Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 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.

Fixed-target invariant energy threshold

Physical scope

Necessary invariant-energy threshold for a massive projectile incident on a massive target at rest in one named inertial laboratory frame; it is a kinematic condition and does not predict a reaction rate or occurrence.

Statement

In a named inertial laboratory frame, consider a mass-specified isolated collision configuration of Finite event collisions and observer quantities whose complete incoming list consists exactly of two massive legs: a projectile of rest mass $m_{1} > 0$ and laboratory momentum $P_{1} = (E_{1} / c, p_{1})$ , and a target of rest mass $m_{2} > 0$ initially at rest, with momentum $P_{2} = (m_{2} c, 0)$ . Assume the adopted balance postulate Isolated event four-momentum balance applies. Let the complete outgoing list be a proposed finite nonempty mass-specified final state, and let $μ \geq 0$ be the sum of its recorded rest masses. Thus its total momentum equals $Q : = P_{1} + P_{2}$ . Define

$σ : = - c^{2} η (Q, Q) .$

Then

$σ = m_{1}^{2} c^{4} + m_{2}^{2} c^{4} + 2 m_{2} c^{2} E_{1} .$

The necessary final-state threshold from System invariant mass and final-state thresholds is $σ \geq μ^{2} c^{4}$ , hence

$E_{1} \geq \frac{(μ^{2} - m_{1}^{2} - m_{2}^{2}) c^{2}}{2 m_{2}}, E_{1} \geq m_{1} c^{2} .$

If the proposed complete final list contains a nonzero null leg, the invariant-mass inequality is strict, $σ > μ^{2} c^{4}$ ; the displayed invariant threshold can then be an unattained endpoint. These are kinematic necessities, not guarantees that the collision or channel occurs.

In SI units $m_{1}, m_{2}, μ$ are kilograms, $E_{1}$ is joules, and $σ$ is joules squared.

Facts & Assumptions

Given: A flat inertial laboratory frame, a mass-specified finite isolated collision datum with exactly the two stated massive incoming legs and a complete finite nonempty outgoing list, and the adopted isolated-balance postulate.

[A1]

Isolated event four-momentum balance: the complete incoming and outgoing four-momentum sums are equal.

[F1]

Finite event collisions and observer quantities: a mass-specified collision configuration has finite nonempty incoming and outgoing lists of nonzero future-causal legs, all at one event $p \in A$ ; a massive leg of recorded mass $a > 0$ has $P = a U$ with $U$ future timelike and $η (U, U) = - c^{2}$ . The recorded mass sum is taken over the complete list, and the laboratory is one inertial chart.

[F2]

Mass shell inversion, observer work and variable mass: for a $C^{2}$ future-timelike worldline on a nonempty open interval, parameterized by one inertial chart's time, with constant mass $a > 0$ and $P = a U$ , the observer components obey $E^{2} = a^{2} c^{4} + c^{2} ∣ p ∣^{2}$ and $E > 0$ .

[F3]

Mechanical momentum and observer energy: in the laboratory time-axis observer, $E_{n} = - c η (P, n)$ and the spatial momentum is the projection of $P$ into $n^{⊥}$ ; a particle at rest has $P = (m c, 0)$ .

[F4]

System invariant mass and final-state thresholds: for the same complete isolated datum satisfying balance, its total invariant squared energy obeys $σ \geq μ^{2} c^{4}$ , strictly if its complete final list contains a nonzero null leg.

Proof

technique · expand the Lorentzian norm in laboratory coordinates and solve the necessary inequality

1.1A1F1F2F3constructalgebra

For a massive leg $P = a U$ of [F1], write $U = (U^{0}, U)$ in the laboratory, with $U^{0} > 0$ by future orientation. The auxiliary affine curve $y : R \to A$ , $y (τ) = p + τ U$ , is smooth and future timelike with proper-time parameter $τ$ , since $η (y^{'}, y^{'}) = - c^{2}$ . Its laboratory time is $t = t_{p} + τ U^{0} / c$ , where $t_{p}$ is the event's time coordinate; thus it has a smooth time parametrization on the open interval $R$ . Assign constant mass $a$ and momentum $a d y / d τ = P$ . All hypotheses of [F2] hold for this representative, so its pointwise shell and positivity apply to the original leg, without assuming smooth motion through the event. The complete incoming list and balance [A1] identify the common total as $Q = P_{1} + P_{2}$ . Its observer components are those of [F3], and the projectile shell just established gives $E_{1}^{2} - c^{2} ∣ p_{1} ∣^{2} = m_{1}^{2} c^{4}$ with $E_{1} > 0$ . The stationary target is $P_{2} = (m_{2} c, 0)$ . With signature $(-, +, +, +)$ , expansion yields $σ = (E_{1} + m_{2} c^{2})^{2} - c^{2} ∣ p_{1} ∣^{2} = m_{1}^{2} c^{4} + m_{2}^{2} c^{4} + 2 m_{2} c^{2} E_{1}$ .

2.1A1F1F4step 1.1algebra∎

Both incoming legs are future timelike by [F1], so their sum $Q$ is future timelike by the system result [F4]. The same complete configuration and balance [A1] therefore satisfy its timelike threshold $σ \geq μ^{2} c^{4}$ . Substituting the expression in step 1.1 and dividing by $2 m_{2} c^{2} > 0$ gives $E_{1} \geq (μ^{2} - m_{1}^{2} - m_{2}^{2}) c^{2} / (2 m_{2})$ . The projectile shell in step 1.1 also gives $E_{1} \geq m_{1} c^{2}$ because $E_{1} > 0$ and $E_{1}^{2} = m_{1}^{2} c^{4} + c^{2} ∣ p_{1} ∣^{2} \geq m_{1}^{2} c^{4}$ . If the complete final list contains a nonzero null leg, the strict case follows from [F4].

Fixed-target invariant energy threshold

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources