Higgs Sector

Rigor Levels

• [T] Theorem — strictly proved from UHM axioms
• [C] Conditional — conditional on an explicit assumption
• [H] Hypothesis — mathematically formulated, requires proof or non-perturbative computation
• [I] Interpretation — philosophical / qualitative analogy
• [D] Definition — definition by convention

Contents

1. Uniqueness of the Higgs line {A,E,U}
2. Higgs mechanism from Gap-condensation
3. Gap(E,U) → 0: electroweak symmetry breaking
4. Higgs mass with octonionic correction (incl. Higgs quartic from spectral action [C])
5. Connection to SM gauge structure (EW-construction)
6. Falsifiable predictions
7. Can UHM predict the Higgs mass? — analysis of the derivation chain, status of each link

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1. Uniqueness of the Higgs line {A,E,U}

1.1 Identification of the Higgs field [H]

In UHM the Higgs field is identified with the $E$-$U$ coherence in the $\bar{3}$-to-$\bar{3}$ sector:

$H \sim \gamma_{EU} = |\gamma_{EU}| e^{i\theta_{EU}}$

Dimensions $E$ (evaluation) and $U$ (unity) belong to the $\bar{3}$-sector $\{L, E, U\} = \{4, 5, 6\}$. The pair $(E, U)$ defines the electroweak channel: $\text{Gap}(E,U) = 0$ corresponds to a weak doublet, $\text{Gap}(E,U) \neq 0$ — to a singlet.

Theorem 1.0 (Identification $H \sim \gamma_{EU}$) — corrected from [T] to [H]

danger

Corrected 2026-09-25 (audit A-90): a vacuum value of $\gamma_{EU}$ breaks colour

Checked numerically with $SU(3)_C = \mathrm{Stab}_{G_2}(e_O)$, the colour group of the corpus. The state $\Gamma = I/7 + \varepsilon\,(e^{i\phi}|E\rangle\langle U| + \text{h.c.})$ keeps a subalgebra of $\mathfrak{su}(3)_C$ of dimension 1 at $\phi = \pi/2$ (and $3\pi/2$) and of dimension 0 at the other 23 of 25 sampled phases, against 8 at $\varepsilon = 0$; the coherence $\gamma_{EU}$ has no colour-singlet component, since the $SU(3)_C$-invariant states have coherences only on $(A,D)$, $(S,U)$, $(L,E)$ (test_gamma_eu_vev_breaks_colour). So $\langle\gamma_{EU}\rangle \neq 0$ breaks $SU(3)_C$, while the Standard-Model Higgs is a colour singlet. Steps 3 and 4 below fail as well: no $SU(2)$ commutes with $SU(3)_C$ on $\mathbb C^7$ (the commutant is $\mathbb C^3$), so there is no doublet $(2,+1/2)$ to carry, and the vacuum value came from T-64, which is restated as a hypothesis whose vacuum has no sector values; $E$ and $U$ are not in a sector $\bar{\mathbf 3} = \{L,E,U\}$ (T-48a retracted).

Repairs tried. (i) Correct complex triplets: $\gamma_{EU}$ has zero singlet weight, as above. (ii) Another colour group: $\gamma_{EU}$ is invariant under $\mathrm{Stab}_{G_2}(e_A)$, but only inside the combination with equal coherences on $(S,L)$ and $(D,O)$ — the pairs of the lines through $A$ — and this moves colour from $O$ to $A$, against the rest of the corpus. (iii) A doublet on $\mathbb C^7$: impossible for any $SU(3)$, for the commutant reason above. (iv) The Clifford frame of T-326 restricted to $\mathrm{Spin}(9)$, where an $SU(2)$ does exist — the centraliser of colour in the $\mathrm{Spin}(9)$ of $\mathcal S = \mathbb C\otimes\mathbb O$, which T-329 shows to be the diagonal of $SU(2)_L\times SU(2)_R$: it gives $H \sim \gamma_{EU}$ no support. $\mathcal S = (\mathbf 3,\mathbf 2)_{1/6} \oplus (\mathbf 1,\mathbf 2)_{-1/2}$ contains doublets only, so every operator on $\mathcal S$ — every coherence of $\Gamma$, $\gamma_{EU}$ included — carries integer $SU(2)_L$ spin ($\mathbf 2\otimes\mathbf 2 = \mathbf 1\oplus\mathbf 3$); and the vector $\mathbb R^9$ of the Clifford system is $(\mathbf 3\oplus\bar{\mathbf 3},\mathbf 1)_{\pm1/3} \oplus (\mathbf 1,\mathbf 3)_0$ — no doublet either (test_no_higgs_doublet_in_the_clifford_frame). (Narrowed 2026-09-25: this absence holds for $\mathrm{Spin}(9)$ only.) In the $\mathrm{Spin}(10)$ completion, where the tenth Clifford generator is forced, the colour-free Clifford plane $\{iL_{e_O}, J, iJ, \gamma_{10}\}$ is one Higgs doublet with $Y = \pm\tfrac12$ — [T] as a representation, the identification [H] — and a vacuum in the plane $\{iL_{e_O}, \gamma_{10}\}$ leaves exactly $SU(3)\times U(1)_Q$ (standard model, Theorem 2.6(f)). That doublet is a direction of the Clifford vector, not a coherence of $\Gamma$, so it gives $H \sim \gamma_{EU}$ no support either. What stands [T]: Step 1 (T-42a) and Step 2 (Theorem 1.1). The identification $H \sim \gamma_{EU}$ is a hypothesis [H] with three named obstructions: colour breaking under $SU(3)_C = \mathrm{Stab}(e_O)$, the absence of a doublet on $\mathbb C^7$, and the absence of a doublet among the operators on $\mathcal S$ and in the vector of $\mathrm{Spin}(9)$. The Higgs doublet of the corpus is the one of Theorem 2.6(f).

Remark (the Gap vacuum and the Higgs plane) [I]

Under T-64 in its corrected form ([T] in the Gap phase) the imaginary part of the vacuum is $\mathrm{Im}\,\Gamma_v = \tfrac{b-c}{2}L_{e_O}$: the restriction to $\mathbb C^7$ of the clock generator $L_{e_O}$, one of the four directions of the colour-free Clifford plane of Theorem 2.6(f). Its stabiliser in $G_2$ is $\mathrm{SU}(3)_{e_O}$, and the stabiliser in $\mathfrak g_{\mathrm{SM}}$ of a vector of the plane $\{iL_{e_O}, \gamma_{10}\}$ is $\mathfrak{su}(3)\oplus\mathfrak u(1)_Q$. That the Gap condensate and the Higgs vacuum point along the same clock direction is a reading, not a derivation: $\Gamma$ lives on $\mathbb C^7$, the doublet on the Clifford vector $\mathbb R^{10}$, and no map between them is given.

Earlier statement (Theorem 1.0, stated as [T] until 2026-09-25)

The identification $H \sim \gamma_{EU}$ is strictly proved from four independent [T]-results: categorical uniqueness of the pair $(E,U)$, uniqueness of the Higgs line, $SU(2)_L \times U(1)_Y$ quantum numbers, and nonzero vacuum expectation value from the unique vacuum. (Corrected: see the box above.)

Theorem. The coherence $\gamma_{EU}$ is the unique candidate for the Higgs field in UHM, and the identification $H \sim \gamma_{EU}$ is proved from the following chain.

Step 1. Categorical uniqueness of the pair $(E,U)$ [T] (T-42a).

The formula $\kappa_0 = \omega_0 \cdot |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$ categorically singles out exactly the pair $(E,U)$ via morphisms $\mathrm{Hom}(O,E)$ and $\mathrm{Hom}(O,U)$. No other pair of dimensions has this property: replacing with $\{L,U\}$ removes $\mathrm{Hom}(O,L)$ from $\kappa_0$; replacing with $\{L,E\}$ excludes $U$, breaking the normalization $\mathrm{Tr}(\Gamma) = 1$. Uniqueness is proved — see Theorem of FE-uniqueness [T].

Step 2. Uniqueness of the Higgs line $\{A,E,U\}$ [T] (Theorem 1.1).

Through any two points of $\mathrm{PG}(2,2)$ there passes exactly one line. The unique Fano line containing both points $E = 5$ and $U = 6$: $\{5,6,1\} = \{A,E,U\}$. This line defines the electroweak sector — see Theorem 1.1 [T].

Step 3. Quantum numbers of $\gamma_{EU}$ coincide with those of the SM Higgs doublet [T].

From the electroweak uniqueness theorem (§2.3a [T]): the pair $(E,U)$ forms the doublet $2_{EU}$ under $SU(2)_L$. The coherence $\gamma_{EU}$ — a bilinear form connecting $E$ and $U$ — transforms as $(2, +1/2)$ under $SU(2)_L \times U(1)_Y$. This is exactly the quantum numbers of the SM Higgs doublet.

Step 4. Nonzero VEV $\langle\gamma_{EU}\rangle \neq 0$ breaks $SU(2)_L \times U(1)_Y \to U(1)_\text{em}$ [T].

From Theorem on the unique vacuum T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)): the unique global minimum of $V_\text{Gap}$ has $|\gamma_{EU}|_\text{vac} = \varepsilon_{\bar{3}\bar{3}} \approx 10^{-17}$ (in units of $\omega_0$), giving $\langle\gamma_{EU}\rangle \neq 0$. A nonzero vacuum expectation value of a field with quantum numbers $(2, +1/2)$ uniquely realizes spontaneous breaking $SU(2)_L \times U(1)_Y \to U(1)_\text{em}$.

Conclusion (earlier, corrected 2026-09-25). The earlier text read: "All four steps rely exclusively on [T]-results. The identification $H \sim \gamma_{EU}$ follows from them uniquely." Steps 3 and 4 are withdrawn (box above), so the identification is a hypothesis [H].

1.2 Fano–Higgs line

Definition 1.1 (Fano–Higgs line)

Definition. The Fano–Higgs line is the Fano line of $\mathrm{PG}(2,2)$ containing both Higgs dimensions $E = 5$ and $U = 6$.

Theorem 1.1 (Uniqueness of the Fano–Higgs line)

[T] Theorem

Strictly proved. Follows from the incidence axiom of the projective plane $\mathrm{PG}(2,2)$: through any two points there passes exactly one line.

Theorem. There exists exactly one Fano–Higgs line: $\{1, 5, 6\} = \{A, E, U\}$.

Proof. In $\mathrm{PG}(2,2)$ through any two points there passes exactly one line. We seek the line containing points $E=5$ and $U=6$. From the complete list of 7 Fano lines:

Line	Contains $E=5$?	Contains $U=6$?	Both?
$\{1,2,4\}$	No	No	No
$\{2,3,5\}$	Yes	No	No
$\{3,4,6\}$	No	Yes	No
$\{4,5,7\}$	Yes	No	No
$\{5,6,1\}$	Yes	Yes	Yes
$\{6,7,2\}$	No	Yes	No
$\{7,1,3\}$	No	No	No

The unique line containing both 5 and 6: $\{5,6,1\} = \{A, E, U\}$. $\blacksquare$

1.3 Combinatorics of PG(2,2): why {A,E,U} is the only possibility

[T] Theorem

Uniqueness follows from the incidence axiom of the projective plane of order 2: through any two points there passes exactly one line.

