Finite spectral triples as towerless intervals

1 Introduction

Between 1979 and 1987 the Kaluza-Klein programme sought the Standard Model in the geometry of extra dimensions. The Higgs field appeared as the internal components of a higher-dimensional gauge field (Manton 1979; Fairlie 1979; Forgács and Manton 1980; Chapline and Manton 1981), and Witten’s search for a realistic Kaluza-Klein theory found that $SU(3)\times SU(2)\times U(1)$ can arise from isometries of the internal space while the fermion quantum numbers are difficult to achieve (Witten 1981). In 1983 he made the difficulty precise (Witten 1985): for a ground state $M^4\times B$ with $B$ compact and all gauge fields coming from isometries of $B$, a theorem of Atiyah and Hirzebruch makes the index of the internal Dirac operator vanish as a character of the isometry group, so the four-dimensional fermion quantum numbers are vectorlike. The result is true in that scope, and the same text names its escapes: elementary gauge fields in the higher dimension, and a unified theory, perhaps a supergravity or superstring theory, that fixes the gauge group. From 1984 attention moved to supersymmetry and superstrings, and the heterotic string took both escapes, with elementary gauge fields whose group the theory fixes. The reviews of 1986 and 1987 record that move and close the programme (Duff, Nilsson, and Pope 1986; Bailin and Love 1987); we read their open-problem sections only through abstracts and later retrospectives (Duff 1994; Duff, Nilsson, and Pope 2025), the second of which counts Witten’s chirality result as one of three reasons the coset programme lost ground, with the large negative cosmological constant and the string models.

Noncommutative geometry entered particle physics with a Kaluza-Klein reading from the start. In 1988 Madore generalized Kaluza-Klein theory by replacing the functions on the internal space with a matrix algebra, for which the internal Laplacian has a finite spectrum (Madore 1989). Connes and Lott wrote the Standard Model as a gauge theory on a spacetime “of Kaluza-Klein type, where the fiber is formed of a two-point set”, with the Higgs field as the connection along the discrete direction (Connes and Lott 1991). In its simplest form the fibre has algebra $\mathbb C\oplus\mathbb C$ and Dirac operator $m\sigma_1$, and its two points lie at Connes distance $1/|m|$. Coquereaux, Esposito-Farèse and Vaillant called their gauging of discrete directions “in a sense, a discrete analog” of the Kaluza-Klein approach (Coquereaux, Esposito-Farèse, and Vaillant 1991), and in 1994 Viet and Wali discretized the fifth dimension (Viet and Wali 1996). In the texts and reference lists we read, none of these founding papers cites the chirality obstruction; the advantage they claim for the discrete direction is finiteness, and chirality is supplied by the grading of the finite Hilbert space.

The two-point space is the oldest layer of the present theory, which adds a real structure, a KO-dimension and a first-order condition (Connes 2006; Chamseddine, Connes, and Marcolli 2007; Chamseddine and Connes 2008); the finite space of the noncommutative Standard Model has KO-dimension 6, so that its product with four-dimensional spacetime has total KO-dimension $10\equiv2\pmod 8$ (Connes 2006; Barrett 2007; Chamseddine, Connes, and Marcolli 2007). We read the two-point space as the minimal spectral truncation of a chiral interval, whose walls carry the chirality that the obstruction excludes from a closed internal manifold, so that the obstruction and the discrete direction are two statements about the same limit. Within the searches recorded in the Supplemental Material, part H (the history of the field, Section 1.4), this link has not been made in print before.

KO-dimension and metric dimension differ: the finite space has metric dimension 0 and KO-dimension 6. So the numbers 9, 10 and 11 below are KO-degrees of factorizations, while the emergence or loss of metric dimension is measured by Weyl laws and requires a singular limit. We make the discrete-dimension reading precise and test it against a proposal suggested by the Kaluza-Klein picture: with massless weak bosons a dimension should open (10 to 11), with infinitely heavy weak bosons and top a dimension should close (10 to 9), and the physical point, with its finite Fermi constant and measured weak angle, lies between these limits. Below we give exact results for an interval model. The maps from KO signs to interval geometries, and from the length $l$ to the measured parameters, are part of the proposal and are not constructed here.

Our model of the fifth dimension is the chiral interval $I_l=[0,l]$, with boundary conditions that place the two chiralities at opposite ends. Table 1 summarizes what the picture gives. The full proofs are in the Supplemental Material; the main text gives their structure.

2 Setting and conventions

Let $H=L^2(I_l;\mathbb C^2)$ with $\psi=(L,R)$, grading $\Gamma=\mathrm{diag}(1,-1)$, and for a real bulk mass $M$ $$(D_M\psi)_L=R'+MR,\qquad (D_M\psi)_R=-L'+ML.$$(1) We use three boundary types: (a) $R(0)=R(l)=0$; (b) $L(0)=L(l)=0$; (c) $R(0)=0$, $L(l)=0$. Each defines a self-adjoint operator with compact resolvent anticommuting with $\Gamma$.

A real structure is an antiunitary $J$ with $J^2=\epsilon$, $JD=\epsilon'DJ$, $J\Gamma=\epsilon''\Gamma J$; KO-dimension 6 means $(\epsilon,\epsilon',\epsilon'')=(+,+,-)$.

A spectral truncation of $(A,H,D)$ to a spectral subspace $PH$ is $(PAP,PH,PDP)$. The map $a\mapsto PaP$ is the compression. For functions it is the classical Toeplitz compression, and its use for truncating spectral triples is due to (D’Andrea, Lizzi, and Martinetti 2014; Connes and Suijlekom 2021). The image $PAP$ is an operator system, in general not an algebra.

3 The two-point space is a truncated interval

Theorem A. Take $M=0$ and type (c). The spectrum of $D_0$ is $\pm(r-\tfrac12)\pi/l$, $r\ge1$. The lowest spectral subspace is spanned by $u_L=\sqrt{2/l}\,(\cos(\pi x/2l),0)$ and $u_R=\sqrt{2/l}\,(0,\sin(\pi x/2l))$; on it $PD_0P=m\sigma_1$ with $m=\pi/(2l)$, $P\Gamma P=\sigma_3$, and the compression of $C(I_l)$ has range the diagonal algebra $\mathbb C\oplus\mathbb C$. Hence the truncation is the Connes-Lott two-point space with $|m|=\pi/(2l)$, and the Connes distance between the two states of the compressed operator system is $2l/\pi$.

The two states pull back to the smeared measures $|u_L|^2dx$ and $|u_R|^2dx$ on $I_l$, which differ from endpoint evaluations. With the inherited real structure, the full diagonal algebra fails the first-order condition: for $e=\mathrm{diag}(1,0)$ one has $e^\circ=e$ and $[[m\sigma_1,e],e^\circ]=m\sigma_1$ (Supplemental Material, part 08).

The discrete version is a staggered chain on $$H_N=\mathrm{span}\{L_1,\dots,L_{N-1},R_2,\dots,R_N\}$$(2) with $B_N=h_N(I-S)$, whose spectrum $2h_N\sin((2r-1)\pi/(4N-2))$ converges to that of $D_0$ and whose sites lie at distance $|i-j|/h_N$; the two-point space is its $N=2$ case, as in dimensional deconstruction (Arkani-Hamed, Cohen, and Georgi 2001).

Proposition A.1 (bare limits). At fixed truncation, $m\to0$ produces no heat dimension, and $m\to\infty$ leaves no light fermions. In the product with any finite KO-6 triple, $$D_F^2=D_0^2\otimes1+1\otimes D_I^2\ge\frac{\pi^2}{4l^2},$$(3) so the literal collapse $l\to0$ of the massless interval has an empty light sector at every fixed cutoff.

4 Bulk mass: the light pair and both limits

Theorem B.

1. Chirality from boundary conditions. For every $M$ and $l$, $\ker D_M$ is $\mathbb C(e^{Mx},0)$ for type (a), $\mathbb C(0,e^{-Mx})$ for type (b) and $0$ for type (c), with indices $+1,-1,0$. For a finite sum of representations $V_\rho$ of a compact group with types $\tau(\rho)$, the equivariant index is $\sum_{\tau=a}[V_\rho]-\sum_{\tau=b}[V_\rho]$, which realizes $[V_L]-[V_R]$.

2. Light pair. For type (c), $M=-\mu<0$ and $\mu l>1$, there is exactly one positive eigenvalue below $\mu$, $$\lambda_0=\mu\,\mathrm{sech}(\kappa_0l),\qquad \tanh(\kappa_0l)=\frac{\kappa_0}{\mu},$$(4) with $\lambda_0=2\mu e^{-\mu l}(1+o(1))$, and the others are $(\mu^2+k_n^2)^{1/2}$ with $k_n\in(n\pi/l,(n+\tfrac12)\pi/l)$. The profiles satisfy $R_0(x)=L_0(l-x)=\sinh(\kappa_0x)$ up to normalization.

3. Truncation. The light pair truncates to Connes-Lott with $D=\lambda_0\sigma_1$, and the Connes distance between its states is $1/\lambda_0$.

4. Decompactification ($l\to\infty$, fixed $\mu$). $\lambda_0\to0$; the normalized profiles converge, uniformly on compacta and in $L^2$, to $\sqrt{2\mu}\,e^{-\mu y}$ at the two walls, exact chiral zero modes of the half-line operators; and $\#\{n:\lambda_n\le E\}=(l/\pi)\sqrt{E^2-\mu^2}+O(1)$ for $E>\mu$.

5. Collapse at fixed light mass. For $m_f>0$ and $l<1/m_f$ there is a unique $\mu(l)$ with $\lambda_0=m_f$. As $l\to0$, with $c=\mu(l)l$ and $L=\log(2/(m_fl))$, $$c=-W_{-1}\!\left(-\frac{m_fl}{2}\right)+o(1)=L+\log L+o(1),$$(5) a statement about this localization model. The next eigenvalue exceeds $(\mu^2+\pi^2/l^2)^{1/2}$, and the truncation stays Connes-Lott with mass $m_f$ at each $l$. At fixed $\mu$ instead, every eigenvalue eventually has $|\lambda|>\mu$, and the smallest is $(\pi/2l)(1+o(1))$.

So in this one-field family the dimension emerges with the tower restored (item 4), and the collapse keeps a finite light sector along the curve $\mu(l)$ (item 5); Theorem C gives another route. The excited levels combine a quantized internal momentum with the bulk mass in quadrature, $\lambda_n^2=\mu^2+k_n^2$, while the light pair has imaginary momentum $k=i\kappa_0$, with $\lambda_0^2=\mu^2-\kappa_0^2$.

5 Boundary Yukawa matrices and towerless textures

Let $Q\in L^2(I_l;\mathbb C^2)\otimes\mathbb C^{N_Q}$ and $U\in L^2(I_l;\mathbb C^2)\otimes\mathbb C^{N_U}$ carry diagonal bulk masses, and for a matrix $X$ impose $Q_R(0)=0$, $U_L(0)=0$, $Q_R(l)=XU_R(l)$, $U_L(l)=-X^\dagger Q_L(l)$. This defines a self-adjoint $D_X$ anticommuting with $\Gamma$. Write $c^Q_i=-m^Q_il$, $c^U_j=m^U_jl$ (positive $c$ localizes a field away from the Yukawa wall), $\phi(c)^2=2c/(e^{2c}-1)$, $\Phi_Q=\mathrm{diag}\,\phi(c^Q_i)$ and $\Phi_U=\mathrm{diag}\,\phi(c^U_j)$. Let $F_Q,F_U$ be the diagonal matrices of values at $x=l$ of the normalized zero modes, so that $F=\Phi/\sqrt l$, and let $\Lambda_Q,\Lambda_U$ be the smallest lowest positive eigenvalues of the single fields with one Dirichlet condition at each end (type (c)).

Theorem C.

1. Index and symmetry breaking. $\ker D_X\cong\ker X^\dagger\oplus\ker X$ in the two chiralities, so $\mathrm{ind}\,D_X=N_Q-N_U$ for all $X$. If a compact group acts, the domain is invariant under the stabilizer $H_X$ of $X$, and $\mathrm{ind}_{H_X}D_X=[\rho_Q]-[\rho_U]$ restricted to $H_X$. All zero modes lift iff $X$ is invertible, which requires $\rho_Q\cong\rho_U$ as $H_X$-representations; for $SU(3)\times U(1)_{\mathrm{em}}$ and leptons without $\nu_R$, a massless $\nu_L$ is forced for every $X$.

2. Interpolation. For one massless flavor the positive spectrum is $\theta/l$ and $(n\pi\pm\theta)/l$, $\theta=\arctan|X|$: the light pair moves from the chiral zero modes of types (a)$\oplus$(b) at $X=0$ to the Connes-Lott mass $\pi/(2l)$ of two copies of type (c) as $|X|\to\infty$.

3. Light spectrum. With $A,B$ the diagonal matrices of $\omega\coth(\omega l)\pm m$, $\omega^2=m^2-\lambda^2$, the light eigenvalues are the fixed points $\lambda=\sigma_k(A(\lambda)^{1/2}XB(\lambda)^{1/2})$, and with $Y_0=F_QXF_U$, $$\sigma_k(Y_0)\bigl(1-\frac{\lambda_k^2}{\Lambda_Q^2}-\frac{\lambda_k^2}{\Lambda_U^2}\bigr)\le\lambda_k\le\sigma_k(Y_0),$$(6) which holds for $\lambda_k$ below $\min(\Lambda_Q,\Lambda_U)$.

4. Towerless limit. As $l\to0$ with the $c$’s and $\hat Y=X/l$ fixed, the spectrum in $[-E,E]$ converges to $0$, with multiplicity $N_Q+N_U-2\,\mathrm{rank}\,\hat Y$, and to the $\pm\sigma_k(M_F)$ with $\sigma_k(M_F)\le E$, where $$M_F=\Phi_Q\hat Y\Phi_U,\qquad D_F=\begin{pmatrix}0&M_F\\ M_F^\dagger&0\end{pmatrix}$$(7) is the Yukawa block of the Chamseddine-Connes finite Dirac operator. All other eigenvalues diverge.

5. Hierarchies. For graded $F=\mathrm{diag}(f_1\le\dots\le f_N)$ and $G$ likewise, let $T_p=\{N-p+1,\dots,N\}$, $f_{T_p}=\prod_{i\in T_p}f_i$ (likewise $g_{T_p}$), $\Delta_p=|\det X_{T_pT_p}|\neq0$, $D_p$ the largest $p\times p$ minor of $X$ in absolute value, $\epsilon^F_p=f_{N-p}/f_{N-p+1}$ (likewise $\epsilon^G_p$), and $K_p=\binom Np-1$. Then $\sigma_1\cdots\sigma_p(FXG)$ lies between $f_{T_p}g_{T_p}\Delta_p$ and $$f_{T_p}g_{T_p}\bigl[\Delta_p+D_p\bigl((1+K_p\epsilon^F_p)(1+K_p\epsilon^G_p)-1\bigr)\bigr].$$(8) If the entries of $\hat Y$ are of order one and its trailing minors are bounded below, the masses follow $$\phi(c^Q_k)\phi(c^U_k)\approx2\sqrt{c^Q_kc^U_k}\,e^{-c^Q_k-c^U_k}:$$(9) order-one differences of localization give exponential hierarchies (Arkani-Hamed and Schmaltz 2000; Gherghetta and Pomarol 2000; Grossman and Neubert 2000).

Item 2 gives a second route to collapse at fixed light mass, with no bulk mass: for one massless flavor and $X(l)=\tan(ml)$, the light eigenvalue is exactly $m$ for small $l$, and the higher levels diverge. Item 4 is its multiflavor form.

The mixing pattern $|V_{ij}|\sim\phi(c^Q_i)/\phi(c^Q_j)$ holds in numerical samples (Python) and is known in the extra-dimensional flavor literature (Huber and Shafi 2001; Agashe, Perez, and Soni 2005). For $M_s=FX_sG_s$ with nonzero trailing minors, there is an upper bound with explicit constants, independent of the right-hand profiles: $|V_{ij}|\le3K_uK_df_i/f_j$ for $i<j$, with $K_s$ built from the ratios of minors (Supplemental Material, part 12, from the referee report). There is no universal lower bound, since identical sectors give $V=1$.

6 Real structures: the choice between 9 and 11

We call the factorization of a KO-6 triple as an odd triple times a point a doubling; this usage is unrelated to fermion doubling.

Theorem D.

1. Factorization criterion. A KO-6 triple $T$ factors as $T_n\times P_k$ with $(s,n,k)=(+1,7,7)$ or $(-1,5,1)$ iff there is a unitary $U$ with $U^2=1$, $U\Gamma=-\Gamma U$, $[U,a]=[U,a^\circ]=[U,D]=0$ and $UJ=sJU$.

2. The real structure does not select $s$. The almost-commutative electrodynamics triple admits involutions of both signs. In a neutral sector with scalar algebra character, the KO-7 factorization always exists, and the KO-5 factorization exists exactly when every singular value of the Majorana block, including zero, has even multiplicity. So the strict light spectrum of the Standard Model representation, with one or three unpaired left-handed neutrinos and no Majorana term, admits only $s=+1$; two unpaired neutral pairs with $D=0$ admit both signs; and while the full algebra keeps $\mathbb H$, neither factorization exists.

3. The interval reflection. On the chiral interval with pointwise reality ($JfJ^{-1}=\bar f$), the real structures are $\mathrm{diag}(\alpha,\beta)K$ with $\alpha=\epsilon'\beta$, of KO-dimension 0 for $\epsilon'=+1$ and an extended sign convention for $\epsilon'=-1$, and the reflection $x\mapsto l-x$ exchanging $L$ and $R$ has sign $\epsilon'$ against $J$. In the internal-first KO-6 product $F_0\times I_l$, the KO-6 conditions force the grading twist $J_F=J_0\otimes J_I\Gamma_I^b$ with $\epsilon'(-1)^b=-1$, and the bare reflection $1\otimes\sigma$ has $s=-1$ for either interval convention.

4. Exact reconciliation. For the truncated interval with scalar algebra, $F_0\times I_l\cong T_5(m)\times P_1$ with $d_5=D_0+m\Gamma_0$ and $j_5=J_0\Gamma_0$: the interval mass enters the odd Dirac operator, and the interval’s two states become the doubling space. With the full compressed two-sheet algebra, the inherited data fail first order and the reflection does not commute with that algebra.

5. No physical selection. Spacetime grading twists ($J_M$ versus $J_M\gamma_5$) do not select the finite involution. A product with fixed $D_F,J_F,\Gamma_F$ and particle number $Q$ admits particle-preserving involutions of both signs. More generally, if $Q$ is a self-adjoint involution commuting with $D$, $\Gamma$, the algebra and its opposite, with $QJ=-JQ$, then particle-preserving involutions come in pairs $U$, $QU$ of opposite sign.

The values 9 and 11 are therefore the KO-degrees, shifted by 4, of the two odd factorizations of one even triple. Replacing $J$ by $J\Gamma$ only changes signs inside the even classification (Dąbrowski and Dossena 2011), and identifying either factorization with the dimension of a parent geometry would need a change of parity in K-homology that these results do not supply.

The two labels differ by sector. Each charged paired sector of the reduced representation is invariant under the algebra, its opposite, $D$ and $J$, and admits both signs, so the formal label 9 is available for the charged fermions sector by sector. The obstruction to $s=-1$ for the whole triple sits in the neutral sector: with one or three neutrinos and no $\nu_R$, only $s=+1$ occurs there, and that neutral involution does not preserve particle number, so with particle preservation imposed on this spectrum neither reduction is available. With $\nu_R$ and pure Dirac masses both signs occur. A Majorana mass breaks $[Q,D]=0$ and the pairing of item 5, and $s=-1$ exists exactly when every singular value of the neutral block has even multiplicity, so a generic Majorana block leaves only $s=+1$.

Chirality at the odd labels. In odd dimension the Clifford volume element is central, so an irreducible spinor has no chirality operator. Witten’s argument for an odd number of extra dimensions ((Witten 1981), pp. 421–423) runs as follows: the reversal $\gamma_i\mapsto-\gamma_i$ of the internal gamma matrices fixes $[\gamma_i,\gamma_j]$ and flips the internal helicity, so on an orientable internal space, with the gauge group acting by coordinate transformations, four-dimensional fermions of opposite helicity carry the same gauge charges, and the low-energy content is vectorlike. That paper treats eleven-dimensional supergravity on $M^4\times M^{pqr}$; the ways out it considers are a non-orientable internal space and torsion. The finite statement here is parallel: in the Standard Model representation of $\mathbb C\oplus\mathbb H\oplus M_3(\mathbb C)$, neither odd factorization exists (item 2). Such arguments depend on their hypotheses: chiral modes can sit on boundaries or defects, and chiral couplings can arise for massive gauge fields linked to non-Killing vector fields (Baptista 2026).

We propose reading the objection as a constraint on limits. The label 9 uses the charged paired sectors, which are vectorlike under the reduced $SU(3)\times U(1)_{\mathrm{em}}$. A neutral Weyl fermion has a real gauge representation and can be written in Majorana form, though it stays a chiral spacetime fermion; for an odd number of them, the obstruction to $s=-1$ is the parity forced by $j^2=-1$. In the type-(c) family as $l\to\infty$ at fixed $\mu$, the pair becomes chiral zero modes of the two half-lines, and a region receding from both walls sees only the bulk operator, with threshold $\mu$ (Theorem B, item 4). This shares a feature with Hořava-Witten theory (Hořava and Witten 1996), where chirality is carried by the boundaries of an odd bulk. The content differs: there, an $E_8$ vector multiplet sits on each boundary, with one ten-dimensional chirality for a common orientation, while here two modes of one bulk Dirac operator have opposite $\Gamma$-chirality. The same-chirality analogue is type (a) at both ends, with opposite localization masses, and unbroken $E_8\times E_8$ allows no bilinear connector between the boundary multiplets (Supplemental Material, part 20). Two parts of the proposal are not constructed here. The assignment of the label 11 to this limit is open, since the interval product of item 4 also has a KO-5 factorization. And we take the physical point, with its finite Fermi constant and measured weak angle, to lie at finite $l$ in the even interior, where the KO-6 structure accommodates chirality, without relating $l$ quantitatively to those parameters.

7 Spectral action: heavy fermions and the gauge sector

Proposition E. Assume the Chamseddine-Connes-Marcolli relation $8M_W^2=\sum N_fm_f^2$ over Dirac masses at the matching scale, with $N_g$ generations, and let $m_{\max}$ be the largest Dirac mass. Then $$\frac{m_{\max}}{2\sqrt2}\le M_W\le\sqrt{N_g}\,m_{\max}.$$(10) Hence a heavy-W limit at fixed fermion masses is incompatible with the relation, and if only the up-type mass of one generation diverges, $m_t/M_W\to\sqrt{8/3}$, the top-mass relation of (Chamseddine, Connes, and Marcolli 2007).

The relation belongs to the truncated heat-kernel expansion, with its coefficient $\mathrm{Tr}\,Y^\dagger Y$ polynomial in the masses and higher terms in $m^2/\Lambda^2$ dropped. It is therefore a statement for $m\ll\Lambda$. The case of fixed cutoff is the following.

Proposition E.1. Let $S=\mathrm{Tr}\,f(D_A^2/\Lambda^2)$ on $\mathbb R^4$, as a relative trace or action density, with a constant vacuum $V_0$ with eigenvalues $m_i$, and let $a_\mu(q)T$ be a transverse vector. For $f(z)=e^{-tz\Lambda^2}$ the quadratic action is exact, $\int_q|a|^2C(q^2)$, with $d_{ij}=m_i-m_j$ and $$\begin{split} C(q^2)=\frac1{8\pi^2}\sum_{ij}|T_{ij}|^2\bigl[&\bigl(\tfrac2t+d_{ij}^2+q^2\bigr)I_{ij}\\ &-\tfrac1t\bigl(e^{-tm_i^2}+e^{-tm_j^2}\bigr)\bigr], \end{split}$$(11) where $I_{ij}=\int_0^1e^{-t[(1-u)m_i^2+um_j^2+u(1-u)q^2]}du$. Its small-mass expansion gives the mass formula of Proposition F and the CCM relation. In the three-generation Standard Model data with only one up-type Dirac mass $m$, $f(z)=e^{-z}$ and $x=m^2/\Lambda^2$:

1. the Taylor mass is $$\frac{M_W^2}{\Lambda^2}=\frac{3[1-(1+x)e^{-x}]}{x[3+w(x)]},$$(12) with $w(x)=3[2-(x^2+2x+2)e^{-x}]/x^3$, so that $M_W^2\sim\tfrac38m^2$ for $x\to0$ and $M_W^2\sim\Lambda^4/m^2$ for $x\to\infty$; a unique positive timelike pole has the same limits, and both obey $M_W^2\le\Lambda^2/e$;

2. $M_Z^2\sim\tfrac23xe^{-x}\Lambda^2$ for $x\to\infty$, so that $M_W^2/M_Z^2$ does not stay at $5/8$, while the neutral-current angle stays at $\cos^2\theta_{\mathrm{NC}}=5/8$ and the ratio of the photon and gluon coefficients stays at $8/3$ for all $x$;

3. the top is separated: the finite trace in the vacuum density is $84+12e^{-x}$, and among single charged species only an up-type quark can be removed while keeping the two values in item 2 (for a down-type quark they become $59/92$ and $46/15$, for a charged lepton $17/28$ and $7/3$);

4. the broken vectors keep a positive kinetic form, so that they do not decouple by becoming heavy. The light space consists of the five other quarks and the leptons, on which $SU(3)\times U(1)_{\mathrm{em}}$ acts with $b_L$ paired with $b_R$; its electroweak $SU(2)$ does not act linearly on it.

For the heat cutoff and the Taylor mass $M_W^2=C(0)/C'(0)$, and for doublets with one massive member and one massless, item 1 is the CCM relation with a form factor. With $\rho(x)=2[1-(1+x)e^{-x}]/x^2$, $$8\kappa M_W^2=\sum_fN_fm_f^2\,\rho(m_f^2/\Lambda^2),\qquad \kappa=\frac{P+\sum_id_iw(x_i)}{6},$$(13) where $P$ is the particle-trace weight of complete massless doublets, $d_i=N_i/2$ that of each massive doublet, and $P+\sum d_i=6$. For the top alone, $\kappa=(3+w)/4$. If all $x_i\to0$, then $\rho,\kappa\to1$, which is the CCM relation. In the top-alone family, $m_t/M_W$ differs from $\sqrt{8/3}$ for every $x>0$, and approaches it only as $x\to0$, so a limit with the relation intact needs $m_t/\Lambda\to0$; the departure is 5% at $m\approx0.45\Lambda$, and $M_W$ has a unique maximum of $0.524\Lambda$ at $m\approx1.42\Lambda$ (Python, for the location). A massive diagonal contribution to the trace is exponentially suppressed above the cutoff, and a mixed loop with a massless partner has $\rho\sim2/x^2$. With no other mass in the vacuum, the W mass then has no source. This regime lies beyond the range of the expansion, and whether the classical action is meaningful for $m>\Lambda$ is a separate question.

The Taylor statement holds for nonnegative cutoffs, including those flat near $0$. With a complete massless doublet retained, the Taylor mass tends to $0$ for any finite set of heavy doublets. With every doublet of equal mass and no light fermions, the poles satisfy $2x<M_W^2/\Lambda^2<2x+2$ and $\tfrac{16}{5}x<M_Z^2/\Lambda^2<\tfrac{16}{5}x+2$, so they diverge at fixed cutoff with $M_W^2/M_Z^2\to5/8$; there the kernel vanishes at fixed momentum and the Taylor kinetic term has the wrong sign for $x>2$. In the top-alone family, a limit with $m_t/M_W\to\sqrt{8/3}$ and all scales diverging exists jointly with $\Lambda\to\infty$ and $m/\Lambda\to0$. For nonnegative $f$ of unbounded support, no finite $c>0$ is stationary along $V=cV_*$, so these are masses about a prescribed background. All of this is classical and bosonic. So in the families treated, a heavy W and Z at fixed cutoff is found only without a massless complete doublet.

Proposition F. Fluctuate the product of spacetime with the chiral interval by a five-dimensional gauge field $(A_\mu,A_5)$ and compress to the light pair. Then:

1. the sheet connections are the weighted averages $A_{L,R}=\int w_{L,R}A_\mu$ and the link is $\phi=m+iB$ with $B=\int_0^ll^{-1}\sin(\pi x/l)A_5\,dx$, so the image is $\{\phi+\phi^*=2m\}$ and not every Connes-Lott fluctuation arises;

2. the fixed compression is not gauge covariant: $A_5=a\cos(2\pi x/l)$ has trivial Wilson line and changes the compressed spectrum;

3. at leading order, in the stated trace and vacuum, the spectral action gives the vector mass matrix $\mathcal M^2=\tfrac32Q^{-1/2}KQ^{-1/2}$. For the minimal two-sheet model its eigenvalues are $0$ and $6m^2$: a massless diagonal vector and a vector of mass $\sqrt6\,m$. With the CCM three-generation trace and one massive colour triplet, $M_W^2=\tfrac38m^2$. Neumann-Dirichlet conditions, which are separate input, give the tower mass $m=\pi/(2l)$.

So in these cases the finite spectral action and the interval tower disagree on the vector mass, and the value depends on the whole fermion trace. With the gauge-covariant projection onto the light pair of the fluctuated operator, the covariance defect disappears, and on a common profile space equality of the two masses becomes an explicit trace-and-overlap sum rule (Supplemental Material, part 14). In the one-fermion Abelian mixed-boundary reduction, it fails at zero bulk mass, where $\mathcal M_{\mathrm{spec}}^2=\tfrac35\mathcal M_{\mathrm{KK}}^2$, and holds at some tuned bulk mass $0<M_*<3\pi/(4l)$, at which the matched fermion has a mass of the order of the tower scale. At leading spectral-action order, if all fermion masses retained in the finite trace are uniformly bounded, then $\mathcal M_{\mathrm{spec}}^2\le6\|V_0\|^2$, so no matching is possible in the collapse. Equality of one mass does not establish equality of the full gauge or scalar actions. In these results the gauge sector ties the W to a fermion at the tower scale, in line with Proposition E.

8 Whole triples and truncations

The results above concern compressions at each $l$. For the limit of the whole triple we use the spectral propinquity $\Lambda_{\mathrm{spec}}$ of Latrémolière (Latrémolière 2022) (doi:10.1016/j.aim.2022.), a metric on metric spectral triples up to unitary equivalence, that is, on triples whose Lipschitz seminorm $L(a)=\|[D,a]\|$ has kernel $\mathbb R1$ and induces the weak* topology on states. A tunnel between two such triples is a quantum compact metric space, carrying a Hilbert module with a D-norm, that maps onto both; its extent is the Hausdorff distance between its state space and theirs, and the propinquity $\Lambda$ is the infimum of extents. Then $\Lambda_{\mathrm{spec}}$ is the infimum of $\sqrt2/2$ and of those $\varepsilon$ for which some tunnel has both extent and separation of the families $e^{itD_j}$, $0\le t\le1/\varepsilon$, below $\varepsilon$. The proofs below use only this definition, written out in Supplemental Material, part 15, Conventions, and facts proved there, chiefly $\Lambda_{\mathrm{spec}}(T_1,T_2)\ge\min(\sqrt2/2,|\mathrm{diam}\,T_1-\mathrm{diam}\,T_2|/2)$ (consequence (c)) and a comparison of norms along a tunnel (Lemma 15.1).

Proposition G. Let $T=(A,H,D)$ be a metric spectral triple of diameter $\delta$. Let $\Sigma\subset\mathbb R$ be finite with $P=1_\Sigma(D)$ of rank $N$, let $|\lambda|\ge g$ on the rest of the spectrum, let $W:PH\to\mathbb C^N$ be unitary and $\Delta=\|WDW^*-D_F\|$. Then $$\begin{split} \min\bigl(\tfrac{\sqrt2}{2},\tfrac\delta2\bigr)&\le\Lambda_{\mathrm{spec}}\bigl(T,(\mathbb C,\mathbb C^N,D_F)\bigr)\\ &\le g^{-2}+\max\bigl(\tfrac\delta2,\Delta\bigr)+\Delta+\sqrt\Delta, \end{split}$$(14) with no anticommutation hypothesis. Conversely, a sequence $T_n$ converges to $(\mathbb C,\mathbb C^N,D_F)$ iff $\delta_n\to0$ and there exist rank-$N$ spectral projections $P_n$ and unitaries $W_n$ with gaps $g_n\to\infty$ and $\Delta_n\to0$. Moreover:

1. the tuned interval of Theorem B satisfies $$\frac l2\le\Lambda_{\mathrm{spec}}\bigl(T_l,(\mathbb C,\mathbb C^2,m_f\sigma_1)\bigr)\le\frac l2+\frac{l^2}{\pi^2+c^2},$$(15) and stays at distance at least $\min(\sqrt2/2,(1/m_f-l)/2)$ from the Connes-Lott triple;

2. the Yukawa family of Theorem C converges at rate $O(l)$ to $(\mathbb C,\mathbb C^{N_Q+N_U},D_F)$;

3. propinquity limits of commutative spaces are commutative, and the number of connected components does not increase, so the finite limits of connected commutative triples are exactly the triples $(\mathbb C,\mathbb C^N,D_F)$;

4. for a symmetric block, the grading and real structure descend to the block with their signs.

So one family has two limiting descriptions, and both hold: the whole triple converges to a point carrying a Dirac block, while the compression at each $l$ is Connes-Lott with distance $1/m_f$. The metric compares whole algebras, and the compression is not a quantum isometry. Item 3 is a statement about this metric on scalar function algebras. The class of metric triples also excludes the Standard Model finite triple, where $D_F$ commutes with the nonscalar subalgebra $\{(\lambda,\lambda,0)\}$ [, Sec. 2.6].

Proposition H. Let $P_0$ be a finite-rank spectral projection, for a symmetric set, of an even triple $(A_0,H_0,D_0,\gamma_0)$, with $V=P_0H_0$, $d=D_0|_V$ and $Q(a)=P_0aP_0$. Let $A_F$ be the C*-algebra generated by $Q(A_0)$.

1. Since $Q(1)=1$, $A_F$ is the bicommutant of $Q(A_0)$ in $\mathrm{End}_\mathbb C(V)$, so its structure is set by the operators commuting with all compressed functions. For the chiral interval with the two lowest levels, $Q(C(I_l))$ is a four-dimensional operator system; the grading commutes with it, and $A_F=M_2(\mathbb C)\oplus M_2(\mathbb C)$. On $\mathbb{CP}^3$ at level one, $A_F=M_4(\mathbb C)$.

2. With a real structure satisfying $J_0aJ_0^{-1}=a^*$, the inherited data satisfy order zero iff $A_F$ is commutative.

3. On the coarse correspondence $H_0\otimes\overline{H_0}$, with $A_0$ acting on the left, $\mathcal D=D_0\otimes1+\gamma_0\otimes\overline{D_0}$ and $J$ the graded flip, one has a real even triple of signs $(+,+,+)$, with the same Connes distance on $A_0$, and its truncation $(A_F,V\otimes\overline V,d\otimes1+\gamma_V\otimes\bar d,J_F,\Gamma_F)$ satisfies order zero and order one. A product with a Clifford point gives the KO-6 signs and $\dim H=2k^2$, with $k=\dim V$.

4. KO-6 is reached in the irreducible form by a parent with two sheets, on $(H_X\otimes\overline{H_Y})\oplus(H_Y\otimes\overline{H_X})$ with $J$ exchanging the summands. For equal dimensions and full matrix algebras, the truncation is the pair $M_k(\mathbb C)\oplus M_k(\mathbb C)$, $J(x,y)=(y^*,x^*)$, of the classification (Chamseddine and Connes 2008). At twistor level one, real functions on $S^4$ and on $\mathbb{CP}^3$ generate the real algebras $M_2(\mathbb H)$ and $M_4(\mathbb C)$, giving that algebra with its bimodule of dimension 32.

Connes and van Suijlekom observe that the C*-algebra generated by $PAP$ is, in many examples, a noninformative full matrix algebra (Connes and Suijlekom 2021). That is the case here apart from the symmetries of the window. The grading commutes with compressed functions and gives the block structure on the interval. The antiunitary $j$, with $j^2=-1$, commutes with the compressions of real functions on $S^4$, which generate the real algebra $M_2(\mathbb H)$, while the complex algebra generated by the same functions is $M_4(\mathbb C)$. So item 1 is modest, and the information sits in those symmetries.

The order conditions in items 3 and 4 are formal: the same tensor square works for any finite even representation, and the two-sheet bimodule is already in [, Remark 2.9]. What is specific here is that the tensor square arises from a commutative parent, namely the product Dirac operator on $X\times X$, with algebra restricted to $C(X)\otimes1$ and $J$ covering the flip. The Clifford point is an input and fails finite orientability. In item 4 the second sheet, the base-function restriction and the window are inputs. The nontrivial grading of the classification is added after truncation as abstract data, since an even parent cannot give a grading anticommuting with compressed functions. In this parent the two sheets are not connected, so the selection of $\mathbb C\oplus\mathbb H\oplus M_3(\mathbb C)$ is not reached there; Proposition I supplies a connector in another parent. The triples are usually not metric, and the eigenvalues of the doubled operator add in quadrature.

Proposition I. Let $E=\mathrm{Hom}(V,W)$ and $E^*=\mathrm{Hom}(W,V)$, with $J(x,y)=(y^*,x^*)$, grading $gx=\gamma_Wx$, $hy=-y\gamma_W$, and the even algebra $\mathcal A^{\mathrm{ev}}$ acting by $\pi_1,\pi_2$, with opposite actions $\rho_1,\rho_2$. Write $c$ for the adjoint map $E\to E^*$.

1. A block $T:E\to E^*$ added to a sheet-diagonal $D_F$ that already has these signs and order one keeps the signs iff $T=cT^*c$ and $hT+Tg=0$. It satisfies order one on an involutive $B\subseteq\mathcal A^{\mathrm{ev}}$ iff $T\pi_1(a)-\pi_2(a)T$ intertwines $\rho_1(b)$ and $\rho_2(b)$ for all $a,b\in B$. The maximal such $B$ are the maximal $\mathcal A(R)=\{a:\pi_2(a)R=R\pi_1(a),\ \pi_2(a^*)R=R\pi_1(a^*)\}$, over odd $R$ with $T=R+cR^*c$.

2. Let $W=W_+\oplus W_-=\mathbb H\oplus\mathbb H$, $V=\mathbb C^4$, and let $w\in W_+$, $v\in V$ be unit vectors. Then, for $\kappa\neq0$, $T_\kappa x=\kappa KxK$, with $K=vw^*$, has complex rank one, and $\mathcal A(T_\kappa)=\{(\lambda\oplus q,\lambda\oplus m)\}\cong\mathbb C\oplus\mathbb H\oplus M_3(\mathbb C)$, the algebra of the classification (Chamseddine and Connes 2008).

3. Let $Q,U$ be massless interval fields valued in $E,E^*$, with the boundary conditions of Theorem C and $X=tS^*$, where $t>0$ and $S=|ce\rangle\langle e|$ for a unit vector $e$ with $ge=e$. The constant multipliers preserving the domain are exactly $B=\mathcal A(S)$. With $\theta=\arctan t$ and $m=\theta/l$, a symmetric cutoff in $(m,(\pi-\theta)/l)$ gives the finite data $(B,E\oplus E^*,m(S+S^*),J,\Gamma)$, of signs $(+,+,-)$, with order zero and order one. For $t(l)=\tan(ml)$, the mass stays $m$ and the next level diverges.

4. The transport $\exp(mxF)$, with $F$ rotating the coupled channels of $Q$ and $U$, carries $D_X$ to the uncoupled operator plus a constant term $mF$ times the Clifford generator, so after a unitary change of domain the wall coupling is a constant matrix connection with relative endpoint rotation angle $\theta=ml$.

5. For scalar multipliers and $X\neq0$, preservation of the domain is the gluing $f_Q(l)=f_U(l)$, and this does not pass to the truncation: in the one-flavor window the compressions generate $\mathbb C\oplus\mathbb C$ with $D=m\sigma_1$ and $J=\sigma_1K$, which fails order one. For separate compressions in unchanged windows, including the level-one twistor windows, scalar gluing on closed sets with empty interior, which allows independent functions off those sets, leaves the image unchanged.

Items 2 and 3 realize the classified connecting block in an exact parent, on $E=\mathrm{Hom}(\mathbb C^4,\mathbb H^2)$ with $\dim H_F=32$. The Hom fibres, the constant matrix multipliers, the grading and the coupled channel are inputs, and deriving them from the commutative twistor parent remains open. On $B$ the block commutes with the algebra, so it supplies no finite one-forms, and the Yukawa block needs further sheet-diagonal terms, which one wall coupling can carry together with it (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026). With $w$ along $\nu_R$ and $v$ along the lepton line, $T$ has the form of the one-generation Majorana block (Chamseddine, Connes, and Marcolli 2007), and the bound $m<\pi/(2l)$ ties that mass to the first level of the tower; a general three-generation Majorana matrix needs further flavour data.

We propose reading the connector as a marked pair of points with a coupling. Since $K=vw^*$, its data are a point of $\mathbb{CP}^3$ on each sheet and $\kappa$. At fixed $|\kappa|$ the line choices are conjugate, since $Sp(2)$ and $U(4)$ act transitively, and $|\kappa|$ remains a modulus. On the twistor window, $w$ lies over a point $p$ of $S^4$, and the compressed height function $5Q(p\cdot x)=p_a\Gamma_a$ can serve as added finite grading after restricting to its even algebra. This evades the inheritance hypothesis of Proposition H and does not derive the grading, and a parent with an interface at those points has not been constructed.

9 Discussion

What the towerless picture gives. The finite Dirac operator of an almost-commutative geometry behaves as the remnant of internal momentum on an interval: its excited levels combine quantized momenta with bulk masses, its light masses are set by localization and boundary couplings, its texture is localization data together with the matrix $\hat Y$, and its chirality is carried by boundary conditions, with an ordinary index that is stable under boundary couplings and an equivariant index read on the stabilizer of the coupling. In the families treated, the emergence limit needs the tower, and the collapse limit keeps a finite light sector along a tuned bulk mass or a scaled coupling. In the reduced representation the labels 9 and 11 attach to sectors: charged paired sectors carry both, and an odd number of unpaired neutrinos carries only 11.

What remains open. The results are spectral and kinematic. The length, the bulk masses and the boundary data are free, and chirality is an input through boundary conditions, as in orbifold constructions. The results reach the Connes-Lott layer and the Yukawa block as compressions. Beyond that layer, Proposition H gives finite algebras at fixed truncation, with order zero and order one on a restricted doubled parent. Proposition I supplies an operator connecting the sheets in a parent with chosen fibres. The derivation of those fibres, the selection of a particular finite algebra, the KO-6 sign and the grading from a commutative parent, and any dynamical origin remain open. Proposition G decides the whole-triple limit in one metric. Proposition E.1 concerns one classical action at quadratic order about a prescribed background. It shows that, for a top alone, the heavy limit of the 10 to 9 proposal keeps the CCM ratio only as $m_t/\Lambda\to0$, hence jointly with $\Lambda\to\infty$; at fixed cutoff, with a light sector retained, the vectors end up light in the families treated. The link between the KO labels and these limits, and between $l$ and measured parameters, remains a proposal. A companion note (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026) carries Theorem C to the whole finite Dirac operator, as the relative position of two gradings at the walls, with the compression as an exact average, for a matrix-valued parent. There the sum rule of Proposition E has the coefficient $(3-d)/24$, with $d$ the spectral dimension of the interval at the cutoff, so the relation $8M_W^2=\sum N_fm_f^2$ is its value at $d=0$.

No-go results and their escapes. Each negative statement above holds in a stated family, at a stated order, and we state it together with its hypotheses and known escapes, as the Coleman-Mandula theorem is stated together with supersymmetry. The list below collects them. We read them as guides to where to go next; none of them closes the question it bears on, and other escapes are possible.

• No net chiral spectrum from an odd bulk: smooth closed product, standard reduction, exact isometries. Escapes: Boundaries and defects (Hořava and Witten 1996); massive non-Killing gauge fields (Baptista 2026); as our proposal, an even interior with odd limits.

• No odd factorization: full chiral Standard Model representation, involution of Theorem D. Escapes: Reduced algebra and representation, with vectorlike sectors (Theorem D).

• Only KO-7: scalar-character neutral sector, a singular value of odd multiplicity, zero included. Escapes: Even multiplicity; $\nu_R$ with Dirac pairing (Theorem D).

• Empty light sector in collapse: massless type-(c) interval, fixed spectral cutoff (Proposition A.1). Escapes: Type-(a) zero modes; tuned bulk mass (Theorem B); scaled boundary coupling (Theorem C).

