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

The CLM theorem is formulated on the admitted CHC branch category defined below. The root and gauge-chiral antecedents are imported from the CHC root, operational phase-link, electroweak-facing, and precision-window layers [citation]; the finite-response and charged-lepton spectrum statements are proved in the present paper within the declared CLM branch grammar. The theorem is a conditional theorem inside that declared CHC branch grammar. It does not assert an unconditional derivation from standard physics alone.

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.

Let

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

The four objects of $\Icat$TeX\Icat are the typed 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 typed 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.

theorem: Faithfulness of admitted CLM response functors. On a charged-lepton branch carrying the recovery interface reference, any admitted finite response functor that preserves and separates the typed left/right singular-spectrum slots, the phase-link/local-commit layers, the three-family cyclic comparison type, and the spinor-deck type is response-reflecting in the sense of the preceding definition.

proof. If two distinct chiral incidences are identified by an admitted functor, then the typed left/right singular-spectrum separation of the charged-lepton recovery map ceases to distinguish the two boundary domains. The statement concerns preservation of the typed 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 normal slots. The minimal normal family-phase support is $\C[(\Z_3)^4]$TeX\C[(\Z_3)^4], and the minimal normal spinor support is $\C[(\Z_2)^4]$TeX\C[(\Z_2)^4].

proof. Response-reflection requires one family comparison coordinate and one spinor deck coordinate for every element of $C\times O$TeXC\times O. Since $|C\times O|=4$TeX|C\times O|=4, fewer than four coordinates identify two typed incidences. More than four coordinates introduce an untyped normal axis. The family coordinate is cyclic of order three, giving $\C[(\Z_3)^4]$TeX\C[(\Z_3)^4]. The spinor deck coordinate is binary, giving $\C[(\Z_2)^4]$TeX\C[(\Z_2)^4].

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 typed 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 typed 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 typed 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: Defect classification. The admissible charged-spinor commit-defect group is canonically

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

and the minimal faithful defect support is $\C[(\Z_2)^5]_{\rm def}$TeX\C[(\Z_2)^5]_{\rm def}.

proof. The five admissible binary defect coordinates are the two boundary orientation parities $O_L,O_R$TeXO_L,O_R, the spinor-deck parity $S$TeXS, the charged $Q=-1$TeXQ=-1 orientation parity $Q$TeXQ, and the commit-side orientation parity $K$TeXK. Independence gives an injective map $(\Z_2)^5\to G_{\rm def}$TeX(\Z_2)^5\to G_{\rm def}, and exhaustiveness in the declared defect grammar gives surjectivity: no additional binary coordinate is admitted without adding a branch-changing or trivial orientation. The regular representation of this group separates all admissible defect characters. Any proper quotient identifies two admissible defect states. Any larger binary support introduces an inadmissible branch-changing or trivial orientation. The minimal faithful support is therefore the regular representation $\C[(\Z_2)^5]_{\rm def}$TeX\C[(\Z_2)^5]_{\rm def}.

theorem: Minimal finite-response object. Up to permutation of equal-type axes, the unique minimal response-reflecting finite object of the charged-lepton branch is

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

proof. The two normal supports are forced by the four-slot corollary. The defect support is forced by the five-orientation theorem. Removing any factor violates response-reflection. Adding any factor introduces an untyped axis. Axis permutations inside equal-type factors preserve the object.

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.

theorem: Exactly two invariant normal null channels. The normal support has exactly two invariant null channels, and the charged-lepton normal quotient removes exactly these two channels.

proof. Each regular summand has a one-dimensional fixed subspace. Since $V_N$TeXV_N is a direct sum of two regular summands, $\dim\Fix(V_N)=2$TeX\dim\Fix(V_N)=2. The two fixed lines are the trivial family-normal and spinor-normal response lines. They do not resolve charged-lepton family order, spinor deck sign, chirality, or commit readability. Every other normal character is nontrivial on at least one group element and is not fixed by the regular action. Hence exactly two invariant null channels are removed.

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.

theorem: Trace quotient invariant.

\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 support has dimension $2^5$TeX2^5, so $\Tr(\TRSD)=32$TeX\Tr(\TRSD)=32. The quotient is $32/95$TeX32/95. The relabelling groups $S_4,S_4,S_5$TeXS_4,S_4,S_5 act by conjugation on the corresponding axis families; trace is invariant under conjugation. Hence the quotient is chart-invariant.

The root action is fixed by a compact biphase cell and the central spinor holonomy class.

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.

theorem: Root action.

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

proof. The biphase torus contributes $4\pi^2$TeX4\pi^2. The charged spinor interface removes one nontrivial central holonomy defect from the $2\pi$TeX2\pi comparison. The preceding lemma fixes the defect action to $\pi$TeX\pi. Therefore $\Sroot=4\pi^2-\pi$TeX\Sroot=4\pi^2-\pi.

theorem: Three-family phase action.

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

proof. The neutral family phase sector is generated by $\Z_3$TeX\Z_3. Its nontrivial primitive characters are $e^{\pm2\pi i/3}$TeXe^{\pm2\pi i/3}. The minimal positive generator action is $2\pi/3$TeX2\pi/3.

The weak-balance selector is the unique element of an admissible selector class. The class is fixed before numerical comparison. Its role is deliberately local: it is the scalar Gram-norm correction associated with the minimal weak leakage line, not a fitted charged-lepton mass relation.

The admissibility conditions below are fixed without reference to charged-lepton masses because they are the unique local scalar conditions compatible with spinor-sheet reversal, neutral normalization at zero leakage, one open leakage line, first closed return on the two-sheet response, four-to-three response compression, and minimal finite-window closure. Thus the selector class is declared and exhausted before downstream CODATA comparison constants are introduced.

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: Minimal leakage amplitude. The leakage amplitude is

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

proof. A leakage quantum crosses one primitive family generator and one compact spinor comparison scale. The primitive family generator has order three. The compact spinor comparison line has nontrivial central phase norm $\pi$TeX\pi. The branch grammar admits the minimal scalar leakage amplitude as the reciprocal product of these two typed comparison norms: one discrete family generator and one compact spinor phase line. The reciprocal generator norm is therefore $1/(3\pi)$TeX1/(3\pi). No charged-lepton mass, Yukawa value, or downstream comparison constant enters this amplitude.

lemma: Restitution coefficient. The quartic closed-return coefficient is $9/16$TeX9/16.

proof. Let $E$TeXE be the four-dimensional boundary-layer incidence space and $F$TeXF the three-dimensional family-mode space. The minimal balanced projection has normalized squared Hilbert-Schmidt ratio

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

The selector corrects the leakage Gram norm, not a signed amplitude. Therefore the balanced compression contributes by Hilbert-Schmidt norm: the four-to-three projection contributes the normalized factor $\rho_{3/4}$TeX\rho_{3/4} on departure and the conjugate factor on return. A closed two-sheet return leaves and returns, so the restitution weight is $\rho_{3/4}^2=(3/4)^2=9/16$TeX\rho_{3/4}^2=(3/4)^2=9/16. This coefficient is fixed by the typed projection geometry of the response object, not by numerical comparison with charged-lepton masses.

theorem: Admissible selector uniqueness. 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. Evenness and minimality restrict $R$TeXR to $1+a x^2+b x^4$TeX1+a x^2+b x^4. The quadratic term is the unique open leakage Gram line. The fourth-order term is not chosen for numerical fit. It is the first typed order at which a leakage path can leave the two-sheet response and return while preserving the branch type. A sixth-order term would encode a second independent closed return and therefore an additional finite-response memory cell, contradicting the minimal closure condition in the admissible class. The unit leakage Gram line fixes $a=-1$TeXa=-1. The four-to-three closed-return lemma fixes $b=9/16$TeXb=9/16. Substituting $x=\epsEW=1/(3\pi)$TeXx=\epsEW=1/(3\pi) gives reference. No free coefficient remains in the admissible class.

The unreduced Planck-root convention is

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

The scale chain is

\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.

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: Admissible affine family phase-density normalization. Within the declared CLM branch, the family phase density is normalized by the admissible affine sheet-to-family-cell rule: it is the ratio of the two spinor-sheet orientations to the $3\times3$TeX3\times3 family-phase chart count. No charged-lepton mass, charged-lepton Yukawa value, electroweak-scale inversion, or downstream comparison constant is an input to this normalization.

lemma: Family phase density. Under the admissible affine sheet-to-family-cell normalization,

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

proof. The declared response-reflecting chart has two spinor-sheet orientations and nine family-phase cells. The normalization rule above therefore gives $\theta_{\rm fam}=2/9$TeX\theta_{\rm fam}=2/9. Uniqueness is only within this declared normalization class.

theorem: Family spectrum uniqueness. The charged-lepton family weights are the diagonal entries of the unique admissible rank-one family spectrum 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. The preceding lemma gives the unique admissible vector at phase density $\theta_{\rm fam}$TeX\theta_{\rm fam}. A positive rank-one spectrum 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. No second admissible spectrum operator exists without breaking rank-one positivity, real conjugation, equal Hilbert--Schmidt deck normalization, or response-reflecting phase density.

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.

theorem: Residual chiral action.

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

proof. In the declared scalar leakage grammar, the minimal residual chiral amplitude contains one inverse three-family generator and two inverse compact spinor phase factors. Thus its amplitude is $1/(3\pi^2)$TeX1/(3\pi^2). The action is the positive quadratic leakage norm, giving reference. This statement is a branch-grammar normalization; it is not a Standard-Model loop calculation, a QED radiative correction, or a fitted charged-lepton mass term.

theorem: Trace action.

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

proof. The spin-frame solid angle is $4\pi$TeX4\pi and the family sector has three cyclic components, giving $4\pi/3$TeX4\pi/3. The trace-defect component is $\DeltaTr=32/95$TeX\DeltaTr=32/95. The residual chiral component is reference. These components live on independent summands of the finite response object or its scalar leakage line, so the action is additive.

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: Admissible finite-response formula grammar. The formula grammar 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.

theorem: Grammar-relative formula uniqueness. Once the declared CHC branch category $\Bcat$TeX\Bcat and the admissible finite-response formula grammar are fixed, 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)

are determined as theorem outputs within that grammar. This is not a uniqueness claim over all possible flavor models, all possible finite grammars, all possible electroweak completions, or all possible ultraviolet completions.

proof. The root term is the area of the unique two-circle compact root cell minus the unique nontrivial central spinor holonomy action. The three-family phase term is the primitive generator action of $\Z_3$TeX\Z_3. The weak selector is the minimal even scalar built from the unique leakage amplitude $1/(3\pi)$TeX1/(3\pi), the leading two-sheet Gram determinant term, and the unique four-to-three finite-slot return coefficient. The family angle is the ratio of the two spinor sheets to the nine family-phase cells. The trace quotient is fixed by the minimal finite response object and exactly two normal null lines. The residual chiral term is the positive square of the unique one-family/two-spinor inverse compact leakage amplitude. The trace action is additive over independent finite response components. No alternative term in the grammar has the same type and lower or equal support without violating response-reflection, selector admissibility, rank-one positivity, or adding an untyped axis.

theorem: Upstream 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: Retrospective-selection exclusion. The admissible finite-response grammar contains no continuous or discrete free coefficient whose value can be chosen after charged-lepton comparison.

proof. The root action is fixed by the biphase cell and the central spinor holonomy class. The phase action is fixed by the primitive $\Z_3$TeX\Z_3 character. The weak selector is the unique element of the admissible selector class. The trace quotient is fixed by the minimal response object and the two invariant null lines. The family spectrum is the unique admissible rank-one real $\Z_3$TeX\Z_3 projector at the response-reflecting density. The residual action and interval width are reciprocal finite-slot norms. Therefore all coefficients are determined before the comparison constants are introduced.

theorem: CHC-CLM mass-loading intervals. 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 comparison value lies inside the corresponding interval.

proof. For each sorted weight $w_i$TeXw_i 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: Interval-inclusion status. The displayed intervals are branch-window intervals generated by the declared finite-response grammar and the Planck-root normalization convention. They are not metrological determinations of the charged-lepton masses, not likelihood intervals from a global electroweak fit, and not a replacement for QED or Standard-Model mass and scattering descriptions. Their role is target-separated downstream inclusion under the declared CHC branch grammar.

The construction is not a Koide-type empirical mass relation fitted to the observed charged-lepton pole masses. It is a conditional finite-response grammar on a declared CHC branch; the observed charged-lepton masses enter only as downstream comparison constants after the interval certificate has been produced [citation].

The theorem returns, within the declared CHC branch, a branch-level electroweak loading scale and charged-lepton loading singular spectrum for the recovery interface reference. These are CHC loading singular values, not renormalized Standard-Model Yukawa parameters. Gauge-field scattering amplitudes, QED perturbation theory, and Standard-Model renormalized amplitudes remain the selected recovery descriptions for their respective domains. The finite-response result is a charged-lepton loading theorem on the declared CHC branch.

No new observational or experimental dataset is introduced by this paper. All definitions, assumptions, finite-response formulas, downstream comparison separation rules, and interval-calculation inputs used in the argument are contained in the text and in cited public references and companion statements. Those materials are bounded-support: they do not assert a Standard Model or QED replacement, do not introduce a new electroweak global likelihood, and do not claim an unconditional derivation from standard physics alone. NIST CODATA constants are used only for the Planck-root convention and for downstream comparison values after the CHC intervals have been computed.

Funding and competing interests..

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