# TGA 2.1 — scaling identities and six-mark membership

Supplementary release, 1 October 2026. Prepared beside the frozen TGA 2.0 verifier. TGA 2.0 files are not rewritten. This is not a general theorem prover, not a source paper, and not a replacement for the production geometry or field-decoder modules.

## Run

Python 3.10 or newer. Standard library only. Place this directory at `tga/2.1` next to the published `tga/2.0` directory. The loader checks the TGA 2.0 hashes before it imports that verifier.

```sh
python3 tga_verifier.py
python3 tga_verifier.py --report graph-report.json
python3 tga_verifier.py --write-html
python3 -m unittest -v test_tga
python3 -O -m unittest -v test_tga
```

`PROOFS.md` is the source derivation. `PROOFS.html` is the rendered page produced by `--write-html`. The JSON report keeps publication status, mathematical justification, execution pointers, and numerical comparisons in separate objects.

## What this release adds

- Geometric scaling `M_r(z) = r z` for `r > 0` on centered six-vectors.
- Softmax `S_β` for a declared `β > 0`, defaulting to `β = λ = φ/(φ+π)`. The configuration keeps `β`, dimensionless coordinate units, and the label order `d2, d1, d, b, b1, b2`.
- Aitchison powering `P_r`, with `P_r ∘ P_t = P_{rt}`, `P_1` the identity, and `P_r` inverse `P_{1/r}`.
- The path identity `S_β ∘ M_r = P_r ∘ S_β`. The proof is cancellation of the common softmax denominator after exponentiation.
- The marked Fourier scaling `D_r`, which leaves the DC bin unchanged and multiplies the other five bins by `r` without discarding phase, and the identity `F ∘ M_r = D_r ∘ F`.
- A path-equality certificate object that admits only those two implemented schemas. An arbitrary equation or a small residual is not a certificate. A path that leaves its domain, including underflow to a zero probability, is reported and not certified. Probabilities are not clipped.
- A six-mark membership test, distinct from the general positive-probability type. Uniform data stay valid probabilities and are not the positive-scale family. Binary64 agreement is not an exact membership certificate. Boundary cases that are not decided by the tolerance are marked uncertain; `s` is not forced into `(0, 1]`.

The admitted-relationship count in the report is the length of the graph's edge list. Inherited TGA 2.0 edges remain in that list with their original support labels.

Centered log-ratio coordinates and Aitchison powering are the general equivalence between centered vectors and positive probabilities. Membership in the six-mark family is a further gap condition. Those two statements are not the same edge.