# Anyons and braiding verification

This directory is the audit trail for the immersive stratum at
`public/strata/anyons-braiding/index.html`. The page contains a pure JavaScript engine between the
markers `ANYON_ENGINE_START` and `ANYON_ENGINE_END`. The verifier also contains an independent
implementation. It does not import the page implementation as its source of expected answers.

## Run it

From the repository root:

```sh
node research/anyons-braiding/verify-anyons-braiding.mjs
```

The verifier requires Node 22 and no packages. It prints every assertion and exits nonzero if any
assertion fails. Its final sections are `PAGE COUPLING`, which extracts and executes the shipped
page's own engine with `node:vm`, and `MUTATION SWEEP`, which deliberately corrupts constants or
logic and requires every corruption to be detected.

To run the downloadable verifier without a repository checkout, save the verifier and the page
in an empty directory, then pass the page filename explicitly:

```sh
curl -fSL https://artwaste.land/strata/anyons-braiding/verify-anyons-braiding.mjs -o verify-anyons-braiding.mjs
curl -fSL https://artwaste.land/strata/anyons-braiding/ -o index.html
node verify-anyons-braiding.mjs index.html
```

With an explicit page filename, the script-source check permits the site's Cloudflare analytics
beacon as well as local scripts. The repository command still requires local script sources.

The page footer links to this README and the verifier. Their public copies are in
`public/strata/anyons-braiding/`; copy these two research files there after changing them.

To syntax-check the executable inline scripts without treating the JSON-LD block as JavaScript:

```sh
perl -0777 -ne 'while (/<script(?:\s[^>]*)?>([\s\S]*?)<\/script>/g) { next if $& =~ /application\/ld\+json/; print "$1\n" }' \
  public/strata/anyons-braiding/index.html > /tmp/anyons-braiding-inline.js
node --check /tmp/anyons-braiding-inline.js
```

## What is computed

The page and verifier compute the following from equations rather than storing their displayed
answers:

- the positive root of `d^2 = 1 + d`, its fusion-polynomial residual, and the ideal Fibonacci
  monodromy `-1/d^2`;
- the Fibonacci and Ising `F` and `R` matrices in the conventions named on the page;
- `sigma1 = R`, `sigma2 = F R F`, inverse generators, unitarity residuals and both numerical
  Yang-Baxter residuals;
- axis-angle SU(2) target matrices and the global-phase-insensitive trace distance;
- every freely reduced word over four signed braid generators through the selected maximum length;
- the cumulative count `1 + 4(1 + 3 + ... + 3^9) = 118097` through length ten;
- the best cumulative Fibonacci and Ising approximation for each cutoff;
- the distinct projective images visited by both representations, using explicit phase
  normalization and eight-decimal deduplication, including Fibonacci growth through length ten
  and saturation of the standard Ising representation at 24 elements;
- whether a selected target belongs to the finite Ising image within a numerical error tolerance
  of `1e-7`;
- endpoint permutations and Artin free-group automorphisms for every braid entered in layer one;
- the one-third-state full-braid phase `2*pi/3` from the stated fraction.

The hardware values are not recomputed experiments. They are named as reported source inputs on
the page. The verifier checks each transcription independently, checks basic cross-relations such
as the opposite reported signs, and checks that the page engine carries the same inputs. This is a
protection against transcription drift, not an independent reanalysis of raw processor or
transport data.

The compiler's finite search shows specific approximations and the finite Ising image. It does not
prove the Fibonacci density theorem. The theorem is sourced separately below. Likewise, the
verifier checks Artin's formulas and braid relation on concrete words but does not re-prove that
Artin's action is faithful for every braid.

The target comparison is not a claim that Fibonacci wins at every finite cutoff. The default
axis-angle target `(57, 31, 73)` is a generic example where it does win through length ten. Targets
inside the 24-element Ising image are reached exactly by Ising within numerical tolerance, and
Fibonacci can tie at shared gates or lose at other Clifford targets. The round-number audit grid is
the Cartesian product of polar angles `{0,30,45,60,90,120,135,150,180}`, azimuths in multiples of
45 degrees from 0 through 315, and turn angles `{0,30,45,60,90,120,135,180,270,360}`. Exactly 216
of its 720 parameter settings land in the Ising image. This count treats repeated axis-angle
descriptions of one gate as separate slider settings.

## Conventions and limitations

Positive Artin generators act by `x_i -> x_i x_(i+1) x_i^-1` and
`x_(i+1) -> x_i`. The first pair of anyons fuses first. The Fibonacci basis order is vacuum, tau;
the Ising order is vacuum, psi. The real symmetric `F` gauges and diagonal `R` gauges are fixed in
the page. Matrix comparison removes global phase. Enumeration omits immediate inverse pairs but
does not quotient by the Yang-Baxter relation, so algebraically duplicate braid words can both be
visited. All complex arithmetic is IEEE 754 double precision.

The browser model contains no gap, temperature, quasiparticle separation, Hamiltonian transport,
leakage, calibration data, decoder or error-correction process. A digital circuit that realizes the
same algebra does not acquire topological protection from that fact alone.

## Sources actually opened

These primary records were opened during scouting or the build. Claims taken from them are marked
on the page either as reported measurements or as theory.

1. J. M. Leinaas and J. Myrheim, "On the theory of identical particles," *Il Nuovo Cimento B* 37,
   1-23 (1977). DOI: `10.1007/BF02727953`. Opened for the configuration-space treatment of
   statistics by spatial dimension. The page uses the modern fundamental-group phrasing and keeps
   the paper's spinless scope explicit.

2. Michael H. Freedman, Michael J. Larsen and Zhenghan Wang, "A modular functor which is universal
   for quantum computation," *Communications in Mathematical Physics* 227, 605-622 (2002).
   DOI: `10.1007/s002200200645`; arXiv: `quant-ph/0001108`. Opened for the density and universality
   result. The finite browser search is presented only as an illustration of that theorem.

3. Trond I. Andersen, Yuri D. Lensky, Kostyantyn Kechedzhi, Ilya K. Drozdov, Andreas Bengtsson and
   Google Quantum AI and Collaborators, "Non-Abelian braiding of graph vertices in a superconducting
   processor," *Nature* 618, 264-269 (2023). DOI: `10.1038/s41586-023-05954-4`; arXiv:
   `2210.10255`. Opened for the 25-qubit grid, opposite postselected signs, circuit-layer counts,
   GHZ-like overlap and purity, and the authors' description of gate-controlled graph defects and
   future error correction. Nature lists the collective author as Google Quantum AI and
   Collaborators; the arXiv record supplies the full collaboration roster.

4. June-Young M. Lee, Changki Hong, Tomer Alkalay, Noam Schiller, Vladimir Umansky, Moty Heiblum,
   Yuval Oreg and H.-S. Sim, "Partitioning of diluted anyons reveals their braiding statistics,"
   *Nature* 617, 277-281 (2023). DOI: `10.1038/s41586-023-05883-2`; arXiv: `2209.15461`. Opened for
   the time-domain correlation claim and the expected full-braid phase at the one-third state. The
   page does not describe this as a tracked spatial exchange or as Fibonacci evidence.

5. Shibo Xu, Zheng-Zhi Sun, Ke Wang, Hekang Li, Zitian Zhu, Hang Dong, Jinfeng Deng, Xu Zhang,
   Jiachen Chen, Yaozu Wu, Chuanyu Zhang, Feitong Jin, Xuhao Zhu, Yu Gao, Aosai Zhang, Ning Wang,
   Yiren Zou, Ziqi Tan, Fanhao Shen, Jiarun Zhong, Zehang Bao, Weikang Li, Wenjie Jiang, Li-Wei Yu,
   Zixuan Song, Pengfei Zhang, Liang Xiang, Qiujiang Guo, Zhen Wang, Chao Song, H. Wang and
   Dong-Ling Deng, "Non-Abelian braiding of Fibonacci anyons with a superconducting processor,"
   *Nature Physics* 20, 1469-1475 (2024). DOI: `10.1038/s41567-024-02529-6`; arXiv: `2404.00091`.
   Opened for the fusion rule, displayed braid matrices, measured quantum dimension and monodromy,
   publication date, 27-qubit implementation, and explicit digital-simulation method.

6. Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh, Michael Mills, Peter E. Siegfried, Maxwell D. Urmey,
   Nathanan Tantivasadakarn, Henrik Dreyer, Ashvin Vishwanath, Ruben Verresen and Mohsin Iqbal,
   "Universal gates from braiding and fusing anyons on quantum hardware," *Nature* 655, 591-597
   (2026). DOI: `10.1038/s41586-026-10709-y`; arXiv: `2601.20956`. Opened for the dated open-edge
   update: a 54-qubit quantum-double state and a universal gate set obtained by combining braiding
   with fusion. The page does not infer an experimental fault-tolerance threshold from this result.

## Bibliographic lead not opened

Emil Artin, "Theorie der Zopfe," *Abhandlungen aus dem Mathematischen Seminar der Universitat
Hamburg* 4, 47-72 (1925). DOI: `10.1007/BF02950718`. This is the historical source associated with
the braid group and faithful free-group representation. We did not open the paper during this
build. The exact action used on the page was supplied by the verified scout specification and was
then checked directly for inverses, noncommutation and the braid relation by the verifier.