Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Preserve these useful restrictions

Chemistry §6.1–6.4 contains a useful conditional factorization result. If H=H_obs tensor I + I tensor H_hidden, the unobserved states are orthonormal, and O=O_obs tensor I, then

⟨ψ⊗χa∣O∣ψ′⊗χb⟩=⟨ψ∣Oobs∣ψ′⟩δab. \langle\psi\otimes\chi_a|O|\psi'\otimes\chi_b\rangle =\langle\psi|O_{\rm obs}|\psi'\rangle\delta_{ab}.

A purely unobserved contribution gives a common shift when all compared states share the same unobserved state and particle content. Under these assumptions it cannot generate orbital-dependent splittings. A mixed interaction is the place to investigate deviations. Do not import the universal chemistry claim or numerical contact bound without separately establishing their premises, range, units, amplitude, and relevant observables.

Appendix B §20a and §50 record that isotropic density overlaps retain the monopole, with the mean traceless quadrupole vanishing. This is a restriction on that reduction, not on all anisotropic states, virtual transitions, or spinor operators. The imported notes do not rely on §50's separate fourth-moment formulas.

Appendix D §49b records the rejection of an old rho-meson comb-root claim: a target-bracketed root among many roots does not select a physical resonance. Preserve this warning rather than the contradictory positive claim still present in Mass §10.

Do not copy these claims as established mathematics

  • Kernel uniqueness: Spectral T2 treats a cross-sector dot square as invariant under independent rotations. It is not, with a fixed identification. The identification and its symmetry must be specified; coefficient rank one also remains a separate condition.
  • Colour connection: Forces §3a defines psi_color as a column in C^3. Then i psi_color-dagger d psi_color is a scalar, not a matrix in su(3). It does not solve the V2 global gauge-bundle problem. A multi-state frame and a compatible connection would be different mathematical data, with their own global determinant condition.
  • CKM normalization: idwt.py STEP 134 verifies cancellation in a vertex ratio and r+(1-r)=1. This does not show that r is the fraction of a conserved total rate. For two exclusive channels whose actual rate ratio is r, normalizing their sum instead gives r/(1+r). A distinct incident-flux or dynamical argument is needed to justify r itself. The full charged-current calculation still requires kinetic metrics and mass bases; a chosen unitary completion is not that calculation.
  • CP closure: CP Completion and Appendix E §47 contain candidate determinant-phase and beat-link arguments. A determinant phase alone does not establish rephasing-invariant CP violation. No numerical CP prediction or spectral-flow closure is promoted by this import.

The review searched the corpus and closely inspected relevant passages; it was not a complete audit of every historical equation or the PDF rendering.