Arithmos — Where the Scripts and the Notes Disagree

This dive was assembled from two kinds of source: a written design record, and the scripts that actually produced the geometry. They do not agree. That is not a criticism — a design document written before the CAD work is supposed to be overtaken by it — but the divergences were never folded back, and a reader coming to the design document cold would build the wrong machine. Collecting them is the most useful thing this dive can do.

Each item below is stated as: what the design record says, what the scripts do, and which one this dive follows.

6.1 The carry finger

The design specification calls for a radial pin, 2 mm in diameter, projecting 3 mm from the drum’s outer surface at the angular position of the digit zero.

The scripts build an axial finger — 2 mm by 5 mm, mounted on the drum’s side face at a radius of 12 mm from the shaft axis. The change was forced when the arm turned out to be eight millimetres short of the radial pin, and it is the historically correct Pascaline arrangement.

This dive follows the scripts. The design document’s specification table was never amended.

6.2 The pinion teeth

The specification calls for a ten-tooth pinion, 20 mm in diameter and 8 mm wide, at module 2.0 — a standard gear tooth form.

The scripts cut ten rectangular grooves, 3 mm wide and 3 mm deep, producing a lantern or pin-wheel profile. The script says so in its own header and calls it a placeholder for the bench test, with real involute teeth deferred to a brass version.

This is a legitimate engineering choice for a rig whose only requirement is discrete catch points every 36 degrees. It is nonetheless a different part from the one specified, and nothing in the project record confirms that the placeholder was ever promoted to the plan.

6.3 The number of shafts

The specification lists nine bodies, including two separate shafts, one for each drum.

The rig was built with eleven bodies — two end plates were added for shaft support — and later reduced to ten when the second shaft was deleted. The deletion was not a simplification for its own sake: the two shafts had been placed on the same axis at the same height and depth, overlapping along thirty-two millimetres. They were modelled intersecting one another, and the collision was found only when the end plates were repositioned.

6.4 The drum spacing

The specification sets the bench-rig drum centres 48 mm apart and gives a reason: the main machine uses 30 mm, and the rig is widened to give the sautoir clearance to swing.

As built, after the shaft correction, the spacing is 20 mm — narrower than the main machine, and less than half the specified figure. The correction script argues that twelve millimetres between drum faces is ample for a two-millimetre arm, which may well be true. The point is that the rig now contradicts the stated reason for its own geometry, and the specification still reads 48.

6.5 The print material

The test protocol says to build version one in PLA.

The export script and its manifest specify PETG throughout, with the note that the builder prints in PETG rather than PLA. The project’s own session notes, meanwhile, still refer to a PLA print and a PLA-versus-brass test sequence.

6.6 The leaf spring

Three plans exist for one part. The specification calls for 0.5 mm spring steel. The export manifest lists it as a PETG part with a note that PETG may be too compliant and a music-wire swap is planned. The session notes say to exclude it from the print entirely and cut a real spring-steel strip instead. All three are defensible; none of them has been reconciled.

6.7 The connector version

The project’s standing notes pin the CAD connector at version 1.4.2 and describe its capabilities in detail. The joint-building script refuses to run below 1.4.3. The animation script requires 1.4.6 or later for the ability to set a joint value at all. The export script accepts 1.4.6 through 1.4.8. The documented version is behind the version the scripts require by at least four releases.

6.8 The session log stops before the work does

The written session log ends at session sixteen. On disk there are a joint-building script, a diagnostic script, two layout-correction scripts, an animation script, a close-up capture script, an export script and twenty-two screenshots, all of which identify themselves as session seventeen, and a manifest generated by a connector version that session sixteen did not have. A reader relying on the log would conclude the project stopped two rounds before it did.

6.9 The one that matters: the rest pose

The remaining divergence is not documentary. It is geometric, and it could not be resolved from the files.

The finger is placed at a radius of 12 mm from the shaft axis, and in the rig’s rest pose it sits directly above the axis — at the very top of its sweep, 70 mm above the base plane. The sautoir arm was lowered so that its underside sits at the same height, which the script describes as putting the arm “exactly at the doigt strike path.”

The difficulty is that the top of a circle is a maximum. From that pose, rotating the drum in either direction can only carry the finger downward, away from the arm. A sautoir has to be lifted, which means the arm must sit somewhere on the rising part of the sweep, with the finger approaching from below. As scripted, the finger starts where it should finish.

The animation does not settle this, because it cannot: the joint system poses parts at commanded values and does not compute contact. Its sequence drives the drum from zero to fifty-four degrees while raising the arm to twenty degrees at thirty-six and dropping it again at fifty-four. Over that same interval, the finger’s height falls from 70 mm to about 65 mm. The frames assert a lift that the geometry does not produce.

There are innocent explanations. The rest pose of a register drum is arbitrary — it corresponds to whichever digit happens to be showing — so “at rest” may simply mean “the carry has just fired,” in which case the finger being at the top is correct and only the animation’s choreography is wrong. Alternatively the arm height, set by a four-millimetre offset chosen to make the two numbers meet, may be the error. The files do not say which, and no note in the project addresses it.

What can be said with confidence is narrower than the project’s own summary suggests. The rig is geometrically complete and free of the collisions and under-reaches that consumed three rounds. The rest pose and the animated strike sequence cannot both be right. And the question is one a printed rig would answer in about ten seconds, which is a fair argument for printing it.

6.10 What has and has not been verified

Verified from the model: every dimension quoted in this dive; the four register drums at 4.6623 cubic centimetres each with upright, right-reading digits; the pinion at 1.5761 cubic centimetres against a prediction of 1.57; the enclosure, base, window and rabbet geometry; the ten exported bench parts and their intended materials.

Not verified: that the carry fires. Nothing has been printed, machined, assembled or cranked. The hundred-cycle gate the project set for itself has not been attempted, in plastic or in brass. The export directory referenced by the manifest does not exist in this tree, so even the mesh files are a record of a run on a different machine.

Not determinable from the files: whether the rest pose or the arm height is the error; whether the lantern pinion is the plan or still a placeholder; and whether the twenty-millimetre drum spacing leaves the sautoir enough room, since the only stated clearance requirement was written for a forty-eight millimetre rig that no longer exists.