Premises of UHM

What this page is

One list of everything the corpus assumes rather than proves, as of 2026-09-28. Each entry names the premise, its status, the results that use it and what is known about deriving it. The status registry remains the canonical record of each result; this page is the canonical record of the inputs. A result that uses none of the premises in sections 2–4 is listed in section 6.

Status letters are those of the registry: [P] postulate (an axiom), [H] hypothesis (formulated, not proven — here a named assumption), [Pr] research programme, here a principle kept open (neither assumed as an axiom nor claimed proven), [D] definition by convention; [I] interpretation and [C] conditional theorem appear only where a row names them, as in the registry. A free parameter is not a statement and has no letter.

1. How to read the list

Three kinds of input are kept apart.

• Axioms fix the mathematical object: the ∞-topos, its metric, the dimension, the scale and the Page–Wootters constraint. Everything labelled "[T] as mathematics" in the corpus uses only these.
• Bridge premises say which part of that object is physical spacetime, matter or a self-model. They are the reason a result reads "[T] as mathematics, [C at (X)] as physics".
• Identification hypotheses of the flavour and vacuum sectors attach numbers to particle data. Several have been refuted in their exact form and survive only in a weaker one.

A premise counts as used when a live registry row or theorem carries [C at (X)] for it, or when its statement is an explicit step of a proof. A premise that has since become a theorem is listed in section 5 and is no longer an input.

2. Axioms [P]

Premise	Statement	Used by	Status and what is known
Metatheory	∞-categories / homotopy type theory as the language; intuitionistic internal logic	every page	outside the theory (honest axiomatics)
A1	reality is the ∞-topos $\mathbf{Sh}_\infty(\mathcal C)$ over $\mathcal D(\mathbb C^N)$	all results	[P]; derivable from the operational basis only through the hypothesis T-186(a) (T-190)
A2	the Grothendieck topology is induced by the Bures metric	the topology, the stratification, T-173, T-190	[P]; its topology is forced (every continuous distance on the compact $\mathcal D$ induces the standard one), and within the CPTP-monotone metrics Bures is canonical (T-187; the maximum-entropy recasting T-189) — what stays postulated is the monotonicity of the enrichment. It is not derivable from (AP)+(PH)+(QG)+(V)+MaxEnt, so it is the third condition of T-190; it follows from the operational reading (O) — the enrichment is the best distinguishability reachable by measurement — which also gives Bures directly (Fuchs–Caves), so the postulated content can be stated as (O) (Lemma M)
A3	$N = 7$	all results	[P]; $N \geq 7$ is [T] (Theorem S); strict necessity needs (Σ₆), section 3 (T-349; (P1₆) until 2026-09-28)
A4	the scale $\omega_0 > 0$	dynamics, calibration	[P]; its value is a free parameter, section 4
A5 constraint	$\hat C\,\Gamma = 0$, the support condition $\mathrm{supp}\,\Gamma \subseteq \ker\hat C$ of Property 2 — the form of the timeless state	the Page–Wootters link of the clock (T-87, step 4), T-190	[P]; the clock register and the tensor factor of A5 are [T] (T-87, steps 1–3); the constraint is not derived (A5)
(QG)'s formalism	states are density matrices, admissible maps are CPTP (definition O3)	all dynamics	[D]; "why quantum theory" stays external (T-188)
Frame decision D-0910	the dynamical frame group is $\Gamma_{\mathrm{oct}}$, 48 physical parameters	the dynamical half of the $G_2$-rigidity (42a), T-334	[D] (uniqueness theorem)

The independent content of the axioms is A1–A4 plus the constraint of A5. Independence of the constraint [T]: A1–A4 admit every density matrix on $\mathcal H_O \otimes \mathcal H_{\text{rest}}$, and $\ker\hat C$ is a proper subspace whenever $\hat C \neq 0$, so a state such as $\lvert\tau_0\rangle\langle\tau_0\rvert \otimes \rho$ with $\hat C(\lvert\tau_0\rangle \otimes \cdot) \neq 0$ satisfies A1–A4 and violates the constraint.

Independence of the monotonicity of A2 [T]: the Hilbert–Schmidt distance — the one in which (V) is written, $P - 1/7 = \lVert\Gamma - I/7\rVert_{\mathrm{HS}}^2$ — is Riemannian, induces the standard topology and gives a stable ε-δ coverage, and every property (AP), (PH), (QG), (V) and MaxEnt is indifferent to it; yet a partial-trace channel on $\mathbb C^7$ moves a full-rank pair $\sqrt2 = 1.41421$ times farther apart in it while their Bures angle keeps the ratio $1.00000$ (Lemma M).

3. Bridge premises

These are the free inputs that remain after four waves of repair (2026-09-25/26). Each has been tested against derivation, and for each the obstruction is stated.

(P) — spacetime is built from the spinor factor [H]

Statement. The tangent vectors of spacetime at a point are the Hermitian forms on the spinor factor $W$ of the fermion field $F = W \otimes_{\mathbb C} \mathcal S_{\mathbb C}$; $\dim W \geq 2$; the causal structure, a quadratic form given up to a factor, is preserved by every transformation of $W$ that preserves UHM's internal structure (Theorem 48e).

Equivalent forms. (P) ⟺ (L) ∧ (W) [T, 48e(f)–(g)]: (L) says that tangent vectors form the observables of a two-level system with amplitudes in a composition subalgebra of $\mathbb O$ containing $e_O$; (W) says that the fermion field is a two-component Weyl field. (P) can be stated with any of 48e(g)(iii)–(vii) in place of the quadratic form: light rays are the pure states of $W$, space is isotropic in every rest frame, the group of $W$ is a Lorentz group, $W$ is a bit (the premise of Masanes and Müller), a Lorentz-invariant Weyl mass term exists.

Used by. The physical reading of 48c — 3+1 dimensions, signature $(1,3)$, rotations that commute with colour — and the second route to the signature in the same page. Every result stated [C at (L)] uses it.

Not derivable [T, 48e(f)]: the commutant of UHM's internal operators on the generation is $\mathbb C'$, so the transformations preserving the internal structure are $\mathrm{GL}(n, \mathbb C')$ for every $n$, and nothing internal fixes $n$. Routes tried and closed: the clock's complex structure, the history state of the depth register, the two slots of a self-model, chirality and the Distler–Garibaldi test, the Masanes–Müller reconstruction, the spinor bundle of T-119's $S^3$ (reasons in 48e). Minimality does not replace it: the smallest spinor factor compatible with T-329 is $n = 1$, which leaves spacetime without a spatial direction (T-347(d)).

(Cl₀) — fermions are vectors of the spinor module [H]

Statement. Fermion fields take values in the spinor module $\mathcal S = \mathbb C \otimes \mathbb O$ of the octonionic Clifford system (Standard Model, §2.6). The second half of the former assumption (Cl) — that the clock breaks the spin group to the largest connected subgroup in which colour is a normal factor — is a theorem (T-329).

Used by. The physical reading of T-326, T-327 and T-329 (each [T] as mathematics, [C at (Cl)] in UHM), of T-332 and T-333, of the joint structure $F = (\mathbf 2, \mathbf{16})$ in 48e(e), and every flavour hypothesis of section 3.4, all of which are stated inside the frame of $\mathcal S$.

