The Science of Representation, by ChatGPT

The following is an edited version of a synthesis written by ChatGPT.

Tectology is the discipline concerned with designing interfaces (representations) for abstract structures that are both faithful to the underlying structure and free from arbitrary or extraneous features. In tectology, a representation is valid if it reflects the intrinsic properties of the abstract structure without introducing artificial asymmetries or discontinuities.

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Core Principles

1. Soundness
   Every component of a representation must be well defined and accurately mirror the corresponding part of the abstract structure.
   Math idea: If \(R = F(S)\) represents an abstract structure \(S\), then every operation \(o_R\) in \(R\) corresponds to a well-defined operation \(o_S\) in \(S\).

2. Completeness
   A representation must provide all the primitives required to fully describe the abstract structure.
   Math idea: No observable property of \(S\) is omitted in \(R\).

3. Equivariance
   Transformations (symmetries) of the abstract structure must correspond one-to-one with transformations of the representation.
   Math idea: For any symmetry \(g\) acting on \(S\) and for \(R = F(S)\), there exists a transformation \(T_g\) acting on \(R\) such that, for every \(s \in S\), \(T_g(F(s)) = F(g(s))\).

4. Minimality
   Among all representations satisfying the above, the chosen representation must be minimal—it contains only the irreducible, essential primitives.
   Math idea: If \(R = F(S)\) and \(R'\) is a proper subrepresentation of \(R\), then \(R'\) fails to capture some necessary aspect of \(S\).

5. Factorization
   The representation must be explicitly presented as a composition of irreducible (atomic) components, each capturing an essential part of the abstract structure.
   Math idea: For any representation \(R = F(S)\), there exists a canonical factorization\(R \cong A_1 \otimes A_2 \otimes \cdots \otimes A_n\), where each atomic component \(A_i\) is irreducible.

6. Compositionality
   The representation should allow its components to be recombined in a structured manner so that the representation of a composite structure is determined by the representations of its parts.
   Math idea: If an abstract structure decomposes as \(S = S_1 \circ S_2\), then the representation satisfies

   \(F(S) \cong F(S_1) \circ F(S_2)\).

7. Orthogonality
   The primitives (or atomic components) in the representation must be independent; no component should redundantly capture the same information as another.
   Math idea: For distinct atomic components \(A_i\) and \(A_j\) in \(R\) (with \(i \neq j\)), the observable effects of \(A_i\)and \(A_j\) are non-redundant—analogous to linear independence.

8. Abstraction
   The representation must capture only the intrinsic aspects of the abstract structure, omitting all inessential or incidental details.
   Math idea: There exists a forgetful functor \(U: \mathcal{R} \to \mathcal{S}\) such that for any representation \(R = F(S)\), we have \(U(R) \cong S\). Equivalently, any operation in \(R\) that is independent of incidental details is defined generically.

9. Continuity
   The representation should vary continuously with the abstract structure. Small changes in \(S\) must induce only small, continuous changes in \(R\).
   Math idea: If \(\{S_t\}\) is a continuous family of structures, then the corresponding representations \(\{F(S_t)\}\) vary continuously (in the appropriate topology), preventing artificial “jumps” when \(S\) is deformed.

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Meta-Principles (Desirable Consequences)

• Canonicity (Uniqueness)
  Conjecture: If a representation satisfies all the above principles, then it is unique up to trivial (unobservable) isomorphism.
  Interpretation: Canonicity is a meta-property rather than a direct design guideline—following the principles should yield a unique representation for any abstract structure.

• Existence
  Conjecture: For every abstract structure \(S\), there exists at least one representation \(R = F(S)\) that satisfies all the tectological principles.
  Interpretation: Not only should the representation be unique when it exists, but a valid representation should exist for every \(S\).

• Self-Consistency of the Principle Set
  Conjecture: The set of tectological principles is non-redundant, orthogonal, and compositional. In particular, no principle can be derived solely from the others, and together they form an abstract structure that “represents itself” faithfully.