• Fixed compression not gauge covariant: local gauge transformations, fixed light projection (Proposition F). Escapes: A projection invariant under the transformation; covariant projection.

• First order fails: two-point compression, inherited $J$ (Theorem A). Escapes: Fluctuations without order one (Chamseddine, Connes, and Suijlekom 2013b); doubled parent (Proposition H).

• Order zero fails: noncommutative $A_F$, $J_0$-invariant window, inherited pointwise reality (Proposition H). Escapes: Doubled or two-sheet parent; another opposite action.

• No matching with bounded fermions: leading action, positive kinetic form, stated trace, common profiles (Proposition F). Escapes: Tower-scale fermion, which removes the bound, while full matching needs more; other order or action.

• $M_W\to0$ at fixed cutoff: massless complete doublet, stated heavy family, Taylor mass, $f(0)>0$, classical quadratic action (Proposition E.1). Escapes: Joint limit $\Lambda\to\infty$; every doublet heavy; a doublet of fixed nonzero mass; $f(0)=0$; quantum effective action.

• No finite nonzero radial vacuum: ray $V=cV_*$, flat background, nonnegative cutoff of unbounded support, no added term (Proposition E.1). Escapes: Compact support (a plateau); sign-changing cutoffs; a potential or constraint; a compact radial direction on the interval, with its tower (“The Finite Dirac Operator as a Boundary Condition on an Interval” 2026).

• Proposed $\Gamma_5$ not inherited: even observables, $g=P\gamma P$, $g^2=1$, coordinates retained (Proposition H). Escapes: Graded prealgebra; added finite data with a restricted even algebra.

• Scalar gluing selects no algebra: closed sets with empty interior, independent functions off them, unchanged separate windows (Proposition I). Escapes: Coupled window; boundary degrees of freedom; restricted multipliers; changed representation.

• Finite limits are points with a Dirac block: connected scalar algebras, this propinquity class (Proposition G). Escapes: Truncation; matrix-valued parent algebras; other notions of convergence.

Relation to the literature. Theorem A: the Kaluza-Klein reading of the two-point space is in Connes-Lott (Connes and Lott 1991), the two points as points of a fifth dimension in (Viet and Wali 1996), the two-point model as an approximation of a manifold in (Dimakis and Müller-Hoissen 1994), and a discrete Kaluza-Klein reading in (Martinetti and Wulkenhaar 2002); compression and spectral truncation are standard (D’Andrea, Lizzi, and Martinetti 2014; Connes and Suijlekom 2021). We did not find their application to the chiral interval, with $|m|=\pi/(2l)$ and $PC(I_l)P=\mathbb C\oplus\mathbb C$. Theorem B: wall modes with exponentially small mixing are classical (Jackiw and Rebbi 1976; Kaplan 1992; Shamir 1993), and collapse of spectral triples to a point is treated in (Farsi and Latrémolière 2026) (doi:10.1112/jlms.70628), Corollary 4.1 (Corollary 3.1 in arXiv:2404.00240v1): a fixed metric triple $(A,H,D)$ with $0\in\mathrm{Sp}(D)$ gives $\Lambda_{\mathrm{spec}}\bigl((A,H,D/\varepsilon),(\mathbb C,\ker D,0)\bigr)\to0$ as $\varepsilon\to0$. This is the case $\Sigma=\{0\}$, $D_F=0$ of Proposition G (Supplemental Material, part 15, Remark 15.4). In Theorem B, $\mu(l)l\to\infty$, so the family is no rescaling of a fixed triple, and its limit keeps the block $m_f\sigma_1$; we did not find it elsewhere. Theorem C: the one-flavor spectrum is in (Csáki et al. 2004), Eq. (5.6), and warped multiflavor textures are in (Casagrande et al. 2008); we did not find a derivation of $D_F$ from localization in noncommutative geometry. Theorem D: products of real triples and the choice $J$ versus $J\Gamma$ are in (Dąbrowski and Dossena 2011; Vanhecke, Silva, and Sigaud 2012); we did not find the factorization criterion or the results on 9 and 11. Proposition E restates (Chamseddine, Connes, and Marcolli 2007), Eqs. (4.16) and (5.27). The top is also separate in that it is unable to form mesons or diquarks (Rivero 2005). Proposition G: the tunnels follow the proof of (Farsi and Latrémolière 2026), Theorem 3.4 (Theorem 2.5 in arXiv:2404.00240v1; Theorem 3.4 in v3 and in print, with the same six hypotheses). That theorem asserts that if $D=D_h+D_v$, with $0$ isolated in $\mathrm{Sp}(D_v)$, satisfies six conditions on commutators, on $\ker D_v$ and on a positive projection of $A$ onto a subalgebra $B$, then the triples with $D_h+D_v/\varepsilon$ converge to $(B,\ker D_v,pD_hp)$, with $p$ the projection on $\ker D_v$. The scalar family $A=B=\mathbb C$ on $\ell^2$, with $D_ve_n=ne_n$ and $D_he_n=-n^2e_n$, meets the six conditions; at $\varepsilon=1/(n+x/n)$, for fixed $x>0$, the eigenvalues $0$ and $x$ persist, so the converse in Proposition G excludes the limit $(\mathbb C,\mathbb Ce_0,0)$ (Counterexample R1, ours; Supplemental Material, part R1, Section 3.1). The step that fails bounds $\|D_v\xi\|/\varepsilon$ by $\|D_\varepsilon\xi\|$ (Eq. (3.11) in print); anticommutation of $D_h$ and $D_v$, which all its applications satisfy, or a uniform relative bound repairs it, and Proposition G uses the gap of $D$ instead. Proposition H: the first-order form $D=\Delta+J\Delta J^{-1}$ appears in the classification of finite triples (Krajewski 1998; Paschke and Sitarz 1998), and the Hilbert-Schmidt bimodule is the coarse correspondence (Connes 1994). Inner fluctuations without order one (Chamseddine, Connes, and Suijlekom 2013b, 2013a) apply to the two-point data: they factorize as $m|\Phi|^2\sigma_1$ for pointwise $J$, with a trivial gauge group, and as a link $m\Phi^2$ of charge 2 for the reflection $J$, whose quadratic term keeps the modulus fixed along a gauge orbit (Supplemental Material, part 19). Proposition I: the algebra $\mathcal A(T)$ and the maximality argument are in (Chamseddine and Connes 2008), Section 4; the parent is what is added here. As mathematics, most ingredients are known; what may be new is their combination in one limiting construction. Statements that something was not found refer to searches that are recorded, with queries, in the Supplemental Material (part L); they do not establish absence.