Not derivable from axioms about Γ [T for the obstruction]: $-1 \in \mathrm{SU}(2)_L$ acts as $-1$ on $\mathcal S$ and as $+1$ on $\mathrm{End}\,\mathcal S$, so every object built from coherence matrices, including their tensor products, has integer weak isospin; doublets are vectors of $\mathcal S$, not operators on it. Stated exactly (T-350(d)–(e)): every action induced by transformations of the holon is single-valued, $\mathrm{Spin}(9)$ acts trivially on $\mathcal D(\mathbb C^7)$ ($9 > 7$), and the centraliser of colour in $\mathfrak u(7)$ is $\mathfrak u(1)^3$. The spinor module itself is built from the holon's axes, $\mathrm{Cl}(\mathbb R^7)\,p_\varphi$ with $p_\varphi = (1+\mathrm{vol})(1+\varphi)/16$, but its two-valued action is Clifford multiplication, which no transformation of the holon induces.

Equivalent form without a named module [T, T-347(b)]: (Cl₀) ⟺ (Mod) — the product of the holon's octonions (T15) acts on matter: a real-linear $\rho:\mathbb O\to\mathrm{End}_{\mathbb R}(F)$ with $\rho(1) = 1$ and $\rho(x)\rho(x) = \rho(x^2)$, commuting with the $i$ of $\mathcal H$. Then $F \cong U \otimes_{\mathbb R} \mathcal S$, and $\mathcal S$, the parallel spinor $\eta_0$, the nine Clifford generators and $\mathrm{Spin}(9)$ are consequences. The model $\mathbb C^7$ of section 7 fails exactly (Mod). Routes closed by T-347: every property of the holon (a), the maximality of $\mathrm{Spin}(9)$ and a faithful representation of the holon's observables (c). Routes closed by T-350: textures of $\Gamma$ and the Finkelstein–Rubinstein and Wess–Zumino mechanisms (a), the spin lift in $\mathbf{Sh}_\infty$ (b), triality (c), chirality and anomaly freedom among induced modules (d), second quantisation of the holon (e).

(W₀) — the spinor factor is complex: not a free premise

T-329 uses only that the spinor factor of the fermion field is a complex space of some dimension, (W₀). It is implied by (P), whose $W$ is complex. Given (Cl₀) it is equivalent to the requirement that one generation be chiral and anomaly-free [T, 48e(f),(h)]: with a real Lorentz factor every fermion space on $\mathcal S$ is anomalous or vector-like (h), and with a complex one it is chiral and anomaly-free for every $n$ (f). (W₀) is therefore a consistency condition of a chiral gauge theory together with the observed chirality, not an independent input.

(MaxΦ) — the self-model's anchor is maximally integrated [Pr]

Statement. The anchor $\rho_a$ of the replacement-form self-model $k\,\mathcal P_\alpha(\Gamma) + R\,\rho_a$ is a state of maximal integration, $\Phi(\rho_a) = 6$ (T-334).

Equivalent forms [T, T-334(6)]: (Eq-V) — the self-model privileges no axis (its atomic reading is $\Gamma_{\mathrm{oct}}$-covariant) and is the most viable such; $\rho_a = D\,uu^\dagger D^\dagger$ with $D$ diagonal unitary; maximal relative entropy of coherence, $C_{\mathrm{rel}}(\rho_a) = \log 7$; coherent purity $s = 6/7$. It splits into two halves: (Eq), uniform diagonal, and (Pure), a pure anchor.

Used by. The choice of $\varphi_J$ among the self-models the axioms allow: the living attractor of an isolated holon in the window (evolution), the isolated-holon half of C27, the collineation-anchored level of the septicity table. The dynamics of each anchor — T-334 (1)–(5), T-335, T-336 — is [T] without it; the premise decides only which anchor a physical holon has.

Not derivable from the axioms: the self-modelling adjunction and the terminal object make $\varphi$ a CPTP left adjoint and leave its anchor open (every anchor gives a channel of the same form). Routes closed in T-334: Curie's principle (gives the family $D((1-t)I/7 + t\,uu^\dagger)D^\dagger$, not $t = 1$), the terminal object (gives (Eq) only), Lawvere and Brouwer (fixed points, not anchors), viability alone. Section 7 adds the routes through the corpus's variational principles.

(Σ₆) — every decomposition is perfectly diagnosable [H]

Statement. Every decomposition of a viable holon into axes covering (AP)+(PH)+(QG) has a grammar of admissible status profiles $\mathcal{C} \subseteq \mathbb{F}_2^N$ with (D1) $d(\mathcal{C}) \geq 3$, (D2) perfect localisation of a single faulty axis (the radius-1 balls around $\mathcal{C}$ partition $\mathbb{F}_2^N$) and (D3) $\lvert\mathcal{C}\rvert > 2$ (Theorem Σ). (Σ₆⁺) adds (D4): the grammar is unique up to relabelling and translation.

Used by. The strict necessity of $N = 7$ — no decomposition with fewer than seven axes — [C at (Σ₆)] (T-349(a)). With (D4): the maximality half of the double extremality and the statement (P1₆) below, [C at (Σ₆⁺)] (T-349(b)). $N \geq 7$ for F1–F7 does not use it.

Status. [H]. For the seven-dimensional frame it is what Step T8 of the bridge asks; Corollary Σ.1 calls the reading of UHM's axes as diagnosable status bits interpretive [I]. It is strictly weaker than the input it replaced — (P1₆) together with P2 for the competitor implies (Σ₆⁺), while the Hamming grammar of length 15 satisfies (Σ₆) and admits no normed division algebra (T-349(c)). No model of a competing decomposition covering the axioms is known, so its independence from A1–A5 is open.

Replaced 2026-09-28: (P1₆) — P1 for a competing decomposition. P1 holds for any decomposition covering (AP)+(PH)+(QG), not only for the seven-dimensional one. It was the premise of the strict necessity of N = 7 from 2026-09-25 to 2026-09-28, and the Hurwitz route that used it also needed P2 for the competitor (P1 alone leaves $\mathbb{H}$, $N = 3$). The chain T1–T15 proves P1 for the frame it starts from, and its Step T8 takes $N = 7$ from Track A; the orientation input (Alt) of that chain is discharged (T15-canon, section 5). (P1₆) is now [C at (Σ₆⁺)]: at $N = 7$ the grammar is $H(7,4)$ and Steps T9–T15 give P1 for the competitor (T-349(b)). The hosting route of the foundations corpus (Foundations of Mathematics, Part XVIII, ch. 11) does not discharge it: it starts from Hurwitz's list (T-349(d)).

Flavour, vacuum and identification hypotheses [H]

Premise	Statement	Used by	Status
(SV)	the sector values of the Gap vacuum ($\varepsilon_{33}$, $\bar\varepsilon = O(10^{-2})$)	T-69, T-70, T-79–T-81, T-99 (conclusion retracted 2026-09-26), T-120b(ii), T-176, T-180, T-185b, T-216, T-219, C35 and the Λ-budget	[H]; as the vacuum of $V_{\text{Gap}}$ it is refuted [✗] by T-64; it survives only as an independent hypothesis, and under (Cl₀)+(GC) it no longer carries a family index (T-332)
(GC)	a generation is a non-trivial real harmonic of the clock register	$N_{\text{gen}} = 3$ in the harmonic reading (T-328), the mixing discussion	[H]; the exact family $\mathbb Z_3$ is refuted by $\lvert V_{us}\rvert \approx 0.224$ — only the broken form survives
(UP)	only up-type fields couple to the Higgs doublet at tree level (holomorphy in one complex doublet)	T-332, the Dirac neutrino mass	[H] at leading order; the exact form is refuted [✗] (it leaves $e$, $\mu$, $\tau$ massless to all orders, T-332(i))
(PQ)	an added Peccei–Quinn sector	the Gap-axion table of dark matter and confinement	[H]; the Clifford content has no Peccei–Quinn symmetry and no spontaneous CP violation (T-333(e)–(h)); strong CP is open [Pr]
(FE)	the electroweak group acts on $\mathrm{span}\{L,E,U\}$ of the system factor	the axis-frame electroweak construction (T-175b, T-219, T-265)	[H]; replaced by (Cl₀) in the Clifford frame, still carried by the axis-frame pages
(SA)	sector asymmetry of the vacuum Gap profile	neutrino generation assignments, T-219	[H]; T-52 retired as a theorem
Higgs identification	$H \sim \gamma_{EU}$ (axis frame); the colour-free plane as the Higgs doublet (Clifford frame, T-329(f))	the Higgs sector	[H] in both frames
T-186(a)	the cohesive route to A1	T-190 (axiomatic closure)	[H]
Physical reading of $M^4$	the reconstructed $M^4 = \mathbb R \times S^3$ (T-118, T-119, T-120) is physical spacetime	T-120 and T-121 as physics	[I], as in the registry rows T-119 and T-120, which are [T] as mathematics; the former reconstruction conditions of T-119 and the aperiodic time parameter are discharged (section 5); T-120b(ii) stays [C at the vacuum symmetry] through T-64 and (SV), and T-87 as a Page–Wootters mechanism uses the A5 constraint of section 2
(HOL)	a composite of holons is itself a holon, with its own dynamics on $\mathcal D(\mathbb C^7)$	the literal reading of CC-5 and of the population rungs	[I]; not derivable (dimension 49, not 7); CC-5 and CC-6 hold at weak coupling without it

4. Free parameters

Parameter	Where	What is known
$\kappa$, weight of the associator cubic in $V_{\text{Gap}}$	T-64, T-331	Free [T]: no derived source carries it (T-331(e)); its weight is $0$ for every functional of the isolated dynamics (T-331(f)). Both phases occur: for $0 < \kappa \leq \mu^2/48$ the vacuum is $I/7$, for $\kappa > \min(7\mu^2/48, \kappa_1)$ the Gap is spontaneous with orbit $S^6$ (T-64)
$\mu^2$, $\lambda_4$	$V_{\text{Gap}}$ (T-64)	free couplings of the potential
regeneration rate $\kappa$, Fano weight $\alpha$	the evolution equation, T-334–T-336, T-346, T-351	Free [T]: the window needs $\kappa \geq 11.83,\ 20.91,\ 42.64$ at $\alpha = 0,\ \tfrac12,\ 1$ for every self-model (T-336); no route fixes the value above it (T-346) — the threshold $\kappa_c$ of $\varphi_J$ has Galois group $S_7$, the attractor has no interior optimum, the fold survives no detuning, every norm balance with $\mathcal{D}_\Omega$ gives $\kappa g_V \le 1$ against the needed $\ge 1.70$, the categorical $\kappa(\Gamma)$ moves the freedom into $\omega_0$, and composition has only trivial fixed points; nor does a population of holons (T-351) — selection on a common resource runs to the fold, or to the edge $\kappa_H$ set by the energy spread of the environment, Hamiltonian coupling only lowers the effective rate, exchange coupling makes regeneration a public good whose stable level is set by a price, and maximal entropy production selects $\kappa \to \infty$
$\omega_0$	A4	the scale; its value differs between holons
phase reference $D$ of $\varphi_J$	T-334(4)	a gauge of the $H$-free dynamics; physical only relative to a non-diagonal $H$
$\bar\theta_{\mathrm{QCD}}$	T-333, T-99 (corrected 2026-09-26)	free in the Clifford content [Pr]; no $G_2$-invariant Gap term fixes it (Confinement §3.1c)

5. Premises discharged — no longer inputs

Former premise	Discharged by	Date
(P1₆): P1 for a competing decomposition, as the premise of the strict necessity of $N = 7$	T-349: replaced by the strictly weaker (Σ₆); (P1₆) itself follows at (Σ₆⁺)	2026-09-28
(Alt): the Fano orientation is the normed one	T15-canon: the unique orientation class invariant under the 168 collineations	2026-09-25
(MP)	T11–T13 (Choi rank, L-unification, forced BIBD)	earlier
(MM): elementary systems can be entangled	48e(b): a theorem inside UHM	2026-09-25
(Q) = (Q1) ∧ (Q2)	replaced by the weaker (L), now (P); (Q) still suffices	2026-09-25/26
second sentence of (Cl)	T-329: the clock's stabiliser	2026-09-25
(RT), the real twirl inequality	proven (T-64, Lemma 3)	2026-09-25
(Col), (Pure)	(Col) follows from (Eq-V); (Pure) is equivalent to maximal viability under (Eq) (T-334)	2026-09-25
(AGG)	Theorem 9.5 at weak coupling (CC-5, CC-6)	2026-09-25
(ND)	CC-7 for almost every anchor	2026-09-25
(CG)	T6, uniform contraction from $S_7$-equivariance	earlier
the open reconstruction conditions of T-119	T-119 restated: the spatial algebra is $C(S^3)$, and all seven of Connes' conditions hold for its Dirac triple (T-120 and T-121 raised to [T] as mathematics with it)	2026-09-25
an aperiodic time parameter	the depth register: $A_{\text{time}} \cong C_0(\mathbb R)$ as its scaling limit (T-118, T-53b)	2026-09-25

6. Results that use no premise of sections 3–4

These are [T] from the axioms of section 2 alone (the metatheory, A1–A4, the constraint where stated, the definitions O3 and D-0910):

• Structure of the primitive. Cohomological monism and local non-triviality; the octonionic structure from (AP)+(PH)+(QG)+(V) through T15 with the canonical orientation (row 41n); $N \geq 7$ (Theorem S); $G_2$-rigidity (42a, T-123); rigidity of the primitive (T-173); the universal property of the kinematic object (T-174); PhysTheory as a Grothendieck construction (T-211); the encodings T-171 and T-172.
• Time. The clock register (T-87, steps 1–3); the depth register and its dissipative arrow (T-53b); the time line as its scaling limit (T-118).
• Dynamics and consciousness. Non-emptiness of the conscious window (T-124); dead isolation for every unital self-model; the anchor theorems T-334 (1)–(5), T-335, T-336 for every anchor; no-signalling of the full dynamics (Theorem 8.5).
• Vacuum. The $G_2$-invariant potentials (T-331) and the vacuum phases of $V_{\text{Gap}}$ as functions of the free $\kappa$ (T-64).
• As mathematics. 48c, 48d, 48e; T-119, T-120; T-326–T-329, T-332, T-333. Their physical readings carry the premises named in section 3.

7. Independence and mergers

Method. A premise X is independent of the others when there is a model in which all the others hold and X fails. Two premises merge when one statement is proven equivalent to their conjunction and is strictly weaker than asserting them separately — that is, when the merged form reduces the number of independent inputs. Numerical witnesses are in website/scripts/check_core_numbers.py (test_spinor_factor_premise_and_fermion_module_premise_are_independent, test_anchor_principle_is_independent_and_attractor_integration_does_not_replace_it, test_fermion_module_premise_is_the_holons_product_acting_on_matter, test_no_holon_property_maximality_or_minimality_gives_the_bridge_premises, test_spinors_from_the_tensorial_holon_triality_kahler_dirac_and_bosonic_textures).

Independence [T]

Fails	Model in which every other premise holds	Why X fails
(P)	$F_3 = \mathbb C^3 \otimes_{\mathbb C} \mathcal S_{\mathbb C}$, with any anchor and any $\kappa$	by 48e(f) every statement about the generation holds for $n = 3$ (anomalies $\times 3 = 0$, chirality); by 48e(g) $\mathrm{Herm}(\mathbb C^3)$ carries no $\mathrm{SL}$-invariant quadratic form (count 0), so no causal structure is preserved
(Cl₀)	$F = W \otimes_{\mathbb C} M$ with $W = \mathbb C^2$ and $M = \mathbb C^7$, the holon's own vectors with $\mathfrak g_2$ and the $i$ of $\mathcal H$	the commutant of $\{\mathfrak g_2, i\}$ on $\mathbb R^{14}$ has dimension 2 (it is $\mathbb C$; without $i$ it is $M_2(\mathbb R)$, dimension 4), so the structure-preserving transformations of $W$ are $\mathrm{GL}(W)$ and (P) holds with $\det$; but $\dim_{\mathbb R}\mathbb C^7 = 14$ is not a multiple of 16, so $M$ is not a $\mathrm{Cl}_7$-module
(MaxΦ)	anchor $\rho_t = (1-t)I/7 + t\,uu^\dagger$, $t = 0.9$, $\alpha = \tfrac12$, $\kappa = 100$	(Eq) holds and a living sink exists in the window, but $\Phi(\rho_t) = 6t^2 = 4.86 < 6$; the unital anchor $I/7$ ($\Phi = 0$) satisfies all axioms and gives dead isolation
(Eq) half	a pure anchor with non-uniform diagonal (amplitudes $1 \pm 0.3$, a sink at $\kappa = 50$, T-335)	the anchor is pure, its diagonal is not $I/7$
(Pure) half	$\rho_t$ above	the diagonal is uniform, the anchor is mixed
$\kappa$	$V_{\text{Gap}}$ with two values of $\kappa$	both are consistent with every other premise and give different vacua (T-64)
A5 constraint	a state off $\ker\hat C$	section 2

(Σ₆) and the flavour hypotheses of section 3.4 are not in the table: no model of a competing decomposition covering the axioms is known, and the flavour hypotheses are stated inside the frame of (Cl₀), so they presuppose it.

(P) and (Cl₀): one sentence, two independent inputs

Proposition [T]. Let (P̂) be the single statement "matter is a Weyl spinor of the clock-commuting Clifford module": the fermion field is $F = W \otimes_{\mathbb C} \mathcal S_{\mathbb C}$ with $\mathcal S$ the spinor module of the octonionic Clifford system, and spacetime's tangent vectors are the Hermitian forms on $W$, $\dim W \geq 2$, with a causal form preserved by every internal-structure-preserving transformation of $W$. Then

1. (P̂) is exactly (P) as the corpus states it, which already names $\mathcal S_{\mathbb C}$; hence (P) as stated contains (Cl₀) and (W₀);
2. (P̂) ⟺ (Cl₀) ∧ (P*), where (P*) is (P) for an arbitrary fermion module $M$ in place of $\mathcal S_{\mathbb C}$ whose internal commutant is $\mathbb C$;
3. (Cl₀) and (P*) are independent: the two models of the table above realise (Cl₀) ∧ ¬(P*) and (P*) ∧ ¬(Cl₀).

Proof. (1) is the wording of 48e. (2): given (Cl₀), (P*) with $M = \mathcal S_{\mathbb C}$ is (P); conversely (P̂) names $\mathcal S$. (3) is the table. $\blacksquare$

So the merger exists as a statement and does not reduce the inputs: the conjunction of two logically independent premises stays two premises. The reason is structural. (Cl₀) is about the internal module, and 48e(f) proves that the internal structure is blind to $W$; (P*) is about $W$, and the $\mathbb C^7$ model shows that it is blind to the internal module. The accounting used on this page is therefore (Cl₀) plus (P) relative to (Cl₀), with (W₀) contained in both readings.

(MaxΦ) and the variational principles of the corpus: no derivation

Proposition. None of the variational principles that the corpus states yields (MaxΦ):

1. The retracted cross-entropy principle (FEP derivation): its minimiser is the projector onto the top eigenvector of $\Gamma$, an intrinsic anchor; intrinsic anchors are spectral and phase-covariant, so they hold no hyperbolic attractor in $\mathcal V_{\mathrm{full}}$ near $H = 0$ (phase-reference obstruction), and they are not constant [T].
2. (MaxEnt) of the operational basis selects the Bures metric (T-189). Applied to the anchor itself it gives $I/7$, $\Phi = 0$, the opposite of (MaxΦ); applied to the atomic reading of the anchor it gives only (Eq) [T].
3. (V), viability: the anchor $\rho_t$ of the table lives in the window with $\Phi(\rho_t) < 6$ [T, numerical witness]. Maximal viability alone picks anchors with non-uniform diagonal (T-334(6)).
4. Maximal integration of the living attractor (a new candidate, not stated in the corpus). For a constant anchor the window attractor at $H = 0$ depends only on $d = \sum_i (\rho_a)_{ii}^2$ and $s = P(\rho_a) - d$, and its integration is $\eta^2 s/d$ with $\eta$ the top root of $\eta = B(P)/A(P)$, $P = d + \eta^2 s$ (T-335). At fixed $d$ it grows with $s$ [T: the root moves right as $s$ grows], so the maximiser is pure. But in the band $\kappa_c(\alpha) < \kappa < \kappa_*(\alpha)$, $\kappa_* \approx 1.012\,\kappa_c$, a pure anchor with slightly non-uniform diagonal beats $uu^\dagger$: at $\alpha = 0$, $\kappa = 16.8$, $d = 1/7 + 10^{-4}$ gives $\Phi_{\mathrm{att}} = 1.25155$ against $1.25148$ [T by the witness]. Above the band $uu^\dagger$ wins on every tested grid (40 diagonals × 3 purities at $\alpha = 0$, $\kappa = 20$), [H] as a global statement. So this principle agrees with (MaxΦ) only above $\kappa_*$, and below $\kappa_c$ — where $uu^\dagger$ has no living attractor and non-uniform anchors do — it contradicts it.

(MaxΦ) therefore stays [Pr]. What the analysis adds: it is the conjunction of two independent halves, (Eq) and (Pure); (Eq) is the terminal object read on the axes and (Pure) is maximal viability under (Eq); a principle about the observable attractor rather than the anchor reproduces it except in a band of relative width $1.2\,\%$ above $\kappa_c$.

Routes from the holon to (Cl₀), (P) and (W): T-347

The independence table shows that (Cl₀) and (P) do not follow from the present axioms. T-347 takes the next question: which property of the holon, which extremal principle or which minimal strengthening would give them. It answers route by route.

Theorem T-347 (the bridge premises of physics against the holon) — [T] as mathematics

(a) No property of the holon decides the bridge premises. (Cl₀), (P) and (W₀) are statements about the fermion field $F$, which A1–A5 do not mention. Each model of the independence table can be built over one and the same holon: any $\Gamma$, anchor, $\alpha$ and $\kappa$, in particular a living holon in the window $2/7 < P \le 3/7$ with the anchor $uu^\dagger$. Viability, the window, the self-model and its anchor, regeneration, the 168 Fano collineations and the $G_2$-rigidity therefore hold or fail in those models exactly as in the intended one. The holon's symmetry does not see the difference either: as a $\mathfrak g_2$-module, $\mathcal S = \mathbb C\eta_0 \oplus \mathcal H$ is the holon's Hilbert space plus a trivial line. What separates $\mathcal S$ from every space built from $\mathcal H$ is the centre of the spin cover. The rotation by $2\pi$ in the plane of two axes is $\exp(\pi L_{e_1}L_{e_2}) = -1$ on $\mathbb O$ and $+1$ on $\mathbb R^7$ and on every tensor built from it. The genuinely spinorial modules of $\mathrm{Spin}(7)$ have dimensions $8, 48, 112, \dots$; the tensorial ones have $1, 7, 21, 27, 35, \dots$.

(b) (Cl₀) is equivalent to the holon's product acting on matter. Let (Mod) be the statement: there is a real-linear map $\rho:\mathbb O\to\mathrm{End}_{\mathbb R}(F)$ with $\rho(1) = 1$ and $\rho(x)\rho(x) = \rho(x^2)$ for every $x$ — the left alternative law of the holon's octonions — that commutes with the imaginary unit $i$ of $\mathcal H$. Then (Mod) ⟺ (Cl₀): $F \cong U \otimes_{\mathbb R} \mathcal S$ for a real multiplicity space $U$, and on every copy the Clifford system of T-326 is forced. The smallest $\rho$- and $i$-stable space containing $\mathcal H$ is $\mathcal S$. The independence model $M = \mathbb C^7$ fails exactly (Mod).

(c) Maximality and faithfulness presuppose (Mod). The Hurwitz–Radon number of $\mathbb R^{14}$ is $\mathrm{HR}(14) = 2$. So the holon's $\mathcal H$ carries at most one complex structure of an anticommuting family (its $i$), and $\mathbb C^7$ has no quaternionic structure. Seven anticommuting complex structures need $8 \mid \dim$: $\mathrm{HR}(8) = 8$, $\mathrm{HR}(16) = 9$, $\mathrm{HR}(14) = \mathrm{HR}(98) = 2$. The maximality of $\mathrm{Spin}(9)$ in T-326(a) — nine generators on $\mathbb R^{16}$ — is a statement about Clifford systems inside a module that (Mod) supplies, and it cannot select the module. Faithful representations of the holon's observable algebra $M_7(\mathbb C)$ have real dimension $14k$. $\mathcal H$ itself ($k = 1$) is faithful and is not a module of $\mathbb O$, and a space that carries both has dimension divisible by $112$. So faithfulness forces nothing.

(d) Minimality gives $n = 1$, not (W). By 48e(f) every statement of T-329 holds on $\mathbb C^n \otimes_{\mathbb C} \mathcal S_{\mathbb C}$ for every $n \ge 1$. The smallest spinor factor compatible with T-329 is therefore $n = 1$: $F_1 = \mathcal S_{\mathbb C}$, sixteen left-handed Weyl fields with $\sum Y = \sum Y^3 = 0$, chiral. But $\mathrm{Herm}(\mathbb C^1) = \mathbb R$, so spacetime would have no spatial direction. Minimality under the extra clause "a spatial direction exists" gives $n = 2$ ($\dim \mathrm{Herm} = 4$). That clause plus Occam's razor is an input of the same kind as the boost clause of (P), not a derivation. The only source of the "2" with content is the boost invariance of 48e(g).

(e) The independence model of (Cl₀) is not chiral. On $M = \mathbb C^7$ the $\mathfrak g_2$-matrices are real, so $M$ is isomorphic to its conjugate and every Weyl field on it is vectorlike. Gauging the $\mathrm U(1)$ of $i$ instead gives $\operatorname{Tr} Q = \operatorname{Tr} Q^3 = 7$ per component of $W$, an anomaly. So (Cl₀) is independent of the axioms and of (P*), not of the observed chirality. Whether chirality and anomaly freedom select $\mathcal S$ among all fermion modules built from UHM's structures was left open here; T-350(d) answers it: they do not, the Clifford action does; on $\mathcal H$ itself the centraliser of colour in $\mathrm U(7)$ is $\mathrm U(1)^3$ and holds no weak $\mathrm{SU}(2)$ (Standard Model, §2.1).

Proof. (a) The premises are about $F$ and $W$, and the holon is a separate factor of each model, so every property of the holon has the same value in all of them. $\mathfrak g_2$ acts on $\mathbb O$ by derivations, which kill $1$. $L_{e_1}L_{e_2}$ squares to $-1$, and the rotation by the angle $\theta$ in the $(e_1, e_2)$-plane lifts to $\exp(\tfrac\theta2 L_{e_1}L_{e_2})$; at $\theta = 2\pi$ this is $-1$, and conjugation by $-1$ is trivial on every operator. The dimensions come from Weyl's formula for $B_3$ with highest weights in $(\mathbb Z + \tfrac12)^3$ and in $\mathbb Z^3$ (Humphreys, Introduction to Lie Algebras and Representation Theory, Springer 1972, §24). (b) On $\mathrm{Im}\,\mathbb O$ we have $x^2 = -\lvert x\rvert^2$, and polarising $\rho(x)^2 = \rho(x^2)$ gives $\rho(a)\rho(b) + \rho(b)\rho(a) = -2\langle a, b\rangle$. So $F$ is a module of $\mathrm{Cl}_{0,7} \cong M_8(\mathbb R) \oplus M_8(\mathbb R)$ (Lawson and Michelsohn, Spin Geometry, ch. I §4). Its two irreducible modules are eight-dimensional and differ in the sign of the volume element: they are $\mathbb O$ with $L$ (volume $-1$) and $\mathbb O$ with $R$ (volume $+1$), and both obey the law because $\mathbb O$ is alternative. With $i$ commuting, each isotypic part of $F$ is $U_\pm \otimes_{\mathbb R} (\mathbb C \otimes \mathbb O)$. Octonionic conjugation $c$ satisfies $cL_{e_k}c = -R_{e_k}$ and commutes with $i$ and with the conjugation $J$. It carries the system $\{iR_{e_k}, J, iJ\}$ to $\{-iL_{e_k}, J, iJ\}$, which generates the $\mathfrak{spin}(9)$ of T-326. So both parts are $\mathcal S$ with the group of T-326, and $F \cong (U_+ \oplus U_-) \otimes_{\mathbb R} \mathcal S$. Conversely, $\mathcal S$ with $\rho = L$ satisfies (Mod). $\mathcal H$ is not $\rho$-stable because $e_k e_k = -1$; its closure is $\mathcal S$ (fact (i) of Standard Model, §2.6). (c) $\mathbb R^n$ carries $\mathrm{HR}(n) - 1$ anticommuting complex structures and no more, with $\mathrm{HR}(2^{4a+b}\cdot\text{odd}) = 2^b + 8a$, $0 \le b \le 3$ (Hurwitz 1923, Radon 1922). A quaternionic structure on $\mathbb C^7$ would be $K = A\kappa$, with $\kappa$ the complex conjugation, and $K^2 = -1$ means $A\bar A = -1$; but $\det(A\bar A) = \lvert\det A\rvert^2 \ge 0$ and $\det(-1_7) = -1$. The remaining statements are dimension counts. (d) 48e(f) and the charge table of T-329(d). (e) The generators of $\mathfrak g_2$ are real antisymmetric matrices, so complex conjugation intertwines $M$ with $\bar M$. The charge of $i$ is $1$ on each of the seven complex dimensions. $\blacksquare$

Numbers (test_fermion_module_premise_is_the_holons_product_acting_on_matter, test_no_holon_property_maximality_or_minimality_gives_the_bridge_premises):

• $L$ and $R$ satisfy $\rho(x)^2 = \rho(x^2)$ for random $x$; the algebra generated by the seven $L_{e_k}$ on $\mathbb R^8$ has dimension $64$ and commutant of dimension $1$; the volume elements are $-1$ for $L$ and $+1$ for $R$; $cL_{e_k}c = -R_{e_k}$ for all seven;
• $\{iR_{e_k}, J, iJ\}$ satisfies the relations of $\mathrm{Cl}(9,0)$ on $\mathbb R^{16}$; the operators anticommuting with the seven $iR_{e_k}$ form a space of dimension $2$; its $\mathfrak{spin}(9)$ lies in $c\,\mathfrak{spin}(9)_L\,c$ to $10^{-9}$; the closure of $\mathcal H$ (rank $14$) under $L_{e_k}$ and $i$ has rank $16$;
• $\mathfrak g_2$ kills $\eta_0$; $\exp(\pi L_{e_1}L_{e_2}) = -1$ on $\mathbb O$, the $2\pi$-rotation is $+1$ on $\mathbb R^7$; spinorial $\mathrm{Spin}(7)$-dimensions $8, 48, 112$, tensorial $1, 7, 21, 27, 35$;
• $\mathrm{HR}(8) = 8$, $\mathrm{HR}(14) = 2$, $\mathrm{HR}(16) = 9$, $\mathrm{HR}(98) = 2$; $\det(A\bar A) > 0$ for five random $A \in M_7(\mathbb C)$; $\operatorname{lcm}(14, 16) = 112$;
• $F_1$: $16$ fields, $\sum Y = \sum Y^3 = 0$; $\dim \mathrm{Herm}(\mathbb C^n) = 1, 4, 9$ for $n = 1, 2, 3$; one $\mathrm{SL}$-invariant quadratic form at $n = 2$; $\operatorname{Tr} Q = \operatorname{Tr} Q^3 = 7$ on $\mathbb C^7$.

What T-347 changes. No status changes. (Cl₀) and (P) stay [H]; the physical readings of T-326, T-327, T-329, T-332 and T-333 stay [C at (Cl)], and that of 48c stays [C at (L)]. What changes is the form of (Cl₀). It no longer names a module or a Clifford system: it says that the product the holon already has (T15) acts on matter, and $\mathcal S$, $\eta_0$, the nine generators and $\mathrm{Spin}(9)$ follow. The premise sits exactly at the centre of the spin cover. The holon, its symmetries and everything built from $\Gamma$ are tensorial, while matter is spinorial, and no property of the holon can take that step (a). For (P), the only non-trivial content is the boost clause; minimality alone gives a spacetime without space (d). The routes asked for on 2026-09-26 are closed: a property of the holon that the models violate by (a), the maximality of $\mathrm{Spin}(9)$ and a faithful representation of the observables by (c), the minimality of the spinor factor by (d).

Spinors from the tensorial primitive: T-350

T-347 placed (Cl₀) at the centre of the spin cover: the holon and its symmetries are tensorial, and matter is spinorial. Physics knows several ways of getting spinors or fermions out of bosonic, tensorial data. T-350 takes each of them to the holon and states what it gives.

Theorem T-350 (spinors from the tensorial primitive) — [T] as mathematics

Let the holon be as in A1–A5, with symmetry group $G_2$ (42a, D-0910). Space is a three-dimensional slice ($S^3$ of T-119, or $\mathbb R^3$ with a constant limit at infinity), and a field theory of $\Gamma$ has a local Lagrangian.

(a) Textures of $\Gamma$ are bosons. A texture is a map from space to a manifold $X$ on which $\Gamma$ lives, and its topological sectors are $[S^3, X]$. For $X = \mathcal D(\mathbb C^7)$ and for the full-rank stratum ($X$ convex) there are no sectors. The rank-$k$ stratum (dimension $14k - k^2 - 1$) retracts onto the Grassmannian $\mathrm{Gr}_k(\mathbb C^7)$, whose $\pi_3$ vanishes; so do $\pi_3(\mathbb{CP}^6)$, $\pi_3(S^6)$ for $S^6 = G_2/\mathrm{SU}(3)$, and $\pi_3(S^7)$ for $S^7 = \mathrm{Spin}(7)/G_2$. An isospectral orbit $\mathrm U(7)/(\mathrm U(n_1)\times\dots\times\mathrm U(n_r))$ has $\pi_3 = \mathbb Z$ only when all eigenvalues are distinct (the full flag manifold $\mathrm{Fl}$; one of the 15 partitions of 7); a $G_2$-valued field has $\pi_3(G_2) = \mathbb Z$. In every sector with $Q = 0$ the constant map is a rotation-fixed point, and $\mathrm{SU}(2)$ has no non-trivial character, so the rotation by $2\pi$ acts as $+1$. For $\mathrm{Fl}$ and $G_2$ the sectors $Q \ne 0$ exist, but $\pi_1$ of each sector is $0$ ($\operatorname{coker}(\mathbb Z^7 \to \mathbb Z) = 0$ for $\mathrm{Fl}$, $\pi_4(G_2) = 0$), so there is no Finkelstein–Rubinstein sign, and $H^5(X;\mathbb R) = 0$ (the cohomology of $\mathrm{Fl}$ is even; that of $G_2$ sits in degrees 3 and 11), so there is no Wess–Zumino term. The two known mechanisms therefore both fail: $\mathrm{SU}(2)$ has $\pi_4(S^3) = \mathbb Z_2$, $\mathrm{SU}(3)$ has $H^5 \ne 0$, the spaces of $\Gamma$ have neither. The dyon route (a charge bound to a monopole) needs $\pi_2$ of the vacuum manifold; the vacua of T-64 are the point $I/7$ and the orbit $S^6$, and $\pi_2(S^6) = 0$.

(b) The spin lift is forced and empty. In $\mathbf{Sh}_\infty(\mathcal C)$ a spin structure on an object with an $\mathrm{SO}(7)$-frame is a lift of its classifying map through $B\mathrm{Spin}(7) \to B\mathrm{SO}(7)$; the obstruction is $w_2$, and lifts form a torsor under $H^1(-;\mathbb Z_2)$. The holon's frames have structure group $G_2$, and $BG_2$ is 3-connected, so $BG_2 \to B\mathrm{SO}(7)$ lifts, and uniquely. Every $G_2$-frame therefore carries one canonical spin structure, whose spinor bundle is the trivial line plus the axis bundle, $\mathbf 8 = \mathbf 1 \oplus \mathbf 7$ as $G_2$-modules. Viability, the self-model, regeneration and the clock cannot choose a lift, because there is exactly one, and that lift cannot tell a spinor from a vector plus a line. Where the holon's own manifold would need a choice, there is none to make: $c_1(\mathrm{Gr}_k(\mathbb C^7)) = 7\sigma_1$ is odd, so the pure-state space $\mathbb{CP}^6$ and every $\mathrm{Gr}_k(\mathbb C^7)$ carry no spin structure ($\mathrm{Fl}$ does: $c_1 = 2\rho$).

(c) Triality: $G_2$ cannot tell $8_v$ from $8_s$. The triples $(A, B, C) \in \mathfrak{so}(8)^3$ with $A(xy) = (Bx)y + x(Cy)$ form a 28-dimensional algebra, and each projection is an isomorphism onto $\mathfrak{so}(8)$ (the three eight-dimensional representations). The triples with $A = B = C$ are the derivations, $\mathfrak g_2$ (dimension 14). The triples with $A(1) = 0$ form $\mathfrak{so}(7)$ (dimension 21); in the slot $A$ it acts as $\mathbf 1 \oplus \mathbf 7$, in the slots $B$ and $C$ irreducibly. The rotation of the axes in the $(e_1, e_2)$-plane is $A$; its partners are $B = \tfrac12 L_{e_1}L_{e_2}$ and $C = \tfrac12 R_{e_1}R_{e_2}$, and $\exp(2\pi\,\cdot)$ is $+1$, $-1$, $-1$. So the holon's octonions, $\mathbb R \oplus \mathbb R^7$ with $G_2$, are at once the vector of one $\mathrm{Spin}(7)$ and the spinor of another; the two agree on $G_2$ and differ on the seven generators outside it. (Mod) is exactly the statement that the axes act on matter through the slot $B$ (or $C$), and its negation that they act through $A$.

(d) No induced action is two-valued; the question of T-347(e) has a negative answer. Call an action on matter induced when it comes from transformations of the holon, that is, factors through $\mathrm U(7)$ acting on $\mathcal H$ (or through $\mathrm{PU}(7)$ acting on $\mathcal D(\mathbb C^7)$). Then:

1. every action of $\mathrm{SO}(k)$, $k \ge 3$, on $\mathcal D(\mathbb C^7)$, and of $\mathrm{Spin}(k)$ in which $-1$ acts trivially on states, lifts to a linear action in which the rotation by $2\pi$ is $+1$: $-1$ acts on $\mathbb C^7$ by a scalar $\pm 1$, a scalar $-1$ needs every summand spinorial, and spinorial representations have even dimension while $7^m$ is odd;
2. $\mathrm{Spin}(9)$, the group of T-326, acts trivially on $\mathcal H$ and on $\mathcal D(\mathbb C^7)$: its smallest non-trivial representation has dimension $9 > 7$ (the spinor has 16). $\mathrm{Spin}(7)$ acts on $\mathbb C^7$ only as $\mathbf 1^7$ or $\mathbf 7$ (its spinor has 8);
3. the centraliser of colour in $\mathfrak u(7)$ is $\mathfrak u(1)^3$ (dimension 3), so no induced action contains the weak $\mathfrak{su}(2)$, on $\mathcal H$ or on any tensor built from it.

So among fermion modules whose internal symmetry is induced there are no doublets at all, and chirality and anomaly freedom have nothing to select from. Among modules on which the holon's product acts, every module is $U \otimes_{\mathbb R} \mathcal S$ (T-347(b)). What selects $\mathcal S$ is the Clifford action, not chirality. (Cl₀) is incompatible with every principle that makes the internal symmetry of matter induced by the holon.

(e) The spinor module is built from the holon's tensors (Kähler–Dirac). On the exterior algebra $\Lambda^\bullet\mathbb R^7$ of the holon's axes (dimension 128), $c(v) = v\wedge{} - \iota_v$ satisfies $c(v)^2 = -\lvert v\rvert^2$. So $\rho(a + v) = a + c(v)$ satisfies (Mod) on $\Lambda^\bullet_{\mathbb C}\mathbb R^7$, with multiplicity $16$. The Fano 3-form $\varphi$ (T15), read as a Clifford element, has eigenvalues $\pm7$ (8 each) and $\pm1$ (56 each), and $p_\varphi = \tfrac1{16}(1 + \mathrm{vol})(1 + \varphi)$ is a $G_2$-invariant primitive idempotent. The left ideal $\mathrm{Cl}(\mathbb R^7)\,p_\varphi$ is 8-dimensional, $G_2$-stable, irreducible under $c$, and $x \mapsto x\,p_\varphi$ is an isomorphism from $(\mathbb O, R)$ that carries $c(e_k)$ to $R_{e_k}$ and the tensorial $G_2$ to the derivations. Its complexification is $\mathcal S$ with the group of T-326. The spinor module is thus a canonical subspace of a tensor space of the holon. What is not tensorial is the action: the rotation of the forms by $2\pi$ is $+1$, the Clifford one $\exp(\pi c(e_1)c(e_2))$ is $-1$, and by (d) the Clifford action is induced by no transformation of the holon. The same space is the fermionic Fock space of the seven axes ($v\wedge$ creates, $\iota_v$ annihilates). In that reading none of the 21 Clifford bivectors $c(e_k)c(e_l)$, and no combination of them, conserves the particle number, and the number-conserving part of $c(e_1)c(e_2)$ is the tensorial rotation. So second quantisation of the holon gives (Mod) only on many-particle space and at the price of pair creation.

Proof. (a) $\mathcal D(\mathbb C^7)$ is convex, so $(1-t)\Gamma + tI/7$ contracts it. The rank-$k$ stratum fibres over $\mathrm{Gr}_k(\mathbb C^7)$ (the support) with the interior of $\mathcal D(\mathbb C^k)$ as fibre, which is convex; $\dim = 2k(7-k) + k^2 - 1 = 14k - k^2 - 1$. For $H = \prod_i \mathrm U(n_i)$ the exact sequence of $H \to \mathrm U(7) \to \mathrm U(7)/H$ gives $\pi_3(\mathrm U(7)/H) = \operatorname{coker}(\pi_3 H \to \pi_3\mathrm U(7))$ because $\pi_2 H = 0$; $\pi_3\mathrm U(n) = \mathbb Z$ for $n \ge 2$ maps onto $\pi_3\mathrm U(7) = \mathbb Z$, and $\pi_3\mathrm U(1) = 0$, so the cokernel is $\mathbb Z$ exactly when every $n_i = 1$. $\pi_3(S^6) = \pi_3(S^7) = 0$. Every map $S^3 \to \mathrm{Fl}$ lifts to $\mathrm U(7)$ (the obstructions lie in $H^2(S^3;\mathbb Z^7) = 0$), and $\mathrm{Map}(S^3, T^7) \simeq T^7$; so $\pi_1 \mathrm{Map}_Q(S^3,\mathrm{Fl}) = \operatorname{coker}(\pi_1 T^7 \to \pi_1\mathrm U(7) \oplus \pi_4\mathrm U(7)) = \operatorname{coker}(\mathbb Z^7 \xrightarrow{\Sigma} \mathbb Z) = 0$, using $\pi_4\mathrm U(7) = 0$. For $G_2$, $\pi_1\mathrm{Map}_Q(S^3, G_2) = \pi_1 G_2 \oplus \pi_4 G_2 = 0$ (Mimura, J. Math. Kyoto Univ. 6, 131 (1967)). The Poincaré polynomial of $\mathrm{Fl}$ is $\prod_{k=1}^{7}(1 + t^2 + \dots + t^{2(k-1)})$, even, with $7! = 5040$ cells; the rational cohomology of $G_2$ is exterior on generators of degrees $2e+1$ for the exponents $e = 1, 5$, that of $\mathrm U(7)$ on degrees $1, 3, \dots, 13$ and that of $\mathrm{SU}(3)$ on $3, 5$. A local Lagrangian adds topological phases only through $\theta$-terms (classes of $H^4(X;\mathbb R)$, which act through $\pi_1$ of the sector) and Wess–Zumino terms (classes of $H^5(X;\mathbb R)$) (Finkelstein and Rubinstein, J. Math. Phys. 9, 1762 (1968); Witten, Nucl. Phys. B 223, 433 (1983)). With both absent the wave functions are functions on a simply connected configuration space, and the rotation by $2\pi$, which acts as the identity map, acts as $+1$. Monopoles are classified by $\pi_2$ of the vacuum manifold (Goldhaber, Phys. Rev. Lett. 36, 1122 (1976), for the statistics of dyons). (b) The fibre of $B\mathrm{Spin}(7) \to B\mathrm{SO}(7)$ is $B\mathbb Z_2$; lifts of a map $Y \to B\mathrm{SO}(7)$ are obstructed by a class in $H^2(Y;\mathbb Z_2)$ and classified by $H^1(Y;\mathbb Z_2)$. $G_2$ is 2-connected ($\pi_1 = \pi_2 = 0$), so $BG_2$ is 3-connected and both groups vanish for $Y = BG_2$. Since $\mathfrak g_2$ kills $\eta_0$, the spinor representation restricted to $G_2$ is $\mathbf 1 \oplus \mathbf 7$ (T-347(a)). For a complex manifold $w_2 = c_1 \bmod 2$, and $c_1(\mathrm{Gr}_k(\mathbb C^n)) = n\sigma_1$ with $\sigma_1$ primitive; $c_1(\mathrm U(n)/T) = 2\rho$. (c) The triality principle (Schafer, An Introduction to Nonassociative Algebras, Academic Press 1966, ch. III §8) and a linear computation: the condition is linear in $(A, B, C)$, its solution space and the stated subspaces are computed as kernels. (d) (1) If $-1 \in \mathrm{Spin}(k)$ acts trivially on $\mathcal D(\mathbb C^7)$, its lift to $\mathrm U(7)$ commutes with everything and squares to $1$, so it is $\pm1$; with $-1$ every irreducible summand is spinorial. Spinorial representations of $\mathrm{Spin}(k)$, $k \ge 3$, have even dimension ($2j + 1$ with $j \in \mathbb Z + \tfrac12$ for $k = 3$, multiples of $2^{\lfloor (k-1)/2\rfloor}$ in general). A homomorphism into $\mathrm{PU}(7)$ from a group whose fundamental group is a 2-group lifts to $\mathrm{SU}(7)$, since $\pi_1\mathrm{PU}(7) = \mathbb Z_7$ has no 2-torsion. (2) Weyl's formula for $B_4$ and $B_3$. (3) Under $\mathrm{SU}(3)_C$, $\mathcal H = \mathbf 1 \oplus \mathbf 3 \oplus \bar{\mathbf 3}$, three inequivalent irreducibles, so the commutant is $\mathbb C^3$; an induced action on a tensor of $\mathcal H$ is the image of $\mathrm U(7)$, and its part commuting with colour is the image of $\mathrm U(1)^3$. The last sentences are T-347(b). (e) Direct computation; the left ideal is the image of right multiplication by $p_\varphi$. On forms, $c(e_1)c(e_2) = e_1\wedge e_2\wedge{} - e_1\wedge\iota_{e_2} - \iota_{e_1}e_2\wedge{} + \iota_{e_1}\iota_{e_2}$; the middle terms preserve the degree and form the tensorial rotation, the outer ones change it by $\pm2$. $\blacksquare$

Numbers (test_spinors_from_the_tensorial_holon_triality_kahler_dirac_and_bosonic_textures):

• $\pi_3$ of the isospectral orbit is non-zero for 1 of the 15 partitions of 7; the Poincaré polynomial of $\mathrm{Fl}$ has 5040 cells and no odd term; degree 5 is absent for $G_2$ (3, 11) and present for $\mathrm U(7)$ and $\mathrm{SU}(3)$; the segment $(1-t)\Gamma + tI/7$ stays in $\mathcal D$; $c_1 = 7$ on every $\mathrm{Gr}_k(\mathbb C^7)$, $2\rho = (6, 4, 2, 0, -2, -4, -6)$;
• triality: 28 triples, each projection of rank 28, $A = B = C$ gives 14, $A(1) = 0$ gives 21 with commutants $2, 1, 1$ in the three slots; the rotation of $(e_1, e_2)$ has partners $\tfrac12 L_{e_1}L_{e_2}$ and $\tfrac12 R_{e_1}R_{e_2}$; $\exp(2\pi\,\cdot) = +1, -1, -1$; on $\mathfrak g_2$, $B = C = A$;
• the smallest non-trivial $\mathrm{Spin}(9)$-dimensions are $9, 16, 36, 44$; $\mathrm{Spin}(7)$-dimensions $\le 7$ are $1, 7$, the smallest spinorial is $8$; the commutant of $\mathfrak{su}(3)_C$ on $\mathbb C^7$ has dimension 3;
• $c(e_k)$ is $e_k\wedge{} - \iota_{e_k}$ and satisfies $\mathrm{Cl}_{0,7}$ on $\mathbb R^{128}$; $\mathrm{vol}^2 = 1$, central; $\varphi$ has spectrum $-7^8, -1^{56}, 1^{56}, 7^8$; $p_\varphi$ has 16 coefficients $\pm\tfrac1{16}$, its left multiplication has rank 8; the ideal has dimension 8, generated algebra 64, commutant 1, $G_2$-commutant 2; $x \mapsto xp_\varphi$ intertwines $c(e_k)$ with $R_{e_k}$ (not with $L_{e_k}$) and $\mathfrak g_2$ with the derivations; the tensorial $2\pi$-rotation is $+1$ on all 128 dimensions and $\exp(\pi c(e_1)c(e_2)) = -1$; the commutators of the 21 bivectors with the degree operator have rank 21; the degree-preserving part of $c(e_1)c(e_2)$ is the tensorial rotation.

What T-350 changes. (Cl₀) stays [H], and the physical readings of T-326, T-327, T-329, T-332 and T-333 stay [C at (Cl)]: no route derives it. Three things change. First, the obstruction is stated exactly. The earlier formula "everything built from $\Gamma$ is tensorial" is too strong. The spinor module is itself built from the holon's axes and Fano form, canonically ((e)). What no construction from the holon gives is a two-valued action: every action induced by transformations of the holon is single-valued, and the group of T-326 cannot act on the holon at all ((d)). Second, the open question of T-347(e) is closed [T]: chirality and anomaly freedom do not select $\mathcal S$. Induced modules have no weak $\mathrm{SU}(2)$, and Clifford modules are all $U \otimes \mathcal S$. Third, the routes through solitons, spin structures and triality are closed with the reasons (a)–(c). The premise is (Mod), the holon's product acting on matter, and T-350 gives two readings of it: the axes act on matter through the spinor slot of triality, which $G_2$ cannot distinguish from the vector slot (c); the Clifford multiplication of the holon's forms, not their rotation, is the internal symmetry of matter (e). Each is a statement about how the holon acts on matter, and A1–A5 say nothing about matter.

Other pairs

• (P) and (MaxΦ) live in different factors (the spinor factor of matter, the anchor on $\mathbb C^7$); the $F_3$ model with anchor $uu^\dagger$ satisfies (MaxΦ) and not (P), the $\rho_t$ model with $n = 2$ satisfies (P) and not (MaxΦ). Independent.
• (Σ₆) and (P): (Σ₆) concerns the decompositions of the state space, (P) the spinor factor of matter; no implication is known either way. Open.
• $\kappa$ and (MaxΦ): $\kappa$ weights a cubic of $V_{\text{Gap}}$, which the isolated dynamics does not see (T-331(f)); the anchor lives in the self-model. Independent.

8. Count

After the four waves of 2026-09-25/26 the free inputs of UHM are: the axioms A1–A4 and the constraint of A5 [P]; two bridge premises of physics, (Cl₀) and (P) relative to it [H]; one principle of the self-model, (MaxΦ) [Pr]; the strict-necessity premise (Σ₆) [H], perfect diagnosability of every decomposition, which replaced the stronger (P1₆) on 2026-09-28 (T-349); the free parameters of section 4, $\kappa$ among them; and the identification hypotheses of section 3.4, three of which, (SV), (GC) and (UP), survive only in weakened form. Every other input used earlier has been discharged (section 5). (Cl₀) has an equivalent form, (Mod), in which the holon's product acts on matter; T-347 closes the routes that would derive it or (P) from a property of the holon, from maximality or from minimality; T-350 closes the routes that would make spinors out of the tensorial holon (textures, the spin lift, triality, second quantisation) and shows that (Cl₀) is incompatible with any principle under which the holon induces the internal symmetry of matter.