Finite spectral windows with an opening tower in seven-dimensional parents

1 Introduction

In the towerless reading of almost-commutative geometry proposed in (“Finite Spectral Triples as Towerless Intervals” 2026), a finite spectral triple is the low-energy window of a parent with an internal geometry, and the labels 9 and 11 are limits of an even interior at finite Fermi constant, where the physical point would lie. In that proposal, the limit towards 11 is the one in which the $W$ and $Z$ become massless and a dimension opens. This note asks what that statement can mean.

It needs a path. On the interval of (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026), the vector masses obey $\sin^2(M_Wl)=\tfrac12\sin^2 2\theta$, so $\theta\to0$ at fixed $l$ makes them vanish with the tower spacing fixed. The criterion we use is a hierarchy along a family indexed by $\varepsilon\to0$, $$m_\varepsilon\ll E_\varepsilon\ll\Delta_\varepsilon\to0,$$(1) with $m_\varepsilon$ the light scale and $\Delta_\varepsilon$ the first discarded level. Then a window at the moving scale $E_\varepsilon$ holds a fixed number of states. The hierarchy alone does not make the count at fixed $E>0$ diverge, so we require that as a separate property, and prove it for each family below. For such a family the two observations have different limits, which reconciles a finite description near the weak scale with a tower at fixed energy.

The results of Sections 2 and 3, and the formulation of (1), follow an audit of this project (see the Provenance). The new content is Section 4 and what rests on it: a window obtained from a local operator on a connected seven-dimensional parent, rather than prescribed.

2 The interval

Let $(H_F,D_*,J_F,\Gamma_F)$ have the signs of KO-dimension six, with $r=\dim H_F$ and an algebra $A_F$ acting through $\pi$, as in (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026). For $0<\alpha<1$ and $l_*>0$, let $D_F=\varepsilon D_*$ and $l_\varepsilon=l_*\varepsilon^{-\alpha}$, and let $D_\varepsilon$ be the derivative on the interval with the boundary conditions that realize $D_F$ exactly (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026). Its signed spectrum is $(n\pi\pm\theta_k)/l$, $n\in\mathbb Z$, with $\theta_k=l\sigma_k$ and $\sigma_k$ the singular values of the block of $D_F$.

This is a direct consequence of the exact formulas (Supplemental Material, part 30): $l_\varepsilon\|D_F\|\to0$, and $\varepsilon\ll\varepsilon^\beta\ll\varepsilon^\alpha$. The finite data remain inputs, the wall data vary by hand, and a fixed Majorana block does not scale as $\varepsilon D_*$, so an application to an electroweak Dirac block needs a reducing sector.

In the five-dimensional Yang-Mills toy of (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026), with universal mass $m_\varepsilon=\varepsilon m_*$ and $\theta_\varepsilon=l_\varepsilon m_\varepsilon\to0$, the light poles are $M_W=l^{-1}\arcsin(\sin2\theta/\sqrt2)$ and $M_Z=l^{-1}\arcsin(2\sin2\theta/\sqrt5)$, so $$M_W^2\simeq2m_\varepsilon^2,\qquad M_Z^2\simeq\tfrac{16}{5}m_\varepsilon^2,$$(3) and the vector poles follow (1). The small-mass coefficients of the finite spectral action, in the family with both members of each doublet at one Dirac mass and no Majorana mass, are $3$ and $24/5$. The leading Taylor quotient of (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026), $M^2=\tfrac12(3-d_{\mathrm{eff}}(s))\,\|[D_F,T]\|^2/\|T\|^2$ with $s=l\Lambda$, connects the two pairs at the level of this coefficient: $d_{\mathrm{eff}}=0$ gives the finite values, and $d_{\mathrm{eff}}=1$ the interval values. It does not identify the spectral-action pole problem with the Yang-Mills toy. Along $\Lambda=E_\varepsilon$ we have $s\to0$, and at fixed $\Lambda$ we have $s\to\infty$; a joint use needs a remainder uniform in $s$. Varying the cutoff of the action is an additional prescription.

A family closer to the geometry below uses localization instead of boundary angles. With a bulk mass $\mu_\varepsilon=\mu_*\varepsilon^\alpha$ and a new length $l^{\mathrm{loc}}_\varepsilon=c_\varepsilon/\mu_\varepsilon$, where $c_\varepsilon=(1-\alpha)\log(1/\varepsilon)+c_0$, the light pair has $m=\mu\,\mathrm{sech}\,\kappa l$ with $\kappa l=c\tanh\kappa l$, so that $$m_\varepsilon\sim2\mu_*e^{-c_0}\varepsilon,\qquad\Delta_\varepsilon\sim\mu_*\varepsilon^\alpha,\qquad\frac{1/\kappa}{l^{\mathrm{loc}}_\varepsilon}\to0.$$(4) The widths and the length both grow, with a shrinking ratio. Writing the profile as $e^{-\kappa x}-re^{\kappa x}$, the opposite-wall coefficient is exactly $r=(m/(\mu+\kappa))^2$, quadratic in the light mass.

