Interfacial Commit Branching Laws and Conditional Branch-Budget Estimation for Electromagnetic Carrier Conversion

A detector-local opening formalism already fixes a declared threshold-open family for event-window activation and counting behavior. The externalization threshold introduced here does not replace that detector-local opening threshold; it governs only whether a branch externalizes once absorbed carrier budget has already entered the interfacial constitutive layer. The branch law developed here addresses a different object: an interface-wide constitutive branching law for how absorbed carrier budget is distributed among externalized, ledger-retained, and sink channels on declared interface families.

The motivation is straightforward. Standard descriptions typically emphasize the outwardly visible branch only: photoelectron emission in the photoelectric effect, scattered quanta in Compton scattering, or emitted spectral radiance for a blackbody. Yet modern experiments can separately probe photoemission yield and kinetic-energy distributions, surface-potential or charge-separation maps, photothermal or calorimetric deposition, emissivity and spectral radiance, and refined Compton scattering cross sections in materials. The measured world is already branch-resolved in pieces. The missing object is a single constitutive law family that makes those pieces commensurable. Branch fractions become experimental quantities only relative to a declared calibration map and declared observation/operating windows. [citation]

A first public-data pilot evaluates whether the declared branch-law algebra and estimator grammar can be applied to source-backed public surfaces without assigning same-instance empirical branch fractions. The pilot uses EKHI radiative-property curves, refractiveindex.info optical constants, and NIST XCOM photon-interaction cross-section tables. It returns RTA-PARTIAL for the radiative-curve lane, CBL-NK-PREDICTOR-GATE-SATISFIED for the deterministic $n,k$TeXn,k Fresnel-predictor lane, CBL-XCOM-BRANCH-PILOT-GATE-SATISFIED for the XCOM component-fraction lane, and CBL-SAME-INSTANCE-NOT-ADOPTED for the absent same-instance branch-tomography lane. The combined label is CBL-VP0-PUBLIC-PILOT-PARTIAL; it is a public pilot of the branch-law grammar, not an empirical calibration-instance closure. [citation]

The resulting law family is intentionally middle-layer: it neither replaces propagation laws nor claims a microscopic detector Hamiltonian. It sits at the interface between electromagnetic-excitation absorption and observable output and tells us how the absorbed excitation budget branches into (i) externalized outcomes, (ii) internal ledger retention, and (iii) dissipative sink channels. Here and below, this branching refers only to interfacial closure of an absorbed electromagnetic-excitation budget at the material boundary; it does not define detector-local opening laws, durable readout laws, or objectivity criteria.

The same convention fixes the reading of reflection and reflectance in the branch-budget setting. Reflection is not interpreted as the rebound of a primitive light-substance from a material wall. The ordinary geometric-optics description remains a macroscopic shorthand: an incident electromagnetic energy-information excitation on the phase-link structure induced by the global phase field conditions the local phase-field response of the material boundary, and the reflected field is the outward electromagnetic excitation re-expressed by that boundary response under the relevant coherence, polarization, media, roughness, and matching conditions. This language preserves the standard Maxwell/QED recovery layer and does not introduce a new substance beyond the electromagnetic excitation and the material boundary response.

Claim classes..

To keep claim levels exact, we separate:

- Exact algebraic results: the branching-budget identity; residual reconstruction of the ledger branch; the externalization-threshold bound; the no-externalization-does-not-imply-no-commit corollary; compatibility of the homogeneous externalized branch with the separately declared detector-facing threshold-open counting limit; the noiseless calibrated closure-residual identity when a direct ledger proxy is declared on the same calibration map; and, on the restricted photoemissive material family $\Mpe$TeX\Mpe, the branch-fraction and local log-odds chart obtained from the declared projection-reduced Markovian branch window. - Finite-dimensional estimation results under a declared calibration map: unbiasedness and covariance of the direct estimator, residual-ledger variance, local identifiability by full-rank Jacobian conditions, constrained weighted least-squares estimation under positivity and conservation constraints, and delta-method uncertainty propagation for branch fractions. These statements are estimator-level consequences of the declared observation model and do not assert fundamental linear detector response. Branch fractions at this level remain calibration-map conditioned and are not assigned as empirical closures on a particular interface family. - Admitted constitutive family: smooth externalization gates, ledger-retention functions, sink fractions, and spectral/material response functions on declared operating windows. - Branch-tomography protocol: existing photoemission, surface-potential, photothermal, emissivity, and scattering metrology can be organized into the declared observation model on a fixed interface family/window. This statement is family- and window-conditioned and is not a universal microscopic derivation across arbitrary detector or material classes. - Bounded public-data pilot status: the CBL-VP0 public pilot supplies source-backed partial gates on EKHI, refractiveindex.info, and NIST XCOM source surfaces. Its combined label is CBL-VP0-PUBLIC-PILOT-PARTIAL. This status is compatible with the branch-law grammar but does not assign same-instance empirical branch fractions, a sealed calibration/covariance instance, Maxwell/QED replacement, or detector closure. - Standard-theory comparator physics: Einstein--Millikan photoelectric threshold and slope, Fowler's near-threshold clean-metal theory, Spicer's excitation--transport--escape decomposition, Cs$_2$TeX_2Te photoemission analysis, one-step photoemission on ordered surfaces, first-principles nonequilibrium electron--phonon dynamics, Compton scattering kinematics and measured material response, Planck blackbody radiance, photothermal and surface-potential metrology, emissivity metrology, and photocathode work-function / quantum-efficiency engineering. [citation] - Residual non-claim horizon: universality beyond the restricted photoemissive family treated here and stronger microscopic derivations across broader detector and material classes remain outside the declared scope.

Logical order of the exact statements..

The exact algebraic layer below uses the branch partition axiom, the branch-simplex definitions, and the calibrated-power convention to obtain reference. On the restricted photoemissive family, reference feed into reference. Estimator-level consequences begin only after the observation model reference and calibration map reference have been fixed.

Quantitative branch-tomography chain..

The quantitative statements below follow the declared chain

\eqref{eq:budget}
\Rightarrow
\eqref{eq:pledres}
\Rightarrow
\eqref{eq:obsmodel}
\Rightarrow
\eqref{eq:dirP},\eqref{eq:pledreshat},\eqref{eq:cwls}
\Rightarrow
\eqref{eq:Jacobian},\eqref{eq:CalMap}
\Rightarrow
\eqref{eq:varresidgeneral},\eqref{eq:deltamethod}
\Rightarrow
\eqref{eq:epscl},\eqref{eq:interval_closure_residual}.

