Non-Orthogonal Analytical Frame

This is a rich and generative idea. Let me try to unpack what you're proposing and then explore whether and how objects could inhabit such a frame.

What you're describing

You're positing an analytical space where:

1. Each point has a unique key (like a primary key in a relational table), defined by its coordinate across all axes.

2. The axes are not perfectly orthogonal — there's some mutual projection, some "leakage" between dimensions.

3. The energy required to orthogonalize them is the binding energy of the frame — the cost of holding the frame apart from its entangled state.

4. This binding energy is memetic — it propagates, replicates, mutates, and is carried by minds/cultures rather than by fields and particles.

So the frame isn't a neutral grid. It's a held structure — held together by a kind of cognitive-cultural tension, and the axes lean into each other because of that.

Why non-orthogonality matters

In a perfectly orthogonal frame, each axis is independent. You can vary one without disturbing the others. Objects are cleanly decomposable: "this object has 3 units of X, 5 of Y, 0 of Z."

When axes are non-orthogonal, varying one necessarily perturbs others. This means:

• Objects can't be specified axis-by-axis. A coordinate in one dimension implies projections onto others.

• The metric is distorted. Distance, similarity, and identity become path-dependent.

• There is no canonical decomposition. You can't uniquely factor an object into independent components.

This is actually closer to how real analytical frames behave (in psychology, semantics, economics) than the idealized orthogonal picture.

The binding energy as memetic

If the binding energy is memetic, then the frame is maintained by the circulation of ideas. The non-orthogonality isn't a fixed geometric fact — it's a dynamic equilibrium sustained by:

• Shared assumptions that link axes together (e.g., "time is money" couples a temporal axis to an economic one)

• Taboos, dogmas, or paradigms that prevent certain rotations of the frame

• Institutional and linguistic scaffolding

To orthogonalize the frame would require dissolving those memes — hence the energy cost. The binding energy is literally the cultural work required to keep the axes entangled.

Can objects inhabit such a frame?

Yes — but they'd have unusual properties. Let me sketch what such objects would look like:

1. They are not points but trajectories or regions.
Because axes are coupled, an object can't be pinned to a single coordinate without ambiguity. It occupies a smear, a manifold, or a path through the frame. Its identity is partly its history of projection.

2. They have frame-dependent identity.
The same object, addressed from different rotations of the frame, yields different keys. Uniqueness of the key holds within a frame, but not across frames. So the primary key is frame-relative — which is a strange but coherent notion.

3. They carry binding energy themselves.
If an object is coherent within the frame, it must participate in the memetic binding. It either reinforces the non-orthogonality (making itself stable) or resists it (making itself costly). Objects become agents in the maintenance or dissolution of the frame.

4. They can be entangled with each other.
Non-orthogonality means that addressing one object can implicitly address another. Objects aren't fully separable. This gives you a natural notion of correlation, allusion, connotation — the "overtones" between axes.

5. They admit gauge transformations.
You could rotate the frame, and objects would transform accordingly — but because the frame is held by memetic energy, such rotations are costly and contested. Objects might be "read" differently under different rotations, and the politics of rotation becomes part of the analytical space.

A formal sketch

Let the frame be a tuple of axes $A_1,\dots ,A_n$ with a metric $g$ that is not diagonal. The off-diagonal terms $g_{ij}$ encode the non-orthogonality. The binding energy is something like:

$$E_b=\sum _{i\mathrm{\ne }j}\mathrm{\mid }{g}_{ij}{\mathrm{\mid }}^2\cdot {\mu }_{ij}$$

where ${\mu }_{ij}$ is the memetic density coupling axes $i$ and $j$. An object is a section of a bundle over this frame — a field, not a point. Its key is a local coordinate, valid only within a neighborhood where the frame is approximately stable.

Orthogonalization is a flow that drives $g_{ij}\to 0$, at cost $E_b$. Memes resist this flow.

What this buys you

This picture lets you model:

• Paradigm shifts as frame rotations that become cheap once binding energy drops

• Conceptual blending as objects that live in the non-orthogonal region between axes

• Ideological rigidity as high binding energy

• Creativity as the ability to navigate non-orthogonal space without collapsing it

• Analytical uniqueness as a local, frame-relative property rather than a global one

The short answer

Yes — objects can inhabit such a frame, but they can't be points. They have to be fields, trajectories, or tensors that participate in the frame's binding. The frame isn't a passive container; it's a maintained structure, and objects within it are partly responsible for its maintenance. The unique key becomes a local gauge rather than a global identifier.

Would you like to push further into any of these — the formal structure, the memetic dynamics, or what such objects would do?