Charged-Lepton Mass Loading from Minimal Finite-Response Slot Completion on a Declared Gauge-Chiral Family in the CHC Framework

The proposed CLM construction uses inputs from the CHC root, operational phase-link, electroweak, and precision papers [citation]. We separate valid representation-theoretic consequences from normalization choices and from the numerical mass ansatz.

definition: Admissible CLM branch category. An admissible CLM branch category $\Bcat$TeX\Bcat consists of objects

(\mathfrak B_{\rm root},\mathfrak B_{\rm EW},\mathfrak W,\Pi_{\rm nt})

with the following structure.

- $\mathfrak B_{\rm root}$TeX\mathfrak B_{\rm root} is a small-gradient CHC root branch with controlled GR recovery and root normalization $\Mroot$TeX\Mroot. - $\mathfrak B_{\rm EW}$TeX\mathfrak B_{\rm EW} is a declared color-singlet $Q=-1$TeXQ=-1 gauge-chiral charged-lepton branch with left/right charged-spinor boundary set $C=\{L,R\}$TeXC=\{L,R\} and three-family cyclic comparison sector. - $\mathfrak W$TeX\mathfrak W is the two-layer operational incidence set $O=\{P,K\}$TeXO=\{P,K\}, with $P$TeXP denoting phase-link propagation and $K$TeXK denoting local commit. - $\Pi_{\rm nt}$TeX\Pi_{\rm nt} is a source partition for which charged-lepton masses, charged-lepton Yukawa values, $G_F$TeXG_F scale inversion, weak-boson mass inversion, Higgs-vacuum lookup values, and atomic metrology inversions have no edge into upstream prediction nodes.

Morphisms in $\Bcat$TeX\Bcat preserve chirality, operational layer, charge, color-singlet status, family cyclic type, and spinor deck type.

The electroweak recovery interface is

m_{\ell_i}^{\rm CHC}=\frac{v_*^{\rm CHC}}{\sqrt2}\,y_{\ell_i}^{\rm CHC},
 \qquad i=1,2,3.

Equation reference is used as the recovery map between a branch-level loading scale, an ordered charged-lepton singular spectrum, and charged-lepton masses.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

theorem: Minimum faithful complex dimension. For a prime $p$TeXp and integer $r\ge1$TeXr\ge1, the minimum dimension of a faithful complex linear representation of the elementary abelian group $G=(\Z_p)^r$TeXG=(\Z_p)^r is $r$TeXr.

proof. Every irreducible complex representation of the finite abelian group $G$TeXG is a one-dimensional character. Hence any $d$TeXd-dimensional representation is a direct sum of $d$TeXd characters $\chi_1,\ldots,\chi_d$TeX\chi_1,\ldots,\chi_d. Identify the character group $\widehat G$TeX\widehat G with the dual vector space $(\mathbb F_p^r)^*$TeX(\mathbb F_p^r)^*. The kernel of the representation is $\bigcap_{j=1}^d\ker\chi_j$TeX\bigcap_{j=1}^d\ker\chi_j, which is trivial exactly when the characters span $(\mathbb F_p^r)^*$TeX(\mathbb F_p^r)^*. Fewer than $r$TeXr vectors cannot span an $r$TeXr-dimensional vector space, so $d\ge r$TeXd\ge r. Conversely, the $r$TeXr coordinate characters form a dual basis and their direct sum is faithful. Thus the minimum is $r$TeXr.

corollary: Dimensions relevant to the proposed response groups. The groups $(\Z_3)^4$TeX(\Z_3)^4, $(\Z_2)^4$TeX(\Z_2)^4, and $(\Z_2)^5$TeX(\Z_2)^5 have minimum faithful complex dimensions $4$TeX4, $4$TeX4, and $5$TeX5, respectively. Their regular representations have dimensions $81$TeX81, $16$TeX16, and $32$TeX32 and are not minimal.

proof. Apply reference to $(p,r)=(3,4),(2,4),(2,5)$TeX(p,r)=(3,4),(2,4),(2,5). The regular representation of a finite group has dimension equal to the group order.

theorem: Finite symmetry does not determine charged-lepton masses. Let a finite abelian group act on three one-dimensional complex character spaces. For every ordered triple of positive numbers $(m_1,m_2,m_3)$TeX(m_1,m_2,m_3), there exists a positive group-invariant mass operator with precisely those eigenvalues. Consequently, the group, its character table, and faithfulness alone do not determine charged-lepton mass ratios.

proof. In a character basis the representation is diagonal,

\rho(g)=\diag(\chi_1(g),\chi_2(g),\chi_3(g)).

