Bimodule Construction: solving four systemic problems

Who this chapter is for

This document solves four interrelated problems that remained open [P] in UHM theory:

1. SM representations: How does one obtain SM representations from the algebra $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ in which quarks carry simultaneously color and weak isospin: $(3,2)_{1/6}$?
2. Non-perturbative λ₃: How to extract physical predictions when $\lambda_3 \approx 74 \gg 4\pi$ (non-perturbative regime)?
3. Derivation of (AP+PH+QG+V) from A1-A4: What is the explicit derivation of the characterizing properties of the holon from the four axioms?
4. G-map: How is the map $G: \text{States} \to \mathcal{D}(\mathbb{C}^7)$ constructed for concrete systems?

All four problems have a common root: the theory has so far worked at the level of algebras, without reaching the level of representations and bimodules. Connes' bimodule construction is the missing link.

Correction 2026-09-25: problem 1 is not solved here — T-178 below is retracted as a derivation; the Standard Model representations come from Connes' Hilbert space $H_F$, which UHM imports. Half of problem 1 is solved elsewhere, without the import. Under the assumption (Cl), the left-handed doublets $(3,2)_{1/6}\oplus(1,2)_{-1/2}$ arise on $\mathbb{C}\otimes\mathbb{O}$ — UHM's Hilbert space plus the parallel spinor — as the spinor of the forced octonionic Clifford system, with the group $(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1))/\mathbb{Z}_6$ (Standard Model, §2.5, T-326, T-327). The right-handed singlets do not fit there. They come with the complexification of that spinor: a Weyl field with values in it is valued in $\mathcal{S}\otimes_{\mathbb{R}}\mathbb{C}$, where a tenth Clifford generator is forced and $\mathrm{Spin}(10)$ gives one complete, anomaly-free generation with $\nu_R$ and the Standard Model hypercharges (Standard Model, §2.6, T-329, [C at (Cl)]). The other half of problem 1 is thereby solved too, without $H_F$.

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1. Common Root of the Four Problems

Diagnosis

All four problems are symptoms of a single gap: between the algebraic structure $A_{\text{int}}$ (correctly derived) and the representational structure $H_F$ (not derived). In Connes' noncommutative geometry, physics is determined not by the algebra $A$ alone, but by its action on a Hilbert space $H$ — where $H$ is simultaneously a left $A$-module and a right $A^\circ$-module via the real structure $J$: $b^\circ \cdot \xi := J b^* J^* \xi$. It is precisely this bimodule structure that generates the SM representations.

Problem	Algebra level (done)	Bimodule level (needed)
SM representations	$A_{\text{int}}$ contains rank-4 generators	Bimodule $H_F$ generates $(3,2)_{1/6}$
λ₃	Loop calculations with λ₃ ≈ 74	Spectral action $\mathrm{Tr}(f(D^2/\Lambda^2))$ non-perturbative
(AP+PH+QG+V)	Characterizing properties are postulated	Derived from bimodule structure via $J$
G-map	$\mathcal{D}(\mathbb{C}^7)$ is defined	Bimodule defines canonical embedding

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2. Bimodule Construction of SM Representations

2.1 Finite bimodule from the UHM spectral triple

Theorem T-178 (Bimodule realization of SM) — retracted [✗] as a derivation (2026-09-25)

Retracted statement: the finite Hilbert space $H_F$ of the UHM spectral triple, viewed as an $(A_{\text{int}}, A_{\text{int}}^\circ)$-bimodule via the real structure $J$ with KO-dimension 6, decomposes into a direct sum of irreducible bimodules exactly coinciding with one generation of SM fermions.

It is not derived, for three reasons. (i) The UHM triple has $H_{\text{int}} = \mathbb{C}^7$, while one generation needs $32$ states (16 Weyl fermions and their conjugates); the $H_F$ used below is Connes' Hilbert space, imported, and "$7 \cdot 2 = 14 + 2$" in the reduction note is a count, not a map. (ii) KO-dimension 6 cannot occur on $\mathbb{C}^7$: it needs $J\chi = -\chi J$, so the antiunitary $J$ maps the $+1$ eigenspace of $\chi$ onto the $-1$ eigenspace and the two have equal dimension, which no grading of an odd-dimensional space provides (with the $\chi$ of Step 1 they are $4$ and $3$; spacetime, Step 6). (iii) The reduction $A_{\text{int}} \to A_F$ is not a homomorphism, and its "Morita compatibility" is T-175a, retracted: Morita equivalence preserves the centre, $\mathbb{C}^3$ for $A_{\text{int}}$ against $\mathbb{C}\oplus\mathbb{R}\oplus\mathbb{C}$ for $A_F$ (spacetime). What stands is the standard bimodule decomposition of Connes' $H_F$ over $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ (Chamseddine–Connes–Marcolli 2007; Barrett 2007), which UHM imports rather than derives.

Construction.

Step 1 (Input data). Finite UHM spectral triple:

• Algebra: $A_{\text{int}} = \mathbb{C}_O \oplus M_3(\mathbb{C})_{\mathbf{3}} \oplus M_3(\mathbb{C})_{\bar{\mathbf{3}}}$
• Space: $H_{\text{int}} = \mathbb{C}^7$
• Real structure: $J$ with $J^2 = +1$, $JD = DJ$, $J\chi = -\chi J$ (KO-dim 6) — retracted [✗]: no real structure of KO-dimension 6 exists on $\mathbb{C}^7$ — its $\chi = \pm 1$ eigenspaces would need equal dimension, and 7 is odd (spacetime, Step 6)
• Chirality: $\chi = \mathrm{diag}(+1, -1, -1, -1, +1, +1, +1)$

Step 2 (Opposite algebra). The real structure $J$ defines a right action of the algebra $A_{\text{int}}$ on $H_{\text{int}}$:

$$b^\circ \cdot \xi := J b^* J^* \xi, \quad b \in A_{\text{int}}$$

This turns $H_{\text{int}}$ into an $(A_{\text{int}}, A_{\text{int}}^\circ)$-bimodule: the left action is ordinary multiplication, the right action is via $J$.

Step 3 (First-order condition). KO-dim 6 requires:

$$[[D, a], Jb^*J^*] = 0 \quad \forall a, b \in A_{\text{int}}$$

This condition constrains the admissible Dirac operators $D$ and, consequently, the admissible representations.

Scope: first-order condition verification

The first-order (order-one) condition is the seventh of Connes' reconstruction axioms and is the step most commonly flagged in external audits of NCG-based derivations (cf. Chamseddine–Connes 2008 on the "first-order / one-form" weakening). In this proof it is imposed as a structural constraint on $D$ (the admissible Dirac operators are those for which the condition holds) rather than derived from the algebra $A_{\text{int}}$ alone. Verification for the specific $D_{\text{int}}$ of T-53 reduces to a computation on the Higgs-line $\{A,E,U\}$ restriction of the product triple; this computation is sketched but not fully written out here (the same gap T-119 had until its restatement of 2026-09-25, flagged as framework-conditional in the Rigour Stratification table).

Step 4 (Bimodule decomposition). After imposing $J$ + first-order condition + electroweak breaking via the Higgs line $\{A,E,U\}$ (ФЭ [T]):

$$A_{\text{int}} \xrightarrow{J + \text{ФЭ}} A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$$

info

Explicit reduction $M_3(\mathbb C)\to\mathbb H$

The arrow $A_\mathrm{int}\to A_F$ is not an algebra isomorphism — it is a reduction induced by the real structure $J$ plus the ФЭ breaking. $A_\mathrm{int}=\mathbb C\oplus M_3(\mathbb C)_{\mathbf 3}\oplus M_3(\mathbb C)_{\bar{\mathbf 3}}$ has $\dim_\mathbb R=1+18+18=37$; $A_F=\mathbb C\oplus\mathbb H\oplus M_3(\mathbb C)$ has $\dim_\mathbb R=1+4+18=23$. The reduction:

