04
◇ part I · foundations

A throat is a finite-radius brane-bulk opening represented by Σ = r - R(Ω,w,t)

The particle analog of the program is the throat: a finite brane-bulk defect, or puncture/open conduit, whose brane-side mouth is only the entrance cross-section. The interior carries support, wall, Maxwell/mixed, and outgoing-port degrees of freedom. Early closures parameterize a throat by a radius and a length; the distributed moving-wall lift uses R(Ω,w,t)R(\Omega, w, t) but remains an effective closure until the promoted parent wall action is supplied.

Physical ontologyEffective wall closureMoving-wall R(Ω,w,t)sector: geometry + matter
◇ effective throat geometry

The throat is a surface/open conduit, not a pointlike zero

Place the mouth near x0x_0 on the brane and use local coordinates (r,Ω,w)(r,\Omega,w), where r=∣x−x0∣r = |x-x_0| and Ω\Omega labels direction on the brane-side mouth sphere. The current ontology represents the finite throat by a level-set or shape-field closure:
◇ throat surface · ontology closure
The mouth is at w=0w = 0 with radius aa. The open-exit condition is R(L)>0R(L) > 0; a hard cap R(L)=0R(L) = 0 is obsolete except as a negative-control toy model.
Σ(X,t)  =  r−R(Ω,w,t),Σ(X,t)  =  0,R(0)  =  a,R(L)>0\begin{aligned} \Sigma(\mathbf X,t) &\;=\; r - R(\Omega,w,t), \\ \Sigma(\mathbf X,t) &\;=\;0, \\ R(0) &\;=\; a,\qquad R(L) > 0 \end{aligned}
Axisymmetric or static reductions may collapse R(Ω,w,t)R(\Omega,w,t) to a simpler radial profile or to the coarse parameters (a,L)(a,L). Those variables are useful branch data, but the current strict parent action does not yet contain an autonomous wall PDE unless an explicit wall/throat action SΣS_\Sigma is added.
  • Static closure. The branch is summarized by a finite mouth radius aa, a finite depth LL, and an open exit R(L)>0R(L)>0.
  • Moving-wall closure. R(Ω,w,t)R(\Omega, w, t) is a dynamical field on the throat's coordinates in an effective wall/throat closure; its strict parent-level promotion is currently open.
◇ winding and charge branch

Winding and electric sign are separate data

The winding integer n∈Zn \in \mathbb{Z} counts the phase change of ψ\psi around the throat axis — the same homotopy invariant introduced in topic 02. Independently, a charged throat branch carries an electric sign:
◇ charge sign · Part I Eq. 4.2
q∗q_* is the microscopic branch coupling; qeffq_\text{eff} is the observable brane charge. The factor 1/Zint1/\sqrt{Z_\text{int}} dresses q∗q_* through the EM localization integral.
ηQ  ∈  {+1, −1},q∗  =  ηQ e∗,qeff  =  q∗ / Zint\begin{aligned} \eta_Q &\;\in\; \{+1,\,-1\}, \\ q_* &\;=\; \eta_Q\, e_*, \\ q_\text{eff} &\;=\; q_* \,\big/\, \sqrt{Z_\text{int}} \end{aligned}
It is a load-bearing feature of the framework that these two pieces of branch data are independent. Circulation (winding) and electric sign are distinct throat-sector labels. In particular:
◇ boundary conditions

Mouth, interior support, and open exit

A throat is specified by boundary data at three locations:
Mouth
w = 0, R(0)=a
The brane-side cross-section. It is the observable entrance geometry, not the whole defect.
Interior support
Σ < 0
The bulk throat/cavity region carrying support, wall, Maxwell/mixed, and outgoing-port degrees of freedom.
Open exit
w = L, R(L)>0
The finite-radius outlet into unconfined bulk. A hard cap R(L)=0 is obsolete except as an explicit negative control.
◇ brane projection of a throat

What the brane observer reads

Applying the projection of topic 03 to a throat produces a pair of localized brane profiles: density/source bookkeeping such as δρ^(r)\widehat{\delta\rho}(r) and transverse current data such as Jw^(r)\widehat{J^w}(r). Topic 05 uses the projected inflow channel in the Newtonian regime; the amplitudes are branch data rather than a completed generic throat theorem.
◇ projected throat data · bookkeeping
These projected quantities are readouts of a frozen branch; they are not the full physical ontology of the throat.
δρ^(r)  =  ∫W(w) (ρ(r,w)−ρ∞) dw,Jw^(r)  =  ∫W(w) Jw(r,w) dw\begin{aligned} \widehat{\delta\rho}(r) &\;=\; \int W(w)\,\big(\rho(r,w)-\rho_\infty\big)\, dw, \\ \widehat{J^w}(r) &\;=\; \int W(w)\,J^w(r,w)\,dw \end{aligned}
The electric far-field sector is a controlled reduction of the localized 4+1 Maxwell system. Under the zero-mode assumptions, the localization profile Z(w)Z(w) produces standard-looking 3+1 Maxwell theory with μ0,eff=μ0/Zint\mu_{0,\text{eff}} = \mu_0 / Z_\text{int} and qeff=q∗/Zintq_\text{eff} = q_*/\sqrt{Z_\text{int}}. The derivation is the job of topic 06.
Projected source profilesZero-mode EM couplingMoving-wall branch amplitudes
◇ mixed channels resurface here

Mixed sector near a throat is not negligible

The suppressed channels of topic 03 — Aw, Jw, Fμw, Ew, Ca=FawA_w,\ J^w,\ F_{\mu w},\ E_w,\ C_a = F_{a w} — are generically expected near a charged throat. The zero-mode ansatz suppresses them only in the controlled far-field brane limit. This is the locus of:
  • brane–bulk leakage and transverse current exchange,
  • plasma non-ideality corrections in dense throat ensembles,
  • outgoing quadrupole normalization targets for moving-throat radiation,
  • conditional g-factor / anomaly channels for bound-throat configurations.
None of these corrections are claimed to be resolved here. They are queued against the moving-throat PDE (topic 11) and the results ledger.
◇ end of part I

What Part II does with this

Part II will run the throat through each of the familiar force sectors. Gravity (topic 05) reads the projected inflow Jw^\widehat{J^w}. Electromagnetism (topic 06) reads the localized Maxwell sector through its zero-mode reduction. Light (topic 08) reads the linearized ambient fluid's ripple. Atoms (topic 09) read reduced bound-throat configurations. Two-body PN (topic 10) reads finite-throat response at large separation. The moving-throat frontier (topic 11) is the effective closure whose strict parent promotion and branch realization remain open.
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