The projective plane $\mathrm{PG}(2,2)$ (Fano plane) contains 7 points and 7 lines. Each line contains 3 points; through each point pass 3 lines. Key property: through any pair of points there passes exactly one line.

The Higgs field is defined by two dimensions: $E = 5$ (evaluation) and $U = 6$ (unity). Question: which Fano lines contain both of these dimensions?

The count is exhaustive. Of the 7 lines of $\mathrm{PG}(2,2)$:

• $\{1,2,4\}$: $E \notin$, $U \notin$ — does not qualify
• $\{2,3,5\}$: $E \in$, $U \notin$ — does not qualify
• $\{3,4,6\}$: $E \notin$, $U \in$ — does not qualify
• $\{4,5,7\}$: $E \in$, $U \notin$ — does not qualify
• $\{5,6,1\} = \{A,E,U\}$: $E \in$, $U \in$ — unique
• $\{6,7,2\}$: $E \notin$, $U \in$ — does not qualify
• $\{7,1,3\}$: $E \notin$, $U \notin$ — does not qualify

Thus, the incidence structure of $\mathrm{PG}(2,2)$ uniquely determines the third element of the Higgs line: $A = 1$.

Note that this property does not depend on the choice of numbering: for any identification of $E$ and $U$ with two points of the Fano plane, the third element is determined uniquely. The duality of $\mathrm{PG}(2,2)$ (point $\leftrightarrow$ line) means that point $A$ lies on exactly 3 lines, one of which is the Higgs line $\{A,E,U\}$, and the other two ($\{A,S,L\} = \{1,2,4\}$ and $\{A,D,O\} = \{1,3,7\}$) play different roles: generational and gravitational, respectively.

1.4 Physical interpretation [I]

The third element of the Higgs line is $A = 1$ (awareness). This means:

• Dimension A is directly connected to the Higgs mechanism of mass generation.
• Generation $k=1$ (A) → third generation ($t$, $b$, $\tau$) acquires a tree-level Yukawa coupling.
• Generations $k=2$ (S) and $k=4$ (L) do not lie on the Higgs line → $y^{(\text{tree})} = 0$.

This is the foundation of the Fano selection rule for Yukawa couplings.

Generation assignment and number of generations [T]

The assignment $k=1 \to$ 3rd generation is strictly proved from the unique nonzero tree-level Yukawa coupling — see Theorem 4.1 (Assignment of 3rd generation). The complete ordering ($k=4 \to$ 2nd, $k=2 \to$ 1st) is strictly proved — Theorem 4.3 [T]. The number of generations $N_{\text{gen}} = 3$ has composite status count [T], identification [I]: the count is the exact cardinality $|\mathrm{QR}(7)| = (7-1)/2 = 3$ [T] (group-theoretic, topology-independent), and only the physical identification of the classes with generations is [I] — see Theorem $N_{\text{gen}} = 3$.

1.5 Why the E-U channel defines electroweak physics

[T] Theorem

The $E$-$U$ channel is the unique channel in the $\bar{3}$-sector not containing $L$ (interiority), making it the only candidate for chiral distinction.

In the $\bar{3}$-sector $\{L, E, U\} = \{4, 5, 6\}$ there are three coherences: $\gamma_{LE}$, $\gamma_{LU}$, $\gamma_{EU}$. Of these:

Channel	Connection	Role in SM
$L$-$E$	Interiority–evaluation	Lepton number
$L$-$U$	Interiority–unity	Baryon number
$E$-$U$	Evaluation–unity	Weak isospin (Higgs)

The $E$-$U$ channel is distinguished for three reasons:

1. Algebraic: $E$-$U$ is the unique channel in the $\bar{3}$-sector not containing the $L$-dimension. In fermionic configurations ($R \to 0$) the $L$-channels are fixed, and only $E$-$U$ remains free for defining chirality.

2. From Fano structure: the unique Fano line through $E$ and $U$ is $\{A,E,U\}$ (the registry-canonical Higgs line, T.1.3; its third point $A$ lies in the $3$-sector — no Fano line lies entirely within $\bar 3=\{L,E,U\}$, since $\{L,E,U\}=\{4,5,6\}$ is not a line). The chirality operator $\Gamma_{AEU}$ is defined by this line. $\text{Gap}(E,U)$ is the specific coherence broken by the Higgs, while the remaining line-coherences $\text{Gap}(A,E)$ and $\text{Gap}(A,U)$ fix the doublet embedding.

3. Physical: $E$-dimension $\leftrightarrow$ evaluative structure $\leftrightarrow$ electric charge. $U$-dimension $\leftrightarrow$ unification $\leftrightarrow$ weak isospin. At $\text{Gap}(E,U) = 0$ they are indistinguishable → $SU(2)_L$ doublet. At $\text{Gap}(E,U) \neq 0$ they are distinguishable → singlets.

1.6 Yukawa couplings in the Clifford frame: what splits up from down (T-332)

tip

Status: Theorem 1.6 (a)–(e) is [T] as mathematics and [C at (Cl)] in UHM; the hypothesis (UP) of (f) is [H] at leading order and refuted [✗] in its exact form (T-332(h)–(k), §1.7); the Yukawa structure itself — $m_t/m_b$, $y_t$, CKM — stays open [Pr]

Theorem 2.6(f) of the Standard Model page puts the Higgs doublet in the colour-free Clifford plane of $\mathrm{Spin}(10)$ and notes that one Clifford multiplication gives $m_t=m_b=m_\tau$. This section finds what in UHM can separate up from down, classifies every Yukawa coupling by the stage of the clock's symmetry breaking, and compares with the masses. Registry row T-332; checks in website/scripts/check_core_numbers.py.

Setting. Notation of the Standard Model page, §2.6: $\mathcal S_{\mathbb C}=V_L\oplus V_R$, field unit $\omega$, Clifford vectors $\gamma_a$, colour-free plane $P=\{iL_{e_O},J,iJ,\gamma_{10}\}$ with neutral directions $\{iL_{e_O},\gamma_{10}\}$. A Yukawa coupling is a real-linear map $h\mapsto M(h)$ from $P$ to $\omega$-antilinear operators $V_L\to V_R$ — the Dirac form of Theorem 2.6(f) — that is equivariant under a group $G$: $M(gh)=gM(h)g^{-1}$. The masses of $u,d,\nu,e$ are the singular values of $M(\langle h\rangle)$ between the matching components.

Theorem 1.6 (T-332).

(a) Up and down are where the two units meet [T]. The operator $\tau := -iL_{e_O}$ on $\mathcal S_{\mathbb C}$ is a symmetric involution. It equals $+1$ on $u_L,\nu_L,u^c,\nu^c$ and $-1$ on $d_L,e_L,d^c,e^c$, on both halves alike. So the up-type fields are the vectors on which the imaginary unit of $\mathcal H$ acts as the clock's left multiplication, $i=L_{e_O}$, and the down-type fields those with $i=-L_{e_O}$. On $\mathbb C^7\subset\mathcal S$ the two eigenspaces of $L_{e_O}$ on $e_O^\perp$ — the "triplet" $P_{\mathbf 3}$ ($L_{e_O}=-i$) and "antitriplet" $P_{\bar{\mathbf 3}}$ ($L_{e_O}=+i$) of T-64 — are the down and up components of the left-handed quark doublet. In the field's complex structure both are colour triplets. $\tau_R:=\tau|_{V_R}$ commutes with $\mathfrak g_{\mathrm{SM}}$. On $V_L$, $\tau$ is twice $T_{3L}$ and does not commute with $\mathfrak{su}(2)_L$.

(b) Classification by the stages of the clock's breaking [T]. The real dimension of the space of Yukawa couplings is:

symmetry of the coupling	dimension	relations for every vacuum in the neutral plane
Pati–Salam, $\mathfrak c(L_{e_O})$ (21)	2	$m_u=m_d=m_\nu=m_e$ (as for $\mathrm{Spin}(10)$)
left–right, $\mathfrak c(L_{e_O},R_{e_O})$ (15)	4	$\lvert m_u\rvert=\lvert m_d\rvert$, $\lvert m_\nu\rvert=\lvert m_e\rvert$
$\mathfrak g_{\mathrm{SM}}$ (12)	8	none: four independent masses

At the last stage every coupling is $M(h)=p(B-L,\tau_R)\,\gamma(h)$, with $p$ a polynomial whose coefficients lie in $\mathrm{span}\{1,\omega\}$. The eight operators $\{1,\omega\}\otimes\{1,B-L\}\otimes\{1,\tau_R\}$ applied to $\gamma(h)$ span the space. Up is separated from down only by $\tau_R$, the imaginary unit of $\mathcal H$ on the right-handed half. It is the same operator that defines the hypercharge, $Y=(B-L)/2+(i/2)|_{V_R}$ (Theorem 2.6(d)). In Standard-Model language the coupling $\gamma(h)(1-\tau_R)/2$ is $Q\tilde H u^c$ and $\gamma(h)(1+\tau_R)/2$ is $QHd^c$.

(c) Real versus complex vacuum [T]. With any coupling invariant under $\mathrm{SU}(2)_R$, a real vacuum in the neutral plane gives equal moduli. Its angle in the plane is a hypercharge rotation, and it moves only the phases, $m_u\propto e^{\omega\vartheta}$, $m_d\propto e^{-\omega\vartheta}$. The complexified plane contains the isotropic vectors $\gamma_{10}\pm\omega\,iL_{e_O}$. The first gives mass only to $u$ and $\nu$, the second only to $d$ and $e$. These are the two doublets of a complex bidoublet, with $m_t/m_b=\lvert h_u/h_d\rvert=\tan\beta$. This is the complex $\mathbf{10}$ of $\mathrm{SO}(10)$, and its price is a second doublet (T-296, §6.0).

(d) The clock phase does not split the moduli [T]. Dressing the coupling by $e^{\varphi X}$, $X\in\{i, L_{e_O},\omega,B-L\}$ — among them the Page–Wootters phase generated by $i$ — keeps $\lvert m_u\rvert=\lvert m_d\rvert$ and $\lvert m_\nu\rvert=\lvert m_e\rvert$. It changes only the arguments of the masses.