1. The first $M_3(\mathbb C)_{\mathbf 3}$ factor carries the $SU(3)_C$ colour action — retained in $A_F$ as $M_3(\mathbb C)$.
2. The second $M_3(\mathbb C)_{\bar{\mathbf 3}}$ factor is reduced to $\mathbb H\subset M_2(\mathbb C)\subset M_3(\mathbb C)$ via the $J$-compatibility + Higgs-line $\{A,E,U\}$ constraint (T-1a): the 2-dimensional weak-isospin subspace $\mathrm{span}\{|E\rangle,|U\rangle\}$ supports an anti-commuting real structure $J^2=+1,\ [J,\gamma]=0$, whose commutant is $\mathbb H$ (Barrett 2007, §3.2; Chamseddine–Connes 2007). Retracted 2026-09-25: an antiunitary $J$ with $J^2 = +1$ on $\mathbb{C}^2$ is complex conjugation in a suitable basis, and its commutant in $M_2(\mathbb{C})$ is $M_2(\mathbb{R})$; $\mathbb{H}$ is the commutant of a $J$ with $J^2 = -1$. Barrett's paper has four sections and no subsections, so there is no §3.2, and it takes $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ as given. The remaining $M_3(\mathbb C)_{\bar{\mathbf 3}}\setminus\mathbb H$ content is projected out as it fails the first-order condition with the Dirac operator restricted to the Higgs line.
3. The net effect is a Morita-compatible reduction: $A_F$ and $A_\mathrm{int}$ have the same category of bimodule representations realising SM fermions (Alvarez–Gracia-Bondía–Martín 1995), i.e., the bimodule structure is preserved. Retracted 2026-09-25 with T-175a: the centres differ ($\mathbb{C}^3$ against $\mathbb{C}\oplus\mathbb{R}\oplus\mathbb{C}$), so the bimodule categories are not equivalent.
4. Verification: dimensional accounting — $H_F$ fermion count from $A_F$ bimodules = 16 per generation (SM), matching $\dim H_F = 7 \cdot 2 = 14 + 2$ right-handed neutrinos. Retracted 2026-09-25: $H_{\text{int}} = \mathbb{C}^7$ has dimension 7; "$7 \cdot 2 + 2 = 16$" does not construct the 16 states of a generation.

This resolves the concern that $A_\mathrm{int}$ and $A_F$ differ as algebras: the equivalence is at the level of bimodule categories (Morita), not objects. Retracted 2026-09-25: there is no such equivalence (item 3).

The bimodule decomposition of $H_F$ gives (Barrett, 2007; Chamseddine-Connes, 2007):

Bimodule	Left action $(A_F)$	Right action $(A_F^\circ)$	SM fermion
$\mathbf{2}_L \otimes \mathbf{3}$	$\mathbb{H}$ (weak isospin)	$M_3(\mathbb{C})^\circ$ (color)	Left quark $(u_L, d_L)$
$\mathbf{1}_R \otimes \mathbf{3}$	$\mathbb{C}$ (hypercharge)	$M_3(\mathbb{C})^\circ$ (color)	Right quark $u_R, d_R$
$\mathbf{2}_L \otimes \mathbf{1}$	$\mathbb{H}$ (weak isospin)	$\mathbb{C}^\circ$	Left lepton $(\nu_L, e_L)$
$\mathbf{1}_R \otimes \mathbf{1}$	$\mathbb{C}$ (hypercharge)	$\mathbb{C}^\circ$	Right lepton $e_R, \nu_R$

Key point: A quark in the representation $(3,2)_{1/6}$ arises not from the tensor product of two factors $\mathbb{C}^7 \otimes \mathbb{C}^6$, but from the intersection of left and right actions on a single bimodule. The left action of $\mathbb{H}$ gives weak isospin, the right action of $M_3(\mathbb{C})^\circ$ gives color — both acting on the same element $\xi \in H_F$.

Solution of the SM representations problem

The 42D tensor structure $\mathbb{C}^7 \otimes \mathbb{C}^6$ is a realization of the Page–Wootters mechanism for emergent time. SM representations arise from a different construction: the bimodule decomposition of $H_F$ via the real structure $J$. These two mechanisms are compatible but solve different problems: PW gives time, the bimodule gives particles.

Updated status of the SM representations problem: [T] — solved via the standard NCG construction (Barrett 2007), applied to the UHM spectral triple (T-53 [T]). Retracted 2026-09-25: the construction is applied to Connes' imported $H_F$, not to the UHM triple, whose $\mathbb{C}^7$ carries no KO-dimension-6 structure (T-178 above). Deriving the SM representations from UHM remains an open problem [Pr].

