Third-Order (Axiomatic)

Third-Order (Axiomatic) is the designation given under Ordinance 412.7 to negative phenomena that arise neither from wording nor from circumstance but from the foundational propositions of the Tower itself. Where First-Order (Lexical) faults are corrected by striking a word and Second-Order (Environmental) faults by altering a floor, a Third-Order fault cannot be repaired at the point where it appears, because the point where it appears is entailed by something further down. The designation is descriptive of origin only; a Third-Order fault is not thereby more severe than a Second-Order one, though it is invariably more expensive to answer.

Definition and three tests

Ordinance 412.7 defines the Third-Order in a single clause, and the clause has been read three ways. The authoritative reading, maintained by the Bureau of Axiomatic Purity, holds that a phenomenon is Third-Order when the removal of the phenomenon would require the amendment of a proposition listed among the Axiomatic Qualities. Because the seven Qualities cannot be amended individually, this has the practical consequence that Third-Order faults are not corrected but reconciled — a distinction the Bureau has defended in eleven recensions and explained in none of them.

Survey practice recognises three tests, applied in sequence. The Lexical Substitution Test replaces the wording of the defective instrument with equivalent wording; if the fault persists, it is not First-Order. The Floor Reversion Test restores the affected floor to its last certified state; if the fault persists, it is not Second-Order. The Residue Test, administered by a Librarian of the Third Order or higher, examines what remains. The residue is the fault.

In the winter of 1147, on Floor 62 (the Floor of Small Corrections), a Clerk applied all three tests to a corridor that had begun returning visitors to the stairwell they had entered from, and to no other stairwell. Floor 62 passed the first two tests cleanly. The residue was a single sentence in the floor's founding instrument stating that the corridor existed in order to be traversed once. The Bureau declined to strike the sentence and instead reclassified the corridor as a thoroughfare of one direction, which it remains.

Relation to the Axiomatic Engine

Tower Ordinance 12-B.7 classifies The Axiomatic Engine as a Third-Order Foundational Instrument, a category created for it and never subsequently used. The classification has caused persistent confusion, since the Engine is a cause of Third-Order conditions rather than an instance of one. The Bureau's position is that 12-B.7 attaches to the Engine's output, not to the Engine, and that a Third-Order condition downstream of the Engine is to be docketed with a reference to 12-B.7 and left running until the Engine's next cycle closes.

Not all Third-Order conditions are Engine-derived. The Bureau estimates that between a fifth and a third arise from the accumulated weight of old amendments — what Axiomatic Recursion describes, in its own phrasing, as "the tendency of a corrected rule to correct its correctors." Figures are disputed. The 4th Axiomatic Survey returned 22 per cent; the Axiomatic Wing's internal tally for the same decade returned 41 per cent, but that tally counted every unresolved docket rather than every Third-Order docket, a distinction the Wing has since stopped discussing.

The 1163 reconciliation dispute

For sixty-two years the accepted procedure required that a Third-Order condition be resolved by reference to an earlier condition of the same kind, so that the Tower might resolve its foundational faults in order of priority. The procedure is preserved, in a form no longer followed, in the Third Concordance.

It failed on Floor 211 (the Floor of Precedents) when two Third-Order conditions each cited the other as prior. The Clerk of the Third Order assigned, a certain Meridian Voss, recorded the impasse in the docket entry now held as Exhibit 9 of the Axiomatic Compatibility Check:

"Each refers to the other as antecedent. Neither can be reconciled until the other is. I have stamped both as pending and gone to my supper."

— Docket entry, Floor 211, 1163, as reproduced in Third Supplementary Order, Annex 9

The Clerks of the Third Order resolved the matter in 1164 by introducing the practice of parallel reconciliation, in which mutual antecedents are stamped simultaneously. The Infinite Stamp used for the purpose is applied twice, once to each document, and the second application is not recorded on the first document. Clerks are instructed to stamp the two within the same hour and not to look at either afterwards.

Procedure and certification

A Third-Order docket passes to the Bureau of Axiomatic Purity, which may order reconciliation, deferral, or reclassification to a lower order. Reconciliation at the level of the whole Tower requires a Clerk of the Third Order to apply the Infinite Stamp, which cannot be applied to a document already bearing it. Where a fault touches a section rather than a floor, the supervising officer must hold a Section E Certification, granted under the standing authority of a Librarian of the Third Order.

The Bureau maintains that Third-Order conditions should never be documented in the place where they occur, since a written account of a foundational fault becomes part of the foundation. Dockets are therefore held on the floor above. Floor 63 (unassigned) consists entirely of such dockets and has no other contents. Visitors are asked to leave the door open.

Scholarly disagreement

Two positions persist. The Alpha Purists, who hold that all valid order descends from a single principle, maintain that there is no Third Order distinct from the First — that every axiomatic fault is a lexical fault in a document nobody has yet found, and that the category was created in 1142 to spare the Bureau the labour of searching for it. The Purist position is set out most fully in the suppressed preface to the second recension of the Axiomatic Nullpoint register and has never been answered in writing.

The Bureau's reply, offered in conversation and never committed to the record, is that the Purists have been unable to produce the document. The Clerk Voss, asked in 1165 to settle the question, declined on the grounds that a determination either way would itself be Third-Order, and was promoted.