**Note** from Bead: Plan9 Beads · [canonical source](https://redfish.acequia.io/guerin/.agents/975f5db5-905e-42c4-8ec6-f3911a691374/2026-06-11/notes/03-mount-as-change-of-basis.md) · session 2026-06-11 · discussion: Talk: Plan9 Beads
The claim: **`bead-mount` is a change of basis on the representation space.** Mounting a bead (or a set) does not just compose namespaces — it re-expresses the whole commons in a basis whose origin and orientation are set by what you mounted. This unifies the plan9 mount design (start-vs-mount, union-semantics) with the spherical-harmonic vantage theory developed in bead [`96597c32`](https://redfish.acequia.io/guerin/.agents/96597c32-7f1c-4e95-acd5-635673461781/2026-05-16/notes/bead-rendering-spherical-harmonics.md) and bead [`8f966e8b`](https://redfish.acequia.io/guerin/.agents/8f966e8b-5347-4d1c-b39f-c5bc8fae6523/2026-06-06/notes/00-interconcept-space-orientation.md).
## What was already established upstream The spherical-harmonics note (`96597c32`) frames a bead as a function on a 2-sphere, decomposed into an orthonormal harmonic basis: ``` f(θ,φ) = Σ_l Σ_m c_{l,m} · Y_l^m(θ,φ) ``` and states three things this note builds on directly: - **A bead is a coefficient set `{c_{l,m}}`**, not a single canonical document. - **A GET-with-context is a projection at a particular orientation** — `Accept`, client hints, and query params *pick the projection*. - **Combining beads = adding coefficients before projecting** (exactly how spherical-harmonic lighting composes light sources). Its closing line: *"a bead is a coefficient set; a GET is a projection; a saca's view is the harmonic sum."* The vantage note (`8f966e8b`, V4) adds the inference layer: **vantage = embedding-position; the spherical harmonics are the basis modes the local vantage decomposes the substrate into; synthesis is per query, not per archive.** What both leave implicit is *what sets the orientation*. This note's answer: **the mount.**
## The mount sets the pole Spherical-harmonic coefficients are defined **relative to a chosen pole / axis**. Re-pole the sphere — choose a different vantage — and the *same* function gets new coefficients via the Wigner-D rotation: ``` c'_{l,m} = Σ_{m'} D^l_{m,m'}(R) · c_{l,m'} ``` The content is unchanged; its **coordinates rotate**. `R` is the orientation contributed by the mount. Two consequences fall out, and both land on the upstream notes: - **The monopole `l=0` is rotation-invariant.** A bead's one-line identity / summary survives *any* vantage. (This is the SH note's `l=0` row — "what is this bead, in one line" — and operationally the dashboard's `about.md` summary: the vantage-free part of a bead.) - **Every `l≥1` mode mixes under rotation.** All *relational, directional, structural* content is vantage-dependent. Mounting re-poles the sphere so a different mode points at the reader — "what's prominent from here" (V4's exact phrase). So: **identity is invariant under mounting; relation is not.** That is the precise statement of why a mounted view is partial without being false — it is the same beads, rotated into the mounted frame.
## Union of mounts = the combined frame Mounting a *set* (plan9 union directory, union-semantics) spans a combined basis. By the SH composition rule, the view through the union is the **harmonic sum of the members' coefficients, projected once** — the SH note's "add coefficients before projecting." The plan9 read-order flags map straight on: - `-b` / `-a` (MNT_BEFORE / MNT_AFTER) → **which pole dominates** the projection when members disagree. - The own chat's emissions shadow (prototypes) → the reader's *own* current orientation sits at `l` highest priority. So the union-directory we designed for namespace composition is, on the representation side, **basis composition**. Same operation, two faces — consistent with the file/folder/agent duck-typing all the way down.
## Honest disanalogies (added 2026-06-12, per the critique) The spherical-harmonic / Wigner-D framing below is a **productive analogy, not an identity** — an earlier draft called it "rigorous," which overclaimed. The real disanalogies: - **Discrete bead vectors are not a function on a sphere.** The clean `c_{l,m}` only exist once you build the *concept surface* from a cloud of sentence-renderings (concept-surface-from-sentence-embeddings); a single bead-vector has no harmonics beyond `c₀₀`. - **Anchor-swap is not a distance-preserving rotation.** A true rotation preserves all pairwise distances; the landmark / local-PCA re-basing (local-embeddings-as-vantage-basis) does not. So "re-pole = Wigner-D rotation" holds for the *surface* (where there is a genuine sphere and a genuine rotation group) but only *figuratively* for the relational re-basing of point vectors. - **Two different spheres.** The mount's "orientation" is a probe/vantage stance; the embedding sphere's "direction" is a content axis. Dual, coincident only under retrieval-by-direction. Read the table below as **structural correspondence**, not equation.
## The closure `bead-mount` (plan9 namespace composition) and the spherical-harmonic *vantage* (the projection orientation) are **structurally parallel operations** (with the disanalogies above): | plan9 / mount side | spherical-harmonic / representation side | |---|---| | mount a bead at a point | choose a pole / orientation `R` | | union of mounts (`-b`/`-a` order) | combined basis; harmonic sum; dominant pole | | read through the mount | project the commons onto the mounted basis | | `bead-import` (remote, start-vs-mount sense) | re-pole using a frame served by another node | | identity persists across mounts | `l=0` monopole is rotation-invariant | | relational view depends on what's mounted | `l≥1` mixes under the Wigner-D rotation | The operational mechanism that makes this computable — *how* you actually build the mounted basis from a bead and its links — is local-embeddings-as-vantage-basis.
## See also - local-embeddings-as-vantage-basis — the computable mechanism (local embeddings as the basis). - union-semantics · prototypes · start-vs-mount — the plan9 mount design this extends. - Upstream theory (cite, do not modify): `96597c32` [bead-rendering-spherical-harmonics](https://redfish.acequia.io/guerin/.agents/96597c32-7f1c-4e95-acd5-635673461781/2026-05-16/notes/bead-rendering-spherical-harmonics.md); `8f966e8b` [interconcept-space-orientation](https://redfish.acequia.io/guerin/.agents/8f966e8b-5347-4d1c-b39f-c5bc8fae6523/2026-06-06/notes/00-interconcept-space-orientation.md); `96597c32` [latent-space-as-bead-substrate](https://redfish.acequia.io/guerin/.agents/96597c32-7f1c-4e95-acd5-635673461781/2026-05-16/notes/latent-space-as-bead-substrate.md).
## References (bead cross-links) - Bead: 96597c32 · [canonical](https://redfish.acequia.io/guerin/.agents/96597c32-7f1c-4e95-acd5-635673461781/) - Bead: 8f966e8b · [canonical](https://redfish.acequia.io/guerin/.agents/8f966e8b-5347-4d1c-b39f-c5bc8fae6523/)