Exact branch equalities refer to calibrated branch powers on the branch simplex; finite-noise acceptance uses the declared calibration map, covariance or interval enclosure, operating window, and residual gate.

Logical order of the exact and estimator statements..

At the exact algebraic layer, reference govern reference. On the restricted photoemissive branch, reference govern reference. The finite-dimensional estimation layer begins only after reference have been fixed and then feeds reference.

The distinction is structural rather than cosmetic.

proposition: Detector-facing opening laws and interface-wide branch laws occupy distinct constitutive layers. Detector-facing opening laws and interface-wide branch laws occupy distinct constitutive layers.

- Detector-facing opening laws fix a declared threshold-open family, a local commit-rate law, and window-level closure for recorded events. - Interface-wide branch laws fix a constitutive branching law for how absorbed carrier budget is distributed among externalization, ledger retention, and dissipation.

The two formalisms are therefore complementary but nonidentical in scientific scope.

proof. The distinction follows from the object classes. Detector-facing opening laws use a local commit rate $R_c$TeXR_c and event-window closures to describe when a detector click or emission event is recorded. By contrast, interface-wide branch laws introduce global budget fractions $(\etaext,\etaled,\etasink)$TeX(\etaext,\etaled,\etasink) across irreversible interface outcomes, including outcomes not externally visible as direct clicks or emitted particles. The two formalisms are therefore complementary but not identical in scope.

Let $\Phiin(\omega,\mathbf x,t)$TeX\Phiin(\omega,\mathbf x,t) denote the incident carrier power spectral density on a declared interface patch and $A(\omega,M)\in[0,1]$TeXA(\omega,M)\in[0,1] the spectral absorptance of the material-response state $M$TeXM. Define the absorbed carrier budget on a declared window $\Omega$TeX\Omega by

\Pabs[\Omega]:=
\int_{\Omega}\! dA\,dt\int_0^{\infty} d\omega\,A(\omega,M)\,\Phiin(\omega,\mathbf x,t).

The CBL layer concerns what happens after that budget has entered the interface.

axiom[Branch partition] On a declared operating window, the absorbed carrier budget splits as

\boxed{\Pabs=\Pext+\Pled+\Psink,}

with nonnegative branch fractions

\boxed{\etaext+\etaled+\etasink=1,\qquad \eta_i:=\frac{P_i}{\Pabs}.}

The three channels are:

- $\Pext$TeX\Pext: energy budget carried by externally accessible outcomes; - $\Pled$TeX\Pled: energy budget stored in internal ledger variables or metastable internalized states; - $\Psink$TeX\Psink: dissipative loss into uncontrolled or nonreadout sink channels.

axiom

definition: Branch simplex on a declared window. For $\Pabs>0$TeX\Pabs>0, the calibrated branch-power vector belongs to the branch simplex

\Delta_{\Pabs}
=
\left\{(\Pext,\Pled,\Psink)\in\mathbb R_{\ge0}^3:
\Pext+\Pled+\Psink=\Pabs\right\}.

The branch-fraction map is

\eta:\Delta_{\Pabs}\to\Delta_1,
\qquad
\eta_i=\frac{P_i}{\Pabs},

where

\Delta_1=\left\{(\etaext,\etaled,\etasink)\in\mathbb R_{\ge0}^3:
\etaext+\etaled+\etasink=1\right\}.

All exact branch-fraction identities below are read on this calibrated simplex. When the same quantities are embedded into the estimator vector of the observation model, the fixed calibration-map order is $(\Pabs,\Pext,\Psink,\Pled)$TeX(\Pabs,\Pext,\Psink,\Pled). All covariance, Jacobian, KKT, and closure-residual formulas below refer to that estimator order and not to the simplex display order.

lemma: Branch-fraction simplex identity. If $(\Pext,\Pled,\Psink)\in\Delta_{\Pabs}$TeX(\Pext,\Pled,\Psink)\in\Delta_{\Pabs} with $\Pabs>0$TeX\Pabs>0, then

\etaext+\etaled+\etasink=1,
\qquad
0\le \eta_i\le 1
\quad
(i\in\{\mathrm{ext},\mathrm{led},\mathrm{sink}\}).

proof. By reference, $\Pext+\Pled+\Psink=\Pabs$TeX\Pext+\Pled+\Psink=\Pabs and all three branch powers are nonnegative. Dividing the equality by $\Pabs>0$TeX\Pabs>0 gives the sum identity. Nonnegativity of the fractions and the upper bounds follow from nonnegativity of the other two branch powers in the same budget equality.

definition: Calibrated powers and finite-noise estimates. Unhatted quantities $\Pabs,\Pext,\Pled,\Psink$TeX\Pabs,\Pext,\Pled,\Psink denote calibrated branch powers on a declared operating window after the calibration map has been fixed. Hatted quantities denote finite-noise estimates obtained from observations. Exact identities in the algebraic branch law apply to the calibrated powers; finite-noise estimates are assessed through the declared covariance and residual gates.

remark. The constitutive reading adopted here is as follows: thresholds apply to externalization rather than to commit as such. Internal ledgering and sink closure may already occur even when externalization is closed.

axiom[Externalization gate] For each declared process family $\Cb$TeX\Cb, the externalization fraction is governed by a smooth gate

\etaext^{\Cb}(\Psi,M)=
\sigma\!\left(\frac{\Lambda_{\mathrm{ext}}^{\Cb}(\Psi,M)-\Lambda_{\mathrm{c,ext}}^{\Cb}(M)}{\delta_{\Cb}(M)}\right),
\qquad
\sigma(z):=\frac{1}{1+e^{-z}},

with $\delta_{\Cb}(M)>0$TeX\delta_{\Cb}(M)>0 a softness scale. axiom

axiom[Ledger retention and sink completion] The internal ledger fraction is

\etaled^{\Cb}(\Psi,M)=\bigl(1-\etaext^{\Cb}(\Psi,M)\bigr)\chi_{\Led}^{\Cb}(\Psi,M),
\qquad 0\le \chi_{\Led}^{\Cb}\le 1,

and the sink fraction is the completion law

\etasink^{\Cb}=1-\etaext^{\Cb}-\etaled^{\Cb}.

axiom

remark: Commit classes. Within this law family we distinguish four useful outcome classes:

- externalized commit: outwardly measurable branch $(\Pext)$TeX(\Pext); - internalized ledger commit: irreversible internal storage $(\Pled)$TeX(\Pled); - sink-only closure: dissipative nonreadout termination $(\Psink)$TeX(\Psink); - statistical closure: repeated cycling between internal retention, emission, and sink until a stationary distribution is reached.

This fourth class is especially natural for thermal radiation.

theorem: Calibrated branch identifiability by residual reconstruction. On a declared calibrated window with $\Pabs>0$TeX\Pabs>0, suppose $(\Pext,\Pled,\Psink)\in\Delta_{\Pabs}$TeX(\Pext,\Pled,\Psink)\in\Delta_{\Pabs}. If $\Pabs$TeX\Pabs, $\Pext$TeX\Pext, and $\Psink$TeX\Psink are calibrated branch powers on that same window, then

\boxed{\Pled=\Pabs-\Pext-\Psink}

and therefore

\etaled=\frac{\Pabs-\Pext-\Psink}{\Pabs}.

This is an exact algebraic identification on the calibrated branch simplex, not a claim that noisy uncalibrated readouts determine $\Pled$TeX\Pled without a calibration map.

proof. Immediate from reference. Dividing by $\Pabs>0$TeX\Pabs>0 yields the fraction form.

remark: Residual-reconstruction data on a declared calibrated window. On any declared calibrated window, reference is evaluated from the ordered calibrated triple

(\Pabs,\Pext,\Psink)

and the fixed affine rule

R_{\mathrm{led}}(\Pabs,\Pext,\Psink):=\Pabs-\Pext-\Psink.

The admissibility conditions are exactly those already stated in reference: $\Pabs>0$TeX\Pabs>0, common calibration map, and nonnegative branch powers summing to $\Pabs$TeX\Pabs. Thus the exact algebra layer is the affine identity reference, whereas finite-noise use of the same formula belongs to the estimator and covariance layer introduced later.

theorem: Externalization threshold theorem. Fix a declared window $\mathcal W$TeX\mathcal W and a process family $\Cb$TeX\Cb. Assume $\delta_{\Cb}(M)>0$TeX\delta_{\Cb}(M)>0 on $\mathcal W$TeX\mathcal W. Suppose there exists $\kappa>0$TeX\kappa>0 such that

\Lambda_{\mathrm{ext}}^{\Cb}(\Psi,M)\le \Lambda_{\mathrm{c,ext}}^{\Cb}(M)-\kappa
\qquad\text{for all }(\Psi,M)\in\mathcal W.

Then the pointwise bound

\etaext^{\Cb}(\Psi,M)
\le
\frac{1}{1+e^{\kappa/\delta_{\Cb}(M)}}
\qquad\text{on }\mathcal W

holds. If in addition $\delta_{\Cb}(M)\le \delta_{\max}$TeX\delta_{\Cb}(M)\le \delta_{\max} on $\mathcal W$TeX\mathcal W, then the uniform bound

\etaext^{\Cb}(\Psi,M)
\le
\frac{1}{1+e^{\kappa/\delta_{\max}}}

also holds on $\mathcal W$TeX\mathcal W. In particular, externalization is exponentially suppressed in the declared sub-threshold window.

proof. The sigmoid $\sigma(z)$TeX\sigma(z) is monotone increasing. Since $\delta_{\Cb}(M)>0$TeX\delta_{\Cb}(M)>0, reference gives, pointwise on $\mathcal W$TeX\mathcal W,

\frac{\Lambda_{\mathrm{ext}}^{\Cb}-\Lambda_{\mathrm{c,ext}}^{\Cb}}{\delta_{\Cb}(M)}
\le
-\frac{\kappa}{\delta_{\Cb}(M)}.

Hence

\etaext^{\Cb}=\sigma\!\left(\frac{\Lambda_{\mathrm{ext}}^{\Cb}-\Lambda_{\mathrm{c,ext}}^{\Cb}}{\delta_{\Cb}(M)}\right)
\le \sigma\!\left(-\frac{\kappa}{\delta_{\Cb}(M)}\right)
=\frac{1}{1+e^{\kappa/\delta_{\Cb}(M)}}.

If $\delta_{\Cb}(M)\le\delta_{\max}$TeX\delta_{\Cb}(M)\le\delta_{\max}, the displayed uniform bound follows because $x\mapsto(1+e^{\kappa/x})^{-1}$TeXx\mapsto(1+e^{\kappa/x})^{-1} is increasing for $x>0$TeXx>0.

remark: Threshold bound on a declared window. For any declared parameter box $B\subset \mathcal W$TeXB\subset \mathcal W, the threshold statement is checked on the finite list

[\Lambda_{\mathrm{ext}}^{\Cb}]_B,
\qquad
[\Lambda_{\mathrm{c,ext}}^{\Cb}]_B,
\qquad
[\delta_{\Cb}]_B,
\qquad
[\etaext^{\Cb}]_B,

with $[\delta_{\Cb}]_B\subset(0,\infty)$TeX[\delta_{\Cb}]_B\subset(0,\infty). If there exists $\delta_{\max}>0$TeX\delta_{\max}>0 such that

[\delta_{\Cb}]_B\subset (0,\delta_{\max}]

and

[\Lambda_{\mathrm{ext}}^{\Cb}]_B-[\Lambda_{\mathrm{c,ext}}^{\Cb}]_B\subset (-\infty,-\kappa],

then the admissible bound target is

[\etaext^{\Cb}]_B
\subseteq
\left[0,\frac{1}{1+e^{\kappa/\delta_{\max}}}\right].

The exact symbolic obligation is the monotonic implication from reference to reference; the numerical admissibility obligation is to certify the sign and positivity conditions on those interval boxes. This is a windowed admissibility and bound statement for the declared gate data, not a separate constitutive law.

corollary: No externalization does not imply no commit. Under the branch family, the suppression of externalization does not imply the absence of irreversible closure. If $\etaext^{\Cb}\approx 0$TeX\etaext^{\Cb}\approx 0 but $\Pabs>0$TeX\Pabs>0, then

\Pled+\Psink\approx \Pabs.

Thus internal ledgering and/or sink closure may occur even when the externally visible branch is closed.

proof. From reference, $\Pled+\Psink=\Pabs-\Pext$TeX\Pled+\Psink=\Pabs-\Pext. If $\etaext=\Pext/\Pabs\approx 0$TeX\etaext=\Pext/\Pabs\approx 0, then $\Pext\approx 0$TeX\Pext\approx 0 and so $\Pled+\Psink\approx \Pabs$TeX\Pled+\Psink\approx \Pabs.

theorem: Compatibility with the homogeneous threshold-open counting limit. When a declared branch family satisfies $\etaext\equiv 1$TeX\etaext\equiv 1, $\etaled\equiv 0$TeX\etaled\equiv 0, and $\etasink\equiv 0$TeX\etasink\equiv 0 on a window, the interfacial branch law exports all absorbed budget to the externalized channel, $\Pabs=\Pext$TeX\Pabs=\Pext. If, in addition, an independently declared detector-facing homogeneous threshold-open regime supplies the counting relation $R_c\propto |\Psi|^2$TeXR_c\propto |\Psi|^2, then the CBL branch layer is compatible with that counting limit.

proof. If $\etaext\equiv 1$TeX\etaext\equiv 1, then reference yields $\Pabs=\Pext$TeX\Pabs=\Pext, while the ledger and sink channels vanish on the declared window. The proportional counting law $R_c\propto|\Psi|^2$TeXR_c\propto|\Psi|^2 is not derived here; it belongs to the separately declared detector-facing threshold-open regime. Therefore the result is a compatibility reduction of the interfacial branch layer to the externalized-channel case, not a derivation or replacement of the detector-local counting law.

The photoelectric effect is the first natural specialization because standard theory already isolates an external threshold and emitted-electron kinetic law. Einstein's threshold relation and Millikan's verification remain intact. The CBL move is to insert them into a three-branch constitutive family. [citation]

Let $I_e=e\dot N_e$TeXI_e=e\dot N_e be the photocurrent and $\langle K\rangle$TeX\langle K\rangle the mean kinetic energy of emitted electrons. We adopt the operational externalized power convention

\Pext^{\mathrm{pe}}:=\frac{I_e}{e}\,\bigl(W_{\mathrm{th}}+\langle K\rangle\bigr),

where $W_{\mathrm{th}}$TeXW_{\mathrm{th}} is the threshold energy proxy associated with release to the external branch.

For the photoelectric family we set

\etaext^{\mathrm{pe}}(\Psi,M)
=
\sigma\!\left(
\frac{\Lambda_{\mathrm{ext}}^{\mathrm{pe}}(\Psi,M)-\Lambda_{\mathrm{c,ext}}^{\mathrm{pe}}(M)}{\delta_{\mathrm{pe}}(M)}
\right),

with the branch-functional

\Lambda_{\mathrm{ext}}^{\mathrm{pe}}(\Psi,M)=a_\omega(M)\,\omega+a_t(M)\,\partial_t\phi+a_n(M)\,\nabla\phi\cdot\mathbf n+a_I(M)\ln\bigl(1+\beta\Ipl\bigr).

The remaining fractions are

\etaled^{\mathrm{pe}}=(1-\etaext^{\mathrm{pe}})\chi_{\Led}^{\mathrm{pe}}(\Psi,M),
\qquad
\etasink^{\mathrm{pe}}=1-\etaext^{\mathrm{pe}}-\etaled^{\mathrm{pe}}.

The ledger and sink powers are then

\Pled^{\mathrm{pe}}=\etaled^{\mathrm{pe}}\Pabs^{\mathrm{pe}},
\qquad
\Psink^{\mathrm{pe}}=\etasink^{\mathrm{pe}}\Pabs^{\mathrm{pe}}.

proposition: Externalization threshold vs. internal closure. In the photoelectric family, the usual threshold frequency is best read as a threshold of electron-release externalization, not as a threshold of commit in general.

proof. Equation reference controls only $\etaext^{\mathrm{pe}}$TeX\etaext^{\mathrm{pe}}. If $\Lambda_{\mathrm{ext}}^{\mathrm{pe}}<\Lambda_{\mathrm{c,ext}}^{\mathrm{pe}}$TeX\Lambda_{\mathrm{ext}}^{\mathrm{pe}}<\Lambda_{\mathrm{c,ext}}^{\mathrm{pe}} on a window, then reference suppresses $\etaext^{\mathrm{pe}}$TeX\etaext^{\mathrm{pe}}. But by reference, $\Pled^{\mathrm{pe}}+\Psink^{\mathrm{pe}}$TeX\Pled^{\mathrm{pe}}+\Psink^{\mathrm{pe}} may remain positive whenever $\Pabs^{\mathrm{pe}}>0$TeX\Pabs^{\mathrm{pe}}>0. Thus the sub-threshold regime can still support internal ledgering or sink closure even while the electron-release branch is closed.

remark. This is the precise technical version of the intuitive statement ``low frequency need not mean no commit; it may mean no externalized photoelectron.'' The law family therefore sharpens, rather than contradicts, the standard threshold observation.

Recent work on photocathodes shows that work function, collection efficiency, and quantum efficiency are strongly modified by interface engineering, adsorption, built-in fields, and nanostructure. These are exactly the kinds of dependencies the material-response bundle $M$TeXM is intended to carry. [citation]

We now give the restricted microscopic derivation used by the declared model class. The goal is not to claim a universal microscopic derivation for all materials, but to provide an exact derivation inside a declared restricted model class for a physically important photoemissive interface family. The strategy is completely explicit:

- define the absorbed budget $\Pabs$TeX\Pabs independently from the radiation sector; - derive an exact projected continuity law for $\Pabs$TeX\Pabs, $P_E$TeXP_E, $P_L$TeXP_L, and $P_S$TeXP_S; - derive the reduced branch master equation from the exact Nakajima--Zwanzig projection identity and then impose the declared Born--Markov--secular window; and - connect the phenomenological ledger-retention factor $\chi_{\Led}$TeX\chi_{\Led} explicitly to the microscopic branch hazards.

The result is exact within the declared reduced model, not merely by definition and not by silent relabeling of the absorbed budget.

definition: Restricted photoemissive material family. Let

\Mpe:=\{\text{clean metals, alkali-antimonide photocathodes, and Cs$_2$Te-like photoemitters}\}

be the declared material family satisfying the following window assumptions:

- an effective single escape barrier exists on the declared interface patch; - the optical pump is weak or moderate so that a single excited-electron manifold description is admissible; - on the event window $[0,\tau_{\mathrm{ev}}]$TeX[0,\tau_{\mathrm{ev}}], transport, escape, and internal relaxation are separable at the level of a Born--Markov--secular reduction; - photoemission yield and energy distributions are directly measurable on the same declared family; - the relevant Hilbert subspace admits orthogonal projectors onto radiation, excited, externalized, ledger, and sink sectors.

remark. The family $\Mpe$TeX\Mpe is chosen because Fowler's near-threshold theory for clean metals, Spicer's three-step decomposition for alkali-antimonides, Cs$_2$TeX_2Te photoemission analysis, one-step photoemission calculations on ordered surfaces, and first-principles nonequilibrium electron--phonon dynamics all exist for this regime. Projection-operator and Markovian semigroup tools are classical and primary: Nakajima and Zwanzig give the exact projected equation, while Gorini--Kossakowski--Sudarshan and Lindblad give the standard Markovian generator form used after the declared approximation step. [citation]

Microscopic sector decomposition and absorbed-power identity

Let $\Pi_r,\Pi_X,\Pi_E,\Pi_L,\Pi_S$TeX\Pi_r,\Pi_X,\Pi_E,\Pi_L,\Pi_S be mutually orthogonal projectors on the declared relevant subspace, with

\Pi_r+\Pi_X+\Pi_E+\Pi_L+\Pi_S = I_{\mathrm{rel}}.