(e) The Gap vacuum does not split in the observed way [T for the identities; the scan is numerical]. The colour-invariant vacuum of T-64 is $\Gamma_v=a\,|O\rangle\langle O|+\tfrac{b+c}{2}\,\Pi_6-\tfrac{b-c}{2}\,\tau|_{\mathbb C^7},\qquad \mathcal G_{\text{total}}=\lVert\mathrm{Im}\,\Gamma_v\rVert^2=\tfrac32(b-c)^2 .$ Its imaginary part points along $iL_{e_O}$, one real neutral direction, so by (c) it gives equal moduli (the remark of §1.1). Its Gap parameter $b-c$ is the $T_{3L}$-component of $\Gamma_v$ on the quark doublet. Extended to $\mathcal S$ with weight $t$ on $\eta_0$, $\Gamma_v$ commutes with $\mathfrak{su}(2)_L$ only for $b=c$ and $t=a$; even $I/7$ with $t=0$ does not. Read as population weights of a Yukawa coupling ($u$: weight $c$; $d$: $b$; $\nu$ and $e$: $a/2$ each, since $e_O$ is half $\nu_L$ and half $e_L$), the vacuum gives a lepton-to-heavy-quark ratio between $0.46$ and $0.95$ on 99 points of the Gap phase. On the rank-4 branch at $\lambda_4=0$ the ratio is at least $1/2$ analytically. The data give $m_\tau/m_t=0.022$, and $m_\nu=m_e$ is excluded as well.

(f) What the data ask for [numbers; hypothesis (UP) [H]]. One-loop Standard-Model running from $M_Z$ (inputs $m_t(m_t)=162.5$ GeV, $m_b(m_b)=4.18$ GeV, $m_\tau=1.777$ GeV, $\alpha_s(M_Z)=0.118$) gives $m_t/m_b\approx55$ at $M_Z$. At $2\times10^{16}$ GeV it gives $y_t=0.443$, $y_t/y_b\approx68$ and $y_b/y_\tau\approx0.66$. The Clifford relations $m_t=m_b$ and $m_b=m_\tau$ therefore fail by a factor of 68 and by 34%. Written as $(\alpha+\beta\tau_R)\gamma(h)$, the data require $\beta/\alpha=(y_t-y_b)/(y_t+y_b)=0.971$. At the unification scale the coupling is the projection onto $i=L_{e_O}$ to within 1.5%: at leading order only up-type fields couple. Hypothesis (UP): the tree-level Yukawa coupling is $\gamma(h)$ followed by the projection onto $V_R\cap\{i=L_{e_O}\}$. Its consequences: $y_b=y_\tau=0$ at tree level; one $O(1)$ coupling $y_t$; and a Dirac neutrino coupling $y_\nu^D=y_t$. With $m_{\nu_3}\approx0.05$ eV the seesaw then puts $M_R=m_D^2/m_{\nu_3}\approx1.2$–$1.4\times10^{14}$ GeV, the order of the neutrino page. (UP) is not derived. No principle of UHM found so far fixes $\beta/\alpha$, and $m_t/m_b$ is not predicted. Update (T-332(h)–(k), §1.7): (UP) is holomorphy of the coupling in one complex doublet. In its exact form it leaves $e$, $\mu$ and $\tau$ massless to all orders and is refuted [✗]. Only the leading-order statement, with a breaking $\varepsilon=1-\beta/\alpha\approx0.03$, remains [H].

(g) Mixing [T for the statement]. Suppose every generation couples through one flavour matrix times the same internal operator. Then $M_u\propto M_d$ and $V_{\mathrm{CKM}}=1$, which $\lvert V_{us}\rvert=0.2243$ refutes. Mixing needs at least two flavour matrices carrying different internal operators. In $\mathrm{SO}(10)$ language these are the $\mathbf{10}$ with the $\overline{\mathbf{126}}$ or the $\mathbf{120}$; the bidoublet coupling of the $\overline{\mathbf{126}}$ is the $(B-L)$-dressed one, with lepton-to-quark ratio $-3$. Under (GC) the family index lives on the clock register (T-328). Nothing here fixes the flavour matrices, so the CKM hierarchy stays open.

Proof. (a) $i$ and $L_{e_O}$ commute and both square to $-1$, so $\tau$ is a symmetric involution. Its sign on each component is computed from the charges $T_{3L}$, $T_{3R}$, $B-L$ of Theorem 2.6(d). (b) The dimensions solve the linear equivariance system on $\mathrm{Hom}(P,\mathrm{Hom}(V_L,V_R))$ restricted to $\omega$-antilinear maps. For the Pati–Salam value, $(\mathbf 1,\mathbf 2,\mathbf 2)$ meets $(\mathbf 4,\mathbf 2,\mathbf 1)\otimes(\mathbf 4,\mathbf 1,\mathbf 2)$ once. For the left–right value, $B-L$ commutes with the algebra. The equal moduli at the left–right stage: the bidoublet occurs once in each sector's Hom-space, and a real bidoublet satisfies $\tilde\Phi=\Phi$, so only one coupling per sector exists. (c), (d) Clifford multiplication by a unit vector is an isometry, and the dressings are unitary and preserve every component. The isotropic vectors are $\gamma_{10}(1\pm\tau)$ on $V_L$. (e) The formula for $\Gamma_v$ is the colour-invariant family of T-64 with $iL_{e_O}=-\tau$ on $\mathbb C^7$. The scan uses the closed-form sector minimum of T-64(e). On the rank-4 branch $s=b=\tfrac14-\mu^2/(384\kappa)$ and $a=1-3s$, so $(a/2)/b\ge\tfrac12$. $\blacksquare$

Witnesses: test_up_and_down_are_where_the_hilbert_unit_meets_the_clock, test_clifford_yukawas_split_up_from_down_only_through_tau_r, test_clock_phase_and_gap_vacuum_dressings_do_not_fit_the_masses, test_the_data_ask_for_an_up_projector_at_one_percent.

Routes tried and what they give. (i) One Clifford multiplication: $m_t=m_b=m_\tau$ — fails by a factor of 68. (ii) The Page–Wootters clock phase: phases only (d). (iii) The associator vacuum of T-331/T-64: a real direction, and its populations give $m_\tau\gtrsim m_t/2$ (e). (iv) Two components of the bidoublet with independent vacua: they split, but $\tan\beta$ is free and a second doublet is needed (c). (v) The $\overline{\mathbf{126}}$/$\mathbf{120}$ channels: they give $B-L$ dressing and mixing, not the up–down split, which in every channel with one real doublet comes from $\tau_R$ alone (b). No route fixes $m_t/m_b$.

Reconciliation with T-296 and with (GC). The real colour-free plane is exactly one doublet. A scalar needs no complexification, unlike the Weyl field of Theorem 2.6(a), so "one doublet" (T-296) is what the Clifford frame gives when the Higgs field is real. With one doublet the split must come from $\tau_R$ — (UP), or any $\beta\ne0$ — and the two-doublet route (c) contradicts T-296. T-296 stays [H] with this new basis. Under (GC) statement (b) holds generation by generation, and the flavour matrices carry the family index. With the family $\mathbb Z_3$ exact, T-328(d) makes $V_{\mathrm{CKM}}$ a permutation matrix; (UP) does not change this. The Fano selection rule "only the generation on the Higgs line couples at tree level" (Yukawa hierarchy §2) belongs to the axis identification $H\sim\gamma_{EU}$, which is [H].

1.7 The hypothesis (UP) is holomorphy, and its exact form is refuted (T-332, continued)

Status: Theorem 1.7 (h), (i), (k) are [T] as mathematics and [C at (Cl)] in UHM; (j) is numerical; the exact hypothesis (UP) is refuted [✗]; its leading-order form, with a breaking of about 3%, stays [H]

The hypothesis (UP) of §1.6(f) says that the tree-level Yukawa coupling is the projection onto the clock-aligned complex structure $i=L_{e_O}$. This section tries to derive it from UHM, finds what it is equivalent to, and shows that in its exact form it cannot hold. Registry row T-332, items (h)–(k); checks in website/scripts/check_core_numbers.py.

Theorem 1.7 (T-332(h)–(k)).

(h) (UP) is holomorphy in one complex doublet [T]. Hypercharge acts on the colour-free plane $P$ as $\tfrac12 j$ with $j^2=-1$, and on maps $V_L\to V_R$ $\tau_R\,\gamma(h)=\omega\,\gamma(jh)\qquad(h\in P).$ Hence $\tfrac12(1\pm\tau_R)\gamma(h)=\gamma(\pi_\pm h)$, where $\pi_\pm=\tfrac12(1\pm\omega j)$ act on $P_{\mathbb C}=P\otimes\mathbb C_\omega$. Each $\pi_\pm$ has real rank 4, one doublet. So (UP) says that the coupling depends on the Higgs field only through $\pi_+h$, the doublet of hypercharge $-\tfrac12$ (the $\tilde H$ of the Standard Model), and does so $\omega$-linearly. The coupling is holomorphic in one complex doublet. This agrees with T-296, since $P_+$ is one doublet, not two. It is the holomorphic alternative to the real reading of §1.6(c). In Standard-Model language (UP) is $y_u\,Q\tilde Hu^c$ with no $QHd^c$ and no $LHe^c$.

(i) The exact form keeps the charged leptons massless [T]. Take one generation, the six fields $Q,L,u^c,d^c,\nu^c,e^c$, and the doublet. Which phase rotations that commute with $\mathfrak g_{\mathrm{SM}}$ keep the coupling $(\alpha+\beta\tau_R)\gamma(h)$? For $\lvert\beta\rvert\neq\lvert\alpha\rvert$, with or without a $(B-L)$ dressing, there are three: hypercharge, $B$ and $L$. Each has zero colour anomaly. For $\beta=\alpha$ — exact (UP) — there are five. The two new ones are the phase of $d^c$, with colour anomaly $\tfrac12$ per generation, and the phase of $e^c$, with neither a colour nor an $\mathrm{SU}(2)_L$ anomaly. The phase of $e^c$ is then an exact symmetry of every $\mathfrak g_{\mathrm{SM}}$ gauge theory whose only chirality-flipping coupling is the (UP) Yukawa. Its only anomaly is with hypercharge, and an abelian anomaly has no instantons. It forbids masses for $e$, $\mu$ and $\tau$ at every order and non-perturbatively. The phase of $d^c$ forbids $m_d$, $m_s$ and $m_b$ at every order of perturbation theory; only QCD instantons break it. So exact (UP) is refuted by $m_\tau=1.777$ GeV [✗]. The pattern "tree-level $y_b=y_\tau=0$, radiative $b$ and $\tau$ masses" is impossible in the Clifford content: loops of the gauge bosons, of the Higgs and of the up-type coupling keep both phases.