For arbitrary $m_i>0$TeXm_i>0, let $M=\diag(m_1,m_2,m_3)$TeXM=\diag(m_1,m_2,m_3). Then $M$TeXM is positive and $M\rho(g)=\rho(g)M$TeXM\rho(g)=\rho(g)M for every $g$TeXg. Thus every positive spectrum is compatible with the same finite symmetry. Further dynamical constraints are necessary to select a spectrum.

Let

\Icat=C\times O,
\qquad C=\{L,R\},\qquad O=\{P,K\}.

The four objects of $\Icat$TeX\Icat are the formally specified boundary-layer incidences

(L,P),\ (L,K),\ (R,P),\ (R,K).

A finite response functor assigns to each incidence a family comparison coordinate and a spinor deck coordinate.

definition: Response-reflecting functor. A functor

\Rfun:\Icat\longrightarrow \mathsf{Rep}_{\C}(G)

is response-reflecting when the induced maps on formally specified hom-sets are injective and type-reflecting: if two morphisms become equal under $\Rfun$TeX\Rfun, then they already agree in chirality, operational layer, family comparison type, and spinor deck type in $\Icat$TeX\Icat.

proposition: Faithfulness under an explicit separation hypothesis. Any finite response functor that is assumed to preserve and separate the stated incidence types is faithful on those incidences. This is a consequence of the separation hypothesis and does not determine a representation dimension.

proof. If two distinct chiral incidences are identified by an admitted functor, then the formally specified left/right singular-spectrum separation of the charged-lepton recovery map ceases to distinguish the two boundary domains. The statement concerns preservation of the formally specified slot in which a branch-level singular value may later be loaded, not preservation of an observed numerical charged-lepton spectrum. If $P$TeXP and $K$TeXK are identified on either boundary, phase-linked propagation and local commit become the same operational morphism, contradicting the stated branch incidence set. If two family comparison morphisms are identified, the three-family cyclic comparison ceases to be represented faithfully. If two spinor deck morphisms are identified, the $2\pi$TeX2\pi spinor sign is not represented on that incidence. Each identification violates one of the type-preservation clauses in $\Bcat$TeX\Bcat. Hence the functor is response-reflecting.

corollary: Four coordinate characters suffice. Four independent order-three coordinates and four independent binary coordinates admit faithful complex representations of dimensions four and four. The group algebras $\C[(\Z_3)^4]$TeX\C[(\Z_3)^4] and $\C[(\Z_2)^4]$TeX\C[(\Z_2)^4] are regular representations of dimensions $81$TeX81 and $16$TeX16, not minimal four-slot supports.

proof. This is reference. The coordinate characters give the upper bound, and the dual-space spanning argument gives the lower bound.

The defect sector is a disjoint support. It is not a subprojection of the normal response support. It records the finite orientation obstruction between normal phase response and charged commit-readable spinor response.

definition: Admissible charged-spinor commit defect. An admissible charged-spinor commit defect is a binary orientation character on a formally specified charged-lepton response boundary satisfying four conditions: it is local to a boundary incidence, it preserves the $Q=-1$TeXQ=-1 color-singlet branch, it distinguishes spinor deck sign from charge orientation, and it distinguishes propagation-side phase comparison from commit-side readout. The defect parity space is an $\Ftwo$TeX\Ftwo-vector space.

definition: Defect orientation basis. Let

\Def_{\rm CLM}=\langle O_L,O_R,S,Q,K\rangle_{\Ftwo},

where $O_L$TeXO_L and $O_R$TeXO_R are the left and right boundary orientation bits, $S$TeXS is the spinor deck bit, $Q$TeXQ is the charged $Q=-1$TeXQ=-1 orientation bit, and $K$TeXK is the commit-side orientation bit.

lemma: Independence. The five defect orientations are independent.

proof. For each generator $g_i\in\{O_L,O_R,S,Q,K\}$TeXg_i\in\{O_L,O_R,S,Q,K\} define a character $\chi_i:\Def_{\rm CLM}\to\{\pm1\}$TeX\chi_i:\Def_{\rm CLM}\to\{\pm1\} by $\chi_i(g_i)=-1$TeX\chi_i(g_i)=-1 and $\chi_i(g_j)=1$TeX\chi_i(g_j)=1 for $j\ne i$TeXj\ne i. If $O_L^{a_1}O_R^{a_2}S^{a_3}Q^{a_4}K^{a_5}=1$TeXO_L^{a_1}O_R^{a_2}S^{a_3}Q^{a_4}K^{a_5}=1, applying $\chi_i$TeX\chi_i gives $(-1)^{a_i}=1$TeX(-1)^{a_i}=1 for every $i$TeXi. Hence each $a_i=0$TeXa_i=0 in $\Ftwo$TeX\Ftwo.

lemma: Exhaustiveness. Every admissible charged-spinor commit defect is a product of $O_L,O_R,S,Q,K$TeXO_L,O_R,S,Q,K.

proof. Let $d$TeXd be an admissible defect. Locality to the charged response boundary restricts $d$TeXd to boundary-side parity, spinor-deck parity, charge parity, and operational readout parity. Boundary-side parity has exactly two formally specified endpoint components because the charged-lepton singular map has left and right domains. Spinor-deck parity has one component because the deck group of a spinor comparison is binary. Charge parity has one component because the branch is fixed to the charged $Q=-1$TeXQ=-1 color-singlet sector. Operational readout parity has one component because the only operational distinction not already carried by normal propagation is the commit-side closure. Thus

d=O_L^{a_1}O_R^{a_2}S^{a_3}Q^{a_4}K^{a_5}.

A sixth independent binary defect would either be trivial on all formally specified boundary, deck, charge, and commit tests, in which case it is not response-reflecting, or it would change one of the branch types fixed in $\Bcat$TeX\Bcat. Hence no sixth admissible binary orientation exists.

theorem: Conditional defect classification. Within the stipulated five-bit definition, the defect group is

G_{\rm def}\cong(\Z_2)^5,

and its minimum faithful complex representation dimension is five.

proof. The five stipulated binary coordinates and the preceding independence lemma identify the defined vector space with $(\Z_2)^5$TeX(\Z_2)^5. The minimum faithful complex dimension is five by reference. The regular representation separates group elements but has dimension $32$TeX32 and is not minimal.

theorem: Minimum dimension under separate-support constraints. If the three response groups are required to act on separate faithful summands, the minimum total complex dimension is

4+4+5=13.

The direct sum of regular representations

\C[(\Z_3)^4]\oplus\C[(\Z_2)^4]\oplus\C[(\Z_2)^5]_{\rm def}

has dimension $129$TeX129 and is a nonminimal choice.

proof. The lower bounds $4$TeX4, $4$TeX4, and $5$TeX5 follow from reference; coordinate-character sums attain all three simultaneously. Dimensions add under the required direct-sum separation. The regular dimensions are the group orders $3^4$TeX3^4, $2^4$TeX2^4, and $2^5$TeX2^5.

Let

V_N=\C[(\Z_3)^4]\oplus\C[(\Z_2)^4].