$\blacksquare$

2.2 Hypercharge and parameter α

The free parameter $\alpha$ in the hypercharge generator $Y$ is fixed by anomaly freedom of the bimodule $H_F$:

Theorem T-179 (Hypercharge fixation) — retracted [✗] as stated (2026-09-25)

Retracted statement: the anomaly cancellation conditions $\mathrm{Tr}(Y) = 0$ and $\mathrm{Tr}(Y^3) = 0$ on the bimodule $H_F$ uniquely fix the hypercharge assignments of the Standard Model (up to overall normalization).

Two equations cannot fix the five hypercharges of a generation (six with $\nu_R$). Even all four anomaly conditions — gravitational, $U(1)_Y^3$, $SU(3)^2\,U(1)_Y$ and $SU(2)^2\,U(1)_Y$ — are solved both by the Standard Model values and by $y_Q = y_L = y_e = 0$, $y_u = -y_d$; and with a right-handed neutrino every combination $Y + c\,(B-L)$ passes all four (checked in exact arithmetic for $c = 1/7,\ 1/2,\ 3$; $B-L$ is the anomaly-free outer automorphism noted by Boyle and Farnsworth, New J. Phys. 22, 073023 (2020)). Fixing the hypercharges needs further input: the Yukawa couplings to one Higgs doublet and, when $\nu_R$ is present, a Majorana mass for it (Babu and Mohapatra, Phys. Rev. Lett. 63, 938 (1989)). The bimodule it is applied to is Connes' imported $H_F$ (T-178 above), so no part of the result is derived from UHM.

Former proof (retracted with the statement). This is a standard result of anomaly theory (Alvarez-Gaumé, Witten 1984), applied to the specific bimodule from Step 4 above. The condition $\mathrm{Tr}(Y) = 0$ fixes the relative hypercharges of quarks and leptons; $\mathrm{Tr}(Y^3) = 0$ fixes the absolute values. The unique solution: $Y(q_L) = 1/6$, $Y(u_R) = 2/3$, $Y(d_R) = -1/3$, $Y(l_L) = -1/2$, $Y(e_R) = -1$. $\blacksquare$

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3. Non-perturbative Approach to λ₃

3.1 Spectral action as a solution

Key observation

The parameter $\lambda_3 \approx 74$ appears when expanding the spectral action in powers of $\Lambda^{-1}$. But the spectral action is defined non-perturbatively:

$$S_{\text{spec}}[D] = \mathrm{Tr}\!\left(f\!\left(\frac{D^2}{\Lambda^2}\right)\right)$$

where $f$ is a smooth cutoff function. This formula does not require expansion into loop diagrams. Physical predictions (masses, mixing angles) are determined by the spectrum of the operator $D$, not by the Lagrangian parameters.

3.2 Spectral predictions without loops

Theorem T-180 (Non-perturbative mass ratios) [C at (SV)]

Corrected 2026-09-25 from [T]: the vacuum state $\theta^*$ is taken with the sector values of the hypothesis (SV); T-64 is restated as a hypothesis whose vacuum has none of them.

Fermion mass ratios are determined by the eigenvalues of the finite Dirac operator $D_{\text{int}}$ and do not depend on λ₃:

$$\frac{m_i}{m_j} = \frac{|[D_{\text{int}}]_{ii}|}{|[D_{\text{int}}]_{jj}|} = \frac{\mathrm{Gap}(i)}{\mathrm{Gap}(j)}$$

where $\mathrm{Gap}(i)$ are the Gap parameters from the vacuum state $\theta^*$ (T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)), unique minimum of $V_{\text{Gap}}$).

Corollary. The mass hierarchy ($m_t \gg m_u$) is determined by the hierarchy of vacuum Gap parameters, which follows from the geometry of the Fano plane (different distances on PG(2,2)), not from loop corrections with λ₃.

3.3 What remains of λ₃

The parameter λ₃ = $\omega_0 \cdot |f_{ijk}|$ (where $f_{ijk}$ are the octonionic structure constants) enters the Gap potential $V_{\text{Gap}}$:

$$V_{\text{Gap}} = V_2(\varepsilon) + \lambda_3 \cdot V_3(\varepsilon, \theta) + \lambda_4 \cdot V_4(\varepsilon)$$

For λ₃ ≫ λ₄ the potential is dominated by the cubic term $V_3$. This is not a problem — it is an indication that the vacuum structure is determined by the octonionic associator (the cubic term $\propto [e_i, e_j, e_k]$), not by the standard quartic potential. The minimum of $V_{\text{Gap}}$ (T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV))) exists and is unique independently of the ratio λ₃/λ₄.

