Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Formal tree scattering and scalar-decay coefficients

Physical scope

Conditional finite formal-tree calculations for unpolarized electron-positron to muon-antimuon scattering and the specified scalar contact and decay amplitudes, interpreted only by the adopted coefficient maps. The electron-mass estimate compares two tree coefficients only; no continuum scattering operator, loop or infrared error, threshold resummation, detector prediction, or empirical estimate is asserted.

Statement

Assume ${A C}_{ω}$ The Axiom of Countable Choice ( ${A C}_{ω}$ ). Use the fixed Cartesian Minkowski chart $x = (c t, x)$ , the geometric metric $η = (-, +, +, +)$ , and the distinct algebraic contraction $h = - η = (+, -, -, -)$ of QFT example units, metric, gauge, and pairing conventions. Let $m, M \geq 0$ be the electron and muon rest energies in joules. In this chart, a future energy four-vector is $q = (E, q) \in R^{4}$ with $E > 0$ ; it has rest energy $M \geq 0$ when $h (q, q) = M^{2}$ . All four-vector components are in joules, and $q$ is $c$ times physical spatial momentum. Let $p_{1}, p_{2}$ be future electron and positron energy four-vectors with rest energy $m$ , and $k_{1}, k_{2}$ future muon and antimuon energy four-vectors with rest energy $M$ , satisfying $p_{1} + p_{2} = k_{1} + k_{2} = P$ and $s = h (P, P) > 4 \max (m^{2}, M^{2}) .$ By Complete two-body phase space and coefficient flux map, choose a proper orthochronous $h$ -Lorentz matrix $Λ \in G L (4, R)$ , meaning $Λ^{T} h Λ = h$ , $\det Λ = 1$ , and preservation of the future causal cone $C_{+} : = {(E, q) : E > 0, h ((E, q), (E, q)) \geq 0}$ , with $Λ P = (\sqrt{s}, 0)$ . Its boost construction supplies such a map because $P$ is future timelike. The center-of-momentum inertial chart has coordinates $x^{C M} = Λ x$ on spacetime and corresponding four-vector components $q^{C M} = Λ q$ ; from here on, four-vector components are in that chart, so $P = (\sqrt{s}, 0)$ . The strict threshold makes $p_{1}$ and $k_{1}$ nonzero. Define the scattering angle $θ \in [0, π]$ by $θ : = \arccos ⁣ (\frac{p_{1} \cdot k_{1}}{∣ p_{1} ∣ ∣ k_{1} ∣}),$ using the Euclidean inner product on $R^{3}$ . Write $g = \sqrt{4 π α} > 0$ , as in QFT example units, metric, gauge, and pairing conventions, and $β_{e} = \sqrt{1 - \frac{4 m^{2}}{s}}, β_{μ} = \sqrt{1 - \frac{4 M^{2}}{s}} .$ For the selected formal one-photon tree coefficient, take equal incoherent weight $1 / 4$ for the four initial spin-label pairs and sum over final spin labels. Take the selected muon model charge to be $- e$ as well. Up to an irrelevant common phase, its amplitude is $M_{t r e e} = \frac{g^{2}}{s} [\overset{ˉ}{v} (p_{2}) γ^{a} u (p_{1})] [\overset{ˉ}{u} (k_{1}) γ_{a} v (k_{2})],$ where the brackets denote the product of the two currents. This selected formal amplitude adopts the current draft QFT-G02 line/vertex convention; it is not an exact amplitude of a constructed continuum theory.

Conditional on this amplitude adoption and Formal diagram coefficient-to-observable adoption, the spin-averaged squared coefficient is $\overline{∣ M_{t r e e} ∣^{2}} = g^{4} [3 - β_{e}^{2} - β_{μ}^{2} + β_{e}^{2} β_{μ}^{2} \cos^{2} θ],$ The distinguishable-particle differential and total cross-section coefficients are $\frac{d σ_{c o e f f}}{d Ω} = \frac{α^{2} (ℏ c)^{2} β_{μ}}{4 s β_{e}} [3 - β_{e}^{2} - β_{μ}^{2} + β_{e}^{2} β_{μ}^{2} \cos^{2} θ],$ $σ_{c o e f f} = \frac{π α^{2} (ℏ c)^{2}}{s} \frac{β_{μ}}{β_{e}} [3 - β_{e}^{2} - β_{μ}^{2} + \frac{1}{3} β_{e}^{2} β_{μ}^{2}] .$ For $δ = 4 m^{2} / s \leq 1 / 2$ , the ratio of this differential coefficient to the coefficient at $m = 0$ with the same $M, s, θ$ belongs to $[1, 1 + 4 δ]$ . This bound is only the incoming-electron-mass effect within the selected tree coefficient.