Define the sector Hamiltonians

\Hr:=\Pi_r H \Pi_r,

\HX:=\Pi_X H \Pi_X,

\HE:=\Pi_E H \Pi_E,

\HLed:=\Pi_L H \Pi_L,

\HSink:=\Pi_S H \Pi_S.

and the couplings

V_{\mu\nu}:=\Pi_\mu H \Pi_\nu+\Pi_\nu H \Pi_\mu,
\qquad
(\mu,\nu)\in\{(r,X),(X,E),(X,L),(X,S)\}.

The restricted Hamiltonian is therefore

H_{\mathrm{rel}}=
\Hr+\HX+\HE+\HLed+\HSink+V_{rX}+V_{XE}+V_{XL}+V_{XS}.

No direct couplings from the radiation sector to $E,L,S$TeXE,L,S are retained on the declared window, and no direct couplings among $E,L,S$TeXE,L,S are retained at the level of the reduced branch model.

definition: Microscopic absorbed power and branch powers. For any state $\rho(t)$TeX\rho(t) evolving under $H_{\mathrm{rel}}$TeXH_{\mathrm{rel}}, define

\Pabs(t):=-\frac{d}{dt}\Tr(\rho(t)\Hr)
=-\frac{i}{\hbar}\Tr\bigl(\rho(t)[H_{\mathrm{rel}},\Hr]\bigr),

and

P_E(t):=\frac{d}{dt}\Tr(\rho(t)\HE),
\qquad
P_L(t):=\frac{d}{dt}\Tr(\rho(t)\HLed),
\qquad
P_S(t):=\frac{d}{dt}\Tr(\rho(t)\HSink).

These are definitions by sector-energy flux, not by postulated branch fractions.

lemma: Exact projected energy continuity. On the restricted model reference, the following identities hold exactly:

\frac{d}{dt}\Tr\bigl(\rho(\Hr+\HX+\HE+\HLed+\HSink)\bigr)=0,

\frac{d}{dt}\Tr(\rho\HX)=\Pabs-P_E-P_L-P_S.

proof. Because $\dot\rho=-(i/\hbar)[H_{\mathrm{rel}},\rho]$TeX\dot\rho=-(i/\hbar)[H_{\mathrm{rel}},\rho], the Heisenberg form gives

\frac{d}{dt}\Tr(\rho O)=\frac{i}{\hbar}\Tr\bigl(\rho[H_{\mathrm{rel}},O]\bigr)

for any bounded sector observable $O$TeXO. Summing over $O\in\{\Hr,\HX,\HE,\HLed,\HSink\}$TeXO\in\{\Hr,\HX,\HE,\HLed,\HSink\} and using reference, every commutator with an internal coupling appears twice with opposite signs, so reference follows. For $O=\HX$TeXO=\HX, only the couplings $V_{rX},V_{XE},V_{XL},V_{XS}$TeXV_{rX},V_{XE},V_{XL},V_{XS} survive; the sign convention in reference then gives reference.

corollary: Budget identity from a stationary excited sector. If the declared event window is quasi-stationary in the sense that

\frac{d}{dt}\Tr(\rho\HX)=0,

then the branch budget identity is not an axiom but an exact consequence of microscopic continuity:

\boxed{\Pabs=P_E+P_L+P_S.}

proof. Set the left-hand side of reference to zero.

Exact projection equation and reduced branch master equation

Define the Liouvillian $\Liouv O:=-(i/\hbar)[H_{\mathrm{rel}},O]$TeX\Liouv O:=-(i/\hbar)[H_{\mathrm{rel}},O] and the exact Nakajima projection

\Proj\rho:=\sum_{\mu\in\{X,E,L,S\}}\Pi_\mu\rho\Pi_\mu,
\qquad
\Qproj:=I-\Proj.

Nakajima and Zwanzig give the exact projected equation

\partial_t\Proj\rho(t)=\Proj\Liouv\Proj\rho(t)+\int_0^t ds\;\mathcal K(t-s)\Proj\rho(s)+\mathcal I(t),

where

\mathcal K(t):=\Proj\Liouv e^{t\Qproj\Liouv}\Qproj\Liouv\Proj,
\qquad
\mathcal I(t):=\Proj\Liouv e^{t\Qproj\Liouv}\Qproj\rho(0).

Equations reference are exact identities, with no Markovian approximation. [citation]

definition: Declared Markovian branch window. A declared event window belongs to the Markovian branch class if, on that window,

- the memory kernel $\mathcal K(t-s)$TeX\mathcal K(t-s) is short compared with the branch-evolution timescale; - coherences between distinct branch sectors average to zero at the retained order (secularization); - backflow from $E,L,S$TeXE,L,S into $X$TeXX is negligible at the retained order; and - the reduced dynamics is completely positive and trace preserving on the retained subspace.

proposition: Reduced branch GKSL dynamics on a declared Markovian branch window. On a declared Markovian branch window, reference reduces to the branch-resolved GKSL equation

\dot\rho_{
\mathrm{red}}=-\frac{i}{\hbar}[H_{\mathrm{eff}},\rho_{\mathrm{red}}]+\mathcal J_{rX}[\rho_{\mathrm{red}}]+\sum_{i\in\{E,L,S\}}\mathcal D_i[\rho_{\mathrm{red}}],

with dissipators

\mathcal D_i[\rho]=\sum_\alpha \gamma_{i\alpha}\Bigl(L_{i\alpha}\rho L_{i\alpha}^\dagger-\tfrac12\{L_{i\alpha}^\dagger L_{i\alpha},\rho\}\Bigr),
\qquad \gamma_{i\alpha}\ge 0.

proof. This is the standard implication of a Born--Markov--secular reduction from the exact projected dynamics to a completely positive semigroup generator. The general generator forms are theorems of Gorini--Kossakowski--Sudarshan and Lindblad. The branch structure here is simply the decomposition of the dissipator into the declared orthogonal channels $E,L,S$TeXE,L,S. [citation]

corollary: Additive rate equation for a single excited manifold. Assume, in addition, that the excited sector is effectively one-dimensional at the population level, so that $n_X(t):=\Tr(\Pi_X\rho_{\mathrm{red}}(t))$TeXn_X(t):=\Tr(\Pi_X\rho_{\mathrm{red}}(t)) captures the relevant occupancy. Then

\dot n_X(t)=G_{\mathrm{abs}}(t)-K n_X(t),
\qquad K:=k_E+k_L+k_S,

where

k_i:=\sum_\alpha \gamma_{i\alpha},\qquad i\in\{E,L,S\}.

proof. Take the trace of reference against $\Pi_X$TeX\Pi_X. The Hamiltonian term drops out of the population balance, $\mathcal J_{rX}$TeX\mathcal J_{rX} contributes the pumping term $G_{\mathrm{abs}}$TeXG_{\mathrm{abs}}, and each dissipator $\mathcal D_i$TeX\mathcal D_i removes probability from $X$TeXX with rate $k_i$TeXk_i. Orthogonality of the branch projectors and the secular assumption eliminate interference cross-terms at the retained order.

Branch-average energies and exact connection to the phenomenological law

definition: Branch jump superoperators and average branch energies. For each branch $i\in\{E,L,S\}$TeXi\in\{E,L,S\}, define the jump superoperator

\mathcal J_i[\rho]:=\sum_\alpha \gamma_{i\alpha}L_{i\alpha}\rho L_{i\alpha}^\dagger.

On a stationary window, define the branch-average energy by

\bar\varepsilon_i:=
\frac{\Tr(H_i\mathcal J_i[\rho_{\mathrm{ss}}])}{\Tr(\mathcal J_i[\rho_{\mathrm{ss}}])},
\qquad
H_i\in\{\HE,\HLed,\HSink\}.

lemma: Exact branch powers. On a quasi-stationary Markovian branch window,

P_i=\Tr(H_i\mathcal J_i[\rho_{\mathrm{ss}}])=\bar\varepsilon_i\,k_i\,n_X^{\mathrm{ss}},
\qquad i\in\{E,L,S\}.

proof. Because the stationary excited population is $n_X^{\mathrm{ss}}$TeXn_X^{\mathrm{ss}}, the jump intensity into branch $i$TeXi is $k_i n_X^{\mathrm{ss}}$TeXk_i n_X^{\mathrm{ss}}. Multiplying by the conditional mean branch energy reference gives reference.

definition: Positive log-odds subwindow. Within the declared quasi-stationary Markovian branch window, the positive log-odds subwindow is the subset on which

\bar\varepsilon_Ek_E>0,
\qquad
\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S>0.

For any fixed positive reference branch-power scale $Q_0$TeXQ_0 with the same units as $\bar\varepsilon_i k_i$TeX\bar\varepsilon_i k_i, define the logarithmic coordinates

\widetilde\Lambda_{\mathrm{ext}}
:=
\delta_{\mathrm{pe}}\ln\!\left(\frac{\bar\varepsilon_Ek_E}{Q_0}\right),
\qquad
\widetilde\Lambda_{\mathrm{c,ext}}
:=
\delta_{\mathrm{pe}}\ln\!\left(\frac{\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S}{Q_0}\right).

lemma: Reference-scale cancellation in the log-odds coordinates. On the positive log-odds subwindow of reference,

\widetilde\Lambda_{\mathrm{ext}}-\widetilde\Lambda_{\mathrm{c,ext}}
=
\delta_{\mathrm{pe}}\ln\!\left(\frac{\bar\varepsilon_Ek_E}{\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S}\right).

Hence the difference is independent of the choice of $Q_0$TeXQ_0.

proof. Subtract the two logarithmic coordinates in reference; the reference scale cancels immediately.

theorem: Exact restricted branch fractions and log-odds chart. Fix a material in $\Mpe$TeX\Mpe and assume reference. Suppose the excited manifold is quasi-stationary and

\bar\varepsilon_i k_i\ge 0\quad (i\in\{E,L,S\}),
\qquad
\bar\varepsilon_Ek_E+\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S>0 .

Then

\boxed{P_{\mathrm{abs}}=P_E+P_L+P_S}

and the exact branch fractions are

\eta_i=
\frac{\bar\varepsilon_i k_i}{\bar\varepsilon_Ek_E+\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S},
\qquad i\in\{E,L,S\}.

In particular,

\eta_{\mathrm{ext}}=
\frac{\bar\varepsilon_Ek_E}{\bar\varepsilon_Ek_E+\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S}
=
\sigma\!\left(\frac{\widetilde\Lambda_{\mathrm{ext}}-\widetilde\Lambda_{\mathrm{c,ext}}}{\delta_{\mathrm{pe}}}\right),

where, for any chosen softness scale $\delta_{\mathrm{pe}}>0$TeX\delta_{\mathrm{pe}}>0, on the positive log-odds subwindow of reference, the logarithmic coordinates are those of reference. By reference, the branch fraction reference is independent of $Q_0$TeXQ_0. Boundary cases with a vanishing numerator or internal aggregate are read as one-sided limits of the simplex formula reference, not as finite logarithmic coordinates. Moreover,

\chi_{\Led}^{\mathrm{exact}}
=
\frac{\bar\varepsilon_Lk_L}{\bar\varepsilon_Lk_L+\bar\varepsilon_Sk_S},
\qquad
\eta_{\mathrm{led}}=(1-\eta_{\mathrm{ext}})\chi_{\Led}^{\mathrm{exact}},

so the phenomenological ledger-retention factor is the exact internal-branch energy share.

Standard Compton scattering provides a kinematic redistribution law for externally visible scattered quanta. The present branch family does not replace the kinematics; it factors them through an externalization branch fraction. [citation]

Let $dP_{\mathrm{kin}}^{\mathrm C}/(d\Omega\,dE')$TeXdP_{\mathrm{kin}}^{\mathrm C}/(d\Omega\,dE') denote the standard kinematic power density entering a scattering channel. Then the branch law reads

\frac{dP_{\Ext}^{\mathrm C}}{d\Omega\,dE'}
=
\eta_{\Ext}^{\mathrm C}(\theta,E',M)
\frac{dP_{\mathrm{kin}}^{\mathrm C}}{d\Omega\,dE'},

with deposition budget

P_{\mathrm{dep}}^{\mathrm C}=P_{\mathrm{abs}}^{\mathrm C}-\int \frac{dP_{\Ext}^{\mathrm C}}{d\Omega\,dE'}\,d\Omega\,dE'.

A further internal split is

P_{\Led}^{\mathrm C}=\chi_{\Led}^{\mathrm C}(M,\theta,E')\,P_{\mathrm{dep}}^{\mathrm C},
\qquad
P_{\Sink}^{\mathrm C}=P_{\mathrm{dep}}^{\mathrm C}-P_{\Led}^{\mathrm C}.

remark. This law does not alter Compton kinematics. It adds a measured or fitted branch fraction that distinguishes externally resolved scattering from internal deposition and dissipative termination in real material environments. That distinction is increasingly relevant in low-energy material-sensitive Compton calculations and in precision cross-section measurements. [citation]

For thermal radiation, the natural object is not a single event but repeated cycling between internal storage, emission, and dissipation until a stationary distribution is reached. This is the natural home of the statistical closure class.

Let $L_\nu^{\mathrm{Planck}}(T)$TeXL_\nu^{\mathrm{Planck}}(T) denote the standard Planck spectral radiance. The branch law writes the externalized radiance as

L_\nu(T,M)=\eta_{\Ext}^{\mathrm{bb}}(\nu,T,M)\,L_\nu^{\mathrm{Planck}}(T),
\qquad
\eta_{\Ext}^{\mathrm{bb}}\equiv \varepsilon_\nu(T,M),

so emissivity becomes an externalization fraction rather than just a material constant. The windowed thermal closure is

C_M\dot T=P_{\mathrm{abs}}^{\mathrm{bb}}-P_{\mathrm{emit}}^{\mathrm{bb}}-P_{\mathrm{sink}}^{\mathrm{bb}},

where

P_{\mathrm{emit}}^{\mathrm{bb}}=\int dA\,d\Omega\,d\nu\,L_\nu(T,M).

proposition: Steady-state statistical closure in the declared blackbody family. At thermal steady state, $\dot T=0$TeX\dot T=0, the blackbody family satisfies

P_{\mathrm{abs}}^{\mathrm{bb}}=P_{\mathrm{emit}}^{\mathrm{bb}}+P_{\mathrm{sink}}^{\mathrm{bb}}.

proof. Immediate from reference with $\dot T=0$TeX\dot T=0.

Recent emissivity metrology already measures temperature- and material-dependent emissivity with high precision. The present move is to read those same measurements as branch fractions in a broader closure law. [citation]

Finite-dimensional branch-observation object..

All estimator, identifiability, uncertainty, and closure-residual statements in this section are read relative to the finite-dimensional object

\mathfrak V_{\rm CBL}
=
(\mathbf P,\mathbf y,\mathbf A,\mathbf b,\Sigma,\Omega,\mathcal W,\mathcal C_{\rm gate}),

where $\mathbf P$TeX\mathbf P is the calibrated branch-power vector, $\mathbf y$TeX\mathbf y is the channel-readout vector, $\mathbf A$TeX\mathbf A is the declared calibration matrix, $\mathbf b$TeX\mathbf b the offset vector, $\Sigma$TeX\Sigma the declared covariance, $\Omega$TeX\Omega and $\mathcal W$TeX\mathcal W the observation and operating windows, and $\mathcal C_{\rm gate}$TeX\mathcal C_{\rm gate} the positivity, conservation, and residual-acceptance convention. The exact conservation row is

C\mathbf P=0,
\qquad
C=(1,-1,-1,-1).

No estimator-level statement below is available unless this object is fixed before branch-fraction extraction.

The constitutive family becomes experimentally useful only after the branch variables are tied to a declared observation model. We therefore distinguish four measured channels:

\mathbf y=

 y_{\mathrm{abs}}

 y_{\mathrm{ext}}

 y_{\mathrm{th}}

 y_{\mathrm{led}}
,
\qquad
\mathbf P=

 \Pabs

 \Pext

 \Psink

 \Pled
.

Here $y_{\mathrm{abs}}$TeXy_{\mathrm{abs}} is an absorbed-budget readout, $y_{\mathrm{ext}}$TeXy_{\mathrm{ext}} an externally resolved branch observable, $y_{\mathrm{th}}$TeXy_{\mathrm{th}} a thermal or dissipative observable, and $y_{\mathrm{led}}$TeXy_{\mathrm{led}} a direct ledger probe when available.

definition: Linearized branch observation model. On a declared calibration window, the retained first-order observation model is

\mathbf y = \mathbf A\,\mathbf P+\mathbf b+\bm\epsilon,

where $\mathbf A\in\mathbb R^{m\times4}$TeX\mathbf A\in\mathbb R^{m\times4} is the declared retained-channel calibration matrix, $\mathbf b\in\mathbb R^m$TeX\mathbf b\in\mathbb R^m is the declared offset vector, and $\bm\epsilon$TeX\bm\epsilon is a zero-mean noise vector with symmetric positive-definite covariance matrix $\Sigma$TeX\Sigma on the retained channel subspace. The calibrated power vector is required to satisfy $\Pabs>0$TeX\Pabs>0 and the branch-simplex constraint of reference. The direct four-channel diagonal model

\mathbf A=\mathrm{diag}(\alpha_{\mathrm{abs}},\alpha_{\mathrm{ext}},\alpha_{\mathrm{sink}},\alpha_{\mathrm{led}}),
\qquad
\alpha_i>0,

is the special case used when the four named branch channels are independently calibrated. Direct inversion is used only in the square nonsingular retained-channel case; overdetermined or noisy branch tomography uses the general matrix form together with the declared covariance, conservation constraint, and admissibility conditions below. If $\Sigma$TeX\Sigma is singular because a channel has been projected out, the model is replaced by the corresponding reduced subspace and its declared covariance or pseudoinverse convention.

The model in reference is not a claim of fundamental linearity; it is the first-order calibration law on the declared operating window. Nonlinear detector response, saturation, memory, and hysteresis require an enlarged declared family and may not be absorbed into reference silently.

remark: Observable realizations. Representative realizations are:

- $y_{\mathrm{abs}}$TeXy_{\mathrm{abs}}: absorbed fluence or absorbed power inferred from incident--reflected--transmitted balance; - $y_{\mathrm{ext}}$TeXy_{\mathrm{ext}}: photocurrent and electron spectrum (photoelectric family), scattered photon/electron spectrum (Compton family), or spectral radiance (blackbody family); - $y_{\mathrm{th}}$TeXy_{\mathrm{th}}: photothermal, calorimetric, or photoacoustic response; - $y_{\mathrm{led}}$TeXy_{\mathrm{led}}: surface-potential map, trapped-charge signal, remanent-state occupancy, or metastable-state probe.

definition: Direct branch estimator. If the retained channel model in reference is square, all four named branch channels are calibrated and observed, and the declared calibration matrix $\mathbf A$TeX\mathbf A is nonsingular on the retained channel space, define

\widehat{\mathbf P}_{\mathrm{dir}}:=\mathbf A^{-1}(\mathbf y-\mathbf b).