For a finite regular representation $\C[G]$TeX\C[G], the invariant subspace under left translation is the uniform line. Let $P_{0,3}$TeXP_{0,3} and $P_{0,2}$TeXP_{0,2} be the rank-one projectors onto the two uniform lines in the two summands of $V_N$TeXV_N.

proposition: Fixed lines in the chosen regular representations. The particular normal representation $V_N$TeXV_N has exactly two invariant lines.

proof. Each regular summand has a one-dimensional fixed subspace spanned by the uniform vector. Fixed subspaces add under direct sums, so $\dim\Fix(V_N)=2$TeX\dim\Fix(V_N)=2. This count is a property of the chosen regular representations, not of faithful representations in general.

definition: Trace operators.

\NRSD=\Id_{\C[(\Z_3)^4]}\oplus\Id_{\C[(\Z_2)^4]}-P_{0,3}-P_{0,2},

\TRSD=\Id_{\C[(\Z_2)^5]_{\rm def}}.

The two supports are disjoint.

proposition: Arithmetic quotient for the chosen regular representation.

\DeltaTr=\frac{\Tr(\TRSD)}{\Tr(\NRSD)}=\frac{2^5}{3^4+2^4-2}=\frac{32}{95}.

proof. The two rank-one projectors have trace one. Thus $\Tr(\NRSD)=3^4+2^4-2=95$TeX\Tr(\NRSD)=3^4+2^4-2=95. The defect regular representation has dimension $2^5$TeX2^5, so $\Tr(\TRSD)=32$TeX\Tr(\TRSD)=32. The quotient is $32/95$TeX32/95. Conjugation preserves these traces for this representation, but does not make the quotient representation-independent.

theorem: The trace quotient is not a group invariant. The number $32/95$TeX32/95 is not determined by the groups $(\Z_3)^4$TeX(\Z_3)^4, $(\Z_2)^4$TeX(\Z_2)^4, and $(\Z_2)^5$TeX(\Z_2)^5 or by faithfulness. Faithful choices exist with different dimensions and different fixed-subspace traces.

proof. By reference, coordinate-character representations have dimensions $4$TeX4, $4$TeX4, and $5$TeX5, unlike the regular dimensions $81$TeX81, $16$TeX16, and $32$TeX32. Moreover, if $V$TeXV is faithful then $V\oplus\C_{\rm triv}^k$TeXV\oplus\C_{\rm triv}^k is faithful for every $k\ge0$TeXk\ge0, while its dimension and fixed-subspace trace increase by $k$TeXk. Therefore no quotient formed from these traces is fixed by the group or by faithfulness alone.

The quantities in this section define a dimensionless normalization ansatz. Topological phase data determine phases modulo $2\pi$TeX2\pi but do not determine an action functional or its relative coefficients.

definition: Root biphase torus. The minimal compact root response cell preserving both the GR-recovery root phase and local charged-response comparison is

T^2_{\rm root}=S^1_{2\pi}\times S^1_{2\pi}.

With unit phase stiffness its area action is $4\pi^2$TeX4\pi^2.

lemma: Central spinor holonomy. For a charged spinor branch, the unique minimal positive action of the nontrivial central holonomy class over a $2\pi$TeX2\pi comparison is $\pi$TeX\pi.

proof. The spin group double-covers the rotation group. A $2\pi$TeX2\pi rotation acts by the nontrivial central element $-1$TeX-1 on a spinor, while a $4\pi$TeX4\pi rotation returns the spinor to the identity sheet. In the phase representation of the central $\Uone$TeX\Uone line, the element $-1$TeX-1 is $e^{i\pi}$TeXe^{i\pi}. The positive representative in $[0,2\pi)$TeX[0,2\pi) is unique and equal to $\pi$TeX\pi. No smaller positive action represents the nontrivial central class.

proposition: Root-action value under the subtraction convention.

\Sroot=4\pi^2-\pi.

proof. The convention assigns $4\pi^2$TeX4\pi^2 to the torus and subtracts the representative $\pi$TeX\pi of the central phase. The displayed value follows arithmetically. Neither topology nor representation theory fixes the subtraction rule.

proposition: Primitive $\Z_3$TeX\Z_3 phase.

\Sph=\frac{2\pi}{3}.

proof. The nontrivial primitive characters of $\Z_3$TeX\Z_3 are $e^{\pm2\pi i/3}$TeXe^{\pm2\pi i/3}. The least positive phase representative is $2\pi/3$TeX2\pi/3; interpreting it as an action is an additional normalization convention.

The weak-balance selector is a polynomial ansatz defined by the coefficients stipulated below. Its algebraic uniqueness within that definition does not imply uniqueness among physical electroweak models.

Evenness and normalization alone leave infinitely many coefficients. The remaining conditions select particular coefficients by convention.

definition: Admissible weak-balance selector. Let $x$TeXx denote the minimal weak leakage amplitude. A weak-balance selector is admissible when it satisfies:

- $R(x)$TeXR(x) is even under spinor-sheet reversal $x\mapsto -x$TeXx\mapsto -x. - $R(0)=1$TeXR(0)=1. - The leading two-sheet leakage Gram determinant has one unit leakage line, so the quadratic term is $-x^2$TeX-x^2. - The first closed return is quartic and carries the squared Hilbert-Schmidt restitution weight of the balanced projection from four boundary-layer incidences to three family modes. - No sixth-order or higher term is admitted in the minimal finite-response closure.

lemma: Reciprocal-product normalization. The leakage amplitude is

\epsEW=\frac{1}{3\pi}.

proof. Under the stipulated rule that the amplitude equals the reciprocal product of the group order $3$TeX3 and the phase representative $\pi$TeX\pi, its value is $1/(3\pi)$TeX1/(3\pi). The rule is not implied by either factor.

lemma: Squared dimension-ratio convention. The quartic closed-return coefficient is $9/16$TeX9/16.

proof. Let $E$TeXE be four-dimensional and $F$TeXF three-dimensional. The adopted normalized dimension ratio is

\rho_{3/4}=\frac{\dim F}{\dim E}=\frac34.

Squaring this ratio gives $(3/4)^2=9/16$TeX(3/4)^2=9/16. A general projection is not determined by the dimensions of its domain and codomain, so the use of this ratio as a dynamical coefficient is a convention.

proposition: Selector fixed within the stipulated quartic class. The unique admissible weak-balance selector is

\rew=1-\epsEW^2+\frac{9}{16}\epsEW^4=1-\frac{1}{9\pi^2}+\frac{1}{144\pi^4}.

proof. The definition restricts $R$TeXR to $1+a x^2+b x^4$TeX1+a x^2+b x^4, stipulates $a=-1$TeXa=-1, and uses the squared dimension-ratio convention $b=(3/4)^2$TeXb=(3/4)^2. Substitution of $x=1/(3\pi)$TeXx=1/(3\pi) gives reference. The uniqueness is therefore definitional within this polynomial class.

The numerical ansatz starts from the measured unreduced Planck-scale interval

\Mroot\in[1.220876,1.220904]\times10^{19}\GeV.

and postulates the exponential scale chain

\Lamref=\Mroot e^{-\Sroot},
\qquad
\LamA=\Lamref e^{-\Sph},
\qquad
v_*^{\rm CHC}=\rew\LamA.

Therefore

\Lamref\in[2022.0364982557,2022.0828723526]\GeV,\notag

\LamA\in[249.0031003510,249.0088110757]\GeV,\notag

v_*^{\rm CHC}\in[246.2175978487,246.2232446898]\GeV.

The finite-group representations do not derive the exponential map or the additive action values appearing in this chain. These equations are therefore inputs to the numerical construction.

Let $V_{\rm fam}=\C[\Z_3]$TeXV_{\rm fam}=\C[\Z_3] and let $\chi(k)=e^{2\pi i k/3}$TeX\chi(k)=e^{2\pi i k/3}. The real response subspace is spanned by the trivial character and the real two-dimensional nontrivial character plane.

definition: Admissible family spectrum operator. A family spectrum operator is admissible when it is a positive rank-one operator on $V_{\rm fam}$TeXV_{\rm fam}, is real under conjugation of the nontrivial character pair, contains one trivial component normalized to unit deck weight, assigns the real two-dimensional nontrivial spinor-deck plane the same Hilbert--Schmidt deck weight as that trivial line, splits that nontrivial deck weight equally between the conjugate amplitudes, and has no coefficient depending on a charged-lepton mass or Yukawa target.

lemma: Real character vector. Every admissible family spectrum operator is generated by a vector

u_\theta(k)=1+\frac{1}{\sqrt2}\left(e^{i\theta}\chi(k)+e^{-i\theta}\overline{\chi(k)}\right)=1+\sqrt2\cos\left(\theta+\frac{2\pi k}{3}\right).

proof. A real vector with one trivial component and one nontrivial conjugate pair has the displayed form for some phase $\theta$TeX\theta and conjugate amplitude $a$TeXa. By admissibility, the trivial component has unit deck weight and the nontrivial real two-dimensional spinor-deck plane carries the same Hilbert--Schmidt deck weight. Since that plane is represented by a conjugate pair and the admissibility clause splits the unit deck weight equally between the pair, each conjugate amplitude has squared norm $1/2$TeX1/2, fixing $a=1/\sqrt2$TeXa=1/\sqrt2. Positivity and rank one then force the operator to be $|u_\theta\rangle\langle u_\theta|$TeX|u_\theta\rangle\langle u_\theta| up to scalar normalization.

definition: Chosen affine phase normalization. The ansatz sets the family phase parameter equal to the ratio of two spinor-sheet labels to a $3\times3$TeX3\times3 chart count.

lemma: Evaluation of the chosen phase. Under the admissible affine sheet-to-family-cell normalization,

\thetafam=\frac{2}{9}.

proof. The stipulated ratio is $2/(3\times3)=2/9$TeX2/(3\times3)=2/9.

proposition: Spectrum of the chosen rank-one ansatz. For $\theta=2/9$TeX\theta=2/9, the diagonal entries of the rank-one operator

P_{\thetafam}=\frac{|u_{\thetafam}\rangle\langle u_{\thetafam}|}{\langle u_{\thetafam},u_{\thetafam}\rangle}.

Equivalently,

w_k=\frac{\left[1+\sqrt2\cos\left(\thetafam+2\pi k/3\right)\right]^2}{\sum_{j=0}^{2}\left[1+\sqrt2\cos\left(\thetafam+2\pi j/3\right)\right]^2}.

The denominator is $6$TeX6, and the sorted weights are

(w_e,w_\mu,w_\tau)=(0.00027135251555,0.05610764538022,0.94362100210423).

proof. For the chosen vector, a positive rank-one operator has diagonal weights proportional to $|u_{\theta_{\rm fam}}(k)|^2$TeX|u_{\theta_{\rm fam}}(k)|^2. Character orthogonality gives

\sum_{k=0}^{2}u_\theta(k)^2=3+2\sum_{k=0}^{2}\cos^2\left(\theta+\frac{2\pi k}{3}\right)=6.

Substitution of $\theta=2/9$TeX\theta=2/9 and ordering gives reference.

theorem: Continuous spectrum non-uniqueness. The symmetry and rank-one positivity conditions admit the continuous family $P_\theta=|u_\theta\rangle\langle u_\theta|/\langle u_\theta,u_\theta\rangle$TeXP_\theta=|u_\theta\rangle\langle u_\theta|/\langle u_\theta,u_\theta\rangle for $\theta\in\R$TeX\theta\in\R. The associated ordered weights are not constant in $\theta$TeX\theta. Hence $\Z_3$TeX\Z_3 character theory does not select reference or charged-lepton mass ratios.

proof. For every real $\theta$TeX\theta, reference is real and $P_\theta$TeXP_\theta is positive with rank one. At $\theta=0$TeX\theta=0 the squared components are $(1+\sqrt2)^2$TeX(1+\sqrt2)^2 and two copies of $(1-1/\sqrt2)^2$TeX(1-1/\sqrt2)^2. At $\theta=\pi/6$TeX\theta=\pi/6 they are $(1+\sqrt{3/2})^2$TeX(1+\sqrt{3/2})^2, $(1-\sqrt{3/2})^2$TeX(1-\sqrt{3/2})^2, and $1$TeX1. The multisets differ, so the normalized ordered weights are not constant. The group fixes the available characters but not a continuous vector in their span.

The charged-lepton singular sum is

Y_\Sigma=e^{-A_\Sigma}.

The independent action components are spin-frame solid angle per family, trace defect, and residual chiral leakage.

proposition: Residual term under the reciprocal-square convention.

\Ares=\left(\frac{1}{3\pi^2}\right)^2=\frac{1}{9\pi^4}.

proof. The stipulated amplitude $1/(3\pi^2)$TeX1/(3\pi^2) has squared norm $1/(9\pi^4)$TeX1/(9\pi^4). Representation theory does not require this amplitude.

proposition: Additive action ansatz.

\Asum=\frac{4\pi}{3}+\DeltaTr+\Ares
=\frac{4\pi}{3}+\frac{32}{95}+\frac{1}{9\pi^4}.

proof. Equation reference follows by adding the three stipulated terms. Direct-sum decomposition does not by itself fix their coefficients or require additive contribution to a physical action.

The finite-window action resolution is

\dA=\frac{1}{2^4 3^4\pi^2},

the inverse normal finite slot volume multiplied by the compact phase-stiffness scale. Hence

Y_\Sigma\in[e^{-(\Asum+\dA)},e^{-(\Asum-\dA)}]
=[0.0108146762568,0.0108163673702].

The charged-lepton singular spectrum is

(y_e,y_\mu,y_\tau)^{\rm CHC}=Y_\Sigma(w_e,w_\mu,w_\tau).

definition: Specified finite-response formula ansatz. The formula formalism consists of the following operations only: finite cardinality, regular-representation trace, central $\Uone$TeX\Uone holonomy action, primitive finite character phase, even two-sheet Gram determinant correction, positive rank-one character projector, exponential action loading, and interval arithmetic. It contains no operation that reads a charged-lepton comparison target.

proposition: Evaluation within the specified formula ansatz. If every normalization and operation in reference is imposed, the formulas

\Sroot=4\pi^2-\pi,
\quad
\Sph=2\pi/3,
\quad
\rew=1-1/(9\pi^2)+1/(144\pi^4),

\thetafam=2/9,
\quad
\DeltaTr=32/95,
\quad
\Ares=1/(9\pi^4),
\quad
\Asum=4\pi/3+32/95+1/(9\pi^4)

follow by substitution. They are not consequences of finite-group representation theory alone.

proof. Each displayed formula is one of the stipulated normalization rules or its direct arithmetic consequence. The earlier propositions evaluate those rules. reference and reference show that alternative faithful representations and rank-one character vectors satisfy the same underlying symmetries.

proposition: Formula-dependency separation. The prediction graph for $v_*^{\rm CHC}$TeXv_*^{\rm CHC}, $Y_\Sigma$TeXY_\Sigma, $(w_e,w_\mu,w_\tau)$TeX(w_e,w_\mu,w_\tau), and $(m_e,m_\mu,m_\tau)^{\rm CHC}$TeX(m_e,m_\mu,m_\tau)^{\rm CHC} contains no charged-lepton target and no Fermi-constant electroweak-scale inversion.

proof. The parent set of the prediction graph is

\{\pi,\Mroot,\Sroot,\Sph,\rew,\thetafam,\DeltaTr,\Ares,\dA,3^4,2^4,2^5\}.

The graph edges are exactly those in reference, reference, reference, reference, reference, and reference. Charged-lepton masses, charged-lepton Yukawa values, $G_F$TeXG_F, weak-boson masses, Higgs-vacuum lookup values, Bohr/Rydberg/Compton inversions, and downstream CODATA comparison targets are absent from the parent set.

theorem: A formula-dependency graph does not establish prior independence. Absence of target masses from the evaluation graph proves only that the final arithmetic does not read those variables. It does not prove that the formula class, constants, or normalization conventions were selected independently of the targets.

