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

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Free boost plus instantaneous relativistic pair potential violates the charge algebra

Statement refuted

Let $P = (T^{*} R)^{2} ≅ R^{4}$ have coordinates $(q_{1}, p_{1}, q_{2}, p_{2})$ , symplectic form $ω = \sum_{a = 1}^{2} d q_{a} \land d p_{a}$ , and canonical bracket ${F, G} = \sum_{a = 1}^{2} (F_{q_{a}} G_{p_{a}} - F_{p_{a}} G_{q_{a}})$ . Let $q_{a}$ be one-dimensional positions in metres, $p_{a}$ momenta in kg m/s, $t$ an external equal-time parameter in seconds, $m_{a} > 0$ in kg, and $c > 0$ in m/s. For $k > 0$ in kg/s $^{2}$ define $E_{a} (p_{a}) = \sqrt{m_{a}^{2} c^{4} + c^{2} p_{a}^{2}}$ , $V (r) = k r^{2} / 2$ with $r = q_{1} - q_{2}$ , $H = E_{1} + E_{2} + V (r)$ , $P = p_{1} + p_{2}$ , and $K = (q_{1} E_{1} + q_{2} E_{2}) / c^{2} - t P$ . The proposed assertion that this Hamiltonian and unchanged free boost satisfy ${K, P} = H / c^{2}$ at every configuration is false.

Example

Assume ${A C}_{ω}$ for the cited Poincare algebra input. Use the phase space, bracket, masses, and functions defined in the Statement, with $k$ having units kg/s $^{2}$ , so $V$ has energy units. The retained free total boost is

$K = \frac{q_{1} E_{1} + q_{2} E_{2}}{c^{2}} - t (p_{1} + p_{2}), P = p_{1} + p_{2} .$

With canonical brackets, ${K, P} = (E_{1} + E_{2}) / c^{2}$ , while the Poincaré charge algebra requires ${K, P} = H / c^{2}$ . They disagree at every configuration with $q_{1} \neq q_{2}$ . The construction is therefore a counterexample to combining this instantaneous potential with the unchanged free boost. It is not a universal no-interaction theorem.

Facts & Assumptions

Given: Independent canonical pairs $(q_{a}, p_{a})$ , masses $m_{a} > 0$ , $c > 0$ , $k > 0$ , and the functions specified above.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only; no full axiom of choice is used.

[L1]

Poincare cotangent moment map and full generator algebra: the current assigned A draft states the Poincare generator algebra; setting K=-J^{01}/c and P=P^1 in its bracket gives the required free identity {K,P}=P^0/c=H/c^2. The carrier exists as a batch-26 draft, but its proof remains conditional on H0/H8 and has no independent approval.

[F1]

Poincare semidirect group, dual pairing and moment components: $J^{01} = x^{0} P^{1} - x^{1} P^{0}$ and $P^{0} = E / c$ on the positive-energy inertial shell, fixing the free boost sign.

Counterexample

technique · direct canonical bracket and counterexample

Given: The free shell energies, translation-invariant potential, unchanged free boost, and [L1] target identity.

1.1F1L1A1algebra

By the canonical product bracket, ${q_{a} E_{a}, p_{1} + p_{2}} = E_{a}$ for each $a$ , since $E_{a}$ is independent of position; also ${t (p_{1} + p_{2}), p_{1} + p_{2}} = 0$ . Hence ${K, P} = (E_{1} + E_{2}) / c^{2}$ .

2.1L1step 1.1algebra∎

The Hamiltonian is $H = E_{1} + E_{2} + k r^{2} / 2$ , so $H / c^{2} - {K, P} = k r^{2} / (2 c^{2}) > 0$ whenever $r = q_{1} - q_{2} \neq 0$ . The required algebra relation in [L1] fails there. Since $k$ has units $k g / s^{2}$ , $k r^{2} / 2$ is energy and the compared charges have matching units. This refutes this proposed free-boost construction only.

Free boost plus instantaneous relativistic pair potential violates the charge algebra

Statement refuted

Example

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources