Signals in the model: receipt in the future cone, relay composition, absence of closed signal loops, and frame independence of the causal signal condition

Clause (i). Suppose a signal is transmitted from $p$ to $q$. By [L1] the model admits this only if $q\in J^{+}(p)$. If $q\mathrm{\ne }p$, then $D=q-p$ is a nonzero element of $\overline{C^{+}}$, so $D^0>0$ and $D^0\ge \mathrm{\mid }{D}_s\mathrm{\mid }$ by [L2]; in the chart $x$ of any inertial frame, $D^0=x^0(q)-x^0(p)$ and $\mathrm{\mid }{D}_s\mathrm{\mid }=\mathrm{\mid }{x}_s(q)-x_s(p)\mathrm{\mid }$ by the affine linearity of the chart, so $x^0(q)>x^0(p)$ and $\mathrm{\mid }{x}_s(q)-x_s(p)\mathrm{\mid }\le x^0(q)-x^0(p)$. Since the chart is arbitrary and the computation used the same criterion available in every inertial chart, every inertial frame's time order places the reception after the emission, and the temporal order of the endpoints of the transmission is the same in every inertial frame.

Clause (ii). Let signals be transmitted from $p$ to $q$ and from $q$ to $r$. By [L1] applied twice, $q-p\in \overline{C^{+}}$ and $r-q\in \overline{C^{+}}$, so $r-p=(q-p)+(r-q)\in \overline{C^{+}}$ by the finite-sums clause of [L2], that is $r\in J^{+}(p)$. If in addition $q\in I^{+}(p)$, that is $q-p\in C^{+}$, or $r\in I^{+}(q)$, that is $r-q\in C^{+}$, the strict case of the same clause gives $r-p\in C^{+}$, that is $r\in I^{+}(p)$. In particular a relay whose second leg is causal does not leave the causal future of the first emission, and if either leg has nonzero displacement then its time-coordinate increment is positive, while the other increment is nonnegative by [L2], so the total increment is positive and the relay cannot return to the first emission event.

Clause (iii). If signals are transmitted from $p$ to $q$ and from $q$ to $p$, then [L1] gives $q-p\in \overline{C^{+}}$ and $p-q\in \overline{C^{+}}$; by the antisymmetry clause of [L4] (equivalently by $\overline{C^{+}}\cap \overline{C^{-}}=\{0\}$ from [L2]) this forces $q-p=0$, that is $p=q$. For a finite chain $p_0\to p_1\to \cdots \to p_n$ with integer $n\ge 0$ (at $n=0$ both closure and the all-zero-leg condition hold vacuously), every leg displacement lies in $\overline{C^{+}}$ by [L1], so every inertial time-coordinate increment is nonnegative, and is strictly positive if that leg displacement is nonzero by [L2]. If the chain is closed, telescoping gives total time-coordinate increment zero; a finite sum of nonnegative reals is zero only when every summand is zero, and [L2] then forces every displacement to be zero. Conversely, if every displacement is zero then $p_0=p_1=\cdots =p_n$, so the chain is closed. Thus only the all-zero chains are closed, and no nontrivial chain returns to its own emission event.

Clause (iv). By the covariance clause of [L4] one has $F(J^{+}(p))=J^{+}(F(p))$, $F(I^{+}(p))=I^{+}(F(p))$ and $p\le q$ iff $F(p)\le F(q)$ for every proper orthochronous affine Lorentz map $F$, and by [L5] these maps are exactly the inertial changes of description of the model. Since [L1] imposes $q\in J^{+}(p)$ as a necessary condition on an admitted transmission, this condition is preserved under every change of description; no sufficiency or separate transmission-existence claim follows. [L6] records independently that the causal classification and the future/past alternative of the specified model are chart independent, in agreement with Interval, cone and orientation invariance across inertial descriptions.

Clause (v). Let $q-p$ be spacelike. By the first clause of [L3], $\mathrm{\mid }{D}^0\mathrm{\mid }<\mathrm{\mid }{D}_s\mathrm{\mid }$ in every inertial chart, so neither $D^0\ge \mathrm{\mid }{D}_s\mathrm{\mid }$ nor $-D^0\ge \mathrm{\mid }{D}_s\mathrm{\mid }$ holds; by [L2] $q\mathrm{\notin }{J}^{+}(p)$ and $q\mathrm{\notin }{J}^{-}(p)$, and by [L1] no signal is transmitted from $p$ to $q$ (nor from $q$ to $p$). By the remaining clauses of [L3] there are inertial frames whose charts assign $x^0(q)-x^0(p)$ either sign, including zero. Conversely, if the temporal order of a pair $p\mathrm{\ne }q$ is observer-independent with the emission $p$ the earlier endpoint, then the displacement is not spacelike by [L3], hence it is nonzero causal by the classification clauses of [L6], and since $p$ is the earlier endpoint the displacement lies in $\overline{C^{+}}$; the model's restriction admits a transmission only in the future direction by [L1], so the restriction is exactly the requirement that transmissions occur only between coincident events or distinct events whose temporal order is observer-independent, with the emission the earlier endpoint. For coincident endpoints, $q=p$ gives $q\in J^{+}(p)$ and equality of every chart coordinate.

Clause (vi). Steps 1.1 to 2.4 are conditional deductions from the adopted postulates [L1], [L5] and the mathematical results [L2], [L3], [L4], [L6]: no actual signal, transmitter, receiver, detector or measurement is described, no statistical or empirical claim is made, and clause (v) is a mathematical exclusion resting on [L3] rather than on any experiment. Every quantification is over the ideal flat model, and no choice principle is used, all arguments being finite-dimensional coordinate algebra with the explicit objects already named.