3 A cylinder with a prescribed window

Let $A$ be the Dolbeault-Dirac operator of $\mathbb{CP}^3$ twisted by $\mathcal O(1)$, with kernel $\mathbb C^4$ and first nonzero level $\Delta_K$, and let $\mathcal D_r=E\partial_x+ZA/r$ on $I_l\times\mathbb{CP}^3$, with the spectral walls of Supplemental Material, part 24, which prescribe finite data on this $\mathbb C^4$, with gradings of type $(2,2)$, so here $r=4$. Taking $r_\varepsilon=r_*\varepsilon^{-\gamma}$ and the scaling of Theorem 1, the next level is at least $\min\{(\Delta_K/r_*)\varepsilon^\gamma,\,(\pi/l_*)\varepsilon^\alpha-\varepsilon\|D_*\|\}$, so a cutoff $E_*\varepsilon^\beta$ with $\max(\alpha,\gamma)<\beta<1$ isolates the window, and the count at fixed energy grows as $l_\varepsilon r_\varepsilon^6E^7$, by Dirichlet and Neumann bracketing. For $\gamma=\alpha$ all seven directions expand homothetically. So (1) is compatible with seven internal dimensions, for $4+7=11$. The window here is imposed by nonlocal walls.

4 A window from a local operator

Let $X$ be the disc bundle of $\Lambda^2_-T^*S^4$, with $G=Sp(2)$, $K=Sp(1)_a\times Sp(1)_b$, and $H=Sp(1)_a\times U(1)_b$. It has zero section $S^4$ and boundary $\mathbb{CP}^3$, and a smooth function has independent values at the two ends. Let $S=G\times_K(1,\mathbf 2)$, let $E=S^3(S)$ with its pull-back to $X$, and let $J$ be the spin-$\frac32$ action on it. On $X$ take $$T_M=D_X^E+i\Phi_M,\qquad\Phi_M=\epsilon Mf(r)\,n\cdot J,$$(5) with $\widehat D_M$ the doubled operator, $Z=-ic(dr)$, and the local wall condition $Zu=-u$ for $T_M$, $Zv=+v$ for $T_M^*$, which gives a self-adjoint elliptic operator (Supplemental Material, part 28, Theorem 28.9). The sign $\epsilon$ is the one that puts the kernel of the end model of $T_M$ at the zero section. We assume the following: the metric is $dr^2+\pi^*g_{S^4}+\rho(r)^2g_{S^2}$, with $S^4$ of radius $1$ and $\rho,f$ smooth and positive for $r>0$, with $\rho=f=r$ on $r\le2e$, and with $\rho=1$, $f=f_0>0$ on a product collar $l-2a\le r\le l$, which is the Fubini-Study metric on $\mathbb{CP}^3$. The connection on $E$ is $Sp(2)$-invariant, in radial parallel gauge. It is either (a) the pull-back throughout, or (b) the pull-back near $S^4$ and, on the collar, the connection preserving the four line summands $\mathcal O(3),\mathcal O(1),\mathcal O(-1),\mathcal O(-3)$.

The proof (Supplemental Material, part 29) is a partition into three regions, and each estimate holds on a fixed manifold, so uniformity over the representations of $Sp(2)$ needs no separate argument.

The zero section. Sections on a tube are $K$-equivariant maps $G\to\Sigma_4\otimes L^2(\mathbb R^3;\mathbb C^2\otimes\mathbb C^4)$. Put $D_H=i\sum\gamma_ae_a$ and $\hat Q=\gamma_5\sigma\cdot\partial_y+\epsilon My\cdot J$. Then $T_M=T_0+R$, with $T_0=D_H+i\hat Q$ and $|R|\le C|y|$ the O’Neill term. The operator $\hat Q$ acts on the values and is $K$-equivariant, so it commutes with the derivatives $e_a$. With the Clifford signs, this gives exactly $$T_0^*T_0=D_H^2+\hat Q^*\hat Q.$$(6) On one chirality of $\Sigma_4$ the fibre operator has kernel $(1,\mathbf 3)\oplus(1,\mathbf 1)$, and its square is at least of order $M$ on the complement and on the other chirality. The projection onto this $K$-type, on both chiralities, commutes with $D_H$, and on its range $D_H$ is a twisted Dirac operator on $S^4$, with $D_H^2=\mathrm{Cas}_G(V)+\tfrac32-\mathrm{Cas}_K$ (Parthasarathy 1972). This vanishes only on the $\mathbf 4$ and is at least $4$ otherwise. Since (6) also bounds $M^2\||y|u\|^2$, the term $R$ is a small perturbation. The stronger localization theorem of (Maridakis 2021) has a hypothesis that fails for this mass, since $(n\cdot J)^2$ depends on $n$, and it is not needed here; nor is the block analysis of (Bismut and Lebeau 1991).

The bulk. For $w$ supported away from both ends, $\|T_Mw\|^2=\|Dw\|^2+\|\Phi_Mw\|^2+\langle w,i[D,\Phi_M]w\rangle\ge\int(M^2f^2/4-CM)|w|^2$, for every representation.

The wall. On the collar in case (b), with $q=\epsilon n\cdot J$ and $m=Mf_0$, $$\begin{aligned} \|T_Mw\|^2={}&\|(\partial_x+mq)w_+\|^2+\|Aw_+\|^2\\ &+\|(\partial_x-mq)w_-\|^2+\|Aw_-\|^2, \end{aligned}$$(7) where $A$ is the sum of line-twisted Dirac operators. Only the channels $Z=-1,\ q>0$ for $T_M$ and $Z=+1,\ q>0$ for $T_M^*$ have a stable profile, and on these the lower bound comes from $A$. On the Fubini-Study metric $D_k^2=C_2+\tfrac32-\tfrac38k^2$ on the $SU(4)$-type with Casimir $C_2$; the normalization passes the Kirchberg limit $R/3$ for $k=0$ (Kirchberg 1986). So on the stable lines $\mathcal O(3)$ and $\mathcal O(1)$ the kernel is exactly one copy of the $\mathbf 4$, in $Z=+1$, and all other values have $D^2\ge3$, for all representations. In case (a) the line summands mix through a bounded term with overlap $\sqrt3/2$ between the profiles, and a Schur complement of order $1/m$ reduces the wall to an elliptic operator on $\mathbb{CP}^3$, with fibre part $-w(\rho)$, $w(\rho)=2(\rho^2+2)^2/(9+2(\rho^2+2)^2)$, for its kernel in the limit, so $-2/3$ at the Fubini-Study radius. At other radii the theorem holds under one kernel condition.

The transition regions lie in the bulk, where the mass controls $\|u\|$ with a factor $M^{-2}$, so the partition costs nothing at the scale of the gap. The end models fix the expected low values, which were written down before a direct check in Python (Table 1).

The identity is an integration by parts, and Agmon estimates from each end give the bound, with rate set by the smallest eigenvalue $\frac12$ of $|n\cdot J|$. The summands of $E$ on $\mathbb{CP}^3$ have indices $4,0,0,-4$, of total zero, as equivariant cobordism invariance requires, since their sum extends over $X$, although the individual lines do not ((Witten 1985), preprint pp. 43-44). The window has index $\mathbf 4-\mathbf 4=0$, so index theory allows $\lambda\neq0$, and the eight states come from the estimates. The $\mathbf 4$ is pseudoreal, and the auxiliary grading $\hat\Gamma$ with zero virtual index says nothing yet about the chiral spectrum of the Standard Model.

This is a statement about compression maps. Generated algebras need not converge when compression maps do, in line with the setting of (Connes and Suijlekom 2021).

5 The joint limit on the disc bundle

Under $g\to R^2g$, with the potential divided by $R$, the operator is unitarily equivalent to $\widehat D_M/R$. So the light mass and the gap both scale as $1/R$, with the compression maps unchanged. Let $G_0=\int_e^{l-a}f\,dr$, $0<\alpha<\beta<1$, $E_*>0$ and $\eta\in(0,1)$.

Item 4 follows from the Weyl law for $\widehat D_0$, since $\widehat D_M-\widehat D_0$ is bounded by $cM_\varepsilon$, which is small against $ER_\varepsilon$. So (1) holds for a fermion window that comes from the operator, with growth of a seven-dimensional internal space at fixed energy. The bound on $m_\varepsilon$ is one-sided, since a lower bound on $\lambda$ is open. The fixed shape expands, so the pointed limit at fixed energy is flat at an interior point and a half-space at a boundary point. Nothing here identifies it with the complete Bryant-Salamon manifold (Bryant and Salamon 1989; Karigiannis and Lotay 2020), whose metric lies outside our class, or with a solution of field equations.

6 The quaternionic defect

Proposition 28.16 of Supplemental Material, part 28 gives the test for the core block: with $B(r)=F^*F=\mathrm{diag}(b_aI_2,b_+,b_-)$, all real multipliers compress to operators commuting with the quaternionic structure $j$ if and only if $b_+=b_-$. Put $\eta_t=\int|b_+-b_-|\,dr/\int\mathrm{tr}\,B\,dr$, and let $P_H(A)=\frac12(A+jAj^{-1})$.

With $j_E$ parallel, the charge conjugation $C$ commutes with $T_M$ and exchanges the two wall conditions, so only the wall breaks the symmetry. On copies of the $\mathbf 4$, $F\mapsto CFj^{-1}$ is a real structure, which fixes the model kernel. Then $\zeta=u-Cu$ obeys $T_M\zeta=\lambda(v-Cv)$ in the interior, with vanishing projection of its cut-off to the model kernel, and it is exponentially small at the core, sourced at the wall. A weighted Cauchy-Schwarz inequality doubles the exponent of Proposition 3, with the integral starting at $2e$. So under this hypothesis, bounded expressions in the core block lie exponentially close to $M_2(\mathbb H)$. An unrestricted generated algebra may use coefficients of size $1/\eta_t$, and it is $M_4(\mathbb C)$ at finite mass when $b_+\ne b_-$ and the $\mathbf 5$ component is nonzero. The wall block generates $M_4(\mathbb C)$, which requires the loss of $C$ there.

7 Relation to prior work

The mechanisms have precedents, which we distinguish from the combination studied here. Chiral localization on defects and boundaries goes back to (Kaplan 1992; Shamir 1993), and gauge symmetry breaking by interval boundary conditions to (Csáki, Grojean, Hubisz, et al. 2004; Csáki, Grojean, Murayama, et al. 2004); a tree-level $m_t=2M_W$ for a suitable representation is in (Cacciapaglia, Csáki, and Park 2006). The square identity with a zero-order cross term is in (Maridakis 2015), in a concentrating-pair setting, and a horizontal and vertical factorization of the Dirac operator with a curvature contraction is in (Kaad and Suijlekom 2017), under other hypotheses. Index statements of Callias type in families are in (Callias 1978; Kottke 2009), and the finite boundary window is a separate question. Kaluza-Klein modes that become massless during gauge restoration are followed in (Iso, Kitazawa, and Suyama 2021), with a different mechanism and geometry. Targeted searches found these and did not establish whether the eight-state theorem has appeared elsewhere; they were not exhaustive.

8 What is missing

The vector sector. On the family of Theorem 5 one needs the kinetic matrix and the mass Hessian of the vector fluctuations, then the momentum-dependent kernel, with the aim $M_W,M_Z\sim c\lambda/R$. The tunnelling eigenvalue alone does not establish this, so the words “massless electroweak limit” apply to the disc bundle only as a target.

The couplings. For a bulk Yang-Mills field with a constant profile, $g_4^{-2}=\mathrm{Vol}(X)/g_{11}^2$, so with fixed microscopic coupling a plain dilation gives $g_4\propto R^{-7/2}$, which must be separated from restoration at finite coupling. Gauge fields from isometries have Killing overlap integrals instead, and non-Killing fields behave differently again (Baptista 2026).

The dynamics. A potential $a\,r^{-p}+b\,\varepsilon^2r^q$ has its minimum at $r\propto\varepsilon^{-2/(p+q)}$, so it would select $\alpha=2/(p+q)$ if $p+q>2$. Nothing has derived such a potential, and adding it by hand derives nothing.

The finite structure. The window gives the graded eight-dimensional space and the limit algebra. The connector of the bimodule of complex dimension 32 with a real structure, the quaternionic grading and the chirality of the physical spectrum remain inputs. The product of two copies of the parent has heat exponent $t^{-7}$, so the bimodule should come from a finite-rank correspondence, to keep seven dimensions.

9 Status

Theorem 1, the vector poles and the cylinder are direct consequences of refereed lemmas, with proofs in Supplemental Material, part 30. Theorems 2, 4 and 5 and Propositions 3, 6 are new here, for the stated class of metrics, with proofs in Supplemental Material, part 29 and Supplemental Material, part 30. They were read by an independent referee, GPT-6.1 Terra, whose reports found errors in case (a) and a missing hypothesis in the exponential defect bound. These are corrected, and the corrections have had no further independent reading. The referee ran no computation. Other metrics, and case (a) at other radii, stay open. Table 1 is a Python computation; by the rule of the project the script is not kept, and the values and the setting are recorded in Supplemental Material, part 29.

Provenance

The formulation of the joint limit and the interval and cylinder statements come from an audit of the repository by GPT-6 Astra, kept in its reviews folder. The proofs of Sections 4 and 5 and of the defect bound are by Claude Opus 5.5, using results of GPT-6 Astra on the operator and its two ends, with referee reports by GPT-6.1 Terra on the earlier lemmas. Claude Sonnet 5.5 rechecked the audit in Python. The two referee reports on lemmas 29 and 30 and this note are in the repository, and referee status is recorded in its ledger.

References