(j) The size of the breaking [numbers]. Write $\varepsilon:=1-\beta/\alpha$. Then $y_b/y_t=\varepsilon/(2-\varepsilon)$ at the scale where the coupling is set. One-loop Standard-Model running with the inputs of §1.6(f) gives $\varepsilon=0.0357$ at $M_Z$, $0.0292$ at $10^{14}$ GeV and $0.0288$ at $2\times10^{16}$ GeV. A breaking without $(B-L)$ dressing gives $y_b=y_\tau$ where it is set. The ratio $y_b/y_\tau$ falls from $1.73$ at $M_Z$ to $0.655$ at $2\times10^{16}$ GeV and passes $1$ at about $6.3\times10^{6}$ GeV. So $b$–$\tau$ equality holds only near $6\times10^{6}$ GeV, ten orders below unification. Set at $2\times10^{16}$ GeV, the down-type breaking needs a dressing $p+q(B-L)$ with $q/p=-0.349$. In $\mathrm{SO}(10)$ language this is a $\overline{\mathbf{126}}$ admixture of $-0.116$ relative to the $\mathbf{10}$. These are one-loop numbers. Nothing found in UHM fixes either $\varepsilon$ or $q/p$.

(k) The routes to a derivation, and where each ends [T for the statements]. (1) The Higgs direction in the real plane $\{iL_{e_O},\gamma_{10}\}$. A real vacuum gives equal moduli (§1.6(c)), not a projection. (2) The Higgs as the isotropic vector $\gamma_{10}+\omega\,iL_{e_O}$. By (h) this is (UP) exactly, and (i) refutes it. (3) The Gap vacuum, $\mathrm{Im}\,\Gamma_v\propto L_{e_O}$ (T-331, T-64). On 94 of the 99 points of the Gap phase scanned in §1.6(e), the vacuum lies on the rank-4 branch: $c=0$, so one component of the quark doublet is not populated at all. Read as population weights, this is an exact quark projection. Which component is empty depends on the sign of $\mathrm{Im}\,\Gamma_v$. The two signs are degenerate vacua exchanged by PT, since the potential is PT-even (T-331). On the other 5 points the smaller weight is up to $0.75$ of the larger. The same reading gives equal $\nu$ and $e$ weights and a lepton-to-quark ratio of at least $0.46$ (§1.6(e)), so it fails for leptons. In its exact form (i) refutes it as well. (4) The self-model or the regenerator coupling only to the clock-aligned part. The clock-aligned part of $\mathcal S_{\mathbb C}$ is $\tau=+1$ (§1.6(a)), so this is the exact projection again, refuted by (i). (5) The Page–Wootters clock as the source of the Hilbert unit. It fixes the sign of $i$ through $H_O\ge0$, and with it which fields are up-type. It does not supply a coupling: by §1.6(d) the clock phase moves only phases. No route gives (UP) as a theorem, and every route that gives it exactly is refuted by (i). What survives is the leading-order statement that the up-type coupling dominates, by $2/\varepsilon\approx68$ at unification. That restates the data and stays [H].

Proof. (h) $j$ is $2\,\mathrm{ad}(Y)$ on the plane, expanded in the Clifford vectors. The identity is checked on the four basis vectors of $P$, and $\pi_\pm$ are idempotents of trace 4 on $P_{\mathbb C}\cong\mathbb R^8$. (i) The phases are $\omega P_f$, with $P_f$ the projector onto the field $f$. The conditions $[X,M(h)]=q_H\,M(jh)$ form a linear system in seven unknowns, the six field charges and $q_H$. Its null space is computed for $\beta/\alpha=0.971$ and $0.5$, for a coupling with $(B-L)$ dressing, and for $\beta=\pm\alpha$. The colour anomaly of $X$ is $\sum q\,T(R)$ over the coloured fields, read from the charge operator $-\omega X$. A symmetry without anomaly under a non-abelian gauge group is not broken by instantons. The hypercharge anomaly $\propto F\tilde F$ integrates to zero on finite-action configurations. (j) The running is that of §1.6(f). The point $y_b=y_\tau$ is found by root finding. The dressing solves $(p+q/3)/(p-q)=y_b/y_\tau$ with $B-L=1/3$ and $-1$. (k)(3) uses the closed-form sector minimum of T-64(e). $\blacksquare$

Witnesses: test_up_projection_is_holomorphy_in_one_complex_doublet, test_an_exact_up_projection_leaves_the_tau_massless_to_all_orders, test_b_tau_and_the_size_of_the_up_projector_breaking.

What this changes. In T-332(f) the exact (UP) is refuted [✗], and the leading-order statement stays [H] as a description of the data. The loop mechanisms for $m_b$ on the Fano selection-rule page (§12.4) and in Yukawa hierarchy §7.3 start from $y_b^{(\text{tree})}=0$ and generate $y_b$ through the retracted cubic $V_3$. The corrected $G_2$-invariant potential has no such vertex, and by (i) the Clifford content cannot generate $y_b$ from $y_b^{(\text{tree})}=0$. The neutrino relation $y_\nu^D=y_t$ uses only the leading order and is unchanged. With all down-type quarks exactly massless, $\bar\theta$ would be unphysical — the massless-quark solution of strong CP. The data exclude that as well; see confinement §3.1b.

---

2. Higgs mechanism from Gap-condensation

Theorem 2.1 (Higgs mechanism from Gap-condensation)

[T] Theorem

The mechanism of electroweak breaking via $\text{Gap}(E,U) \to 0$ is a consequence of the uniqueness of the minimum of $V_{\text{Gap}}$ in the $\bar{3}$-sector: $\varepsilon_{\bar{3}\bar{3}} \approx 10^{-17}$ is determined uniquely from positive definiteness of the Hessian (theorem on the unique vacuum [T]).

Theorem. Spontaneous electroweak symmetry breaking arises from Gap-condensation in the $\bar{3}$-to-$\bar{3}$ sector:

(a) The Higgs field is identified with the $E$-$U$ coherence:

$H \sim \gamma_{EU} = |\gamma_{EU}| e^{i\theta_{EU}}$

(b) VEV (vacuum expectation value):

$\langle H \rangle = \langle |\gamma_{EU}| \rangle e^{i\langle\theta_{EU}\rangle} \neq 0$

Nonzero VEV breaks $SU(2)_L \times U(1)_Y \to U(1)_\text{EM}$:

• $SU(2)_L$: 3 generators → 2 broken ($W^+$, $W^-$) + 1 linear combination broken ($Z$)
• $U(1)_Y$: 1 generator
• $U(1)_\text{EM}$ = diagonal subgroup (photon) — unbroken

(c) Mass of the $W$-boson:

$M_W = \frac{g}{2} v, \quad v = \langle |\gamma_{EU}| \rangle \cdot \mu_\text{phys}$

where $g$ is the electroweak coupling constant, $\mu_\text{phys} = \mu \cdot \omega_0$.

2.1 Potential in the E-U channel

The potential $V_\text{Gap}$ projects onto the $E$-$U$ channel:

$V_{EU}(\gamma_{EU}) = \mu^2 |\gamma_{EU}|^2 + \lambda_4 |\gamma_{EU}|^4 + \lambda_3 \bar{A} |\gamma_{EU}|^3 \cos(\text{phase})$

At $\mu^2 < 0$ (low-temperature regime): minimum at $|\gamma_{EU}| = v \neq 0$. This is the standard Higgs mechanism applied to the Gap potential. Higgs mass = second derivative of $V_{EU}$ at the minimum.

note

Status of parameter $\lambda_3$ [T]

The parameter $\lambda_3 = 2\mu^2/(3|\bar{\gamma}|) \approx 74$ is a geometric coefficient of the spectral action (T-74 [T]), not a perturbative coupling constant. Physical observables are defined non-perturbatively via the self-consistent vacuum $\theta^*$ (T-79 [C at (SV)]). UV-finiteness (T-66: field-space [T], order-by-order [C]) ensures structural correctness. Loop estimates are approximations to $\theta^*$, giving the right order of magnitude (error $\lesssim \times 5$). For details — see Yukawa Hierarchy.

⚠ C7: $\lambda_3 \approx 74 \gg 4\pi$ — non-perturbative regime. All loop computations with $\lambda_3$ are formally unreliable and downgraded to [H]. See warning.

2.2 Origin of $M_H \approx 125$ GeV from Gap-condensation [C]

[C] Conditional

The parameter $\lambda_4$ is determined from the Chamseddine–Connes spectral action with RG correction (see theorem on Higgs quartic [C]). Conditionality: free parameter $f_0$ in the spectral action. The octonionic correction from $V_3$ additionally modifies $M_H$.

Progress: from fitting to computation

In early versions the parameter $\lambda_4 \approx 0.13$ was adjusted from the condition $M_H \approx 125$ GeV. The spectral action (theorem on Higgs quartic [C]) determines $\lambda_4$ through the spectrum of the finite Dirac operator $D_{\text{int}}$. The remaining free degree is the parameter $f_0$, fixed by calibration to $M_H^{\text{exp}}$.

In the Standard Model the Higgs mass $M_H \approx 125$ GeV is a free parameter, fixed experimentally. In UHM the parameter $\lambda_4$ is determined by the spectral action through the spectrum $D_{\text{int}}$ (theorem on Higgs quartic [C]), and the Higgs mass arises from the structure of the Gap potential:

(a) The Higgs mass is determined by the curvature of $V_{EU}$ at the minimum:

$M_H^2 = \frac{\partial^2 V_{EU}}{\partial |\gamma_{EU}|^2}\bigg|_{v} = 2\lambda_4 v^2 + \frac{3\lambda_3^2 \bar{A}^2}{4\mu^2}$

(b) The first term, $2\lambda_4 v^2$, is the standard contribution from the quartic potential $V_4$. At $v = 246$ GeV and $\lambda_4 \approx 0.13$ we get $\sqrt{2\lambda_4} \cdot v \approx 125$ GeV — coincidence with SM.

(c) The second term, $\delta M_H^2 = 3\lambda_3^2 \bar{A}^2 / (4\mu^2)$, is the octonionic correction from the cubic potential $V_3$. It is absent in the SM and is a direct consequence of the $\mathbb{O}$-structure.

(d) Numerical estimate of the correction (at typical values of Gap parameters):

$\delta M_H^2 \approx \frac{3 \cdot (73.8)^2 \cdot (0.047)^2}{4 \cdot 16.6} \approx 0.54 \; (\text{in Gap units})$

This correction is small compared to the main term, but is nonzero and gives rise to a falsifiable deviation from SM (see section 6). (The stated inputs give $0.54$; $\approx 5.5$ would require $\bar A\approx0.15$, the confinement-sector coherence, rather than the average $0.047$.)

(e) Mechanism for fixing $\lambda_4$: the Chamseddine–Connes spectral action determines $\lambda_4$ via the coefficient $a_4$ and the spectrum $D_{\text{int}}$ (theorem on Higgs quartic [C]). RG evolution from the cutoff scale $\Lambda$ to $v_{\text{EW}}$ brings $\lambda_4(\Lambda) \approx 0.20$ to the observed $\lambda_4(v) \approx 0.13$ (Shaposhnikov–Wetterich result 2010). The remaining free parameter $f_0$ in the spectral action is fixed by calibration. Once it is determined from other observables, $M_H$ will become a full prediction of the theory.

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3. Gap(E,U) → 0: electroweak symmetry breaking

3.1 Connection of Gap(E,U) to particle quantum numbers

$\text{Gap}(E,U)$ defines the weak isospin of elementary fermions:

• $\text{Gap}(E,U) = 0$ → doublet of $SU(2)_L$
• $\text{Gap}(E,U) \neq 0$ → singlet of $SU(2)_L$

3.2 Fermionic representations from Γ-configurations

Theorem 3.1 (Quarks and leptons as Gap-configurations) [C]

[C] Conditional

The identification of fermions with Gap-configurations is conditional on the correctness of the identification of SM quantum numbers with Gap structure (gauge correspondence hypothesis).

Theorem. Elementary fermions are identified with degenerate ($R \to 0$) configurations $\Gamma$, classified by $SU(3)_C \times SU(2)_L \times U(1)_Y$ quantum numbers:

(a) Left quark doublet $Q_L = (u_L, d_L)$:

$\Gamma_{Q_L}: \quad \text{Gap}(A,L) = \text{Gap}(S,E) = 0 \; (\text{color bonds}), \quad \text{Gap}(E,U) = 0 \; (\text{weak isospin})$

Quantum numbers: $(3, 2)_{1/6}$

(b) Right $u$-quark $u_R$:

$\Gamma_{u_R}: \quad \text{Gap}(A,L) = \text{Gap}(S,E) = 0, \quad \text{Gap}(E,U) \neq 0$

Quantum numbers: $(3, 1)_{2/3}$

(c) Left lepton doublet $L_L = (\nu_L, e_L)$:

$\Gamma_{L_L}: \quad \text{Gap}(\{A,S,D\}, \{L,E,U\}) = \text{Gap}_\text{max} \; (\text{colorless}), \quad \text{Gap}(E,U) = 0$

Quantum numbers: $(1, 2)_{-1/2}$

(d) Right electron $e_R$:

$\Gamma_{e_R}: \quad \text{Gap}(\{A,S,D\}, \{L,E,U\}) = \text{Gap}_\text{max}, \quad \text{Gap}(E,U) \neq 0$

Quantum numbers: $(1, 1)_{-1}$

3.3 Mechanism: why Gap(E,U) → 0 in the vacuum

Justification. Of the three candidates for zero Gap in the $\bar{3}$-sector ($L$-$E$, $L$-$U$, $E$-$U$), the pair $(E,U)$ is distinguished because:

1. The unique Fano–Higgs line $\{A,E,U\}$ passes through both points.
2. On this line lies $A$ = the generation with a tree-level Yukawa → maximal coupling to the mass mechanism.
3. The vacuum configuration minimizes $V_\text{Gap}$, and the minimum is reached at $\text{Gap}(E,U) \to 0$ in the $\bar{3}$-sector. $\varepsilon_{\bar{3}\bar{3}} \approx 10^{-17}$ from the unique vacuum → Gap(E,U) ≈ 0 — minimum of $V_{\text{Gap}}$ in the $\bar{3}$-sector [T] (see theorem on unique vacuum).

Hypercharge is determined by the total Gap in the $O$-sector:

$Y = \frac{1}{3}\left(\sum_{i \in 3} \text{Gap}(O,i) - \sum_{j \in \bar{3}} \text{Gap}(O,j)\right)$

3.4 Anomaly cancellation

Theorem 3.2 (Anomaly cancellation)

[T] Theorem

Anomaly cancellation for one generation is the standard SM result, automatically satisfied for Gap-configurations.

Theorem. The set of fermionic representations satisfies the gauge anomaly cancellation condition:

$\sum_\text{fermions} Y^3 = 0, \quad \sum_\text{fermions} Y = 0$

Proof. For one generation:

$Q_L(1/6)^3 \times 6 + u_R(2/3)^3 \times 3 + d_R(-1/3)^3 \times 3 + L_L(-1/2)^3 \times 2 + e_R(-1)^3 \times 1 = 0$

Fermionic representations from Gap-configurations form the same structure as one SM generation — anomalies cancel by construction. $\blacksquare$

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4. Higgs mass with octonionic correction

Theorem T-70 (Canonical definition of $f_0$) [C at (SV)]

Corrected 2026-09-25 from [T]: Steps 2, 3 and 5 take the unique vacuum and its five Hessian eigenvalues from the sector form of T-64; the corrected T-64 has a vacuum with none of these sector values (see Gap thermodynamics §14); the formula holds conditional on the sector-vacuum hypothesis (SV).

[C at (SV)] Theorem

In UHM the moment $f_0$ of the spectral action is uniquely determined through the vacuum effective action of the Gap theory on $(S^1)^{21}$:

$f_0 \Lambda^4 = \frac{1}{7}\left[V_{\mathrm{Gap}}^{\min} + \frac{1}{2}\zeta'_{H_{\mathrm{Gap}}}(0)\right]$

where $V_{\mathrm{Gap}}^{\min}$ is the potential value at the vacuum minimum (T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV))), and $\zeta'_{H_{\mathrm{Gap}}}(0)$ is the log-determinant of the Hessian at the vacuum.

Proof.

Step 1 (Field-space finiteness → finite functional integral). The Gap partition function on the compact target $(S^1)^{21}$ is finite — field-space finiteness [T]; full order-by-order UV-finiteness is structural [C] (T-66). Therefore the functional integral $Z = \int [D\theta] \exp(-S_{\mathrm{Gap}}[\theta])$ is finite and well-defined without regularization ambiguity. The quantum effective action $\Gamma_{\mathrm{eff}} = -\ln Z$ is a finite, concrete quantity.

Step 2 (Unique vacuum → loop expansion). From T-61, T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)): the potential $V_{\mathrm{Gap}}$ has a unique global minimum with positive definite Hessian $H_{\mathrm{Gap}}$. Expansion:

$\Gamma_{\mathrm{eff}} = V_{\mathrm{Gap}}^{\min} + \frac{1}{2}\ln\det(H_{\mathrm{Gap}}) + O(\text{two-loop})$

Step 3 (Determinant regularization). Zeta-regularized determinant: $\ln\det(H_{\mathrm{Gap}}) = -\zeta'_{H_{\mathrm{Gap}}}(0)$. From T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)): all eigenvalues $\lambda_i > 0$ (5 positive on the orbit space), so $\zeta'_{H_{\mathrm{Gap}}}(0) = -\sum_{i=1}^{5}\ln\lambda_i$.

Step 4 (Identification with $f_0$). Coefficient $a_0$ of the spectral action: $f_0 \Lambda^4 \cdot 7$ = vacuum energy density of the internal space = $\Gamma_{\mathrm{eff}}$. Therefore:

$f_0 = \frac{\Gamma_{\mathrm{eff}}}{7\Lambda^4} = \frac{1}{7\Lambda^4}\left[V_{\mathrm{Gap}}^{\min} + \frac{1}{2}\zeta'_{H_{\mathrm{Gap}}}(0)\right]$

Step 5 (Uniqueness). All quantities on the right-hand side are uniquely determined: $V_{\mathrm{Gap}}^{\min}$ from T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)), $\zeta'_{H_{\mathrm{Gap}}}(0)$ from a finite sum over 5 eigenvalues, $\Lambda = \omega_0$. $f_0$ is not a free parameter, but a definite function of the vacuum quantities. $\blacksquare$

Numerical estimate [C]

From T-64 [T] (corrected to the $G_2$-invariant potential; its vacuum has no sector values — hypothesis (SV)), Hessian eigenvalues: $\lambda_1 = 18\mu^2$ (confinement), $\lambda_{2,3} = 6\mu^2(1 + O(\varepsilon^2))$ (spatial), $\lambda_{4,5} = 12\mu^2(1 + O(\varepsilon))$ (O-modes). With $\mu^2 \approx \omega_0^2/7$: $f_0 \approx 2.2/\omega_0^4$. Numerical value [C] — depends on exact $\varepsilon_i$.

Theorem (Higgs quartic from spectral action) [C]

[C] Conditional

$\lambda_4$ is determined through the spectrum of the finite Dirac operator $D_{\text{int}}$. The parameter $f_0$ is canonically determined [T] (theorem above); the numerical value of $\lambda_4$ depends on exact sectoral $\varepsilon_i$ [C].

Theorem. The Higgs quartic self-coupling is determined through the coefficient $a_4$ of the spectral action:

$\lambda_4 = \frac{\pi^2}{2f_0\Lambda^4} \cdot \frac{\mathrm{Tr}(D_{\text{int}}^4)}{[\mathrm{Tr}(D_{\text{int}}^2)]^2}$

This is the standard result of Chamseddine–Connes–Marcolli (2007, Thm 11.2) for the NCG Standard Model. Applicability to the UHM triple is verified:

Proof.

Step 1 (Applicability check). The finite spectral triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$ of UHM (theorem T-53 [T]) satisfies the premises of the Chamseddine–Connes–Marcolli theorem:

1. Algebra $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ — corresponds to NCG Standard Model.
2. Dirac operator $D_{\text{int}}$ — finite-dimensional, self-adjoint — corresponds.
3. Higgs field as internal fluctuation $A_{\text{int}}$: $H = A + JAJ^{-1}|_{E\text{-}U}$ — corresponds.

Step 2 (Spectral action). The spectral action $S = \mathrm{Tr}(f(D/\Lambda))$ (see quantum gravity) expands as:

$S = f_0 \Lambda^4 a_0 + f_2 \Lambda^2 a_2 + f_4 a_4 + O(\Lambda^{-2})$

The coefficient $a_4$ contains the term $\mathrm{Tr}(D_{\text{int}}^4)$, generating the quartic Higgs potential.

Step 3 (Computation). From sectoral values (hypothesis (SV) [H]; T-61 restated):

$\mathrm{Tr}(D_{\text{int}}^2) \approx 6\omega_0^2\varepsilon_0^2, \qquad \mathrm{Tr}(D_{\text{int}}^4) \approx 6\omega_0^4\varepsilon_0^4 + \text{sectoral corrections}$

Step 4 (RG evolution). The bare $\lambda_4(\Lambda)$ is too large. RG running from $\Lambda$ to $v_{\text{EW}}$:

$\lambda_4(v) = \lambda_4(\Lambda) + \frac{1}{16\pi^2}\left(24\lambda_4^2 - 6y_t^4 + \ldots\right) \ln\frac{v}{\Lambda}$

At $y_t \approx 1$ (quasi-IR fixed point [T]): RG brings $\lambda_4$ to the observed $\approx 0.13$ from $\lambda_4(\Lambda) \approx 0.20$ [C] — standard Shaposhnikov–Wetterich result (2010). $\blacksquare$

Status: [C] — $\lambda_4$ determined through spectrum $D_{\text{int}}$ + RG. Parameter $f_0$ is canonically determined [C at (SV)] (T-70). The conditionality [C] remains only for the numerical value — depends on exact sectoral $\varepsilon_i$.

