Table of Recursive Weights

Definition and scope

The Table of Recursive Weights is a standing schedule of mass-equivalents maintained by the Scriptorium of Minor Weights, recording the weight that a given quantity acquires by being weighed a second time. It is distinct from, and frequently confused with, Citation Weights, which measure the authority a statement accrues when cited; the Table of Recursive Weights measures only the additional mass. The distinction was settled in the Eleventh Concordance and reopened in the Twelfth.

The Table is inscribed on nine laminae of cold-rolled brass, each 210 mm × 340 mm, hinged along the long edge and stored in a cedar case beneath the Scriptorium's calibration bench. Entries are added in a single hand; corrections are added in a second hand, at an angle, and are never erased.

Principle

A first weighing of an object yields its base weight. A second weighing, conducted under identical conditions by the same operator on the same instrument, yields a greater figure. The difference is the recursive increment. A third weighing yields a further increment, smaller than the second by a fixed ratio. The ratio is given in the Table as 0.3125, though three recensions of the Scriptorium's own calibration tables record it as 0.3125, 0.3125, and "approximately one third."

The increments do not accumulate without limit. The Table provides an asymptote for each registered object class; once an object approaches its asymptote, further weighings are recorded as saturated and no further increment is assigned. Brass instruments saturate early. Paper does not saturate within a working lifetime.

Dispute over the Second Recension

The Twelfth Concordance, Annex 3, holds that recursive increment is a property of the act of weighing and therefore belongs to the instrument, not the object. On this reading the Table should be organized by scale rather than by item. The Scriptorium has never implemented this reading, on the grounds that its existing laminae would have to be re-cut and the cedar case is not wide enough.

The opposing view, associated with the clerk Halvern Pace and recorded in his 1891 petition, holds that increment is a property of the object's willingness to be re-encountered, and that certain catalogued items — chiefly those listed in Recursive Nullities — return a negative increment. Pace's petition was upheld in part and is cited in Annex A under Error 7.

"The scale does not remember the object. The object remembers the scale. Whichever of these is true, the second figure is heavier, and the second figure is what we file."

— Halvern Pace, petition of 1891, clause 4

Relation to the Office of Weights

The Office of Weights has never fully discharged the regressive chain of certificates produced by the Axiom of Recursive Validation, and the Table of Recursive Weights is one of the instruments through which that chain propagates. A certificate of weight filed against a recursively-weighted entry itself becomes subject to recursion upon re-filing. The Scriptorium acknowledges this in the marginal note to entry 4,411: "see also, indefinitely."

Cross-reference with the Tome of Jurisprudential Weights and Measures is required biannually. The last completed cross-reference is dated three years before the current one, and the intervening two have been certified as complete by an officer whose name does not appear on the certificate.

Calibration and maintenance

Recalibration of the laminae is performed quarterly, aligned with the Resonance Alignment overseen by the Keeper of the Tide-Table of the Bronze Gears. Harmonic vibration introduced during the Alignment causes measurable drift in the recursive ratio for approximately eleven days. During this period the Scriptorium accepts weighings but does not record increments, a practice known locally as "letting the brass settle."

The Table is consulted most heavily on Floor 1147, where the stairwell narrows enough that returning expeditions must weigh their inventories before filing them with the Bureau of Ontological Weights and Measures. Visitors are reminded that the cedar case may be opened only by a certified clerk, and that no object may be weighed twice in the same afternoon for the purpose of demonstrating the principle.

See also

• Taxonomy of Qualia Weights
• Index of Operative Chains
• Registry of Minor Impressions
• Master Orthographic Table Document