Lemma 2: The Static Resistance Count

Layer 1 — Manifold capacity: 21² × 4 = 1764

The manifold has 21 geometric channels (AP10, AP28) and resolves into 4 spacetime dimensions (AP10, AP20). Each channel must express through each dimension, giving 21 × 4 = 84 dimensional expressions.

The proton integrates all 21 channels. It is not a 1-channel object like the electron.

Each of the 84 dimensional expressions couples to each of the 21 channels, generating the full resistance network: 84 × 21 = 21² × 4 = 1764.

This is the dominant term. It says: the proton resists 1764 times more than the bare break because it integrates the full channel network across all spacetime dimensions.

Layer 2 — Face projection: 21 × 3 = 63

The proton forms under SU(3) geometry (AP19): three faces corresponding to three colours. The 21 channels must anchor to these three specific faces.

This is the resistance cost of projecting the full manifold onto the 3-colour strong-force substrate. Each channel anchored to each face: 21 × 3 = 63.

This layer is a constraint ON the manifold capacity (Layer 1). The channels are already counted. Layer 2 adds the resistance of locking them to three specific faces rather than leaving them unanchored.

Layer 3 — Exchange matrix: 3² = 9

By Axiom S, the three faces must maintain continuous mutual exchange to remain bounded. The full interaction matrix of three faces is 3 × 3 = 9 pathways.

This maps exactly to U(3): 8 gluon generators + 1 colour singlet required for total confinement.

This is the smallest static term. It counts only the face-to-face exchange overhead, not the channel structure within each face.

Uniqueness of the construction

All three-term sums of structural products (products of powers of {21, 3, 4}) that equal 1836 were enumerated:

Among them, exactly one is hierarchically nested — each successive term using fewer body of work numbers and corresponding to a distinct geometric feature (manifold capacity, face projection, exchange matrix). This hierarchical decomposition is the one used here; whether it is the unique admissible decomposition under a precise enumeration constraint is exactly what KS-30.4 holds open, and the constraint set and the count of admissible decompositions under it are not exhibited in this paper.

The construction is not arbitrary. It is the hierarchical decomposition this paper adopts; its uniqueness among hierarchically nested decompositions is asserted, not exhibited (KS-30.4).