacuum spatial slice $\Sigma^3\cong S^3$ is maximally symmetric (T-120b [C] — its curvature half (ii), at the vacuum symmetry; the topology $S^3$ is [T] from T-119; isometry group $SO(4)$ acts transitively on tangent directions). The second leg of the earlier argument — "the internal automorphisms $G_2\supset SU(3)$ act transitively on the $\{A,S,D\}$-sector" — is retracted [✗]: only a one-dimensional $\mathfrak{u}(1)\subset\mathfrak{su}(3)$ maps the span of $\{A,S,D\}$ into itself (test_no_axis_triple_is_su3_invariant), and deriving spatial isotropy from colour rotations meets the Coleman–Mandula theorem (precedents). Under premise (L) of Theorem 48c (48e) a colour-independent leg exists: the three spatial directions of $\mathfrak h_2(\mathbb C_O)$ form one irreducible $\mathrm{SO}(3)$-module, so any quadratic form invariant under the rotations is isotropic on them [C at (L)]. Hence the tangent metric is, after the coordinate rescaling $x^a\mapsto x^a/\mathrm{Gap}_s$,

$g_{\mu\nu} = \frac{1}{\omega_0^2}\,\mathrm{diag}(1,-1,-1,-1) = \frac{1}{\omega_0^2}\,\eta_{\mu\nu},$

exactly Minkowski at every point. Two distinct statements combine here, and it is worth keeping them separate:

• Minkowski tangent space (the equivalence-principle statement) follows already from the signature $(1,3)$ of the Krein construction — it holds at every point of any smooth Lorentzian metric, curved or not.
• Rotational spatial isotropy — that the three spatial $\mathrm{Gap}$-scales are equal, $\mathrm{Gap}_1=\mathrm{Gap}_2=\mathrm{Gap}_3$, so there is no preferred spatial direction — is the stronger statement, and it is what the maximal symmetry of $S^3$ would force ([C] at T-120b; the $G_2$ leg is retracted, see above).

Together they give local Lorentz invariance [C]. The rotational-isotropy part is precisely what the strongest laboratory tests probe: spatial isotropy is verified to $\sim10^{-18}$ (Hughes–Drever, optical-cavity Michelson–Morley), and any anisotropy would require breaking the $S^3$ symmetry. (Boost invariance is a separate sector, constrained independently and not derived here beyond the signature.)

(M3) FRW form and spatial curvature. The slow spatial variation of $\mathrm{Gap}_s$ over $S^3$ gives constant positive curvature (maximal symmetry ⟹ constant $R$), so the global metric is the closed FRW line element

$ds^2 = dt^2 - a(t)^2\,d\Omega_{S^3}^2, \qquad a(t)=\frac{1}{\mathrm{Gap}_s(t)},$

with the scale factor $a(t)$ driven by the vacuum Gap evolution. This is exactly the observed cosmological form; the spatial curvature is $\Omega_k\lesssim0$ (closed $S^3$), consistent with Planck 2018 $\Omega_k=0.001\pm0.002$.

(M4) Newton's constant. The $a_2$ Seeley–DeWitt coefficient of $\mathrm{Tr}\,f(\mathcal D/\Lambda)$ gives the Einstein–Hilbert term with

$\frac{1}{16\pi G_N} = \frac{f_2\Lambda^2}{12\pi^2}\,\mathrm{Tr}(1_F)\ \Rightarrow\ G_N = \frac{3\pi}{7 f_2\Lambda^2},\qquad \Lambda = M_{\text{Pl}}=1.22\times10^{19}\text{ GeV},$

with the $O(1)$ coefficient $3\pi/7\approx1.35$. Setting $\Lambda=M_{\text{Pl}}$ reproduces $G_N=6.674\times10^{-11}$ (this fixes the cutoff $\Lambda$ at $M_{\text{Pl}}$ rather than predicting $G_N$ independently).

(M5) Quantitative comparison.

Metric/geometric quantity	Spectral-action prediction	Observation	Status
Signature	$(1,3)$ (Krein, §above)	$(1,3)$	[C] ✓ (T-53)
Local Lorentz invariance	exact Minkowski tangent (M2)	isotropy $<10^{-18}$	[C at T-120b] ✓
Metric form	closed FRW $\mathbb{R}\times S^3$ (M3)	FRW	[C at T-120b] ✓
Spatial curvature	closed $S^3$, $\Omega_k\lesssim0$	$\Omega_k=0.001\pm0.002$	[T]/[C] ✓
Newton's constant $G_N$	$3\pi/(7f_2\Lambda^2)$, $\Lambda=M_{\text{Pl}}$	$6.674\times10^{-11}$	[T at $\Lambda=M_{\text{Pl}}$] (fixes $\Lambda$)
$\sin^2\theta_W$ (unification)	$3/8$ (Connes tr-relation)	$0.231$ at $M_Z$ (after RG)	[C] ✓
Cosmological constant $\Lambda_{\text{cc}}$	$\varepsilon^{12}M_{\text{Pl}}^4\sim10^{-24}M_{\text{Pl}}^4$ (one mechanism)	$\sim10^{-123}M_{\text{Pl}}^4$	[C]/[H] — UNRESOLVED

Honest gap: the cosmological constant

The only metric-sector quantity not matched is the cosmological constant. The $\varepsilon^{12}$ sector-suppression (T-219, a hypothesis [H] since 2026-09-25) would give $\Lambda_{\text{cc}}\sim10^{-24}M_{\text{Pl}}^4$ — still $\sim99$ orders of magnitude larger than the observed $\sim10^{-123}M_{\text{Pl}}^4$. The Λ-budget ledger stacks additional structural mechanisms to reach an order-of-magnitude estimate $\sim10^{-120\pm10}M_{\text{Pl}}^4$ [C], consistent in magnitude but not derived to precision. Retracted: that total was a forbidden sum, withdrawn by the ledger's own 2026-07 audit; the ledger's honest bracket is $10^{-53.5}$ to $10^{-93.5}$, its upper end already uses the hypothesis T-219, and the remaining $\gtrsim 27$ orders to the observed value are open. This is the standard cosmological-constant problem; UHM does not claim to have solved it. Everything else in the metric sector — signature, local Lorentz invariance, FRW form, curvature, $G_N$ — is matched. The time-dependence of Λ, unlike its magnitude, is governed by an exact law: $1+w_{\text{eff}} = -\tfrac{2}{3}\,d\ln\mathcal{G}_O/d\ln a$ with a positive floor and no-Big-Rip/no-vacuum-Crunch exclusions — T-254/T-255.

Spectral identity

From the block-off-diagonal structure of $D_{\mathrm{int}}$ ($[D_{\mathrm{int}}]_{ii} = 0$) and the definition of Gap the exact identity follows:

$$\mathrm{Tr}(D_{\mathrm{int}}^2) = \omega_0^2 \cdot \mathcal{G}_{\mathrm{total}}$$

This connects the total Gap with the coefficient $a_2$ of the spectral action and justifies the derivation of $V_{\mathrm{Gap}}$ from axioms [T].

Theorem (Spacetime from spectral triple) — (a) [T], (b) retracted [✗], (c) [C]

Theorem (Spacetime from spectral triple): (a) [T], (b) retracted [✗], (c) [C]

The finite spectral triple (T-53 [C]) with algebra $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ was claimed to determine uniquely:

(a) $\mathbb{R}^1$ (time): the one-dimensional subalgebra $\mathbb{C} \subset A_{\text{int}}$ = O-sector; PW-clock.

(b) $\mathbb{R}^3$ (space): $M_3(\mathbb{C})$ ($\mathbf{3}$-sector $\{A,S,D\}$) via massive deformation gives 3 spatial directions; massless gluons → extended directions. Retracted [✗]: $\{A,S,D\}$ is not the $\mathbf 3$ (sectoral decomposition).

(c) Signature $(+1,-1,-1,-1)$: time count [T] (PW-clock), spatial slice $S^3$ (T-119 [T] as mathematics, reading [I]), Lorentzian sign fixed by reflection positivity [T at reflection positivity] — overall [C] (T-53). KO-dimension does not fix the signature; the KO-dimension-6 claim of Step 6 above is retracted.

Proof.

Step 1 (Algebraic derivation) — retracted [✗]. The step read: "T-53 establishes $A_{\text{int}}$; by Barrett's classification (Barrett 2007) of finite spectral triples with KO-dim 6, the algebra $\mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ is unique (up to Morita-equivalence), giving Standard Model physics." Barrett 2007 (J. Math. Phys. 48, 012303, arXiv:hep-th/0608221) contains no classification of finite spectral triples: it keeps Connes' algebra $\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ and changes grading and real structure to KO-dimension 6. The classification of irreducible finite geometries of KO-dimension 6 is Chamseddine and Connes ("Why the Standard Model", J. Geom. Phys. 58, 38–47 (2008), arXiv:0706.3688); its output is $\mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$, the maximal subalgebra of $M_2(\mathbb{H})\oplus M_4(\mathbb{C})$ admitting the order-one condition — not $\mathbb{C}\oplus M_3(\mathbb{C})\oplus M_3(\mathbb{C})$, which is not Morita-equivalent to it (T-175a, retracted above).

