Planetary Gear CSG Confirms the Law: Computational Geometry as a New Domain

A new domain: computational geometry

IRDME has been tested across molecular biology, neuroscience, software systems, formal mathematics, finance, and AI model architecture. The question naturally arises: does the Functional Proximity Law extend to computational geometry — systems where nodes are geometric parts and edges are spatial or operational relationships?

The first test was M_GEOM_CSG_1: a planetary gear assembly modeled as a Constructive Solid Geometry (CSG) graph. Three layers were defined:
• d1 (structural): geometric containment — which parts are subcomponents of which assembly
• d2 (functional / CSG): CSG boolean operations — union, difference, intersection between parts
• d3 (dynamic): motion coupling — which parts constrain the motion of which other parts under gear rotation

Pre-registered before any analysis (hash 1dd686ee, commit 5c45bbe).

Result: r = 0.668, p = 0.042, Δr = 0.935 — CONFIRMED

r(d1 ↔ d2) = 0.668 (p = 0.042)

r(d1 ↔ d3) = −0.267

Δr = r(d1 ↔ d2) − r(d1 ↔ d3) = 0.935

The structural containment layer (d1) agrees with the CSG operation layer (d2) far more than with the motion coupling layer (d3). This is the pre-registered prediction: layers encoding functionally similar relationships should agree more strongly than layers encoding different relationships.

Why the hubs align in d1 and d2

In a planetary gear, the central shaft and the carrier plate are the assembly hubs in d1: they connect to, contain, or support the most other parts. They are also the hubs in d2 (CSG operations), because those parts participate in the most boolean operations — differencing holes for bolts, unioning flanges, intersecting tolerances.

In d3 (motion coupling), the dominant hub is the ring gear, which constrains the orbital motion of all planet gears simultaneously. The ring gear is structurally a single outer shell — not a hub in d1 or d2 — but it is the kinematic hub of the system. This is why d1 ↔ d3 is negative: the assembly center (central shaft) is not the motion center (ring gear).

The BC3b boundary

The mathematics boundary condition (BC3) was established when certain formal mathematical structures denied the law. The underlying mechanism was a mismatch between the resolution of one layer (containing atomic-level symbolic definitions) and another (containing holistic proof references). The law requires both layers to operate at comparable structural resolutions.

M_GEOM_CSG_1 prompts a refinement: BC3b distinguishes continuous geometric operations from discrete boolean algebra.

CSG operations on a physical gear — union of surfaces, difference of volumes, intersection of tolerances — produce gradients. Some parts participate in more CSG operations than others; hub rank is meaningful and varies continuously. The Functional Proximity Law can operate here.

In discrete boolean circuits, operations are uniform: every gate receives exactly the inputs specified by the circuit, with no notion of "more" or "fewer" operations of the same type. Hub-rank gradients in the structural layer may not survive into the operation layer in the same way.

A parallel experiment in digital circuits

A boolean circuit experiment was run in parallel on a priority arbiter circuit with n = 19 nodes (pre-reg hash 1ff07ec9). Result: 4/4 hypotheses CONFIRMED, r = 0.8741.

This does not contradict BC3b. A priority arbiter has a clear hub-rank gradient: the arbiter logic node is the hub in both the structural layer (it connects to all request lines) and the functional layer (it resolves priority for all requests simultaneously). The gradient exists; the law confirms.

BC3b is not "digital circuits always deny." It is more precise: when a boolean circuit's operations are structurally uniform — no node participates in significantly more operations than others — the hub-rank gradient flattens. That is the condition under which BC3b applies.

Computational geometry is now a confirmed domain

M_GEOM_CSG_1 extends the confirmed domain list to include computational geometry via CSG. The mechanism is the same as in software systems, biological networks, and neuroscience: hubs in the structural assembly layer co-occur with hubs in the functional operation layer, because the parts that hold the system together are also the parts that most operations depend on.

The motion layer (d3) diverges because rotational kinematics have a different hub than static assembly. This is the law working correctly in both directions: confirming where functional similarity is high, and disagreeing where functional similarity is low.