The on-shell equations also give $P_{a} \overset{ˉ}{v} (p_{2}) γ^{a} u (p_{1}) = 0, P_{a} \overset{ˉ}{u} (k_{1}) γ^{a} v (k_{2}) = 0.$ These are the two current contractions for this finite external-spinor calculation; they do not assert an all-orders or loop Ward theorem.

For the separate scalar conventions of Formal diagram coefficient-to-observable adoption, let $m_{s} \geq 0$ be a scalar rest energy, let the incoming $r_{1}, r_{2}$ and outgoing $l_{1}, l_{2}$ lie on the positive $m_{s}$ -shell, meaning each $(E, q)$ has $E = \sqrt{∣ q ∣^{2} + m_{s}^{2}} > 0$ , and require $r_{1} + r_{2} = l_{1} + l_{2} = P_{s}$ with $s = h (P_{s}, P_{s}) > 4 m_{s}^{2}$ . Evaluate in a proper orthochronous center-of-momentum chart taking $P_{s}$ to $(\sqrt{s}, 0)$ as in [F4]. Let $λ \in R$ be the dimensionless constant quartic amplitude parameter. The identical-final full-sphere and distinguishable total coefficients are respectively $σ_{c o e f f}^{i d e n t i c a l} = \frac{λ^{2} (ℏ c)^{2}}{32 π s}, σ_{c o e f f}^{d i s t i n g u i s h a b l e} = \frac{λ^{2} (ℏ c)^{2}}{16 π s} .$ For a parent at rest of energy $M_{0} > 2 m_{s}$ and a constant identical daughter amplitude $G_{E}$ in joules, take the daughters on the positive $m_{s}$ -shell with total $P_{0} = (M_{0}, 0)$ . The rate coefficient is $Γ_{c o e f f} = \frac{∣ G_{E} ∣^{2}}{32 π ℏ M_{0}} \sqrt{1 - \frac{4 m_{s}^{2}}{M_{0}^{2}}} [s^{- 1}] .$ If an energy width is named, its convention is $Γ_{E} : = ℏ Γ_{c o e f f}$ in joules. None of these coefficient statements implies an exact survival law. The QFT-G02 postulate is a current draft carrier; its use here retains draft status and does not imply independent acceptance.

Facts & Assumptions

Given: ${A C}_{ω}$ , the masses and strict thresholds above, the finite Dirac spinor normalizations and traces, the two-body flux/phase-space map, and the stated physical coefficient adoptions.

[F1]

QFT example units, metric, gauge, and pairing conventions: $h = - η$ , energy components and rest energies are joule-valued, $s$ has unit $J^{2}$ , $g^{2} = 4 π α$ , and $ℏ c$ has unit $J m$ .

[F2]

Example observable types and scoped meanings: a formal coefficient is not an evaluated exact probability or error bound without a separate adoption; an empirical estimate is a finite-data report with its source conditions and uncertainty.

[F3]

Clifford traces and normalized spin completeness: the finite spin sums are $\sum_{s} u_{s} {\overset{ˉ}{u}}_{s} = p̸ + M$ and $\sum_{s} v_{s} {\overset{ˉ}{v}}_{s} = p̸ - M$ ; the two- and four-gamma traces, on-shell equations, adjoint, and current conjugation identity use $h = - η$ and the stated finite spinor normalization. The supplier's matrix definition carries ${A C}_{ω}$ .

[F4]

Complete two-body phase space and coefficient flux map: its boost construction sends any future-timelike $P$ to $(\sqrt{s}, 0)$ by a proper orthochronous $h$ -Lorentz transformation; transformations fixing $(1, 0)$ act as proper spatial rotations. Equal-mass two-body density is $d Φ_{2} / d Ω = β / (32 π^{2})$ , equal-mass incoming flux is $F = 2 s β$ , and the coefficient map is $(ℏ c)^{2} \overline{∣ A ∣^{2}} d Φ_{2} / (F S_{f})$ ; the decay map is $\int ∣ G ∣^{2} d Φ_{2} / (2 ℏ M_{0} S_{f})$ .

[F5]

Scalar symmetry factors and decay coefficient units: for the adopted constant quartic amplitude with positive-shell scattering inputs, common total, and center-of-momentum evaluation as stated, the identical scalar total coefficient is $λ^{2} (ℏ c)^{2} / (32 π s)$ and the distinguishable result is twice that; for positive-shell identical daughters summing to the stated rest parent, the constant-amplitude decay coefficient is $∣ G_{E} ∣^{2} \sqrt{1 - 4 m_{s}^{2} / M_{0}^{2}} / (32 π ℏ M_{0})$ .

[A1]

Formal diagram coefficient-to-observable adoption: the displayed QED and scalar coefficient maps are adopted as model-specific leading formal readings; they are not exact continuum observables or empirical estimates.

[A2]

The selected QED one-photon tree amplitude, initial spin weights, and final spin sum are exactly those in the Statement. This fixes the finite formal calculation and supplies no LSZ or continuum scattering construction.

[A3]

Explicit formal QED vertex/free-line convention is the current batch-53 QFT-G02 draft. It adopts a species-charge vertex $i q_{n} γ^{a}$ (electron and selected muon $q_{n} = - e_{n}$ ), the fermion line $i (p̸ + μ) / (h (p, p) - μ^{2})$ only for $μ > 0$ and nonzero denominator, and the Feynman-gauge photon line $- i h_{a b} / (h (k, k) + i 0)$ . This channel has no internal fermion line. A time-ordered free line uses the actual Countable-Choice-dependent $H_{μ}$ carrier. The postulate's carrier is current and its interface was inspected; its physical adoption remains draft.

Proof

1.1givenF1F3A2A3algebra

Put $L = ℏ c > 0$ . By [F1], the selected photon's natural momentum is $κ = P / L$ , so $h (κ, κ) = s / L^{2} > 0$ . Thus [A3] supplies photon factor $- i h_{a b} / (s / L^{2} + i 0)$ , whose ordinary boundary value at this regular point is $- i L^{2} h_{a b} / s$ . For each external energy vector $q$ , use the same orthonormal range bases as in [F3] and define natural-normalized columns $u_{n} (q / L) = u (q) / \sqrt{L}$ and $v_{n} (q / L) = v (q) / \sqrt{L}$ . Indeed the energy and rest-energy parameters both scale by $1 / L$ , leaving the projections in [F3] unchanged, while their normalization $\sqrt{2 E}$ scales by $1 / \sqrt{L}$ . Each natural current is therefore the corresponding energy-normalized current divided by $L$ . The product of two currents cancels the photon factor's $L^{2}$ ; the two charge- $- e_{n}$ Dirac vertices, with $e_{n} = g$ by [F1], give the coefficient $g^{2} / s$ up to a common phase. This agrees with the selected amplitude adoption [A2], and the phase disappears in its modulus square. No internal fermion line or loop product is used.

1.2givenF3algebra

Summing the final spin labels and averaging the four initial pairs, [F3] reduces the squared currents to $L_{e}^{a b} = \frac{1}{4} Tr [(̸ p_{2} - m) γ^{a} (̸ p_{1} + m) γ^{b}], L_{μ, a b} = Tr [(̸ k_{1} + M) γ_{a} (̸ k_{2} - M) γ_{b}] .$ The terms with three gamma matrices vanish. Expanding the remaining terms with the two- and four-gamma traces in [F3] gives $L_{e}^{a b} = p_{2}^{a} p_{1}^{b} + p_{2}^{b} p_{1}^{a} - \frac{s}{2} h^{a b}, L_{μ, a b} = 4 (k_{1 a} k_{2 b} + k_{1 b} k_{2 a} - \frac{s}{2} h_{a b}),$ because $p_{1} \cdot p_{2} + m^{2} = s / 2$ and $k_{1} \cdot k_{2} + M^{2} = s / 2$ follow by expanding the two total-momentum squares.

1.3givenF3algebra

The adjoint equations from [F3] give $\overset{ˉ}{v} (p_{2}) ̸ p_{2} = - m \overset{ˉ}{v} (p_{2})$ and $̸ p_{1} u (p_{1}) = m u (p_{1})$ , so contraction with $P = p_{1} + p_{2}$ cancels the two terms. Similarly, $\overset{ˉ}{u} (k_{1}) ̸ k_{1} = M \overset{ˉ}{u} (k_{1})$ and $̸ k_{2} v (k_{2}) = - M v (k_{2})$ cancel in the outgoing current. This proves both displayed current contractions, including when either mass is zero; no general off-shell or loop Ward statement is inferred.

1.4givenF1F5algebra

The declared positive-shell scalar inputs, common total and center-of-momentum evaluation meet [F5] exactly; the decay daughters have total $P_{0}$ and satisfy its strict rest-parent threshold. Apply [F5] to these declared scalar amplitudes and thresholds. Its identical-final factor is $S_{f} = 2$ on the full labeled sphere, while the distinguishable case has $S_{f} = 1$ , so the total scattering coefficients have the stated factor of two. The decay result uses the same identical-particle factor and the parent-rest coefficient map. Since $[G_{E}] = [m_{s}] = [M_{0}] = J$ and $[ℏ] = J s$ , $[Γ_{c o e f f}] = J^{2} / (J s J) = s^{- 1}$ ; $(ℏ c)^{2} / s$ has unit $m^{2}$ .

2.1step 1.2F1F4algebra

In this chart $P = (\sqrt{s}, 0)$ gives $p_{2} = - p_{1}$ and $E_{1} + E_{2} = \sqrt{s}$ . The equal mass-shell equations and $E_{i} > 0$ imply $E_{1} = E_{2} = \sqrt{s} / 2$ , hence $∣ p_{1} ∣ = \sqrt{s / 4 - m^{2}} = \sqrt{s} β_{e} / 2 > 0$ ; the same argument gives ${\tilde{E}}_{1} = {\tilde{E}}_{2} = \sqrt{s} / 2$ and $∣ k_{1} ∣ = \sqrt{s} β_{μ} / 2 > 0$ . Thus $θ$ is defined. If $Λ_{1}, Λ_{2}$ both send $P$ to rest, $Λ_{2} Λ_{1}^{- 1}$ fixes $(1, 0)$ and so acts by a proper spatial rotation by [F4]; it preserves the Euclidean dot product, making $θ$ independent of the selected chart's residual orientation. Contracting the tensors in 1.2 and using the mass-shell dot products gives $L_{e}^{a b} L_{μ, a b} = 4 {2 [(p_{2} \cdot k_{1}) (p_{1} \cdot k_{2}) + (p_{2} \cdot k_{2}) (p_{1} \cdot k_{1})] + s^{2} - s (p_{1} \cdot p_{2} + k_{1} \cdot k_{2})} .$ The cross terms with one metric contribute $- s (p_{1} \cdot p_{2} + k_{1} \cdot k_{2})$ , and the metric-metric term contributes $s^{2}$ ; together these equal $s (m^{2} + M^{2})$ . The mixed dot products are $p_{2} \cdot k_{1} = p_{1} \cdot k_{2} = s / 4 + ∣ p_{1} ∣ ∣ k_{1} ∣ \cos θ$ and $p_{2} \cdot k_{2} = p_{1} \cdot k_{1} = s / 4 - ∣ p_{1} ∣ ∣ k_{1} ∣ \cos θ$ . Substituting and multiplying by $g^{4} / s^{2}$ gives the stated spin-averaged squared coefficient.

3.1step 2.1F1F4algebra

The strict threshold gives $β_{e}, β_{μ} > 0$ . For distinguishable muon labels, [F4] gives $d Φ_{2} / d Ω = β_{μ} / (32 π^{2})$ and $F = 2 s β_{e}$ . Inserting the squared coefficient derived in step 2.1 into its map and using $g^{4} = (4 π α)^{2}$ yields the displayed differential coefficient. The sphere integrals are $\int d Ω = 4 π$ and $\int \cos^{2} θ d Ω = 4 π / 3$ (integrate $2 π u^{2}$ over $- 1 \leq u \leq 1$ ), giving the displayed total coefficient.

4.1step 3.1algebra

If $δ = 4 m^{2} / s \leq 1 / 2$ , set $B_{0} = 2 - β_{μ}^{2} + β_{μ}^{2} \cos^{2} θ, Q = 1 - β_{μ}^{2} \cos^{2} θ .$ Then $B_{0} \geq 1$ and $0 \leq Q \leq 1$ . The ratio of the full differential coefficient to its $m = 0$ value is $R = \frac{B_{0} + δ Q}{B_{0} \sqrt{1 - δ}} \geq 1.$ Moreover $R - 1 = \frac{1}{\sqrt{1 - δ}} - 1 + \frac{δ Q}{B_{0} \sqrt{1 - δ}} \leq 2 \sqrt{2} δ \leq 4 δ,$ since $1 / \sqrt{1 - δ} - 1 \leq \sqrt{2} δ$ and $B_{0} \sqrt{1 - δ} \geq 1 / \sqrt{2}$ . This compares only the two formal tree coefficients.

5.1givenF1F2A1step 1.4∎

The units and formal/empirical types in [F1]–[F2] identify the results as formal coefficients with area or inverse-time units. The adopted interpretation in [A1] gives only the named leading-coefficient reading; neither an exact continuum observable, a measured estimate, nor an exponential survival law follows.

Formal tree scattering and scalar-decay coefficients

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources