Numerical Sequencing

Numerical Sequencing is the discipline governing the arrangement of numbered entries into ordered series, and the branch of administrative practice concerned with establishing whether a given number may lawfully precede another. It differs from Hermetic Sequencing in that it proceeds from premises that are, at least on their face, causal: a sequence is held to exist because a number was assigned before another number was assigned. Practitioners insist the distinction is fundamental. It is not clear that anyone else does.

Origins and Early Recensions

Numerical Sequencing is first attested in the second recension of the Treatise on Acausal Taxonomies, whose marginalia — the so-called Fourteen Annotations of the sub-archivist Halvern Vess — already treat number order as a matter of record rather than arithmetic. The early form was simple: entries were sequenced by the order in which they were written into a register, and any number appearing out of turn was struck through in red ochre and rewritten at the foot of the page. This produced a great many strikethroughs and very few sequences.

The discipline was formalised under Concordance 7.4.13, which granted authority for sequencing submissions and permitted procedural re-sequencing where a foundational error was later discovered. The practical effect was that any sequence could be corrected, provided it had first been completed, and that any completed sequence could be re-opened, provided it had first been corrected. The Bureau of Sequential Integrity has twice attempted to resolve the circularity and twice withdrawn the attempt on procedural grounds.

The Sequence Claim

A sequence submitted for verification must be accompanied by a Sequencing Submission — a declaration of the order the sequence believes itself to follow. The declarant's belief is not examined. What is examined is whether the entries, read in the order the declarant has stated, conform to that statement. Sequencing Audits conducted on this basis have never failed, a fact the Audit Office of the Hermetic Carrels records without comment in each annual return.

Where the declaration and the entries do not conform, the submission is not rejected but re-sequenced: the entries are rearranged until they agree with the declarant, on the authority of Clause 5, which provides that the statement of an order outranks the order stated. In Year 268 the entire catalogue of the Porphyry Stacks was re-sequenced in this manner, leaving the shelves in an arrangement no visitor has since been able to locate deliberately and which staff describe as "the declared one."

Interaction with Numerical Precedence

Numerical Sequencing is frequently confused with Numerical Precedence, and the confusion is not discouraged. Precedence determines rank; Sequencing determines order. The two coincide only when a series happens to be ordered by rank, and the Committee for Numerical Sanity has held, in an unpublished opinion of Year 401, that this coincidence is "not to be presumed, nor discouraged, nor remarked upon in the presence of the numbering clerks." The application and interpretation of Precedence remain, as established, under that Committee's purview; Sequencing merely arranges.

Disputes arise where a number carries both a rank and a position. The celebrated case of Entry 44 turned on whether a lesser number in higher position should precede a greater number in lower position when the greater number had been numbered first. The Principal Reconciliers empanelled to hear it retired to the Sub-level Θ reading rooms for eleven winters and returned with a sealed finding that has not been opened. The finding stands at the head of the shelves it concerns, and is dusted weekly.

Instruments and Marks

The working instruments of the discipline are few, and are an inventory in the strict sense:

• the sequence slate, on which provisional numbers are chalked and re-chalked;
• the red ochre stick, for striking;
• the seal of the sequence, held in duplicate by two clerks who are not permitted to confer;
• the numeral ruler, marked in forty-two graduations, of which twenty-nine are used.

The numeral ruler has not been revised since Year 412, owing to a dispute as to whether a ruler marked from zero should count the zero. The question was referred to the Bureau of Numerical Anomalies and, because the audit in question was a dual-authorization matter, returned to the referring desk and thence, by an error in the routing, to the reading room of Floor 202, where the file remains. Clerks in the numbering office keep their own rulers and do not lend them.

Disputes and Amendments

The chief doctrinal quarrel in Numerical Sequencing concerns the status of the Seventh Addendum to the First Concordance, which appears to require that all sequences be read backwards before being accepted forwards. The Bureau of Sequential Integrity holds that this applies only to sequences of more than forty entries; the Committee for Numerical Sanity holds that it applies to all sequences, on the ground that a sequence of one has nowhere to go forward to and nowhere to go back from and is therefore, in the Committee's phrase, "the only honest case." Protocol 12-A is silent on the question, its alphanumeric guidance having been drafted for entries that can be spelled.

Practitioners on Floor 202 resolve the matter locally by reading short sequences backwards twice, which satisfies both readings and neither office. The practice was noticed during an inspection and entered in the record as a minor deviation. No amendment followed. In Year 419 the Decree of Aberrant Sequencing touched the question obliquely through its amendment of Title IX, and the marginal note appended to the Decree by an unidentified hand — "let it be read both ways, and let the reading be logged" — was later adopted verbatim into the standing instructions of the numbering office, where it has settled the matter for all practical purposes.

Numerical Sequencing continues to be taught in the Carrels from the back of the slate. Students are advised not to ask where the sequence begins until the third term, and are given no answer in the first.