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---
## Open propagation reorganizing into localized cores
ASD explains pair creation as nonlinear reorganization of open transverse propagation modes and participating objects into conjugate closed topological cores and surrounding fields. The final manuscript supplies threshold derivations for nuclear recoil and two-photon configurations; formation paths and rates remain calculations of the same coupling. All participating quantities must first share one background energy–momentum calibration.
ASD associates photons with open transverse propagation and electron and positron candidates with localized closed orientation. Pair creation studies incident excitations, participants and background jointly reorganizing in an interaction region, transferring propagation energy into conjugate cores, accompanying fields and remaining motion or radiation.
Energy remains a property of these states, not an independent material. The final organization must retain finite energy and stability after leaving the interaction region. Conservation can permit it without supplying a formation path. The manuscript first specifies participants and kinematics, then leaves nonlinear core formation to evolution under the same coupling.
Lines and arrows show a conceptual candidate sequence, not a computed trajectory. Initial and final total charge are zero: $N_{\rm em}^{\rm in}=0$, with candidate pair sum $-1+1=0$. A fixed-amplitude model can develop local positive and negative concentrations in this total sector; variable amplitude can admit other reconnection paths.
## Kinematic inputs and the meaning of a threshold
In the same approximately uniform background, the manuscript uses upstream [ASD dynamics](/en/theory/dynamics) for massive matter and photon propagation:
$$
E_a^2=m_a^2c^4+c^2|\mathbf p_a|^2,
\qquad \mathbf p_\gamma=\frac{E_\gamma}{c}\mathbf n_\gamma.
$$
$m_a$ is intrinsic mass, $\mathbf p_a$ mechanical momentum and $\mathbf n_\gamma$ a propagation unit vector. All quantities use the same background and energy calibration. Invoking these upstream relations does not make them a first derivation in Chapter 4.
Define the algebraic combination $\mathcal S=E_{\rm tot}^2-c^2|\mathbf P_{\rm tot}|^2$, in energy-squared units. Conservation fixes initial and final totals. At threshold, the final state has no additional relative motion for its total momentum. Under the specified dispersion relations, a common-velocity final state gives the squared sum of masses times $c^4$. This is a kinematic calculation without introducing frame transformations as a new ontological premise.
## A photon and a recoil participant
Consider $\gamma+Z\to e^-+e^++Z$, where $Z$ is a recoil nucleus or another participant, initially at rest and retaining the same ground-state mass $M$. Initially $E_{\rm tot}=Mc^2+E_\gamma$ and $\mathbf P_{\rm tot}=\mathbf p_\gamma$, giving
$$
\mathcal S=(Mc^2+E_\gamma)^2-c^2|\mathbf p_\gamma|^2
=M^2c^4+2Mc^2E_\gamma.
$$
The final threshold includes this participant and two masses $m_e$, so $\mathcal S_{\rm th}=(M+2m_e)^2c^4$. Equate the expressions, subtract $M^2c^4$ and divide by $2Mc^2$:
For $M\gg m_e$, this approaches $2m_ec^2$, with finite recoil raising the threshold. The participant is not consumed: its field and surroundings provide coupling, and its motion carries recoil. Internal excitation or an additional field changes the complete final state and account, so the same threshold cannot then be used unconditionally.
Using measured electron mass gives a conditional threshold with standard kinematic form, not a distinct ASD numerical prediction. Cross sections, polarization and core formation still require explicit sources and nonlinear dynamics.
## Why the two-photon angle matters
Take two free incident photons with energies $E_1,E_2$ and angle $\vartheta$, in a uniform background with no other energy–momentum participant. From $\mathbf n_1\cdot\mathbf n_2=\cos\vartheta$,
$$
c^2|\mathbf p_1+\mathbf p_2|^2=E_1^2+E_2^2+2E_1E_2\cos\vartheta,
$$
and subtraction from $(E_1+E_2)^2$ gives
Head-on photons have $\vartheta=\pi$ and require $E_1E_2\ge m_e^2c^4$. Strictly co-propagating photons give zero on the left and cannot form a free pair through this two-photon process alone. A large incoming energy is therefore insufficient without the momentum account.
A single free photon similarly has $\mathcal S=0$, while a pair requires at least $4m_e^2c^4$. It cannot spontaneously split into a free pair in a background with no other exchange. Any participating background, boundary or external field must be included in the system.
## From an allowed final state to actual formation
The manuscript proposes reorganization of local orientation and topological density during field overlap. Stable final states, reachable conjugate structures, accessible barriers and allocation of remaining energy are separate conditions. Increased intensity changes incident flux and nonlinear interaction without guaranteeing creation in every configuration.
Multiphoton processes require all incoming energies and momenta. [Existing multiphoton experiments](https://link.aps.org/doi/10.1103/PhysRevLett.79.1626) provide phenomena to recover quantitatively, not direct measurements of the candidate ASD topological formation mechanism. This page retains participants and conditional thresholds without inventing rates, cross sections or formation animations.
[OP-MAT-007](/en/open-problems/matter/creation-annihilation) requires time evolution, forward and reverse processes, incident flux and final phase space from the same coupling. [Creation requirements](/en/tests/matter#mat-pair-production) connect the deliverable to observations. Reaching a threshold is one step in a complete particle-reaction problem.
## Intrinsic photon energy and one reaction calibration
Section 4.5 distinguishes two energy comparisons. Different incident photons in one spatial state share a local action scale, $E_\gamma=h_{\rm eff}\nu_\gamma$. Tracking the same photon through a static background without absorption, scattering or other real exchange instead uses $\lambda'=K\lambda$, $\nu'=\nu/K$, $h'=Kh$ to preserve intrinsic propagation energy $E_\gamma^*$. Pair creation in a nuclear field is a real energy–momentum exchange; free-propagation invariance does not hold each part’s energy fixed through the reaction.
The thresholds above therefore use the matched $E_\gamma=c|\mathbf p_\gamma|$, matter dispersion and total initial/final quantities in one approximately uniform background. Changing a moving source’s emission conditions is different from a moving absorber sampling an already emitted photon. Source, recoil and channels must share one calibration before the threshold calculation. [Mass and local action](/en/theory/matter/mass#material-action) explains the interface; the action-scale law is not an extra energy source.