Cross-references

• Spectral triple: Theorem (UHM Spectral Triple) — finite triple $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$; its KO-dimension-6 claim is 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)
• Spectral action: Quantum Gravity — $S = \mathrm{Tr}(f(D_A/\Lambda))$, Einstein equations [T]
• Unique vacuum: T-61 — sectoral values $\varepsilon$

---

Theorem 4.1 (Higgs mass) [C]

[C] Conditional

The formula for the Higgs mass contains $\lambda_4$, determined from the spectral action (theorem on Higgs quartic [C]), and the octonionic correction from $V_3$. Parameter $f_0$ is canonically determined [C at (SV)] (T-70); conditionality [C] — only numerical value through $\varepsilon_i$.

Theorem. The Higgs mass is determined as the second derivative of the potential $V_{EU}$ at the minimum:

(a) Formula:

$M_H^2 = 2\lambda_4 v^2 + \frac{3\lambda_3^2 \bar{A}^2}{4\mu^2}$

First term — standard (from $V_4$). Second — octonionic correction from $V_3$.

Proof. The potential $V_\text{Gap}$ projects onto the $E$-$U$ channel:

$V_{EU}(\gamma_{EU}) = \mu^2 |\gamma_{EU}|^2 + \lambda_4 |\gamma_{EU}|^4 + \lambda_3 \bar{A} |\gamma_{EU}|^3 \cos(\text{phase})$

At $\mu^2 < 0$: minimum at $|\gamma_{EU}| = v \neq 0$.

Higgs mass = second derivative of $V_{EU}$ at the minimum:

$M_H^2 = \frac{\partial^2 V_{EU}}{\partial |\gamma_{EU}|^2}\bigg|_{v} = 2\lambda_4 v^2 + \frac{3\lambda_3^2 \bar{A}^2}{4\mu^2}$

$\blacksquare$

Free parameters

$\lambda_4$ and $f_0$ are two free parameters of the spectral action, not derivable from $\Omega^7$. The prediction of $M_H$ is parametric, not absolute.

4.1 Octonionic correction

Theorem 4.2 (Deviation from SM) [C]

[C] Conditional

The quantitative estimate $\delta\lambda/\lambda_\text{SM} \sim O(10^{-2}\text{--}10^{-3})$ depends on the octonionic parameters of the Gap potential ($\lambda_3$, $\bar{A}$, $\mu$). Parameter $\lambda_4$ is determined from the spectral action [C]; the octonionic correction is an additional contribution.

Theorem. The octonionic structure predicts a deviation from the standard Higgs mass relation:

(a) In SM: $M_H^2 = 2\lambda v^2$ (one parameter $\lambda$).

(b) In UHM: $M_H^2 = 2\lambda_4 v^2 + \delta M_H^2$, where:

$\delta M_H^2 = \frac{3\lambda_3^2 \bar{A}^2}{4\mu^2} \approx \frac{3 \cdot (73.8)^2 \cdot (0.047)^2}{4 \cdot 16.6} \approx 0.54$

(c) Octonionic correction to $\lambda_\text{eff} = \lambda_4 + \delta\lambda$:

$\frac{\delta\lambda}{\lambda_4} = \frac{3\lambda_3^2 \bar{A}^2}{8\lambda_4 \mu^2 v^2}$

(d) Falsifiable prediction: with improved precision in measuring the Higgs triple vertex (HL-LHC, FCC), the effective self-coupling $\lambda_\text{eff}$ differs from the SM value by:

$\frac{\delta\lambda}{\lambda_\text{SM}} \sim \frac{\lambda_3^2 \bar{A}^2}{\lambda_4 \mu^2} \sim O(10^{-2} \text{--} 10^{-3})$

— at the percent level, potentially accessible at FCC-hh.

4.2 Origin of the octonionic correction

The octonionic correction from $V_3$ has the following structure:

1. $V_3 = \lambda_3 \sum_{(i,j,k) \notin \text{Fano}} |\gamma_{ij}||\gamma_{jk}||\gamma_{ik}| \sin(\theta_{ij} + \theta_{jk} - \theta_{ik})$ — the cubic octonionic potential.

2. Projection onto the $E$-$U$ channel gives the contribution $\lambda_3 \bar{A} |\gamma_{EU}|^3$, where $\bar{A}$ is the average product of coherence moduli in other channels.

3. This cubic term is absent in the standard model and is a direct consequence of the octonionic ($\mathbb{O}$) structure of the theory.

4. Physically: $V_3$ is responsible for the breaking of $PT$-symmetry (the Gap arrow), and its contribution to the Higgs mass connects the electroweak sector to the global octonionic structure of the dimension space.

4.3 Connection to the Fano selection rule and octonionic structure constants

[T] Theorem

The Yukawa coupling of generation $k_n$ to the Higgs field $\gamma_{EU}$ is proportional to the octonionic structure constant $f_{k_n,E,U}$, which is nonzero if and only if $(k_n,E,U)$ forms a Fano line.

The octonionic correction to the Higgs mass is directly connected to the Fano selection rule. The tree-level Yukawa coupling of generation $k_n$ to the Higgs field is determined by:

$y_n^{(\text{tree})} = g_W \cdot \varepsilon_{k_n, E, U}^{\text{Fano}} \cdot \sin\!\left(\frac{2\pi k_n}{7}\right) \cdot |\gamma_{\text{vac}}^{(EU)}|$

where $\varepsilon_{ijk}^{\text{Fano}} = 1$ if $(i,j,k)$ is a Fano line, and $0$ otherwise. Equivalently: $y_{abc}^{(\text{tree})} \propto f_{abc}$, where $f_{abc}$ is the structure constant of the algebra $\mathbb{O}$, associated with the multiplication table: $e_a e_b = f_{abc} \, e_c + \delta_{ab}$.

For the three generations $k \in \{1, 2, 4\}$:

Generation	$k$	Triple $(k,E,U)$	Fano line?	$f_{k,5,6}$	$y^{(\text{tree})}$
Third (heavy)	$1$	$(1,5,6)$	Yes: $\{A,E,U\}$	$1$	$\neq 0$
Second	$2$	$(2,5,6)$	No	$0$	$= 0$
First	$4$	$(4,5,6)$	No	$0$	$= 0$

Consequence for Higgs mass. The Higgs mass is generated by a loop with a virtual $t$-quark (the only fermion with $y^{(\text{tree})} \neq 0$). Radiative corrections to $M_H^2$ from the top quark:

$\delta M_H^2 \Big|_{\text{top}} = -\frac{3 y_t^2}{8\pi^2} \Lambda^2 + \ldots$

In UHM the role of the UV cutoff $\Lambda$ is played by the scale $\mu_\text{phys}$ — the physical unit of Gap coherence. The octonionic correction from $V_3$ partially compensates the quadratic divergence, since the cubic potential modifies the vacuum structure. This is the germ of a solution to the hierarchy problem from within the Gap formalism.

4.4 Parity breaking from $V_3$ and stability of the chiral vacuum

[C at (SV)] Theorem

Dynamical stability of the chiral vacuum follows conditional on the sector-vacuum hypothesis (SV): Step 2 uses the unique sector vacuum with positive-definite Hessian (hypothesis (SV); the corrected T-64 gives a different vacuum) and Step 3 the barrier of T-69, both conditional on (SV) since 2026-09-25 (earlier stated as proved from [T]-results).

Correction 2026-09-26 (T-166, T-99 corrected): the corrected vacuum selects no chirality

The theorem below is kept for the retracted cubic $V_3$, [C at (SV)]. Its step 1 has no carrier in the $G_2$-invariant potential, which has no PT-odd term (T-331). The corrected vacuum $\Gamma_v$ is invariant under $\Theta_v\otimes1$, an element of $\mathrm{Spin}(10)$ that exchanges $V_L$ and $V_R$ (T-333(a)), so the Gap vacuum does not distinguish left from right. What holds instead, [T] as mathematics and [C at (Cl)] in UHM: chirality is a property of the fermion module. The $\mathbf{16}$ is chiral and forced (T-329), and a left–right flip changes the $\mathrm{SU}(2)_L\times\mathrm U(1)_Y$ representation, so it needs a Yukawa mass insertion — no vacuum barrier is involved.

The cubic potential $V_3$ (and the associated orientational $V_\varphi$-contribution) ensures dynamical stability of chiral distinction in the $E$-$U$ channel:

(a) In the $\bar{3}$-sector $V_\varphi$ takes the form:

$V_\varphi^{(\bar{3})} = \lambda_\varphi \cdot \varphi_{LEU} \cdot |\gamma_{LE}||\gamma_{EU}||\gamma_{LU}| \cdot \sin(\theta_{LE} + \theta_{EU} - \theta_{LU})$

(b) $PT$-property: $V_\varphi \to -V_\varphi$ under $PT$-transformation ($\theta \to -\theta$). This creates an asymmetry of the minimum of $V_\text{Gap}$ in the $E$-$U$ channel.

(c) Energy difference between the left ($\text{Gap}(E,U) = 0$) and right ($\text{Gap}(E,U) \neq 0$) fermionic vacua:

$\Delta V = V_\varphi^{(\pi)} - V_\varphi^{(0)} = 2\lambda_\varphi |\gamma_{LE}||\gamma_{LU}| \cdot |\gamma_{EU}|$

(d) Without $V_3$, chirality would be unstable to radiative corrections. The $PT$-odd potential prevents relaxation of a left-handed fermion into a right-handed one, ensuring the observed parity violation in weak interactions.

Proof:

Step 1. $V_3$ is the unique $PT$-odd term in $V_{\mathrm{Gap}}$ [T] (T-99, step 2). It distinguishes chiral vacua: $\theta = 0$ and $\theta = \pi$ give different signs of the cubic combination $\sin(\theta_{ij} + \theta_{jk} - \theta_{ik})$.

Step 2. The vacuum of $V_{\mathrm{Gap}}$ is unique with positive definite Hessian — hypothesis (SV) [H] (T-64, corrected to the $G_2$-invariant potential, gives a vacuum unique up to $G_2$ — [T] for every $\kappa > 0$ off the transition lines — but not the sector one). No flat directions → the chiral minimum is non-degenerate.

Step 3. Topological barrier [C at (SV)] (T-69): $\Delta V \geq 6\mu^2 > 0$ prevents tunneling between chiral vacua.

Conclusion. $V_3$ selects the chiral vacuum (step 1), the Hessian ensures local stability (step 2), the topological barrier — global protection from tunneling (step 3). $\blacksquare$

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5. Connection to SM gauge structure

5.1 Gauge boson mass hierarchy

Theorem 5.1 (Mass hierarchy from Gap hierarchy) [T]

[T] Theorem

The gauge mass hierarchy follows from the Fano–electroweak (FE) construction [T]: uniqueness of the pair $(E,U)$ is proved from $\kappa_0$ [T] — see uniqueness theorem. The identification of Gap sectors with SM gauge groups is determined uniquely.

Theorem. The scale hierarchy of gauge bosons is determined by the Gap hierarchy of the vacuum:

(a) Massless ($\text{Gap} = 0$ in the corresponding sector):

• Gluons: $\text{Gap} = 0$ in $3$-to-$\bar{3}$ → confinement (nonlinear dynamics as $\text{Gap} \to 0$)
• Photon: $\text{Gap} = 0$ for the diagonal $U(1)_\text{EM}$ combination

(b) Electroweak scale ($\text{Gap} \sim 10^{-17}$ from Planck):

• $W^\pm$, $Z$: $\text{Gap}(E,U) \sim v/M_\text{Planck} \sim 10^{-17}$

(c) Planck scale:

• $G_2$-extra: $\text{Gap} \sim 1$ → mass $\sim M_\text{Planck}$

Corollary. The mass hierarchy $M_\gamma = 0 \ll M_W \ll M_{G_2}$ follows from the Gap hierarchy $0 \ll 10^{-17} \ll 1$ in the corresponding coherence sectors.

Note

In early versions this section included the GUT scale with $X$, $Y$ leptoquarks ($M_X \sim v_\text{GUT}$), based on the embedding $SU(5) \subset SU(6)$ from the 42D Page–Wootters extension. Within the Fano–electroweak (FE) construction the electroweak sector is derived directly from the Fano geometry of the $\bar{3}$-sector without invoking $SU(5)$-GUT, and the prediction of $X$, $Y$-leptoquarks is not a consequence of the (FE)-framework. The question of the existence of a GUT scale remains open.

5.2 Complete table of gauge fields

Field	Group	Number	Mass	Gap source	Status
Gluons $g$	$SU(3)_C$	8	0 (confinement)	$\text{Gap}_{3\to\bar{3}} \approx 0$	[T]
$W^\pm$, $Z$	$SU(2)_L$	3	$M_W$, $M_Z$	$\text{Gap}(E,U) \sim 10^{-17}$	[T]
Photon $\gamma$	$U(1)_\text{EM}$	1	0	Diagonal $U(1)$	[T]
$G_2$-extra	$G_2/SU(3)$	6	$M_{G_2} \sim \mu_\text{phys}$	$\text{Gap}^{(O)} \sim 1$	[C]

Note on leptoquarks

In the previous version the table included $X$, $Y$-leptoquarks ($SU(5)/\text{SM}$, 12 fields, $M_X \sim v_\text{GUT}$). These particles are specific to the $SU(5)$-GUT embedding and do not follow from the Fano–electroweak (FE) construction. They have been removed from the main table.

5.3 Electroweak sector: Fano–electroweak (FE) construction [T]

Replacement of the former SU(6) derivation

In early versions the electroweak sector was derived from the Page–Wootters extension $\mathcal{H}_\text{total} = \mathbb{C}^7 \otimes \mathbb{C}^6 = \mathbb{C}^{42}$, where the $6D$-factor carried $SU(6)$-symmetry, and via the embedding $SU(5) \subset SU(6)$ (analogue of the Georgi–Glashow model) $SU(2)_L \times U(1)_Y$ was extracted. This approach had a rank problem ($\text{rank}(G_2) = 2 < \text{rank}(SM) = 4$) and led to spurious predictions ($X$, $Y$-leptoquarks).

The Fano–electroweak (FE) construction replaces the $SU(6)/SU(5)$ derivation, extracting the electroweak structure directly from the geometry of the $\bar{3}$-sector of the Fano plane.

In the (FE)-construction the electroweak sector $SU(2)_L \times U(1)_Y$ arises from the structure of the $\bar{3}$-sector $\{L, E, U\}$ of the plane $\mathrm{PG}(2,2)$:

(a) $SU(2)_L$ is identified with the group acting on the doublet $(E, U)$ at $\text{Gap}(E,U) = 0$. The uniqueness of the Higgs line $\{A, E, U\}$ [T] guarantees unambiguity in the choice of the electroweak channel.

(b) $U(1)_Y$ is determined by the total Gap in the $O$-sector (see section 3.3):

$Y = \frac{1}{3}\left(\sum_{i \in 3} \text{Gap}(O,i) - \sum_{j \in \bar{3}} \text{Gap}(O,j)\right)$

(c) $SU(3)_C$ — still from the $G_2$-stabilizer ($G_2 \supset SU(3)$, decomposition $14 \to 8+3+\bar{3}$) [T].

Advantages of (FE) over $SU(6)/SU(5)$:

• Does not require additional structure ($SU(6)$ from 42D)
• Does not generate $X$, $Y$-leptoquarks as a mandatory prediction
• The electroweak sector is tied to the same Fano geometry as the Higgs mechanism
• The rank problem ($\text{rank}(G_2) = 2 < 4 = \text{rank}(SM)$) is resolved: the missing generators are taken from the HS-projection of the $\bar{3}$-sector [T], not from an external $SU(6)$

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6. Falsifiable predictions

6.0 Prohibition of a second Higgs doublet [H]

Corrected 2026-09-25 from [T] to [H]: step (i) takes "$\langle\gamma_{ij}\rangle \neq 0$ only for the $\kappa_0$ pair" from T-64, which never stated it and is now a hypothesis, and the whole argument presupposes the identification $H \sim \gamma_{EU}$ of Theorem 1.0, now a hypothesis with a colour-breaking obstruction. The exclusion of 2HDM spectra is a prediction of that hypothesis, not a theorem.

New basis (T-332, §1.6). In the Clifford frame the colour-free plane of $\mathrm{Spin}(10)$ is exactly one real doublet, so a real Higgs field gives one doublet without reference to $\gamma_{EU}$. The price is the up–down split. With one real doublet it must come from the operator $\tau_R$ (the imaginary unit of $\mathcal H$ on $V_R$), that is from a coupling that breaks $\mathrm{SU}(2)_R$. The alternative is the complex bidoublet — two doublets with $m_t/m_b=\tan\beta$ — which this prohibition excludes. The data require the $\tau_R$-coefficient $\beta/\alpha=0.971$ (T-332(f)). T-296 stays [H]. A charged Higgs would now refute the real-plane reading together with it.

[H] Structural prohibition (T-296)

UHM forbids a second Higgs doublet. The categorical uniqueness that selects the pair $(E,U)$ simultaneously excludes every other scalar candidate.

Theorem (no-2HDM). In UHM there is exactly one condensing scalar channel — $\gamma_{EU}$. No second Higgs doublet (and hence no 2HDM spectrum $H^\pm, A^0, H^0$ of the MSSM type) exists.

Proof. (i) Condensation requires the $\kappa_0$-channel: the vacuum theorem T-64 gives $\langle\gamma_{ij}\rangle \neq 0$ only for the pair singled out by $\kappa_0 = \omega_0|\gamma_{OE}||\gamma_{OU}|/\gamma_{OO}$, whose morphism content is exactly $\mathrm{Hom}(O,E)\cdot\mathrm{Hom}(O,U)$ (T-42a). (ii) The only other $\bar 3$-pairs are $(L,E)$ and $(L,U)$; neither enters $\kappa_0$ ($\mathrm{Hom}(O,L)$ is absent from it), so neither acquires a VEV. (iii) By incidence ($\lambda=1$) the pair $(L,U)$ lies on the single line $\{D,L,U\}$, already exhausted as the Color-U Yukawa channel of the 2nd generation (selection rules) — it is a mass channel, not a scalar sector. $\blacksquare$

Falsification. Discovery of a charged Higgs $H^\pm$ or of a second CP-even/odd neutral scalar of doublet type at the LHC/HL-LHC would refute the categorical uniqueness of $(E,U)$ — i.e. strike at $\kappa_0$ itself, not at a peripheral fit. UHM stakes the entire class of 2HDM/MSSM Higgs sectors on this.

6.1 Deviation of the Higgs triple vertex [C]

[C] Conditional

The quantitative prediction depends on the octonionic parameters of the Gap theory ($\lambda_3$, $\bar{A}$) and the spectral action parameter $f_0$.

Prediction. The effective Higgs self-coupling differs from the SM value:

$\frac{\delta\lambda}{\lambda_\text{SM}} \sim O(10^{-2} \text{--} 10^{-3})$

Test: HL-LHC (precision $\sim 50\%$ on triple vertex), FCC-hh (precision $\sim 5\%$).

6.2 Connection of Higgs mass to octonionic structure [C]

In the SM the Higgs mass $m_H \approx 125$ GeV is a free parameter. In UHM:

$m_H^2 = 2\lambda_4 v^2 + \delta m_H^2(\lambda_3, \bar{A}, \mu)$

The first term is determined by the spectral action (theorem on Higgs quartic [C]). The octonionic correction $\delta m_H^2$ connects the Higgs mass to the octonionic potential parameters. When $f_0$ is fixed from other observables (quark masses, CKM elements), the Higgs mass becomes computable — this is a potentially powerful prediction.

6.3 Mass hierarchy problem [H]

Corollary. The mass hierarchy problem ($M_W / M_\text{Planck} \sim 10^{-17}$) reduces to the question: why does the Gap-vacuum have such different values in different sectors? Answer: sectoral values $\varepsilon_X$ are determined by the unique minimum of $V_{\text{Gap}}$ (theorem on unique vacuum [T]).

Hypothetical solution via RG evolution: at the Planck scale all $\text{Gap} \sim O(1)$ (democratic initial condition). RG flow from Planck to IR: different sectors flow with different anomalous dimensions:

Sector	Anomalous dimension	Gap at IR scale
$3$-to-$\bar{3}$ (color)	$\Delta_{3\bar{3}} = 0$ (marginal)	$\sim 0$ (confinement)
$\bar{3}$-to-$\bar{3}$ (EW)	$\Delta_{\bar{3}\bar{3}} = \Delta_3 = 5/42$	$\sim 10^{-17}$ (EW scale)
$O$-to-$3$ (gravity)	$\Delta_{O3} \gg 1$ (IR-relevant)	$\sim 1$ (Planck scale)

The difference in anomalous dimensions is determined by Fano combinatorics: the number of Fano lines passing through a pair $(i,j)$ affects $\Delta_{ij}$.

6.4 Dynamical dark energy [T at the O-channel; P for the non-O residue]

The drift of the dark-energy equation of state is now derived at the state level: $1 + w_{\text{eff}} = -\tfrac{2}{3}\,d\ln\mathcal{G}_O/d\ln a$ with a positive floor and a three-branch shape classification — the Λ-drift law, T-254/T-255. The Higgs-sector channels considered here ($\bar{3}$-to-$\bar{3}$ — non-O) contribute to that drift only through the sector-suppressed correction $O(\mathcal{G}_{\text{non-O}}/\mathcal{G}_O) \sim 10^{-3}$ (sector Gap bound [T]) — a subdominant channel. The earlier ansatz is kept for the record:

$w(z = 0) = -1 + \delta w, \quad \delta w = \frac{\kappa \cdot \langle|\gamma|^2\rangle}{V_\text{Gap}} \sim \frac{\kappa \cdot \epsilon^2}{\mu^2 \text{Gap}^2}$

What remains open [P] is the RG-scale ↔ $H(t)$ bridge for the non-O channels; the O-channel drift needs no such bridge — it passes through the M3 identification $a = 1/\mathrm{Gap}_s$ directly. The old numerical ansatz $w_a \sim -10^{-2}$ is superseded by the T-255 shape constraints (the DESI quadrant requires the oscillatory branch with a genuine $-1$-crossing).

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6.5 Chirality tunneling rate [C at (SV)]

Theorem T-185b [C at (SV)]: Chirality stability prediction

The chiral vacuum is stable against tunneling with a lifetime vastly exceeding the age of the universe:

$\tau_{\text{chiral}} \sim \frac{1}{\mu} \exp\!\left(\frac{B}{\hbar}\right) \gg \tau_{\text{universe}} \approx 4.4 \times 10^{17}\;\text{s}$

where $B \geq \pi\sqrt{12}\,\mu \approx 10.88\,\mu$ is the WKB bounce action through the barrier $\Delta V \geq 6\mu^2$ (T-69 [C at (SV)]).

Derivation. The WKB tunneling rate between the chiral vacua $\theta = 0$ and $\theta = \pi$:

$\Gamma_{\text{tunnel}} = \mu \cdot \exp\!\left(-\frac{B}{\hbar}\right), \quad B = \int_0^{\pi} \sqrt{2\Delta V(\theta)}\,d\theta \geq \pi\sqrt{2 \cdot 6\mu^2} = \pi\sqrt{12}\,\mu$

In physical units with $\mu \sim M_{\text{Planck}}$: the exponent $e^{10.88 \cdot M_{\text{Planck}} / T_{\text{eff}}}$ is astronomically large for any $T_{\text{eff}} \ll M_{\text{Planck}}$.

Falsifiable prediction. Observation of spontaneous chirality flipping (a right-handed neutrino appearing from a left-handed one without a mass insertion) at any sub-Planckian energy would falsify the topological protection theorem T-69 [C at (SV)] and the cubic potential $V_3$ (T-99 [T]). (Since 2026-09-26: step 2 of T-99 is a property of the retracted $V_3$ only, and T-99's conclusion is [✗]; what forbids an L→R flip without a mass insertion in the corrected frame is gauge invariance of the forced $\mathbf{16}$ — box in §4.4.)

Status. [C at (SV)] — follows from T-69 [C at (SV)] (topological barrier), the hypothesis (SV) [H] (unique vacuum with positive Hessian; the corrected T-64 does not give its values), and step 2 of T-99 ($V_3$ is the unique $PT$-odd term — true of the retracted cubic only; the corrected potential is PT-even, T-331); corrected from [T] on 2026-09-25.

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7. Can UHM predict the Higgs mass?

7.1 Problem statement

Experimental value: $M_H^{\text{exp}} = 125.20 \pm 0.11$ GeV (PDG 2024). In the Standard Model $M_H$ is a free parameter. In Chamseddine–Connes noncommutative geometry (NCG) the Higgs mass is computed from the spectral triple. Question: can UHM do the same?

7.2 Derivation chain for $M_H$ in UHM

The full chain from axioms to $M_H$ consists of five links:

Link	Statement	Status	Dependency
(1) Spectral triple	$(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$ exists [T] (T-53); "KO-dim = 6" is 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)	[T] (T-53), without the real structure	Axioms
(2) Spectral action	$S = \mathrm{Tr}(f(D_A/\Lambda))$ expands in Seeley–DeWitt series	[T] (T-65)	(1)
(3) $f_0$ canonically determined	$f_0 = \Gamma_{\text{eff}} / (7\Lambda^4)$ through Gap theory vacuum	[C at (SV)] (T-70)	(2) + sector vacuum (SV) [H]
(4) $\lambda_4$ from $D_{\text{int}}$ + RG	$\lambda_4 = \frac{\pi^2}{2f_0\Lambda^4} \cdot \frac{\mathrm{Tr}(D_{\text{int}}^4)}{[\mathrm{Tr}(D_{\text{int}}^2)]^2}$, RG: $\Lambda \to v_{\text{EW}}$	[C]	(3) + numerical $\varepsilon_i$
(5) $M_H$ from potential	$M_H^2 = 2\lambda_4 v^2 + \delta M_H^2(\lambda_3, \bar{A}, \mu)$	[C]	(4) + octonionic correction

Verdict: [C] — conditional on numerical values of sectoral parameters $\varepsilon_i$ determining the spectrum $D_{\text{int}}$.

7.3 Why $g_4^* = 4\pi^2/63$ is NOT the Higgs quartic

Common error

The Wilson–Fisher fixed point of the Gap theory $g_4^* = 4\pi^2/63 \approx 0.627$ is not the Higgs quartic $\lambda_H$ of the Standard Model. The naive identification gives $M_H = \sqrt{2 g_4^*} \cdot v \approx 275$ GeV — an incorrect result (the printed "$0.063$"/"$87$ GeV" in earlier drafts mis-evaluated $4\pi^2/63$ by a factor of 10; the correct value $0.627$ still fails to reproduce $M_H=125$ GeV, so the conclusion "$g_4^*\neq\lambda_H$" stands).

Distinction:

	Gap quartic $g_4^*$	Higgs quartic $\lambda_H$
Theory	(0+1)D Gap on $(S^1)^{21}$	4D QFT on $M^4$
Number of fields	21 coherences	1 doublet (4 real fields)
Factor in $\beta$	63 (from combinatorics $\binom{21}{2} \cdot 3$)	$\sim 24$ (loop with $W$, $Z$, $t$)
IR value	$4\pi^2/63 \approx 0.627$	$\approx 0.13$ (from $M_H = 125$ GeV)
Origin	Wilson–Fisher RG fixed point of Gap	Spectrum $D_{\text{int}}$ + SM RG running

Connection between them: $g_4^*$ determines the IR value of the quartic coupling of the Gap potential $V_{\text{Gap}}$. The Higgs quartic $\lambda_H$ is determined by the projection of $V_{\text{Gap}}$ onto the $E$-$U$ channel via the spectral action, and then evolves under 4D SM RG equations.

7.4 Comparison with Chamseddine–Connes NCG

In the Chamseddine–Connes–Marcolli (CCM) approach the history of predicting $M_H$ went through three stages:

(a) Tree level (CCM 2007): $M_H = \sqrt{8\lambda_H} \cdot v$ with $\lambda_H$ from $\mathrm{Tr}(D_{\text{int}}^4)/[\mathrm{Tr}(D_{\text{int}}^2)]^2$. With top quark dominance:

$M_H^{(\text{tree})} \approx \frac{M_t}{\sqrt{2}} \approx \frac{173}{\sqrt{2}} \approx 122 \text{ GeV}$

However, without RG correction the exact Chamseddine–Connes formula (2012) gave $\sim 170$ GeV — an incorrect result.

(b) With RG running (Shaposhnikov–Wetterich 2010): RG evolution from $\Lambda_{\text{GUT}}$ to $v_{\text{EW}}$ reduces $\lambda_H(\Lambda) \approx 0.20$ to $\lambda_H(v) \approx 0.13$, giving $M_H \approx 125$ GeV. But this fixes $\Lambda_{\text{GUT}}$, not predicts it.

(c) With scalar field $\sigma$ (Chamseddine–Connes–van Suijlekom 2013): introduction of the $\sigma$-field from internal fluctuations changes the boundary condition at $\Lambda$, leading to $M_H \approx 126$ GeV — the first correct prediction from NCG.

UHM position: the octonionic correction from $V_3$ plays a structurally analogous role to the $\sigma$-field in CCM-2013. The cubic potential $V_3$ modifies the effective Higgs potential, shifting the tree-level value of $M_H$ closer to the experimental value. However, the exact numerical value of the correction depends on vacuum parameters $\varepsilon_i$, which have not yet been computed.

7.5 What is needed for a full prediction

For converting $M_H$ from [C] to [T] one needs:

1. Numerical solution of vacuum equations on $(S^1)^{21}/G_2$: determine exact values of $\varepsilon_i$ for all 5 orbital parameters (task C16 in the status registry).

2. Computation of $f_0$: substitute $\varepsilon_i$ into the canonical formula T-70 and find the numerical value of $f_0$.

3. Computation of $\mathrm{Tr}(D_{\text{int}}^4)$: determine $\lambda_4(\Lambda)$ from the spectrum $D_{\text{int}}$ with known $\varepsilon_i$.

4. SM RG running: evolution $\lambda_4(\Lambda) \to \lambda_4(v_{\text{EW}})$ — standard procedure containing no additional free parameters.

5. Octonionic correction: compute $\delta M_H^2$ from Gap parameters.

All formulas are defined [T]; the task is computational [C]. This is analogous to the situation in lattice QCD, where the formulas are exact, but numerical predictions require computation.

7.6 Final assessment [C]

[C] Conditional

UHM determines the Higgs mass through chain (1)–(5), in which links (1)–(3) have status [T], and links (4)–(5) — status [C] due to incomplete computation of sectoral parameters $\varepsilon_i$. No additional postulates or hypotheses are required: the task is purely computational.

Summary:

• Can UHM in principle predict $M_H$? Yes — the formulas are fully determined.
• Does it predict now? No — requires solving task C16 (numerical computation on $(S^1)^{21}/G_2$).
• Naive $g_4^* \to M_H$: incorrect ($g_4^* \neq \lambda_H$), gives $\sim 275$ GeV.
• Status: [C] — conditional on computation of $\varepsilon_i$.
• Comparison with NCG: UHM reproduces the CCM structure, but adds the octonionic $V_3$-correction, analogous to the $\sigma$-field of Chamseddine–Connes–van Suijlekom.

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Connection to other sections

• Uniqueness of the Higgs line: Foundation of the Fano selection rule → Yukawa Mass Hierarchy
• Three generations: Generation line $\{A,S,L\}$ orthogonal to Higgs line → Three Fermion Generations
• CKM matrix: Mismatch of $Y^u$ and $Y^d$ via conjugate Higgs → CKM Matrix
• Spectral triple: Finite $(A_{\text{int}}, H_{\text{int}}, D_{\text{int}})$ → Spacetime [T]; the former "with KO-dimension 6" is 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)
• Spectral action: $S = \mathrm{Tr}(f(D/\Lambda))$, determines $\lambda_4$ → Quantum Gravity
• Unique vacuum: Sectoral values $\varepsilon$ — hypothesis (SV) [H] (T-61 restated) → Gap Thermodynamics

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Related documents:

• Standard Model from G₂
• Fano Selection Rules
• Yukawa Hierarchy
• Supersymmetry from G₂