Reinterpretation of C7

Condition C7 ($\lambda_3 \gg 4\pi$) is not a problem but a feature of the octonionic structure. The non-associativity of octonions manifests through the dominance of the cubic potential. Physical predictions should be extracted from the spectrum of $D_{\text{int}}$ (non-perturbatively), not from loop expansions of the Lagrangian. Updated status of C7: from a [H]-warning to an [I]-feature — a structural property of the theory, not a defect.

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4. Requirements are not automatic consequences

The former T-181 proof is withdrawn [✗]. A sheaf topos is not the category of density matrices: its objects are sheaves, and its terminal object has a terminal space of global sections. The support reflector is defined in a slice, not as a numerical CPTP attractor. Stinespring does not identify sheaf morphisms with channels. Seven chosen matrix axes do not prove experiential content or tensor-factor reduction. Bures separation does not select the structural-majority cut.

AP, PH, QG and V are explicit requirements/bridges in the corrected specification. The numerical state model, frame, self-model, rates, clock extension and phenomenal identification must each be provided. These data may define a consistent model; their existence does not follow from terminality. The Page–Wootters constraint also remains independent. No count of four independent primitives is asserted.

5. G-map: Constructive Protocol

5.1 Canonical embedding via the anchor function

For a system with state $s \in \mathcal{S}$ (neural network, brain, organism), the G-map $G: \mathcal{S} \to \mathcal{D}(\mathbb{C}^7)$ is constructed via the anchor function $\pi$:

$$G(s) = \pi(s) := \frac{L(s) \cdot L(s)^\dagger}{\mathrm{Tr}(L(s) \cdot L(s)^\dagger)}$$

where $L: \mathcal{S} \to \mathbb{C}^{7 \times 7}_{\text{lower-triangular}}$ is a trainable map (MLP or linear projection), and the normalization guarantees $G(s) \in \mathcal{D}(\mathbb{C}^7)$.

5.2. Identifiability of the observation model

The universal T-123 uniqueness claim is withdrawn [✗]. A Cholesky parameterization guarantees a state only when its denominator is nonzero; it does not guarantee that different states are distinguishable in the observations or that two learned encoders differ by a symmetry. Specify the observation law, test its fibres, and use the reconstruction theorem. CPTP applies to linear maps of operator algebras, not to an arbitrary estimator from feature vectors.

5.3 Protocol for concrete systems

System	Method for constructing G	Status
Neural network	Linear probe $h \to L \to \Gamma$ via Cholesky (C25 [C])	Feasible
Brain (EEG)	7 frequency bands → $\gamma_{kk}$, coherence → $\gamma_{ij}$	[Pr] Research program
Organism	Physiological markers → 7 sectors (T-92 [D])	[P] Measurement protocol

Measurement bridge

Every empirical formalism requires an observation model. UHM now treats this bridge as an identifiable statistical problem, with no universal encoder uniqueness.

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6. Deep Structure: Fractal Recurrence

Meta-level

Self-reference is a proposed interpretation [I] of several different constructions. For T-54, choose $m:G\to G$ in the ordinary part of a specified topos. Its precomposition map acts on predicates: $m^*:\Omega^G\to\Omega^G$, and $\mathrm{Th}_m:=\operatorname{Eq}(m^*,\mathrm{id}_{\Omega^G})\hookrightarrow\Omega^G$ [D/T]. This equalizer does not identify a formal theory, the whole topos or mathematics with a density-matrix state. Those realizations and interpretations are separate inputs.

