UHM Correspondence with Fundamental Physics

Section Status

Section Status

The main results are formalized and proven [T]: L-unification, reduction of the evolution equation to the von Neumann equation (Theorem 3.1), emergent geometry ($M^4$), Einstein equations, SM gauge group. No-signalling is proven as the marginal identity of Theorem 8.1; for the full nonlinear dynamics it holds in the non-selective reading of §8.5, which the axioms force, so it is [T] (Theorem 8.5, §8.8; this box read [C] until 2026-09-25), and §8.7 proves that no modification of $\mathcal{R}$ keeping the viability threshold gives a second route. The category equivalence of §3.3 and the computational bound of §8.6 are retracted; in place of the bound, the ideal dynamics decides satisfiability in linear time (Theorem 8.6 [T]). An earlier version of this box listed "reduction to QM" and "no-signaling" among the [T] results without these limits; that is retracted. Open directions: concrete SM parameters, non-perturbative partition function.

Contents

1. Categorical Structure of Connections
2. L-Unification: the Logical Origin of Physics
3. Reduction to Quantum Mechanics
4. Emergent Geometry
5. Connection to General Relativity
6. Gauge Symmetries and the Standard Model
7. Correspondence of 7 Dimensions to Physical Structures

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1. Categorical Structure of Connections

L-unification as the foundation

The entire categorical structure connecting UHM with physics is based on L-unification — the derivation of Lindblad operators from the subobject classifier Ω. This provides a unified logical foundation for all physical theories.

1.1 Hierarchy of Physical Categories

Definition 1.1 (Category hierarchy). UHM generates the following commutative diagram of categories:

      Sh_∞(C)
         │
         │ Ω (classifier)
         ▼
                    π_QM
      Hol ─────────────────────▶ QM
       │                         │
       │ π_Class                 │ ℏ→0
       ▼                         ▼
    DensityMat ────────────────▶ ClassMech
                    ℏ→0
       │
       │ π_Space [T] (T-119, T-120)
       ▼
    Riem (M⁴ = ℝ × Σ³)

Key role of Ω:

• The ∞-topos $\text{Sh}_\infty(\mathcal{C})$ contains the classifier Ω
• Lindblad operators are derived from Ω: $L_k = \sqrt{\chi_{S_k}}$
• All physical dynamics is determined by the logical structure of Ω

where:

• $\mathbf{Hol}$ — category of Holons
• $\mathbf{QM}$ — category of quantum-mechanical systems
• $\mathbf{DensityMat}$ — category of density matrices
• $\mathbf{ClassMech}$ — category of classical mechanical systems
• $\mathbf{Riem}$ — category of Riemannian manifolds ($M^4$ assembled at T-120 [T] as mathematics (restated T-119, 2026-09-25))

1.2 Forgetful Functor

Definition 1.2 (Forgetful functor).

$$\mathcal{U}: \mathbf{Hol} \to \mathbf{DensityMat}$$

is defined on objects:

$$\mathcal{U}(\mathbb{H}) := \Gamma_{\mathbb{H}}^{(7)}$$

and on morphisms:

$$\mathcal{U}(f: \mathbb{H}_1 \to \mathbb{H}_2) := \Phi_f$$

where $\Phi_f$ is the CPTP channel induced by morphism $f$.

[T] Theorem 1.1 (Functoriality of forgetting). $\mathcal{U}$ is a functor preserving identities and composition.

Proof: Direct consequence of the definition of morphisms in $\mathbf{Hol}$ as CPTP channels preserving structure. ∎

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2. L-Unification: the Logical Origin of Physics

Central result

L-unification is the key achievement of UHM, showing that Lindblad operators $L_k$ (which define dissipative dynamics) are derived from the subobject classifier Ω, not postulated.

This means: physical dynamics has a logical origin.

2.1 Dependency Hierarchy

[T] Theorem 2.0 (Derivation chain). Fundamental physical objects are derived in the following order:

Definitions:

1. Ω — subobject classifier of the ∞-topos $\text{Sh}_\infty(\mathcal{C})$
2. χ_S: Γ → Ω — characteristic morphism for the subobject $S \hookrightarrow \Gamma$
3. L_k = √χ_{S_k} — Lindblad operators, where $\{S_k\}$ are atoms of the classifier
4. ℒ_Ω — logical Liouvillian constructed from $\{L_k\}$
5. φ — self-modeling operator from the dynamics of ℒ_Ω

2.2 Logical Liouvillian

[T] Theorem 2.0.1 (Logical Liouvillian). Dissipative dynamics is defined via the logical structure of Ω:

$$\mathcal{L}_\Omega[\Gamma] = -i[H_{eff}, \Gamma] + \sum_k \gamma_k \left( L_k \Gamma L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \Gamma\} \right)$$

where $L_k = \sqrt{\chi_{S_k}}$, $\{S_k\}$ are atoms of Ω.

Proof: See Axiom Ω⁷. ∎

2.3 Physical Interpretation

[T] Theorem 2.0.2 (Dissipation as logical uncertainty). The dissipative term $\mathcal{D}[\Gamma]$ reflects the logical uncertainty of the state relative to the structure of distinctions of Ω:

$$\mathcal{D}[\Gamma] = \sum_k \gamma_k \cdot \text{(interaction of Γ with atom } S_k \text{ of classifier Ω)}$$

Physical consequence: Decoherence is not external noise, but the internal logical dynamics of the system.

2.4 Constructive Algorithms

L-unification provides computable formulas:

/// χ_S: Γ → Ω for the subobject S.
public pure fn characteristic_morphism<const N: Int>(
    gamma: &StaticMatrix,
    s:     &Subspace,
) -> StaticMatrix<Complex, N, N>
{
    let p_s = projector_onto_subspace(s);
    p_s.matmul(&gamma).matmul(&p_s)
}

/// L_k = √χ_{S_k} for atoms of Ω. For basis projectors √P = P, so L_k = χ_k.
public pure fn lindblad_from_omega<const N: Int>(_gamma: &StaticMatrix)
    -> [StaticMatrix<Complex, N, N>; N]
{
    (0..N).map(|k| {
        let mut chi_k = StaticMatrix<Complex, N, N>.zeros();
        chi_k[k, k] = Complex.one();        // atom = basis projector
        chi_k
    }).to_array()
}

See: Constructive Algorithms

2.5 Connection to Physical Theories

Physical theory	How L-unification explains it	Status
Quantum decoherence	Dissipation = logical uncertainty relative to Ω	[T]
Second law of thermodynamics	$dS/dt \geq 0$ for the unital part $\mathcal{L}_0$ (Hermitian Lindblad operators); with regeneration the Lyapunov functional is the free energy (T-261)	[T]
Measurement in QM	Reduction = projection onto atom χ_{S_k}	[T]
Arrow of time	Monotone in the parameter $t$ of the dissipative semigroup; not supplied by the ▷-clock, but by the depth register, along whose readings the purity of the unital primitive part falls strictly (T-53b, emergent time Theorems 11.1–11.2)	[T] in $t$ and as emergent (it read "[C] as emergent" until 2026-09-25)

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3. Reduction to Quantum Mechanics

Connection to L-unification

Reduction to standard QM occurs when the logical structure Ω trivializes: at $R_\varphi \to 0$ the system loses its capacity for self-modeling, and the dissipative dynamics ℒ_Ω reduces to purely unitary.

3.1 Limit Functor

[T] Theorem 3.1 (Reduction to the Schrödinger equation). Let $\mathbb{H}$ be a Holon with $R_\varphi \to 0$. Then the evolution equation with emergent internal time τ:

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

reduces to the von Neumann equation:

$$\frac{d\rho}{dt} = -i[H, \rho]$$

for mixed states, or to the Schrödinger equation:

$$i\hbar\frac{d|\psi\rangle}{dt} = H|\psi\rangle$$

for pure states $\Gamma = |\psi\rangle\langle\psi|$.

Proof:

1. At $R_\varphi \to 0$ the system has no significant self-modeling
2. The regenerative term $\mathcal{R}[\Gamma, E] \propto \kappa(\Gamma) \to 0$ as $\kappa_0 \to 0$, where $\kappa_0 = \|\mathrm{Nat}(\mathcal{D}_\Omega, \mathcal{R})\|$ — categorical derivation
3. The dissipative term $\mathcal{D}[\Gamma] = \mathcal{L}_\Omega[\Gamma] + i[H_{eff}, \Gamma] \to 0$ for isolated systems (the logical structure Ω "freezes")
4. The unitary term remains: $\frac{d\Gamma(\tau)}{d\tau} = -i[H_{eff}, \Gamma]$, where $H_{eff}$ is the effective Hamiltonian
5. For $\Gamma = |\psi\rangle\langle\psi|$: $\frac{d|\psi\rangle\langle\psi|}{dt} = |d\psi\rangle\langle\psi| + |\psi\rangle\langle d\psi|$
6. Substituting into the equation: $i\hbar\frac{d|\psi\rangle}{dt} = H|\psi\rangle$ ∎

Interpretation via L-unification: Unitary QM is the limit in which the logical structure Ω is fully determined and admits no uncertainty (all $\chi_{S_k}$ are trivial).

3.2 Category of Quantum-Mechanical Systems

Definition 3.1 (Category QM).

$$\mathrm{Ob}(\mathbf{QM}) = \{(\mathcal{H}, H, \rho_0) : \mathcal{H} \text{ is a Hilbert space}, H = H^\dagger, \rho_0 \text{ — initial state}\}$$ $$\mathrm{Mor}_{\mathbf{QM}}((H_1, \rho_1), (H_2, \rho_2)) = \{U : U^\dagger U = I, U\rho_1 U^\dagger = \rho_2\}$$

3.3 Reduction Functor

Definition 3.2 (Reduction functor).

$$\pi_{\text{QM}}: \mathbf{Hol}_{R \to 0} \to \mathbf{QM}$$ $$\pi_{\text{QM}}(\mathbb{H}) := (\mathcal{H}_{\mathbb{H}}, H_{\mathbb{H}}, \Gamma_{\mathbb{H}})$$

warning

Retracted: Theorem 3.2 (Equivalence of categories $\mathbf{Hol}_{R=0} \simeq \mathbf{QM}$) [✗]

An earlier version stated as [T] that $\pi_{\text{QM}}|_{\mathbf{Hol}_{R=0}}$ is an equivalence of categories. It is not. Essential surjectivity fails: $\mathbf{QM}$ contains a qubit and the state $I/7$ with $P = 1/7$, while objects of $\mathbf{Hol}$ are seven-dimensional with $P > 2/7$, and unitary isomorphisms preserve dimension and purity (under $R = 1/(7P) \geq 1/7$ the category $\mathbf{Hol}_{R=0}$ is even empty). "At $R = 0$ morphisms are unitary" is not justified: the replacement channel onto a fixed point of self-modelling is a non-unitary morphism of $\mathbf{Hol}$. Full faithfulness was asserted without a functor on morphisms. What holds is an identification by definition of the unitary part of $\mathbf{Hol}$ with the full subcategory of $\mathbf{QM}$ on seven-dimensional systems with $P > 2/7$ [D]. Details: reduction to QM, §4.2.

3.4 Taxonomy of Physical Systems via L-Unification

[I] Theorem 3.3 (Classification by $R$ and structure of Ω). (Status [I]: under the master definition $R = 1/(7P) \in [1/7, 1]$ the row $R = 0$ describes no state; the table is a classification scheme. An earlier label [T] is retracted.)

Parameter $R$	Structure of Ω	Dynamics	Physical system
$R = 0$	Trivial (all χ_S defined)	$\frac{d\Gamma}{dt} = -i[H, \Gamma]$	Unitary QM (quarks, leptons, bosons)
$R \ll 1/3$	Partially defined	$\frac{d\Gamma}{dt} = -i[H, \Gamma] + \mathcal{L}_\Omega[\Gamma]$	Open QM (atoms in a medium)
$R \geq 1/3$	Reflexive (Ω models itself)	Full equation with $\mathcal{R}[\Gamma, E]$	Living systems (cells, organisms)

Physical consequence: The difference between "dead" and "living" matter lies in the structure of the logical classifier Ω: living systems are capable of modeling their own logical structure.

3.6 Discreteness of Time and Page–Wootters

Connection to L-unification

In Axiom Ω⁷, time is derived from the Page–Wootters mechanism via the temporal modality ▷ on the classifier Ω.

$$\tau_n = \rhd^n(\text{now}), \quad n \in \mathbb{Z}_7$$

The discreteness of time is a consequence of the finite structure of Ω.

[T] Theorem 3.4 (Discreteness of internal time). For a finite-dimensional system with $\dim(\mathcal{H}_O) = N$, internal time takes values from the cyclic group:

$$\tau \in \mathbb{Z}_N = \{0, 1, 2, \ldots, N-1\}$$

For UHM with $N = 7$: $\tau \in \mathbb{Z}_7$.

Proof: Follows from the finite-dimensionality of the clock algebra $\mathcal{A}_O \cong M_7(\mathbb{C})$. ∎

Physical consequences:

Consequence	Formula	Status
Quantum of time (chronon)	$\delta\tau = 2\pi/(7\omega_0)$	[T] Corollary
Continuous limit	$N \to \infty \Rightarrow \tau \in \mathbb{R}$	[T] Proven
Discrete ∞-groupoid	$\mathbf{Exp}^{disc}_\infty$ for $N < \infty$	[T] Formalized

Connection to the 42D formalism:

Full Page–Wootters state space:

$$\mathcal{H}_{total} = \mathcal{H}_O \otimes \mathcal{H}_{6D}, \quad \dim = 7 \times 6 = 42$$

The minimal 7D formalism is obtained via diagonal embedding — see Coherence Matrix.

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4. Emergent Geometry

Connection to L-unification

Spatial geometry emerges from the structure of distinctions defined by classifier Ω. The metric reflects the "logical distance" between configurations Γ.

4.1 Space as a Structure of Distinctions

[T] Theorem (Spatial metric, T-119). In the thermodynamic limit $M \to \infty$, the macroscopic algebra of observables in the $\{A,S,D\}$-sector is commutative (T-117 [T]). By Gelfand–Naimark duality it is isomorphic to $C(\Sigma^3)$ for the unique smooth compact 3-manifold $\Sigma^3$.

The metric on $\Sigma^3$ is induced by the Connes distance from the spectral triple. See Emergent Manifold $M^4$.

4.2 Pre-metric on the State Space

[T] Theorem 4.1 (Frobenius metric). The space $\mathcal{D}(\mathcal{H})$ of density matrices with metric

$$d_F(\rho_1, \rho_2) := \|\rho_1 - \rho_2\|_F = \sqrt{\mathrm{Tr}((\rho_1 - \rho_2)^2)}$$

is a complete metric space.

Proof: The Frobenius norm is the Hilbert–Schmidt norm, inducing a complete metric on $\mathcal{L}(\mathcal{H})$. Restriction to $\mathcal{D}(\mathcal{H})$ (a closed subset) preserves completeness. ∎

4.3 Information Geometry

[T] Quantum Fisher metric (standard result). The natural Riemannian metric on $\mathcal{D}(\mathcal{H})$ is the quantum Fisher metric:

$$g_{ij}^{(F)}(\rho) = \frac{1}{2}\mathrm{Tr}\left(\rho\{L_i, L_j\}\right)$$

where $L_i$ are logarithmic derivatives: $\partial_i \rho = \frac{1}{2}\{\rho, L_i\}$. The unique monotone Chentsov metric on the space of quantum states (Petz, 1996).

4.4 Emergent Dimensionality

[C] Theorem (Dimension 3+1, T-119 + T-120).

The dimension of macroscopic space is derived under named conditions (the heading read [T] until 2026-09-25):

• $\dim(\Sigma^3) = 3$ — from the rank count of T-119, Step 2c′ (T-119 [T]); the axis triple $\{A,S,D\}$ is not an $SU(3)$ sector (row 48a, retracted), and reading the colour triplet as space is [I]
• Lorentzian signature $(+,-,-,-)$ — [C] (registry row T-53): one time direction [T] (PW clock), three spatial directions at T-119 ($S^3$), the sign at reflection positivity (bounded-below PW generator / Osterwalder–Schrader; Krein route). KO-dimension does not fix the signature, and the KO-dimension-6 claim for $\mathbb{C}^7$ is retracted
• Product $M^4 = \mathbb{R} \times \Sigma^3$ — T-120 [T] as mathematics (restated T-119, 2026-09-25) (an earlier line derived it "from the sector decomposition $7 = 1_O \oplus 3 \oplus \bar{3}$ (T-120 [T])"; the axis-labelled decomposition is retracted, row 48a)

See Emergent Manifold

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5. Connection to General Relativity

tip

Status: Einstein equations [T] on the product triple; the derivation of $M^4$ [C]

The Einstein equations are obtained from the spectral action (T-65 [T]), and the cosmological constant is computed (T-65 [T]); the manifold $M^4$ on which they live is assembled from the categorical structure (T-120 [T] as mathematics since the restatement of T-119 on 2026-09-25; before it, under one condition — the open reconstruction axioms of T-119); the aperiodic clock, a second condition until 2026-09-25, is the depth register (T-118 [T]). An earlier version read "fully formalized [T] … the manifold $M^4$ is derived (T-120 [T])"; retracted with the status of T-120.

5.1 Emergent Manifold

[T] Theorem (Product of spectral triples, T-120) — as mathematics since the restatement of T-119 on 2026-09-25, which computes the spatial spectrum $S^3$; the reading of $M^4$ as physical spacetime is [I]. The heading read [T] until early 2026-09-25, then [C] at the open reconstruction axioms of T-119 (the aperiodic clock, T-118, named here then, is the depth register, [T]). In the thermodynamic limit the effective spectral triple factorizes:

$$(C^\infty(M^4) \otimes A_{\text{int}},\; L^2(M^4,S) \otimes H_{\text{int}},\; D_{M^4} \otimes 1 + \gamma_5 \otimes D_{\text{int}})$$

where $M^4 = \mathbb{R} \times \Sigma^3$ is assembled from the categorical structure under these conditions, not postulated. See Emergent Manifold.

5.2 Einstein Equations

[T] Theorem (Spectral action, T-65). The Chamseddine–Connes spectral action for the product $M^4 \times F_{\text{int}}$ reproduces:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

with $G_N = 3\pi/(7 f_2 \Lambda^2)$. Details: Einstein Equations.

5.3 Cosmological Constant

[T] The cosmological constant is computed from the Gap of the O-sector: $\Lambda_{\text{Gap}} > 0$ (T-71 [T]), which determines the vacuum topology $\Sigma^3 \cong S^3$ (T-120b: the topology $S^3$ [T] from T-119, the curvature [C at the vacuum symmetry]). Details: Cosmological Constant.

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6. Gauge Symmetries and the Standard Model

Section Status

$SU(3)_C$ is the stabiliser of the $O$-direction in $G_2 = \mathrm{Aut}(\mathbb{O})$ [T]; the electroweak factor $SU(2)_L \times U(1)_Y$ comes from the Fano-electroweak construction [C at (FE)], and its uniqueness is [H]. The former sentence — the whole group $SU(3) \times SU(2) \times U(1)$ "derived from $G_2$ via the sector decomposition and spectral triple [T]" — is retracted [✗]: $\mathrm{rank}\,G_2 = 2 < 4$ (registry row T-275), and the axis-labelled decomposition is retracted (row 48a). Specific parameters (masses, mixing angles) — partially derived, partially remain [P].

6.1 Symmetries of the Coherence Matrix

[T] Theorem 6.1 (Unitary symmetry group). The symmetry group of $\Gamma$:

$$\text{Sym}(\Gamma) := \{U \in U(7) : U\Gamma U^\dagger = \Gamma\}$$

is isomorphic to the stabilizer of $\Gamma$ in $U(7)$.

Proof: Direct consequence of the definition. ∎

6.2 Gauge Group from $G_2$

Gauge group: $SU(3)_C$ [T]; $SU(2)_L \times U(1)_Y$ [C at (FE)]; the former theorem is retracted [✗] (2026-09-25). (FE) belongs to the axis frame; in the Clifford frame it is replaced by (Cl₀), under which the whole group is the normaliser of colour in $\mathrm{Spin}(9)$, [C at (Cl)] (T-326, Premises of UHM).

Former statement, "[T] Theorem (Gauge group, T-53 + sector decomposition)": from $G_2 = \mathrm{Aut}(\mathbb{O})$ and the sector decomposition $7 = 1_O \oplus 3 \oplus \bar{3}$,

$$G_2 \supset SU(3) \xrightarrow{\text{Gap hierarchy}} SU(3)_C \times SU(2)_L \times U(1)_Y$$

Retracted: symmetry breaking leads from a group to a subgroup, and $SU(3)$ (rank 2) has no subgroup $SU(3) \times SU(2) \times U(1)$ (rank 4); neither has $G_2$, since $\mathrm{rank}\,G_2 = 2 < 4$ (I. Todorov, M. Dubois-Violette, Int. J. Mod. Phys. A 33, 1850118 (2018), eq. (4.2); registry row T-275). What holds: $SU(3)_C = \mathrm{Stab}_{G_2}(e_O) \subset G_2$ [T]; $SU(2)_L \times U(1)_Y$ from the Fano-electroweak construction on the Page–Wootters system factor [C at (FE)], acting on a different tensor factor, so that the ranks add to $2 + 2 = 4$; the uniqueness of this group is [H].

Details: $G_2$-structure, Standard Model.

6.3 Particles as Configurations Γ

Elementary particles are degenerate ($R \to 0$) configurations $\Gamma$. Three generations of fermions are derived from the triadic Fano structure [T]. Details: Three Generations of Fermions.

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7. Correspondence of 7 Dimensions to Physical Structures

Connection to L-unification

Each of the 7 dimensions has a dual role: physical (as an operator) and logical (as an aspect of classifier Ω).

7.1 Full Correspondence Table

[T] Theorem 7.1 (Physical operators of dimensions).

Dimension	Operator	Physical role	Status
A (Articulation)	Projector $P: P^2 = P, P^\dagger = P$	Quantum measurements, subspace selection	Formalized
S (Structure)	Hamiltonian $H: H^\dagger = H$	Energy spectrum, stationary states	Formalized
D (Dynamics)	$U(\tau) = e^{-iH_{eff}\tau}$, Lindblad operators $L_k$	Unitary evolution in internal time, $H_{eff}$ — effective Hamiltonian	Formalized
L (Logic)	Commutator $[A, B]$, anticommutator $\{A, B\}$	Lie algebras, Heisenberg uncertainty	Formalized
E (Interiority)	$\rho_E = \mathrm{Tr}_{-E}(\Gamma)$	Reduced density matrix	Formalized
O (Foundation)	$\vert 0\rangle\langle 0\vert$, $E_0 = \frac{1}{2}\hbar\omega$	Vacuum, zero-point oscillations	Formalized
U (Unity)	$\mathrm{Tr}(\cdot)$, $P = \mathrm{Tr}(\Gamma^2)$	Normalization, purity measure	Formalized

7.2 Algebraic Structure

[T] Theorem 7.2 (Algebra of dimensions). The operators of dimensions form an algebra:

$$\mathcal{A}_{\text{dim}} := \text{span}\{P_A, H_S, U_D, [,]_L, \rho_E, |0\rangle\langle 0|_O, \mathrm{Tr}_U\}$$

with commutation relations determined by the quantum-mechanical algebra of operators.

7.3 Connection to Symmetry Groups

[T] Theorem (Symmetry group, T-53). The full automorphism group $G_2 = \mathrm{Aut}(\mathbb{O})$ acts on the 7 dimensions. The stabilizer of the $O$-direction is $SU(3)$, determining the gauge structure. Each dimension has a dual role: physical (as an operator) and logical (as an aspect of classifier Ω).

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8. No-Signaling

What is proven and what is not

Proven [T]: the regenerative term of a holon $A$ leaves the unconditioned reduced state of a distant system $B$ unchanged, $\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = 0$ (Theorem 8.1), and so do local unitaries at $A$ (Corollary 8.1). Not proven, and false under the measurement rule the corpus adopts: that the full nonlinear dynamics forbids signalling. If $A$ performs a projective measurement with the Lüders update of measurement, Theorem 2.1, step 4, the state of $B$ becomes one of the conditional states $\rho_B^{(k)}$ with probabilities $p_k$, and the state-dependent regenerative term of $B$ acts on each of them; for a nonlinear term the resulting ensemble depends on what $A$ chose to do (§8.5). No-signalling of the full dynamics holds in the non-selective reading of NS2, and that reading is not a choice: the axioms force it (Theorem 8.5, §8.8), so no-signalling of the full dynamics is [T], at the price named in §8.5 (the "Everett phone"). Updated 2026-09-25: the box said that no-signalling holds "only in the non-selective reading … [C]". An earlier version of this box said that no-signalling is a consequence of the CPTP structure of $\varphi$ and that the nonlinearity "does not violate" it; that is retracted.

8.1 Problem Statement

Introducing nonlinearity into quantum mechanics typically violates the no-signaling principle (Gisin, 1990; Polchinski, 1991). The UHM evolution equation contains a nonlinear regenerative term $\mathcal{R}[\Gamma, E]$, where the nonlinearity arises from $\kappa(\Gamma)$ and $\varphi(\Gamma)$.

The fundamental difference of UHM from Weinberg's nonlinear QM:

Property	Nonlinear QM (Weinberg)	UHM
Defined on	Wave functions $\vert\psi\rangle$	Density matrices $\Gamma$
Extension to $A \otimes B$	Not canonical	$\varphi_A \otimes \mathrm{id}_B$ (CPTP)
Ensemble dependence	Yes (different decompositions → different evolution)	The map depends on $\Gamma$ alone, and a remote measurement leaves $B$'s marginal — the argument of $B$'s dynamics — unchanged (§8.8); branch-by-branch evolution (§8.5) is the selective reading, which is not a dynamics of UHM
Domain of applicability	All quantum systems	Only autonomous L2+ systems

8.2 Canonical Extension of Regeneration to Composite Systems

[T] Definition 8.1 (Canonical extension).

For a composite system $A \otimes B$, where $A$ is an autonomous holon:

$$\tilde{\mathcal{R}}_A[\Gamma_{AB}] := \kappa_A(\Gamma_A) \cdot \left((\varphi_A \otimes \mathrm{id}_B)(\Gamma_{AB}) - \Gamma_{AB}\right) \cdot g_V(P_A)$$

where $\Gamma_A = \mathrm{Tr}_B(\Gamma_{AB})$.

8.3 Central Theorem

[T] Theorem 8.1 (Regeneration of $A$ leaves the marginal of $B$ unchanged).

For two spatially separated autonomous holons $A$ and $B$ with joint state $\Gamma_{AB}$:

$$\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = 0$$

Proof:

$$\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = \kappa_A \cdot g_V(P_A) \cdot \left(\mathrm{Tr}_A[(\varphi_A \otimes \mathrm{id}_B)(\Gamma_{AB})] - \mathrm{Tr}_A[\Gamma_{AB}]\right)$$

For a CPTP channel $\varphi_A$ with Kraus representation $\varphi_A(\cdot) = \sum_m K_m (\cdot) K_m^\dagger$:

$$\mathrm{Tr}_A[(\varphi_A \otimes \mathrm{id}_B)(\Gamma_{AB})] = \mathrm{Tr}_A\left[\sum_m (K_m \otimes I_B)\Gamma_{AB}(K_m^\dagger \otimes I_B)\right]$$ $$= \mathrm{Tr}_A\left[(\sum_m K_m^\dagger K_m \otimes I_B)\Gamma_{AB}\right] = \mathrm{Tr}_A[(I_A \otimes I_B)\Gamma_{AB}] = \Gamma_B$$

Therefore: $\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = \kappa_A \cdot g_V(P_A) \cdot (\Gamma_B - \Gamma_B) = 0$. ∎

The identity concerns the unconditioned marginal $\Gamma_B = \mathrm{Tr}_A\Gamma_{AB}$. An earlier title of this theorem, "No-signaling in UHM", claimed more than it proves; see §8.5 for what a measurement at $A$ does.

[T] Corollary 8.1 (Invariance under local operations).

For any local unitary operation $U_A$ by Alice, the contribution of $\tilde{\mathcal{R}}_A$ to Bob's state remains zero:

$$\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[(U_A \otimes I_B)\Gamma_{AB}(U_A^\dagger \otimes I_B)]] = 0$$

regardless of changes in $\kappa_A$ and $\Delta F_A$.

[T] Theorem 8.2 (Full evolution of subsystem B).

The reduced state $\Gamma_B(\tau) = \mathrm{Tr}_A[\Gamma_{AB}(\tau)]$ obeys:

$$\frac{d\Gamma_B}{d\tau} = \mathrm{Tr}_A[\mathcal{L}_{lin}[\Gamma_{AB}]] + \mathcal{R}_B[\Gamma_B]$$

where $\mathcal{R}_B[\Gamma_B] = \kappa_B(\Gamma_B) \cdot (\varphi_B(\Gamma_B) - \Gamma_B) \cdot g_V(P_B)$ — depends only on the local state $\Gamma_B$.

8.4 No-Signaling Conditions (NS1–NS3)

The proof rests on three structural conditions:

Condition	Statement	Follows from
NS1 (Locality of φ)	$\tilde{\varphi}_A = \varphi_A \otimes \mathrm{id}_B$	Autonomy (A1), categorical structure
NS2 (Locality of κ)	$\kappa_A(\Gamma_{AB}) = \kappa_A(\mathrm{Tr}_B(\Gamma_{AB}))$	Definition of κ₀ via local coherences
NS3 (CPTP φ)	$\varphi$ is a CPTP channel	Definition of φ

NS2 makes $\kappa_A$ a function of the unconditioned marginal. Whether that marginal is updated when a distant partner is measured decides between the two readings of §8.5.

8.5 Ensemble Independence

[D] Theorem 8.3 (The evolution map is a function of $\Gamma$).

The UHM evolution is defined on the density matrix $\Gamma$, not on its ensemble decomposition.

Proof: All components of the equation ($H_{eff}$, $\mathcal{D}_\Omega$, $\kappa$, $\varphi$, $g_V(P)$) are functions of $\Gamma$, not of any specific decomposition $\Gamma = \sum_i p_i |\psi_i\rangle\langle\psi_i|$. ∎ (This holds by the definition of the terms, hence [D].)

Retracted: "two preparations of the same Γ evolve identically" and "resolving the Gisin problem"

An earlier version concluded from Theorem 8.3 that two different preparations of the same $\Gamma$ evolve identically, and the conclusion of this page called that "resolving the Gisin problem". Both are retracted. A preparation that is a proper mixture — a coin toss, or a measurement on a distant partner with the Lüders update — produces in each run one of the states $\rho_k$, and the regenerative term acts on that state; the ensemble then evolves as $\sum_k p_k\,\Phi_t(\rho_k)$, not as $\Phi_t(\sum_k p_k \rho_k)$, and for a nonlinear $\Phi_t$ the two differ. This is the scenario of N. Gisin ("Weinberg's non-linear quantum mechanics and supraluminal communications", Phys. Lett. A 143, 1 (1990)): $A$ measures one half of an entangled pair in a basis of her choice, the conditional ensemble at $B$ depends on that choice, and a nonlinear local evolution at $B$ turns the difference into different statistics. A map that depends on $\Gamma$ alone can still be nonlinear, and C. Simon, V. Bužek and N. Gisin proved that with Hilbert-space states, the trace rule and no signalling the dynamics must be linear and completely positive (Phys. Rev. Lett. 87, 170405 (2001)).

Example (regression check in website/scripts/check_core_numbers.py). $A$ is a qutrit, $B$ a holon, the joint state $\tfrac{1}{\sqrt 3}(|0\rangle|e_0\rangle + |1\rangle|e_1\rangle + |2\rangle|e_2\rangle)$. If $A$ does not measure, the state of $B$ is $\rho_B = \tfrac13(|e_0\rangle\langle e_0| + |e_1\rangle\langle e_1| + |e_2\rangle\langle e_2|)$ with $P = 1/3$, and the viability gate $g_V(P) = \mathrm{clamp}(7P - 2, 0, 1)$ equals $1/3$. If $A$ measures in the basis $\{|0\rangle, |1\rangle, |2\rangle\}$, the states of $B$ are the pure $|e_k\rangle\langle e_k|$ with $g_V = 1$. With $\kappa$ fixed, the initial drift of $B$'s averaged state is $\tfrac{\kappa}{3}(\rho_* - \rho_B)$ in the first case and $\kappa(\rho_* - \rho_B)$ in the second: $B$'s statistics depend on whether $A$ measured. With $\kappa$ depending on $\mathrm{Coh}_E$, two measurement bases of $A$ that both leave $B$'s branches pure already give different drifts.

What would restore no-signalling. (a) The non-selective reading [C]: the nonlinear terms act on unconditioned marginals and never on remotely conditioned sub-ensembles — NS2 kept after a remote measurement. Then no signal passes, but the evolution inside one branch depends on the branches that did not occur; J. Polchinski, who built this construction for Weinberg's nonlinear quantum mechanics, named the price — a channel between branches of the wave function, the "Everett phone" ("Weinberg's nonlinear quantum mechanics and the Einstein–Podolsky–Rosen paradox", Phys. Rev. Lett. 66, 397 (1991)) — and the reading also gives up the Lüders update for remote partners that the corpus uses. (b) Convex quasi-linearity: J. Rembieliński and P. Caban argued that deterministic nonlinear evolutions mapping a mixture to a mixture of the images, $f(\lambda\rho_1 + (1-\lambda)\rho_2) = p\,f(\rho_1) + (1-p)\,f(\rho_2)$ for some $p \in [0,1]$, do not allow signalling and "cannot be ruled out by a standard argument" ("Nonlinear evolution and signaling", Phys. Rev. Research 2, 012027 (2020)), and built a nonlinear extension of the Lindblad generator of this kind ("Nonlinear extension of the quantum dynamical semigroup", Quantum 5, 420 (2021), arXiv:2003.09170); A. Kent gave another route ("Nonlinearity without superluminality", Phys. Rev. A 72, 012108 (2005)). The regenerative flow of UHM is not convex quasi-linear: on 30 random pairs of pure states, with the canonical $\kappa(\Gamma)$, $g_V$ and a dephasing linear part, the image of a mixture lies off the segment between the images of its components by 4–26 % of that segment's length (numerical check). Route (b) is therefore not available as the term is written. Whether $\mathcal{R}$ can be recast so as to take route (b) is settled in §8.7: it cannot while the threshold at $P = 2/7$ is kept, and the Rembieliński–Caban no-signalling itself rests on branch weights that change with $B$'s time. Route (a) is the one the axioms take: §8.8 proves that the argument of a holon's dynamics is its unconditioned marginal and that the branch-by-branch evolution of this section is not a UHM dynamics, so no-signalling of the full dynamics is [T] (Theorem 8.5). Updated 2026-09-25: this paragraph gave "[C] under the non-selective reading (a)" and left the reading open. What survives of the retracted "two preparations of the same $\Gamma$ evolve identically" is its correct form: preparations that leave $B$ with the same marginal leave it with the same future marginal (Theorem 8.5 (ii)).

8.6 Computational Bound

Retracted: Theorem 8.4 (Absence of computational speedup) [T]

An earlier version stated as a theorem that the nonlinear regenerative term provides no computational speed-up beyond BQP, on four grounds: (1) $\mathcal{R}$ is active only for L2+ systems, qubits have $R \approx 0$; (2) each regeneration step requires $\Delta F > 0$; (3) $\varphi$ does not increase quantum information; (4) decoherence suppresses exponentially small differences. None of the four bounds what a nonlinear evolution can compute, and the claim collides with D. S. Abrams and S. Lloyd, who showed that generic deterministic nonlinear quantum evolution solves NP-complete and #P problems in polynomial time by amplifying exponentially small differences between states ("Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems", Phys. Rev. Lett. 81, 3992 (1998), arXiv:quant-ph/9801041). (1) restricts which systems are active, not what an active system can do; holons are seven-level systems, and $R = 1/(7P)$ never vanishes. (2) is an energy cost, not a bound on complexity. (3) constrains the channel $\varphi$, while the nonlinearity sits in the scalar weights $\kappa(\Gamma)\,g_V(P(\Gamma))$. (4) states the question rather than answering it: whether the nonlinear term amplifies small differences faster than decoherence erases them. The gate $g_V$ makes the regenerative flow bistable — for $\kappa$ above the existence threshold the dead state $I/7$ (T-148) and the attractor $\rho_*$ (T-96) both attract, and trajectories that start on either side of the boundary between their basins end a finite distance apart however close they started — which is the kind of amplification Abrams and Lloyd use. The theorem is retracted.

[T] Theorem 8.6 (Abrams–Lloyd amplification runs on marginals). Let a holon regenerate toward the self-registering $\varphi_s$ (φ operator) with a Hamiltonian diagonal in the frame, a constant rate $\kappa > 0$ and the Fano dissipator. Then:

1. $F = \mathrm{diag}(\tfrac12, \tfrac12, 0, \ldots, 0)$ is stationary, and on the invariant line $p = (\tfrac12 + \varepsilon, \tfrac12 - \varepsilon, 0, \ldots, 0)$ the flow is exactly

$$\dot\varepsilon = \frac{2\kappa}{7}\,\frac{\varepsilon\,(1 - 4\varepsilon^2)}{(1 + 4\varepsilon^2)^2},$$

so $F$ is a saddle with unstable rate $\mu = 2\kappa/7$, and $\varepsilon$ grows from $\varepsilon_0$ to $0.4$ in time $\tfrac{7}{2\kappa}\bigl(\ln(1/\varepsilon_0) + O(1)\bigr)$. 2. For a Boolean function $f$ on $n$ bits with a fraction $s$ of satisfying inputs, a circuit of $\mathrm{poly}(n)$ standard gates — Hadamards on an input register and a coin qubit, a reversible circuit for $f$, and a controlled exchange of $|e_1\rangle, |e_2\rangle$ in the holon — leaves the holon's marginal at $\mathrm{diag}(\tfrac12 + \tfrac s2, \tfrac12 - \tfrac s2, 0, \ldots)$. By Theorem 8.5 the holon's regeneration acts on this marginal. If $s = 0$ the holon stays at $F$; if $s \geq 2^{-n}$ it reaches $p_1 \geq 0.9$ within time $\tfrac{7}{2\kappa}(n\ln 2 + O(1))$. One measurement of the holon in the frame distinguishes the two cases with bounded error.

Hence the ideal regenerative dynamics decides satisfiability in time linear in $n$, and a bound "UHM computes no more than BQP" for it would imply NP $\subseteq$ BQP.

Proof. (1) For diagonal $\Gamma$ the Hamiltonian, the dissipator and $\mathcal{P}_\alpha$ contribute nothing, and $\Gamma^2/P$ keeps zero entries zero, so the line is invariant and $\dot p_1 = \kappa\,g_V(P)\,R\,(p_1^2/P - p_1)$. With $p_1 = \tfrac12 + \varepsilon$: $P = \tfrac12 + 2\varepsilon^2 > 3/7$, so $g_V = 1$; $R = 2/(7(1 + 4\varepsilon^2))$; and $p_1^2/P - p_1 = \varepsilon(1 - 4\varepsilon^2)/(1 + 4\varepsilon^2)$. Integrating $d\varepsilon/\dot\varepsilon$ gives the time. (2) The coin puts the holon in $|e_1\rangle$ on one branch; on the other it is $|e_2\rangle$ unless $f(x) = 1$, when it is exchanged to $|e_1\rangle$; averaging over the uniform input and the coin gives the marginal. At $p_1 = 0.9$ against $\tfrac12$, one frame measurement errs with probability at most $0.5$ on one side and $0.1$ on the other; repetition makes the error small. $\blacksquare$

Witness (test_abrams_lloyd_amplification_runs_on_marginals): $\kappa = 1$, $H = \mathrm{diag}(0, 0.3, \ldots, 1.8)$; the Jacobian at $F$ has exactly one positive eigenvalue, $2/7$ to $10^{-6}$; for $n = 10, 20, 30, 40$ the times to $p_1 \geq 0.9$ are $29.55, 53.80, 78.05, 102.30$ — steps of $24.25$ against $10\ln 2 \cdot 7/2 = 24.26$; the start with $s = 0$ has not moved ($10^{-12}$) at $t = 102.3$. The same construction works at any stationary state with a real positive eigenvalue whose eigendirection a preparation can reach, such as the threshold point of the bistable gated flow of §8.7 between $\lambda = 0.6$ and $0.8$; there the preparation must hit the point to within $2^{-n}$, which rotations synthesised to that precision (Solovay–Kitaev) allow in $\mathrm{poly}(n)$ gates.

Open question [H], narrowed. The theorem concerns the ideal dynamics: exact rates, exact frame-diagonal $H$, no noise in the flow. A perturbation of the flow of size $\eta$ moves the saddle by $O(\eta)$, and the amplification then separates only inputs with $s \gg \eta$. Whether UHM with noise of a fixed size computes beyond BQP is open. Updated 2026-09-25: the open question read "whether the regenerative term … permits a speed-up beyond BQP"; for the ideal dynamics it does, whenever the regeneration keeps a holon alive — with the canonical unital $\varphi_{\mathrm{coh}}$ an isolated holon is dead and flows to $I/7$ (dead isolation), and the construction does not apply.

8.7 The measurement reading: three options

Status of this subsection

Updated 2026-09-25: the choice described in this subsection is made by the axioms, not by the author — §8.8 proves that the regenerative term acts on the unconditioned marginal (Option A), so no-signalling of the full dynamics is [T]. Option C remains a consistent restricted model, but no-signalling does not require it. The rest of the subsection stands as the analysis of the alternatives. Research programme [Pr] (superseded by §8.8): the corpus has to choose how a measurement at a remote partner enters the regenerative term. Proven [T] (complete proofs below, numerical witnesses in website/scripts/check_core_numbers.py): selective no-signalling forces the local dynamics to be affine; neither an affine flow nor a normalised-linear (Rembieliński–Caban) flow can keep the viability gate or its bistability. Hence no modification of $\mathcal{R}$ that keeps the threshold behaviour at $P = 2/7$ is non-signalling in the selective reading: Option B below is closed, and the choice is between A and C.

Setting. $A$ and $B$ share $\Gamma_{AB}$; $B$ evolves by a flow $\Phi_t$ on $\mathcal{D}(\mathcal{H}_B)$, possibly nonlinear. $A$ performs a measurement with outcomes $k$. In the selective reading each run leaves $B$ in the conditional state $\rho_k$, the fraction $p_k$ of runs with outcome $k$ is fixed by $A$'s records, and $B$'s statistics at time $t$ are those of $\sum_k p_k\,\Phi_t(\rho_k)$. In the non-selective reading $\Phi_t$ acts on the unconditioned marginal $\rho_B = \sum_k p_k \rho_k$, and conditioning on $k$ is applied only to the joint state at readout.

Steering. For a purification of $\rho_B$ with a large enough partner, every finite decomposition $\rho_B = \sum_k p_k \rho_k$ is produced by some measurement of $A$ (E. Schrödinger, Proc. Cambridge Philos. Soc. 32, 446 (1936); L. P. Hughston, R. Jozsa, W. K. Wootters, "A complete classification of quantum ensembles having a given density matrix", Phys. Lett. A 183, 14 (1993)). Checked on 20 random measurements of a random seven-dimensional $\rho_B$: the branches average back to $\rho_B$ to $10^{-15}$.

[T] Proposition (selective no-signalling forces affinity). $B$'s statistics are independent of what $A$ does, for every $\Gamma_{AB}$ and every measurement of $A$, in the selective reading, if and only if every $\Phi_t$ is affine: $\Phi_t(\sum_k p_k\rho_k) = \sum_k p_k\,\Phi_t(\rho_k)$. For a flow generated by a vector field $X$, this holds for all $t$ if and only if $X$ is affine.

Proof. If $A$ does nothing, $B$ ends in $\Phi_t(\rho_B)$; if $A$ measures, in $\sum_k p_k \Phi_t(\rho_k)$. By steering every decomposition occurs, so no-signalling is the stated identity for all decompositions, which is affinity; the converse is immediate. An affine flow has an affine generator (differentiate at $t = 0$), and an affine vector field generates an affine flow (a linear inhomogeneous ODE). $\blacksquare$ This is the argument of Gisin (1990); Simon, Bužek and Gisin (2001) add complete positivity. Witness: for the linear-rate term $\bar c\,(\bar\varphi(\Gamma) - \Gamma)$ with fixed $\bar\varphi = \varphi_{\mathrm{coh}}$ ($k = 0.8$, $\alpha = 0.4$) the selective and unconditioned drifts agree to $8 \cdot 10^{-16}$ over 20 steered decompositions; for the gated term $g_V(P)(\rho_* - \Gamma)$ on a $\rho_B$ with $P = 0.277 < 2/7$ they differ by $0.66$ in Frobenius norm — the unconditioned gate is shut, the pure branches open it.

[T] Proposition (the gate is neither affine nor quasi-linear). Let $X$ be a vector field on $\mathcal{D}(\mathbb{C}^7)$ generating a flow $\Phi_t$.

1. If $X$ is affine and vanishes on a non-empty open subset of $\mathcal{D}(\mathbb{C}^7)$ — such as $\{P < 2/7\}$, which is the Frobenius ball $\|\Gamma - I/7\|_F^2 < 1/7$ around $I/7$, because $P = \|\Gamma - I/7\|_F^2 + 1/7$ — then $X \equiv 0$.
2. If every $\Phi_t$ is affine and $\sigma \neq \tau$ are fixed points, every point of the segment $[\sigma, \tau]$ is fixed; neither is an isolated attractor.
3. Let $\Phi_t(\rho) = e^{tL}\rho / \mathrm{Tr}(e^{tL}\rho)$ with $e^{tL}$ positive — the class of J. Rembieliński and P. Caban, generator $\dot\rho = L\rho - \rho\,\mathrm{Tr}(L\rho)$. If this generator vanishes on a non-empty open set of states, it vanishes everywhere; and of two fixed states at most one attracts along the segment joining them.

Proof. (1) The trace-one states have non-empty interior in the real affine hyperplane of trace-one Hermitian matrices; an affine map vanishing on an open subset of it vanishes on the hyperplane. (2) $\Phi_t((1-s)\sigma + s\tau) = (1-s)\sigma + s\tau$ by affinity. (3) The generator vanishes at $\rho$ exactly when $L\rho = \mu\rho$ with $\mu = \mathrm{Tr}(L\rho)$. If this holds on a ball of states, then for two linearly independent states of the ball their mixtures, which lie in the ball, are eigenvectors too, which forces equal eigenvalues; the ball spans the operator space, so $L = \mu\,\mathrm{id}$ and the generator is zero. For fixed states $L\sigma = \mu\sigma$, $L\tau = \mu'\tau$, the point $(1-s)\sigma + s\tau$ goes to the point with

$$s(t) = \frac{s\,e^{\mu' t}}{(1-s)\,e^{\mu t} + s\,e^{\mu' t}},$$

so for $\mu' > \mu$ the state $\sigma$ repels along the segment, for $\mu' < \mu$ the state $\tau$ does, and for $\mu' = \mu$ the whole segment is fixed. $\blacksquare$

Witnesses: the segment law holds to $8 \cdot 10^{-16}$ for a random normalised-linear generator on $\mathbb{C}^7$, whose flow carries all seven basis states to the one with the largest eigenvalue. The canonical gated flow $\gamma(I/7 - \Gamma) + \kappa\, g_V(P)(\rho_* - \Gamma)$ with $\gamma = 0.3$, $\kappa = 10$ is bistable: on the segment $(1-\lambda)I/7 + \lambda\rho_*$ the starts with $\lambda \le 0.6$ ($P \le 0.252$) end at $I/7$, those with $\lambda \ge 0.8$ ($P \ge 0.336$) end at a living state with $P = 0.4275$. The same flow with the gate removed has one attractor (three starts end within $10^{-15}$ of each other).

What the Rembieliński–Caban evolutions escape, and what they do not. In their argument the weights of the branches change with $B$'s time: $\lambda \mapsto \bar\lambda(t) = \lambda\,\mathrm{Tr}\,\phi_t(\rho_a)/\mathrm{Tr}\,\phi_t(\rho)$ (J. Rembieliński, P. Caban, Phys. Rev. Research 2, 012027 (2020), eqs. (20), (24); Quantum 5, 420 (2021), eqs. (13), (17)). Their qubit example is a local filter, $\rho \mapsto A\rho A^\dagger/\mathrm{Tr}(A\rho A^\dagger)$ with $A = e^{gt\,\sigma\cdot e/2}$ (checked to $3 \cdot 10^{-17}$ against their eq. (14)), and $\bar\lambda(t)$ is the frequency of branch $a$ among the runs in which the filter succeeds. With $A$'s recorded frequencies held at $\tfrac12$, $B$'s Bloch component along $e$ at $gt = 1$ is $0.668$ when $e\cdot\zeta = \tfrac12$ against $0.762$ when $e\cdot\zeta = 0$ — Gisin's signal returns. Their no-signalling therefore holds either for a heralded, post-selected filter, which is linear quantum mechanics with post-selection, or in a reading in which the evolved weights are not $A$'s frequencies, which is a form of Option A. It is not an independent route for a deterministic law that acts in every run. A. Kent's construction ("Nonlinearity without superluminality", Phys. Rev. A 72, 012108 (2005)) likewise changes which state of an entangled subsystem the nonlinear law acts on — a non-standard definition of that state — rather than the law; for UHM it is a variant of Option A.

Option A — non-selective reading [T]: forced by the axioms (§8.8; the label was [C] before 2026-09-25). $\mathcal{R}$ stays as written: $\kappa$ and $g_V$ are evaluated on the unconditioned marginal, also after a remote measurement (NS2 without update). What is proven: no remote operation changes $B$'s marginal (Theorems 8.1, 8.2 [T], given the reading and no interaction term between $A$ and $B$). Cost: (i) the Lüders update of measurement, Theorem 2.1, step 4 cannot be applied to a system entangled with a holon before the holon's regeneration acts; selection becomes a readout of the final joint state, $\mathrm{Tr}[(\Pi_k \otimes Q)\,\Gamma_{AB}(t)]/p_k$, which differs from $\mathrm{Tr}[Q\,\Phi_t(\rho_k)]$; (ii) the rate at which a branch regenerates depends on the branches that did not occur — J. Polchinski's "Everett phone" (Phys. Rev. Lett. 66, 397 (1991)); in the qutrit example of §8.5 each pure branch regenerates with $g_V = 1/3$, not $1$; (iii) a holon that has observed an outcome regenerates as if it had not, so a recorded outcome is not a fact for the holon's own dynamics — more exactly, if the record is held in the holon's own degrees of freedom, the regeneration acts on the whole record-bearing state, not branch by branch. Kept: the gate, the thresholds and the attractor results (T-96, T-98, T-125, T-127, T-148, T-149), and Corollary T-221.2, whose Loc becomes [T] (the corollary stays [C] only through OW, at T-120).

Option B — a modified $\mathcal{R}$ non-signalling in the selective reading [✗] while the threshold is kept. By the two propositions, a modification is non-signalling in the selective reading only if it is affine, and an affine or normalised-linear term cannot close below $P = 2/7$ or keep the dead state $I/7$ and the living attractor both attracting. Making $\kappa$ and $g_V$ affine in $\Gamma$ does not help: an affine scalar times the linear field $\varphi(\Gamma) - \Gamma$ is quadratic. What survives of B is a term with state-independent rates, which is Option C.

Option C — linear dynamics only. No-signalling in both readings [T]. $\mathcal{R}_{\mathrm{lin}}(\Gamma) = \bar c\,(\bar\varphi(\Gamma) - \Gamma)$ with constant $\bar c \ge 0$ and a fixed CPTP $\bar\varphi$ (fixed $k$ and $\alpha$ in $\varphi_{\mathrm{coh}}$, or the replacement $\Gamma \mapsto \bar\rho$ with a fixed target); $\bar c(\bar\varphi - \mathrm{id})$ is a GKSL generator, and a local GKSL generator does not signal in either reading. Kept: the direction $\rho_* - \Gamma$ and its optimality (T-261, with $\kappa_{\mathrm{eff}} = \bar c$), the fixed point of the regenerative term, the Lüders update of Theorem 2.1 without restriction, and the Abrams–Lloyd concern of §8.6 disappears, since linear CPTP dynamics contracts the trace distance between states. Lost: the gate as a dynamical switch and the bistability (Proposition, items 1–3); viability $P > 2/7$ becomes a condition on parameters of a unique attractor — the balance formula (T-98) at fixed $\kappa$ — rather than a basin; the $\Gamma$-dependence of the self-model through $k = 1 - R(\Gamma)$; and, since $\varphi_{\mathrm{coh}}$ and $\mathcal{L}_0$ are both unital, $I/7$ is stationary for an isolated holon, so by item 2 a living stationary state could not be an isolated attractor: an isolated living attractor needs a non-unital input (a fixed target $\bar\rho \neq I/7$ or the environmental backbone of T-148), and "a dead holon stays dead while a living one with the same parameters stays alive" is no longer available.

Result	Option A (non-selective)	Option C (linear)
No-signalling of the full dynamics	[T]; the reading is forced (Theorem 8.5)	[T]
Lüders update at a partner entangled with a holon (measurement, Thm 2.1 step 4)	only as final readout	unrestricted
Gate $g_V$, bistability, "dead stays dead"	kept	lost
Attractor results T-96, T-98, T-125, T-127	kept	T-98 as a parameter condition; stability of the unique attractor
Genesis T-148, T-149	kept	through a non-unital environment only
T-261 (direction, BKM gradient)	kept	kept with constant $\kappa_{\mathrm{eff}}$
Corollary T-221.2 (Loc)	Loc [T] (Theorem 8.5); the corollary [T] — OW is the single topos, not the emergence of $M^4$, so it no longer passes through T-120 (the corollary read [C] through OW (T-120) until 2026-09-25; T-120 itself is [T] as mathematics since)	Loc [T]
§8.6, speed-up beyond BQP	ideal dynamics: satisfiability in linear time [T] (Theorem 8.6); with noise open [H]	absent (linear CPTP)

One question remains open [Pr]: under A, a relativistic statement of "unconditioned" — relative to which hypersurface the marginal is taken — which Kent's construction suggests but which UHM would have to state on the spacetime it derives (T-120 [T] as mathematics; its reading as physical spacetime [I]). (Until 2026-09-25 a second open question was which of A and C the corpus adopts; §8.8 answers it.)

8.8 The reading is forced

Setting. In UHM every system has a state, a density matrix, and every law is a map of joint states. A holon's flow enters the joint evolution through the canonical extension (Definition 8.1), whose scalars $\kappa$, $g_V$, $k$ are read on the holon's own state. Measurement is not an extra postulate: it is the decoherence channel of the classifier atoms, $L_k = \sqrt{\chi_{S_k}}$ (measurement, §2.3), which in the limit of fast decoherence is $\Gamma \mapsto \sum_k P_k\Gamma P_k$, and with a record $\omega \mapsto \sum_k (P_k \otimes |k\rangle\langle k|)\,(\omega \otimes |0\rangle\langle 0|)\,(P_k \otimes |k\rangle\langle k|)$ after the recording unitary — a channel.

[T] Theorem 8.5 (The measurement reading is forced; no-signalling of the full dynamics).

1. Only the marginal is a state of $B$. Let $S$ assign to each joint state $\omega$ of $B \otimes E$ a state $S(\omega)$ of $B$ such that $S(\rho \otimes \tau) = \rho$ and $S((\mathrm{id}_B \otimes \Lambda)(\omega)) = S(\omega)$ for every channel $\Lambda$ on $E$ — nothing done to the rest changes what $B$ is. Then $S(\omega) = \mathrm{Tr}_E\,\omega$.
2. No-signalling. Let $B$ be a holon whose generator acts through the canonical extension, with no interaction term between $B$ and $A$. For every sequence of channels applied to $A$ — unitaries, measurements with records kept at $A$, discarding — the marginal $\Gamma_B(t)$ is the same function of $t$ as when $A$ does nothing, and so are the statistics of every measurement on $B$.
3. The selective reading is not a UHM dynamics. The conditional state $\rho_{B|k}$ is not a function of the joint state: after a measurement at $A$ the joint state $\sum_k p_k\,\rho_{AB|k} \otimes |k\rangle\langle k|$ is the same whichever outcome occurred. A regenerative term applied to $\rho_{B|k}$ needs, besides the state, a selected outcome — a stochastic reduction, which is not among the axioms. The Lüders formula of measurement, Theorem 2.1, step 4 is the conditional readout $\mathrm{Tr}[(P_k \otimes Q)\,\omega]/p_k$ of the joint state, not a replacement of $\Gamma_B$ inside the dynamics.

Proof. (1) With $\Lambda$ the replacement channel $X \mapsto \mathrm{Tr}(X)\,\tau$: $S(\omega) = S((\mathrm{id} \otimes \Lambda)\omega) = S(\mathrm{Tr}_E\,\omega \otimes \tau) = \mathrm{Tr}_E\,\omega$. In categorical terms the state of a part is the joint state followed by the discarding map of the rest, the unique map to the terminal object; $\mathrm{discard} \circ \Lambda = \mathrm{discard}$ for every channel because a map to the terminal object is unique. B. Coecke showed that this terminality coincides with non-signalling in any process theory with explicit causal structure ("Terminality implies non-signalling", QPL 2014, arXiv:1405.3681). (2) By Theorem 8.2 and trace preservation of $A$'s generator, $\dot\Gamma_B = \mathcal{L}_B[\Gamma_B]$: an autonomous equation whose right side is locally Lipschitz ($\kappa$ smooth, $g_V$ Lipschitz, $1/P \leq 7$). A channel $\Lambda$ on $A$ at time $t_0$ leaves $\Gamma_B(t_0)$ unchanged, since $\mathrm{Tr}_A(\Lambda \otimes \mathrm{id})\omega = \mathrm{Tr}_A\,\omega$; by uniqueness of solutions $\Gamma_B(t)$ is unchanged for $t \geq t_0$. Measurements with records are channels on $A \otimes \text{record}$. (3) As stated: the joint state does not determine $k$. $\blacksquare$

Consequences. The non-selective reading of §8.7 (Option A) is the content of the axioms, and the no-signalling statement of UHM is [T]. Its costs listed under Option A become derived facts: Lüders conditioning is a readout, the rate of a branch depends on the branches that did not occur — J. Polchinski's "Everett phone" (Phys. Rev. Lett. 66, 397 (1991)) — and it is also what lets the amplification of Theorem 8.6 run on a marginal. The relativistic question of §8.7 — relative to which hypersurface the marginal is taken — stays open [Pr].

Witness (test_a_distant_holon_marginal_ignores_every_local_operation): $B$ a holon with $\varphi_s$ and $\kappa = 3$, $A$ a qubit, the pair entangled with $P(\rho_B) = 1/2$ (gate open). Four actions at $A$ — none, a $Z$ measurement, an $X$ measurement, a rotation — give $B$ marginals at $t = 5$ that agree to $5 \cdot 10^{-16}$ and equal the single-holon flow of the marginal to $4 \cdot 10^{-16}$; evolving the two conditional states of a $Z$ measurement separately and averaging — the selective reading — gives a state $0.016$ away.

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9. Summary Table of Correspondences

Physical theory	Connection to UHM	Status	Reference
L-unification	Dissipation from logical structure Ω: $L_k = \sqrt{\chi_{S_k}}$	[T] Proven	§2
Quantum mechanics	The equation reduces to the von Neumann equation when $\mathcal{D}$ and $\mathcal{R}$ vanish (Theorem 3.1); the category equivalence $\mathbf{Hol}_{R=0} \simeq \mathbf{QM}$ is retracted (§3.3)	[T] (Theorem 3.1)	§3
Schrödinger equation	$\frac{d\Gamma(\tau)}{d\tau} = -i[H_{eff},\Gamma]$	[T] Proven	Theorem 3.1
Lindblad equation	$\mathcal{L}_\Omega[\Gamma]$ — logical Liouvillian from Ω	[T] Formalized	evolution.md
Thermodynamics	$dS_{vN}/dt \geq 0$ for the unital part $\mathcal{L}_0$; with regeneration the free energy is the Lyapunov functional (T-261). (An earlier entry read "$dS_{vN}/dt \geq 0$ from the structure of ℒ_Ω"; that is false for non-unital channels and is retracted.)	[T] Proven	emergent time §7.1
Decoherence	Logical uncertainty relative to Ω	[T] Formalized	§2.3
Marginal identity	$\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = 0$	[T] Proven	§8, Theorem 8.1
No-signalling of the full dynamics	Holds in the non-selective reading, which the axioms force: the argument of a holon's dynamics is its marginal, and a selective (Lüders-conditioned) regeneration is not a UHM dynamics (Theorem 8.5; [C] before 2026-09-25)	[T]	§8.8
Ensemble independence	The evolution map is a function of $\Gamma$; the physical reading "same $\Gamma$, same evolution" is retracted	[D]	§8.5
Computational bound	"≤ BQP" retracted; the ideal dynamics with a living regenerator decides satisfiability in linear time (Theorem 8.6); with noise open	[T] / [H]	§8.6
Space	$\Sigma^3$ from Gelfand–Connes, $M^4 = \mathbb{R} \times \Sigma^3$	[T] as mathematics (T-119 restated 2026-09-25: spectrum computed; formerly [C] at the first-order condition and Poincaré duality)	T-119, T-120
Time	Cyclic clock τ ∈ ℤ₇ via modality ▷ on Ω [T]; the aperiodic parameter of the dynamics is assumed [C] (T-53b)	[T] / [C]	emergent-time.md
Discreteness of time	$\tau \in \mathbb{Z}_7$ from the structure of Ω	[T] Corollary	§3.6
GR / Einstein	Spectral action → $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$	[T] Proven	T-65
Standard Model	$SU(3)_C = \mathrm{Stab}_{G_2}(e_O)$; $SU(2)_L \times U(1)_Y$ from (FE); the former "$G_2 \supset SU(3) \to SU(3)_C \times SU(2)_L \times U(1)_Y$" is retracted (rank)	[T] / [C at (FE)]; uniqueness [H]	SM

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Conclusion

Key Achievement: L-Unification

L-unification shows that physical dynamics has a logical origin:

$$\Omega \xrightarrow{\chi_S} L_k = \sqrt{\chi_{S_k}} \xrightarrow{} \mathcal{L}_\Omega \xrightarrow{} \text{Lindblad equation}$$

This means: physics is a consequence of the structure of logical distinctions.

What Has Been Formalized [T]

1. L-unification: Lindblad operators $L_k = \sqrt{\chi_{S_k}}$ are derived from classifier Ω
2. Logical Liouvillian: $\mathcal{L}_\Omega[\Gamma]$ defines dissipation via the logical structure
3. Reduction to QM: with the dissipator and the regenerator switched off, the UHM equation is the von Neumann equation (Theorem 3.1); the claim that UHM contains quantum mechanics as a category equivalence is retracted (§3.3)
4. Thermodynamics: entropy grows under the unital part of ℒ_Ω; the full flow has the free energy as Lyapunov functional
5. Metric on states: The Frobenius norm defines a complete metric
6. Discreteness of time: $\tau \in \mathbb{Z}_7$ from the temporal modality ▷ on Ω
7. Marginal identity: $\mathrm{Tr}_A[\tilde{\mathcal{R}}_A[\Gamma_{AB}]] = 0$ — regeneration of $A$ does not change $B$'s unconditioned marginal; no-signalling of the full dynamics is [C] (§8.5)
8. Ensemble independence: the evolution map is defined on $\Gamma$ [D]; the earlier claim that this resolves the Gisin problem is retracted (§8.5)
9. Computational bound: retracted; whether $\mathcal{R}$ gives a speed-up beyond BQP is open [H] (§8.6)
10. Emergent geometry: $M^4 = \mathbb{R} \times \Sigma^3$ assembled from categorical structure (T-117—T-120) — [T] as mathematics since the restatement of T-119 (2026-09-25); [C] at its open reconstruction axioms before
11. Einstein equations: The spectral action reproduces $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$ (T-65)
12. Gauge group: $SU(3)_C$ from $G_2 = \mathrm{Aut}(\mathbb{O})$ [T]; $SU(2)_L \times U(1)_Y$ from (FE) [C at (FE)]. The former item — the whole group from $G_2$ (T-53) — is retracted (rank $G_2 = 2 < 4$)

Open Directions

1. Standard Model parameters: Specific values of masses and mixing angles from the vacuum configuration $\Gamma_{\text{vac}}$
2. Non-perturbative partition function: The limiting transition $Z_N \to Z$ as $N \to \infty$ [P]
3. Quantum gravity: The strong-field limit and quantum corrections to the spectral action

Open problem: concrete parameters of the Standard Model

UHM derives the structure of the Standard Model: the gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ from $G_2$ [T], three generations of fermions from the Fano plane [T], and the Einstein equations from the spectral action [T]. However, specific parameters are only partially computed:

Parameter	Status in UHM	Reference
Number of generations (3)	[T] Derived	Three Generations
Yukawa mass hierarchy	[T] Derived	Yukawa Hierarchy
Electron mass $m_e$	Not derived	Requires $\Gamma_{\text{vac}}$
Fine structure constant $\alpha$	Not derived	Requires non-perturbative analysis
Exact CKM/PMNS angles	Partial	CKM Matrix

This limitation is not unique to UHM: string theory, loop quantum gravity, and IIT also do not derive all SM parameters from first principles.

$G_2$-Manifolds and M-Theory

Compactification 11 → 4 + 7 [I]

In the structural derivation of N=7, the group $G_2 = \text{Aut}(\mathbb{O})$ arises. In M-theory, $G_2$-manifolds play a central role:

M-theory compactification [I]: 11-dimensional M-theory admits a compactification $M^{11} = M^4 \times X^7$, where $X^7$ is a compact $G_2$-manifold (holonomy = $G_2$). This gives:

• 4 non-compact dimensions → observable spacetime
• 7 compact dimensions with $G_2$-holonomy → internal degrees of freedom
• $\mathcal{N} = 1$ supersymmetry in 4D (the unique exceptional holonomy preserving exactly 1/8 of supercharges)

Numerical coincidence [I]:

• UHM: 7 Holon dimensions, $G_2$-symmetry
• M-theory: 7 internal dimensions, $G_2$-holonomy
• Dimensions coincide: $11 - 4 = 7 = \dim(\text{Im}(\mathbb{O}))$

Decomposition 42 [I]: $\dim(\mathcal{H}_{total}) = 42 = 7 \times 6$ in UHM. In M-theory: $42 = \binom{9}{2} + 6$ arises in a number of contexts.

Bridge [T] — closed with the canonical orientation (T15, T15-canon)

This is a substantive analogy, resting on theorems T1–T15, which close the bridge with the canonical orientation of the Fano lines (T15-canon; registry row 41n). The formal connection between the 7D structure of UHM and the $G_2$-compactification of M-theory is an open problem. Bridge [T] (T15; it read "Bridge [T] (closed, T15)" and "fully closed" until 2026-09-25 without the orientation step, then [C at (Alt)] the same day until T15-canon).

Potential consequences [I]:

• If the connection is physical, the $G_2$-manifold determines the gauge group and mass spectrum in 4D
• Singularities of the $G_2$-manifold → non-perturbative effects (condensates)
• Joyce metric on $X^7$ → internal metric of the space of dimensions

More: structural derivation →

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Related documents:

• Axiom Ω⁷ — L-unification: Ω → χ_S → L_k → ℒ_Ω → φ
• Coherence Matrix — definition of $\Gamma$, connection between formalisms
• Evolution — equation $d\Gamma(\tau)/d\tau$ with derivation of $H_{eff}$
• Emergent Time — Page–Wootters mechanism, temporal modality ▷
• Emergent Manifold $M^4$ — derivation of $M^4$ from categorical structure (T-117—T-121)
• Dimension O — clock algebra $H_O$, $V_O$, $\mathcal{A}_O$
• Dimension L — logical dimension, L = Ω ∩ Γ
• Constructive Algorithms — computation of χ_S, L_k, ℒ_Ω
• Spacetime — emergence
• Categorical Formalism — functor F, $\mathbf{Exp}^{disc}_\infty$
• Minimality Theorem — proof of 7D
• Coherence Cybernetics — L-unification in CC
• Theory Boundaries — open questions