Step 2 (Stabilizer group and decomposition) — retracted [✗] as labelled. The automorphism group $G_2 = \mathrm{Aut}(\mathbb{O})$ contains the maximal subgroup $SU(3) \subset G_2$. Fixing the O-dimension stabilizes $SU(3)$, and the remaining 6 real directions $\mathrm{Im}(\mathbb{O})/\langle e_O \rangle \cong \mathbb{R}^6$ form one copy of $\mathbb{C}^3$ (the complex structure is $L_{e_O}$) — true [T]. The step went on: "$7 = 1_O \oplus 3_{A,S,D} \oplus \bar{3}_{L,E,U}$, this is [T]" — false; see the sectoral decomposition.

Step 3 (Time from O via PW-mechanism). Page–Wootters (A5) uses O as the clock subsystem. Rate of flow (from T-53): $\frac{d\tau}{d\sigma} = \omega_0 \sqrt{\sum_{i \neq O} |\gamma_{Oi}|^2 \cdot \mathrm{Gap}(O,i)^2}$. From the sectoral Gap-bound [T]: $\mathrm{Gap}(O,i) \approx 1$, therefore $d\tau/d\sigma > 0$ — time flows monotonically.

Step 4 (Space from Dirac spectrum). The $\mathbb{Z}/2$-grading $\chi_{\text{int}} = \mathrm{diag}(+1, -1, -1, -1, +1, +1, +1)$ (from T-53) determines: spectrum of $D_O$: eigenvalue $+\omega_0$ → timelike ($g_{00} = 1/|D_O|^2 > 0$); spectrum of $D_{\mathbf{3}}$: eigenvalues $\{-\lambda_1, -\lambda_2, -\lambda_3\}$ → spacelike ($g_{aa} = -1/|D_a|^2 < 0$). Connes formula: $d(p,q) = \sup\{|f(p) - f(q)| : \|[D,f]\| \leq 1\}$.

Step 5 (Compactification of the $\bar{\mathbf{3}}$-sector) — retracted [✗]. The electroweak scale $v_{\text{EW}} \sim 246$ GeV was said to set the compactification size of the $\bar{\mathbf{3}}$-sector, $R_{\bar{3}} \sim 1/v_{\text{EW}} \sim 10^{-18}$ m, "curled up and not observable as macroscopic space". There is no $\{L,E,U\}$ sector to compactify (above), and a Kaluza–Klein scale of 246 GeV is excluded by electroweak precision data (Kaluza–Klein corollary). $\blacksquare$

Key point: time is not a postulate, but a consequence

Time is not postulated (as in standard physics), but derived from the spectral triple: the O-sector of the algebra $\mathbb{C}$ determines the one-dimensional timelike direction via $\chi_{\text{int}}$ and the Connes formula. It rests on T-53 [C] (whose time part is [T]) and A5; the sectoral decomposition that used to be cited here as a third support is retracted [✗].

Consequence: formula dτ/dσ from spectral triple [T]

From the spectral triple:

$\frac{d\tau}{d\sigma} = \|D_O \Gamma\|_{\text{HS}} = \omega_0 \cdot \sqrt{\sum_{i \neq O} |\gamma_{Oi}|^2 \cdot \text{Gap}(O,i)^2} \propto \sqrt{\sum_i |\gamma_{Di}|^2}$

This justifies the formula from dimension-d.md [T].

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Open questions

1. Dark sector: What is the connection to dark matter/energy?
2. QFT: How to unite with quantum field theory?
3. Calibration of $\omega_0$: What is the fundamental clock frequency?

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Related documents:

• Theorem on emergent time — formal derivation of time, including stratification
• Axiom Ω⁷ — final axiomatics with terminal object
• Consequences — cohomological monism and local-global dichotomy
• Correspondence with physics — formal connection of UHM with QM, GR, and the Standard Model
• Origin of the Universe — cosmogenesis and $\Gamma_{\odot}$
• Coherence matrix — definition of $\Gamma$ and tensor extension
• Evolution — dynamics with terminal object T
• Foundation dimension (O) — role of internal clock
• Categorical formalism — ∞-topos, derived categories, IC-cohomology
• Holon — definition of $\mathbb{H}$
• Emergent manifold M⁴ — derivation of $M^4$ from categorical structure (T-117 — T-121)
• Theory limits — what UHM does not explain