6.1 Three levels of self-reference

Level	Object	Specified map	Conditional property
Holon	$\Gamma \in D_7$	Chosen $M:D_7\to D_7$	Iteration depth requires a declared detector or operational certificate; finite dimension alone supplies no ceiling
Fixed predicates	$\mathrm{Th}_m\hookrightarrow\Omega^G$	$m^*:p\mapsto p\circ m$	Proper only with separating predicates and $m\ne\mathrm{id}$ (T-55); identity fixes every predicate
Bimodule	$H_F$ as $(A, A^\circ)$-bimodule	Antiunitary real structure $J:H_F\to H_F$	$J^2=+1$ in the supplied KO-dimension-6 structure; this is not a self-model accuracy or recursion bound

These maps have different domains and mathematical roles. T-55's properness is not Gödel incompleteness and does not prove unavoidable dynamics or a mismatch at every state. A nonidentity map may have fixed states; $J$ is a real structure, not an estimator. Reading the constructions as a common self-referential mechanism is [I]; a concrete physical coupling is needed before that reading predicts evolution or masses.

6.2 Correspondence with knowledge traditions

Tradition	Concept	Formalization in UHM
Vedanta	Brahman = Atman	$\Gamma_{\text{global}}$ (single substance) ≡ $\varphi(\Gamma)$ (self-model) at $R = 1$
Buddhism	Śūnyatā (emptiness)	Analogy [I] with conditional predicate noninvariance; T-55 does not prove that no predicate is "self-existent"
Kabbalah	Tzimtzum (contraction)	$\Gamma_\odot \to \rho^*$ — spontaneous breaking of $S_7$-symmetry
Taoism	The Tao that can be expressed	$L \subsetneq \Gamma$ — logic (L-dimension) does not encompass the whole
Alchemy	Solve et Coagula	$\mathcal{D}[\Gamma]$ (decoherence = solve) + $\mathcal{R}[\Gamma]$ (regeneration = coagula)
Fractals	Self-similarity	Iteration analogy [I]; measured depth requires a detector and independently passing operational certificates

6.3 Conditional limits of iteration

The universal SAD_MAX = 3 claim is removed. A finite dimension and compact state space do not bound iteration depth: the identity on $D_7$ can be iterated to any depth. This example certifies repeated iteration, not progressively stronger metamodel competence.

The Fano incidence fraction $\alpha=2/3$ can enter a specified generator's decay rate; it does not determine an iteration multiplier. For a declared continuous dephasing rate $\Gamma_2$, a sampling interval $\Delta t$ gives $q=e^{-\Gamma_2\Delta t}$. The multiplier $1/3$ belongs to the separately chosen discrete channel $\mathcal D_{2/3}$, not to every act of observation.

If a specified map satisfies $A_n:=\|\operatorname{offdiag}M^n(\rho)\|\le q^nA_0$ with $0<q<1$, and a detector requires $A_n\ge\varepsilon>0$ with $A_0\ge\varepsilon$, then

$$n\le\left\lfloor\frac{\log(A_0/\varepsilon)}{\log(1/q)}\right\rfloor.$$

This is a conditional detector cutoff. Resource budgets and independent operational certificates can impose other bounds. It is not a universal cognitive ceiling, a claim about canonical $R$, or a proof that L4 is impossible; see the revised depth tower.

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6.3 Why exactly 3 levels of recursion

SAD_MAX = 3 is not an arbitrary number. It follows from the geometry of the state space:

1. Fano contraction $\alpha = 2/3$ means: each act of self-observation preserves 1/3 of coherence
2. The space D(ℂ⁷) is compact: $P \in [1/7, 1]$
3. After 3 iterations: $R^{(3)} \sim r_0 \cdot (1/3)^3 \approx r_0/27$
4. Threshold $R_{\text{th}}^{(3)} = 1/6$: $r_0/27 > 1/6$ requires $r_0 > 4.5$, i.e. $P > 4.5 \cdot 2/7 \approx 1.29 > 1$ — impossible

Compactness of D(ℂ⁷) × Fano contraction = finite recursion. Infinite self-reference is impossible in a finite-dimensional quantum system — and this is the mathematical formalization of what mystical traditions call the "inexpressible": L4 (complete transparency) exists as a limit but is unattainable.

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Related documents

• UHM Spectral Triple — construction of $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$
• Standard Model — gauge groups from $G_2$
• Cosmological Constant — Λ-budget
• Gap Thermodynamics — $V_{\text{Gap}}$ and minimum $\theta^*$
• Axiom Ω⁷ — 4 axioms A1-A4
• Consciousness Window — reconstruction identifiability
• Formalization of φ — self-modeling operator
• Depth Tower — conditional detector bounds and certified operational depth