The Standard Argument, Once

Whose it is. What it needs. What it gives. And the two premises the corpus supplies to it: the finite speed and the sign.

Claim. On premises (i)–(iv) — no privileged site (§1), homogeneity and isotropy (§3), a finite invariant speed (§3), and consistent composition of readings — the transformations between two sites’ readings are the Lorentz transformations, with c as the invariant. The quantity every site reads the same is the interval, with signature (−, +, +, +). The sign — Lorentz rather than Euclidean — is fixed by R. Time is the one direction that cannot be reversed. No reading may rotate it into space and back. KS-D.4 is paid at the structural register.

Whose argument. Ignatowsky’s, Physikalische Zeitschrift 11, 972 (1910), and its modern statements. Lévy-Leblond’s “One more derivation of the Lorentz transformation”, American Journal of Physics 44, 271 (1976), is the cleanest. Lévy-Leblond starts from the principle of relativity: equivalent inertial readings; transformations that form a group. He constrains those transformations by homogeneity of spacetime, isotropy of spacetime, the group law, and a causality condition. What survives is the Lorentz transformations and their Galilean contraction. The invariant speed is derived there. It may be infinite. This paper’s packaging is slightly different. Stronger in one place. Weaker in none. Premise (i) is the principle of relativity, from §1. Premise (ii) is homogeneity and isotropy, from §3 and AP10. Premise (iii) is a finite invariant speed, from §3. So the Galilean contraction is dead before the theorem runs. Premise (iv) is composition. Causality is R. It is used only to fix the sign. The corpus adds nothing to the algebra of the theorem. It cites it as it stands. What it adds is why the premises hold, and which sign R forces.

Givens. Premises (i)–(iii) from §§1, 3. Premise (iv), consistent composition: a site’s reading of a second site’s reading of an event is that site’s reading of the event; the maps between readings compose and invert. Stated, as the standard argument states it. AP20’s monoid is the record composing, not the readings. §4 (counts). §5 (velocity). AP10 §0.3 and §4 (R’s irreversibility carries the (−) character; KS-D.4). The author’s “yes” to that reading, 9 September.

Theorem — the Lorentz group on the corpus’s premises. Premise (i) is §1. Premise (ii) is §3’s second theorem. Premise (iii) is §3’s first. Here the corpus is stronger than the standard argument needs. The invariant speed is not a finite constant that happens to exist. It is the unit of distance itself. So the Galilean case — the constant infinite — is excluded by §3 before the theorem runs. Premise (iv) is composition. On these the theorem gives the Lorentz transformations, with the tick as the invariant speed. It gives the invariant interval between two events: minus the count squared plus the crossing record squared, read the same at every site. With the transverse faces, signature (−, +, +, +). Time dilation is then §4’s sentence made exact. Each of two sites in uniform relative settling reads the other’s count as the slower, by γ. Neither is wrong (A.8). Two couplings of the same two sites need one of them to change its settling between them. That is accelerated settling. It is the next paper’s. Length contraction is §3’s. The crossing record between two sites at rest to each other, read from a site settling past them, is shorter. The composition of velocities keeps §5’s bound. Two settlings compose to a settling that never reaches one grain per click.

Theorem — the sign, from R. The theorem’s four premises leave one case open besides the Lorentzian. The standard argument closes it with a fifth: causality. No transformation may reverse the order of events along a site’s own count. Here that premise is R. This is where it comes from. The open case is the Euclidean. There the transformations are rotations mixing distance and count freely. A reading could take a site’s count into another site’s distance and back. An event later in one reading could be earlier in another, with no bound. Axiom R forbids it. The record is written in one direction. It cannot be unwritten (AP20; AP50 §9: the order is total and well-founded). A reading is a coupling to held records (§1). It cannot read what has not landed (AP51 §5). So no reading can place an event’s count before a record it depends on. The transformations must preserve the order of the record along every site’s count. They may mix distance and count. They may not reverse the count. That is the hyperbolic case. The Lorentzian sign. The (−) belongs to the one direction that is R’s. This is AP10 §4’s reading. The author ruled it again on 9 September. Now it stands on the argument rather than beside it. The transformations so obtained are maps between readings. They do not unwrite the record. They do not make the record’s order a readable now (§6).

What this does to KS-D.4. KS-D.4 names two things. That the signature is read from the axioms’ reversibility structure. That the formal metric-level derivation is owed. The first is paid here. The Lorentz group follows on premises each derived or ruled from the axiom. The sign follows from R. The second is paid as far as the premises allow, in Appendix A. It writes the derivation out on the page, in the corpus’s terms, from linearity to the invariant. What it does not do is derive the crossing record as the record algebra’s own distance function, from that algebra alone. It takes it from §3’s ruling. KS-D.4 is therefore paid at the structural register. It is written out at the formal register. It remains live at the formal register, narrowed to that one identification: that the crossing record is distance, taken from §3’s ruling and not from the record algebra. It is one identification, and it is the whole of the formal debt: a derivation of AP10’s distance function from the record algebra is what KS-D.4 has owed since it was written. It is the only debt this paper does not close. The author has ruled it a lemma to be derived, in a paper of its own, at his word of 11 September 2026: the record algebra’s distance function from the algebra, and the crossing record shown to be it. Until that paper, KS-D.4 stays live at the formal register as the gate. The dated note on AP10 says exactly this.

Grade. The argument: Standard, cited, and written out in Appendix A. The premises: the corpus’s, as §§1–5 grade them. The sign: Derived from R. KS-D.4: paid at the structural register; written out at the formal; one identification named as remaining.

Kill switches

KS-REL.4 — The sign. Fires if a reading is shown that reverses a site’s count: an event read before a record it depends on. Or if the transformations between sites are shown to mix distance and count as rotations rather than boosts. LIVE — STRUCTURAL.

KS-D.4 (AP10) — PAID at the structural register and written out (Appendix A). LIVE at the formal register, narrowed to one identification: the record algebra’s own distance function shown to be the crossing record. By dated note.

The mathematics is a century old. It belongs to whoever did it. The premises are the sand’s.