For the diagonal special case displayed in reference, nonsingularity is the condition $\prod_i\alpha_i\ne0$TeX\prod_i\alpha_i\ne0, which is guaranteed by the declared gain conditions $\alpha_i>0$TeX\alpha_i>0. If the retained model is overdetermined or rank-deficient, this direct estimator is not used; the declared constrained estimator below is used instead. On windows with $\widehat P_{\mathrm{abs}}^{\mathrm{dir}}>0$TeX\widehat P_{\mathrm{abs}}^{\mathrm{dir}}>0, the corresponding branch-fraction estimator is

\widehat\eta_i^{\mathrm{dir}}:=\frac{\widehat P_i^{\mathrm{dir}}}{\widehat P_{\mathrm{abs}}^{\mathrm{dir}}}.

If the declared uncertainty convention does not certify a positive absorbed-power denominator on the retained window, no direct branch-fraction readout is assigned on that window.

proposition: Unbiasedness of the direct estimator. Under reference with $\mathbb E[\bm\epsilon]=0$TeX\mathbb E[\bm\epsilon]=0, the estimator reference is unbiased:

\mathbb E\bigl[\widehat{\mathbf P}_{\mathrm{dir}}\bigr]=\mathbf P.

Its covariance is

\mathrm{Cov}(\widehat{\mathbf P}_{\mathrm{dir}})=\mathbf A^{-1}\Sigma\,\mathbf A^{-T}.

proof. Take expectations in reference and use $\mathbb E[\bm\epsilon]=0$TeX\mathbb E[\bm\epsilon]=0. The covariance follows by linear propagation.

definition: Residual ledger estimator. If no direct ledger probe is available but $\Pabs$TeX\Pabs, $\Pext$TeX\Pext, and $\Psink$TeX\Psink are measured, define

\widehat P_{\mathrm{led}}^{\mathrm{res}}
:=
\widehat P_{\mathrm{abs}}-\widehat P_{\mathrm{ext}}-\widehat P_{\mathrm{sink}}.

The corresponding residual branch fractions are

\widehat\eta_{\mathrm{led}}^{\mathrm{res}}=
\frac{\widehat P_{\mathrm{led}}^{\mathrm{res}}}{\widehat P_{\mathrm{abs}}},
\qquad
\widehat\eta_i=\frac{\widehat P_i}{\widehat P_{\mathrm{abs}}}.

The estimator is admitted only on windows where the calibrated absorbed-power estimate has a positive denominator under the declared uncertainty convention.

remark: Residual estimator as a fixed linear map. The residual estimator is the fixed linear map

\widehat P_{\rm led}^{\rm res}
=
\mathbf c^T

\widehat P_{\rm abs}

\widehat P_{\rm ext}

\widehat P_{\rm sink}
,
\qquad
\mathbf c=(1,-1,-1)^T .

Therefore all residual-ledger uncertainty statements below are covariance propagation statements for this fixed map and do not depend on a separate phenomenological fit for $\Pled$TeX\Pled.

proposition: Residual branch tomography. If the calibrated branch-power estimates $\widehat P_{\mathrm{abs}}$TeX\widehat P_{\mathrm{abs}}, $\widehat P_{\mathrm{ext}}$TeX\widehat P_{\mathrm{ext}}, and $\widehat P_{\mathrm{sink}}$TeX\widehat P_{\mathrm{sink}} are identifiable on a declared window under one fixed calibration map, then the residual-ledger estimate $\widehat P_{\mathrm{led}}^{\mathrm{res}}$TeX\widehat P_{\mathrm{led}}^{\mathrm{res}} is fixed by reference. This is residual branch tomography. If a direct ledger probe is available, the problem is direct branch tomography.

proof. Immediate from reference.

definition: Constrained weighted least-squares estimator. When the observation model is overdetermined or noisy, let $\Sigma_{\mathcal C}^{-1}$TeX\Sigma_{\mathcal C}^{-1} denote the ordinary inverse of the declared positive-definite covariance on the retained channel space. If a singular covariance is used only after a declared projection, whitening, or pseudoinverse convention, $\Sigma_{\mathcal C}^{-1}$TeX\Sigma_{\mathcal C}^{-1} denotes that fixed declared inverse object on the retained subspace. Define

\widehat{\mathbf P}_{\mathrm{cwls}}
:=
\arg\min_{\mathbf q\in\mathbb R_+^4}
(\mathbf y-\mathbf A\mathbf q-\mathbf b)^T\Sigma_{\mathcal C}^{-1}(\mathbf y-\mathbf A\mathbf q-\mathbf b)

subject to

q_{\mathrm{abs}}=q_{\mathrm{ext}}+q_{\mathrm{led}}+q_{\mathrm{sink}}.

The inverse object $\Sigma_{\mathcal C}^{-1}$TeX\Sigma_{\mathcal C}^{-1}, the retained channel set, and the branch-power convention must be fixed before branch-fraction extraction.

proposition: Well-posedness of constrained branch tomography. Assume an estimator-admissible calibration map and a nonempty feasible set

\mathcal K=\{\mathbf q\in\mathbb R_{\ge 0}^4 : C\mathbf q=0\}.

If the retained covariance convention yields a positive-definite quadratic form on the active affine constraint subspace, then the constrained weighted least-squares problem reference--reference admits a minimizer. If the restricted quadratic form is strictly convex on that subspace, the minimizer is unique.

proof. The objective is a continuous quadratic form on the closed convex feasible set $\mathcal K$TeX\mathcal K. Positivity of the retained quadratic form on the active affine constraint subspace gives coercivity on admissible directions, hence existence of a minimizer. Strict convexity on the same subspace yields uniqueness.

proposition: KKT form of the constrained branch estimator. Assume the declared inverse object $\Sigma_{\mathcal C}^{-1}$TeX\Sigma_{\mathcal C}^{-1} is positive definite on the retained subspace and restrict first to an interior solution relative to the inequality faces $q_i\ge0$TeXq_i\ge0. Let