proof. For any fixed constants $c$TeXc chosen after inspecting a target, an evaluation $f(c)$TeXf(c) has a directed graph whose parent is $c$TeXc and which contains no target node. Thus graph separation is compatible with retrospective selection. In the present construction the continuous family $P_\theta$TeXP_\theta from reference and the representation choices from reference provide explicit alternative constants. Prior independence requires a timestamped or otherwise auditable pre-specification and an out-of-sample test, neither of which follows from graph topology.

proposition: Arithmetic interval inclusion. Equations reference, reference, reference, and reference give center

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center Each displayed reference value lies inside the corresponding arithmetic interval.

proof. For each sorted weight $w_i$TeXw_i in the chosen $\theta=2/9$TeX\theta=2/9 ansatz, compute

I_i=\frac{1000}{\sqrt2}[v^-Y^-w_i,v^+Y^+w_i],

where $[v^-,v^+]$TeX[v^-,v^+] is the interval in reference and $[Y^-,Y^+]$TeX[Y^-,Y^+] is the interval in reference. Direct interval multiplication gives the displayed intervals. The comparison constants are used only after the intervals have been computed.

remark: Statistical status. The displayed ranges propagate only the chosen Planck-scale input interval through fixed formulas. They are not confidence or credible intervals and include no uncertainty for the discrete representation choice, $\theta$TeX\theta, action normalizations, truncation, radiative corrections, or model selection. Interval inclusion is therefore not a calibrated significance test.

The numerical relation must be treated as a phenomenological ansatz until its choices are fixed independently and evaluated on data not used in their design [citation].

Equation reference has the algebraic form of a Yukawa mass relation, but the finite-group construction does not derive its scale, couplings, renormalization scheme, or radiative evolution. The quantities in reference are therefore ansatz parameters, not renormalized Standard-Model Yukawa couplings.

A finite Abelian response representation can organize charged-lepton labels, but neither its dimension nor compactness of the root scalar target determines three mass eigenvalues. Winding sectors supply global scalar boundary data and a gradient-energy bound; they do not generate Yukawa entries or a protected spectral index. The mass-prediction no-go theorem therefore remains valid after the compact-target extension.

Any proposed phase-dependent mass functional must also pass the inverse variational test: the Fr\'echet derivative of its scalar loading representative must be formally self-adjoint under the declared boundary conditions. Even a valid action becomes predictive only if one frozen parameter vector controls more mass, mixing, and precision observables than its sensitivity rank. The resulting left-null relations are the testable content. Assigning independent slot weights to the three masses saturates the rank and restates the measured spectrum rather than deriving it.

The closure test for the charged-lepton mass loading is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of charged-lepton masses, ratios, phase derivatives, and precision residuals. Let $a$TeXa range over the independent constitutive inputs comprising finite response slots, Yukawa action, spurion coefficients, and renormalization data.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if one gauge-invariant variational mass action with a finite spurion representation determines all three generations.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because assigning an independent response function to each mass slot reproduces the spectrum but predicts no mass relation. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The original minimal-representation premise is false: the relevant minimum faithful dimensions are $4$TeX4, $4$TeX4, and $5$TeX5, not $81$TeX81, $16$TeX16, and $32$TeX32. The value $32/95$TeX32/95 is an arithmetic property of a selected regular representation and changes under other faithful representations. The cyclic character construction likewise contains a continuous parameter $\theta$TeX\theta not fixed by $\Z_3$TeX\Z_3 symmetry.

Most decisively, finite abelian symmetry permits an arbitrary positive invariant diagonal mass operator. It can impose selection rules but cannot determine charged-lepton eigenvalues without additional dynamics. The reported numerical intervals remain correct evaluations of the specified formulas, yet they are not a mass prediction and carry no statistical significance. A viable successor must supply a Lagrangian and symmetry-breaking mechanism, derive the mass matrix and renormalization-scale dependence, pre-specify all normalization choices, and test genuinely held-out observables.

No new observational or experimental data are introduced. NIST CODATA values are used for the Planck-scale input and for the displayed mass comparison. No likelihood or covariance analysis is performed.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.