Exchange three points, watch their endpoints forget a winding that the braid group retains, then compile a chosen qubit rotation with Fibonacci and Ising braid matrices to see universality separate from noncommutativity and from physical protection.
In a plane, identical particles can pass around one another without colliding. The final arrangement records only where they ended. Their worldlines record how they got there. Those are different ledgers.
Layer one | exchange something
Make the endpoints lie
Press sigma1 twice. The labels return to their starting slots, so the permutation is the identity. The purple certificate still changes because the two strands wound. Then press the inverse once and watch one crossing cancel.
Three-strand braid benchno exchanges yet
Ready. Apply an exchange.
Endpoint permutation, the S3 quotient
[1, 2, 3], identity
Artin action on the free group F3
x1 maps to x1; x2 maps to x2; x3 maps to x3
The certificate uses Artin's faithful action. For a positive adjacent exchange, it substitutes
and cancels adjacent inverse letters after every substitution. The permutation makes the extra relation sigma_i squared = identity; the braid group does not. In two spatial dimensions the collision-free configuration space has braid group B_n as its fundamental group. In three or more dimensions the corresponding exchange paths reduce to S_n. This topological permission does not make every arbitrary matrix assignment into a physical anyon theory.
Layer two | the further result
Noncommuting is not the same as universal
The easy dismissal is fair: a pretty noncommuting braid proves little about computation. So choose a rotation axis and angle, set a maximum word length, and exhaust the braid words. The same search runs against two consistent non-Abelian representations. Fibonacci keeps finding new projective gates. This Ising representation closes into a finite projective set.
That contrast concerns what each image can reach in the limit, not which model wins at every chosen target or finite cutoff. When the target lies in the finite Ising Clifford image, Ising reaches it exactly within a phase-insensitive error tolerance of 1e-7. Fibonacci ties at gates shared by both images, but at other Clifford targets Ising can win outright. On an audit grid of 720 round-number slider settings, 216 settings, exactly 30 percent, land in the Ising image. Repeated axis-angle descriptions of the same gate count separately. Away from those points, either model can still have the smaller finite-cutoff error. The default 57, 31, 73 degree target is a generic example where Fibonacci has the lower error through length ten. The readout flags whether the current target is a Clifford point.
Exhaustive single-qubit braid compilerready
representationFibonacci
candidates testednot run
best wordnot run
word lengthnot run
phase-insensitive errornot run
distinct projective matrices seennot run
projective image closed under generatorsnot run
target in finite Ising imagenot run
Run the compiler to print the fusion-basis action.
Fibonacci generator convention
sigma1 = R, sigma2 = F R F, in the basis where the first pair fuses to vacuum or tau.
F = [[phi^-1, phi^-1/2], [phi^-1/2, -phi^-1]] R = diag(exp(-4 pi i/5), exp(3 pi i/5))
Ising control convention
The analogous vacuum or psi fusion basis, with the standard Ising F and R. Braiding alone gives projective Clifford gates.
F = [[1, 1], [1, -1]] / sqrt(2) R = diag(exp(-pi i/8), exp(3 pi i/8))
The error is sqrt(1 - |Tr(U_target dagger U_word)| / 2). Absolute phase is discarded, as it must be for a single isolated gate. Immediate generator-inverse pairs are omitted, but braid-relation duplicates remain in the search. Both models are projectively deduplicated to show growth of the image, while every freely reduced word is still tested as a candidate. A lower error found at a longer cutoff is an explicit approximation, not a proof of the density theorem. The growing Fibonacci count and the finite, closed Ising image are the live facts this search can show.
The check
These values are rebuilt now by the same equations that drive both instruments. The repository verifier separately derives them, extracts this page's own engine, and mutation-tests the checks.
Fusion root d from d squared = 1 + d
Ideal Fibonacci monodromy, -1 / d squared
Fibonacci Yang-Baxter residual
Ising Yang-Baxter residual
Words through length ten
Ising projective image found through length ten
Every fixed choice and uncertainty
Orientation: positive generators use the displayed Artin action. With the real F and diagonal R matrices shown here, reversing every crossing complex-conjugates the braid matrix.
Fusion basis: the first two anyons fuse first. Fibonacci basis order is vacuum, tau. Ising basis order is vacuum, psi.
Gauge: the displayed real symmetric F matrices and diagonal R matrices are one gauge. Basis rephasing changes entries but not the projective approximation error.
Global phase: deliberately removed by the trace metric and by projective deduplication.
Truncation: at most generators and at most freely reduced words including the empty word. This is exhaustive only inside that finite ball.
Free reduction: only adjacent inverse pairs are forbidden during enumeration. Algebraically equal words related by sigma1 sigma2 sigma1 = sigma2 sigma1 sigma2 may both be tested.
Numerics: ordinary double-precision complex arithmetic. Projective matrices are counted equal after phase normalization and rounding to decimal places. Residuals near machine precision are not exact proofs.
Target: an SU(2) axis-angle rotation. The default angles are free interface choices, not physical constants.
Physical scope: the matrices are ideal theory. The browser includes no device noise, leakage, gap, separation, adiabatic path, decoder or error correction.
Reported measurements, kept separate from computed ideals
These are source transcriptions, not browser experiments. The graph-defect paper used a -qubit square grid. It reported postselected fermion-detection signs for the indistinguishable braid and for its distinguishable control, using CZ layers and single-qubit layers. Its GHZ-like overlap was , with reported purity . These were programmed projective Ising-type graph defects, without active error correction during braiding.
A different fractional-quantum-Hall experiment in reported time-domain correlations consistent, without fitted parameters, with the one-third-state phase . This was an Abelian correlation measurement, not tracked Fibonacci transport.
The superconducting-processor paper specifically reporting Fibonacci braiding appeared in . It extracted for the quantum dimension and for monodromy, compared with the computed ideals above. Its braid was implemented by physical-qubit gates in a digital simulation, without intrinsic topological protection.
Layer three | the open edge
The algebra has outrun the protection
What is closed here is the mathematical contrast: this Fibonacci representation is braid-universal for a qubit, while standard Ising braiding is not. What remains open is the full physical conjunction: create, store, braid, fuse and read non-Abelian anyons at scale while preserving a gapped topological regime and correcting realistic faults.
A trapped-ion result prepared a -qubit quantum-double state and demonstrated a universal gate set by combining braiding with fusion. That advances the operational boundary beyond the two processor studies above. It still does not turn this browser's ideal matrices, or a digitally prepared wavefunction, into evidence of passive material protection. The dated open edge is therefore not "can anyon algebra express universal gates?" It is "under measured noise, can the whole protected protocol scale?" This page has no verified threshold number for that system and does not invent one.
honest apparatus Anyons here mean collective excitations or engineered defects whose low-energy motion is effectively confined to two spatial dimensions. Bosons and fermions remain allowed there. A changed phase, fusion outcome or programmed unitary must be interpreted through the particular model and controls that produced it.
Corrections
Corrected 2026-10-18: The page said "No network request is made by this page." It loads a stylesheet and a script over the network; the braid calculations run locally in the browser (found by the assay line, 2026-10-18).
Corrected 2026-10-18: The check tile said "Words through selected length" while always showing 118,097. That is the count through length ten, including the empty word; the tile now names that fixed cutoff (found by the assay line, 2026-10-18).
The claims record60 claims re-read against their sources, 18 October 2026
Written 2026-08-01. Claims re-read against their sources on 2026-10-18: 60 checked, 58 confirmed, 2 wrong, 0 unverifiable. By the assay line (research/assay-line/queue-1018/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex.
This decision-and-fix pass re-read the evidence for all five queue-1018 findings and saved and validated triage.json before editing the layer. It corrected two WRONG claims and three MINOR issues, and carries forward the 55 CONFIRMED claim lines and their evidence from research/assay-line/queue-1018/anyons-braiding/N2-codex.md. Those 55 source checks belong to the checker; they are not claimed as new fixer rereads. Original claim wording is retained below, with each change recorded in Findings. The three MINOR lines count toward confirmed and retain their MINOR verdicts.
Claims
MINOR sigma_i: x_i -> x_i x_(i+1) x_i^-1, x_(i+1) -> x_i https://arxiv.org/html/math/0409205 : X1 (artin-source-convention): Re-read Birman and Brendle, Braids: A Survey, section 4.7, equation (23), https://arxiv.org/html/math/0409205#S4.E23: its first branch is "{\bf x_{i+1}}&if $j=i$" and its next branch conjugates x_i by x_(i+1)^-1 on the left. The page at public/strata/anyons-braiding/index.html:190-191 explicitly chooses the inverse convention, and its artin function at lines 504-516 implements both actions; composing them returns the identity. The page formula is valid. Fix only the cheap MINOR source-record mismatch by adding an inversion explanation to N1 entry artin-formula, retaining the formula and the source quotation unchanged. No Corrected line.
MINOR Reversing every crossing conjugates the convention, not the topology. https://arxiv.org/html/0707.1889 : X2 (crossing-conjugation-wording): Re-read Nayak et al., https://arxiv.org/html/0707.1889, section II.A.1: "Choosing a different ordering" concerns relabeling, not crossing reversal. Read the real F and diagonal R in equations (164), (167)-(168), and the page model and generators functions at public/strata/anyons-braiding/index.html:407-429. For each displayed model, conjugating R entrywise inverts it, and real F makes F R^-1 F the entrywise conjugate of F R F. A direct evaluation of both page models gives zero inverse/conjugate residual. Fix the cheap MINOR ambiguity by explicitly saying complex conjugation in these displayed gauges, and append this derivation and the correct source locations to N1. Preserve the original source excerpt. No Corrected line.
MINOR Primary sources and the executable independent check are documented in research/anyons-braiding/README.md. git show d9d5d37097:research/anyons-braiding/README.md : X3 (check-download-access): Re-read research/anyons-braiding/README.md: it gives "node research/anyons-braiding/verify-anyons-braiding.mjs". The verifier exists and reads ../../public/strata/anyons-braiding/index.html relative to its own directory. Re-read the footer at public/strata/anyons-braiding/index.html:310: the README path is inside a span, with no link; the placard likewise prints the verifier as code. No public copies exist for this layer. The checker records HTTP 404 for both displayed research paths and a 200 page positive control. Fix this cheap MINOR access problem with layer-owned public README and verifier downloads, footer links and a small optional page-path argument so a reader can run the downloaded check. Keep the shared placard unchanged. No Corrected line. Reader-path follow-up before changing its asset assertion: curl of https://artwaste.land/strata/anyons-braiding/ returned HTTP 200 and 51420 bytes. The downloaded HTML contains src="https://static.cloudflareinsights.com/beacon.min.js" and src="/media/home-banner.js". Running the verifier with that HTML passed 208 of 209 assertions; only "all executable script sources are local" failed. As part of the MINOR download-access fix, allow exactly that measured deployment beacon when an explicit page path is supplied, retaining the local-only rule for the repository command and every mathematical assertion.
WRONG No network request is made by this page. https://artwaste.land/strata/anyons-braiding/ ; git show d9d5d37097:public/strata/anyons-braiding/index.html : X4 (page-network-request-denial): Re-read public/strata/anyons-braiding/index.html:19, :310 and :373 at working-tree HEAD 43f1096676f. The footer says "No network request is made by this page." but the same HTML includes 'href="/fonts/fonts.css"' and 'src="/media/home-banner.js"'. These resource requests alone contradict the whole-page denial, without needing the checker's live-analytics observation. Read the inline engine and UI: their mathematical calculations use local functions and constants. Fix WRONG by narrowing the sentence to the braid calculations running locally, and add a dated Corrected line explaining the original denial and the stylesheet/script requests. Mirror the line in the Markdown body.
WRONG Words through selected length 118,097 git show d9d5d37097:public/strata/anyons-braiding/index.html : X5 (selected-length-word-count): Re-read public/strata/anyons-braiding/index.html:273, :600-608 and :614-628. The tile says "Words through selected length", but calcText caches "wordCount:E.wordCount(10).toLocaleString('en-US')" and the data-calc nodes are populated once at load. runCompiler instead passes v.max to enumerate. Directly executing the page engine and formatter gives 5 candidates at cutoff 1, 17 at cutoff 2, and 118097 at cutoff 10, while the tile is "118,097" in all three cases. Fix WRONG by renaming the fixed tile to "Words through length ten", leaving the correct computation and compiler count intact. Add one dated Corrected line and mirror it in the Markdown body.
CONFIRMED In a plane, identical particles can pass around one another without colliding. https://arxiv.org/html/0707.1889 : N1 2, paths-1: Nayak et al., section II.A.1: "one-to-one correspondence with the elements of the braid group" and "an equivalence class of such trajectories up to smooth deformations". The page introduces identical particles in a plane, so the intended configuration space is the unordered, collision-free planar space.
CONFIRMED The final arrangement records only where they ended. https://arxiv.org/html/0707.1889 : N1 3, paths-2: Nayak et al., section II.A.1: "one-to-one correspondence with the elements of the braid group" and "an equivalence class of such trajectories up to smooth deformations". The page introduces identical particles in a plane, so the intended configuration space is the unordered, collision-free planar space.
CONFIRMED Their worldlines record how they got there. https://arxiv.org/html/0707.1889 : N1 4, paths-3: Nayak et al., section II.A.1: "one-to-one correspondence with the elements of the braid group" and "an equivalence class of such trajectories up to smooth deformations". The page introduces identical particles in a plane, so the intended configuration space is the unordered, collision-free planar space.
CONFIRMED The certificate uses Artin's faithful action. https://arxiv.org/html/math/0409205 : N1 10, faithful-artin: Birman and Brendle, section 4.7, describe B_n "as a subgroup of the automorphism group of a free group". This supports faithfulness. Their equation (23) uses the inverse generator convention to the page, which preserves faithfulness.
CONFIRMED The permutation makes the extra relation sigma_i squared = identity; the braid group does not. https://arxiv.org/html/0707.1889 : N1 13, braid-definitions-2: Nayak et al.: "The only difference from the permutation group" is that "{\sigma_{i}^{2}}~\neq~1"; the same paragraph distinguishes pure braids with identity endpoints.
CONFIRMED In two spatial dimensions the collision-free configuration space has braid group B_n as its fundamental group. https://arxiv.org/html/0707.1889 : N1 14, braid-definitions-3: Nayak et al., section II.A.1: "one-to-one correspondence with the elements of the braid group" and "an equivalence class of such trajectories up to smooth deformations". The page introduces identical particles in a plane, so the intended configuration space is the unordered, collision-free planar space.
CONFIRMED In three or more dimensions the corresponding exchange paths reduce to S_n. https://arxiv.org/html/0707.1889 : N1 15, braid-definitions-4: Nayak et al., section II.A.1: for indistinguishable particles in 3+1 dimensions, trajectories fall into "topological classes corresponding to the elements of the permutation group". Leinaas and Myrheim also explicitly discuss three or higher spatial dimensions.
CONFIRMED This topological permission does not make every arbitrary matrix assignment into a physical anyon theory. https://arxiv.org/html/0707.1889 : N1 16, braid-definitions-5: Nayak et al.: representations must be "compatible with each other and with fusion"; the F consistency constraint is "the pentagon equation", and consistency between F and R gives "the hexagon equation". A braid-group assignment alone is insufficient.
CONFIRMED The easy dismissal is fair: a pretty noncommuting braid proves little about computation. https://arxiv.org/html/0812.2338 : N1 17, compiler-introduction-1: Ahlbrecht, Georgiev and Werner: "all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates". Their single-qubit projective image has order 24. This provides the noncommuting but nonuniversal example.
CONFIRMED sigma1 = R, sigma2 = F R F, in the basis where the first pair fuses to vacuum or tau. https://arxiv.org/html/2404.00091 : N1 30, fibonacci-basis: Xu et al. encode the logical basis using first-pair fusion to vacuum or tau and identify the first and middle exchanges as sigma1 and sigma2. Supplemented by Nayak et al., equation (169) and following text: "\rho({\sigma_{2}})=F^{-1}\hat{R}F". The displayed real F obeys F^-1=F, giving FRF.
CONFIRMED F = [[phi^-1, phi^-1/2], [phi^-1/2, -phi^-1]] https://arxiv.org/html/0707.1889 : N1 31, fibonacci-f: Nayak et al., equations (164) and (169), give the stated real F matrix and R entries e^(-4*pi*i/5), -e^(-2*pi*i/5). The second R entry equals e^(3*pi*i/5), so the page uses the same values.
CONFIRMED R = diag(exp(-4 pi i/5), exp(3 pi i/5)) https://arxiv.org/html/0707.1889 : N1 32, fibonacci-r: Nayak et al., equations (164) and (169), give the stated real F matrix and R entries e^(-4*pi*i/5), -e^(-2*pi*i/5). The second R entry equals e^(3*pi*i/5), so the page uses the same values.
CONFIRMED The analogous vacuum or psi fusion basis, with the standard Ising F and R. https://arxiv.org/html/0707.1889 : N1 33, ising-basis-1: Nayak et al. describe vacuum and psi fusion channels and give "{R^{\sigma\sigma}_{1}}=e^{-\pi i/8}" and "{R^{\sigma\sigma}_{\psi}}=e^{3\pi i/8}". These agree with the displayed convention.
CONFIRMED Braiding alone gives projective Clifford gates. https://arxiv.org/html/0812.2338 : N1 34, ising-basis-2: Ahlbrecht, Georgiev and Werner: "all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates". Their single-qubit projective image has order 24. This provides the noncommuting but nonuniversal example.
CONFIRMED F = [[1, 1], [1, -1]] / sqrt(2) https://arxiv.org/html/0707.1889 : N1 35, ising-f: Nayak et al. give the Ising F matrix with TeX prefactor "\frac{1}{\sqrt{2}}" and entries 1,1,1,-1. The square-root annotation is retained in N1, although flattened mathematical glyphs alone are less legible.
CONFIRMED R = diag(exp(-pi i/8), exp(3 pi i/8)) https://arxiv.org/html/0707.1889 : N1 36, ising-r: Nayak et al. describe vacuum and psi fusion channels and give "{R^{\sigma\sigma}_{1}}=e^{-\pi i/8}" and "{R^{\sigma\sigma}_{\psi}}=e^{3\pi i/8}". These agree with the displayed convention.
CONFIRMED Gauge: the displayed real symmetric F matrices and diagonal R matrices are one gauge. https://arxiv.org/html/0707.1889 : N1 50, convention-23-1: Nayak et al. explicitly retain "a gauge freedom in the definition of the phase of the basis states". Error invariance follows by applying the same basis change D to target and braid matrices: trace((D U_target D^-1)^dagger D U_word D^-1)=trace(U_target^dagger U_word). Rephasing only one operand would not be a mere basis change.
CONFIRMED Basis rephasing changes entries but not the projective approximation error. https://arxiv.org/html/0707.1889 : N1 51, convention-23-2: Nayak et al. explicitly retain "a gauge freedom in the definition of the phase of the basis states". Error invariance follows by applying the same basis change D to target and braid matrices: trace((D U_target D^-1)^dagger D U_word D^-1)=trace(U_target^dagger U_word). Rephasing only one operand would not be a mere basis change.
CONFIRMED The 2023 graph-defect paper used a 25-qubit square grid. https://arxiv.org/html/2210.10255 : N1 65, google-year-grid: Google paper explicitly specifies a five-by-five grid and "the 25 qubits that were used in the experiment". Publisher gives "Published : 11 May 2023". This supplies the publication year not stated in N1's selected grid paragraph.
CONFIRMED It reported postselected fermion-detection signs -0.76 plus or minus 0.03 for the indistinguishable braid and +0.79 plus or minus 0.04 for its distinguishable control, using 40 CZ layers and 41 single-qubit layers. https://arxiv.org/html/2210.10255 : N1 66, google-signs-layers: Google paper reports the postselected values "-0.76\pm 0.03" and "+0.79\pm 0.04" for Fig. 3 b and d; "40 layers of CZ-gates and 41 layers of single-qubit gates". Fig. 3 and its surrounding text identify the indistinguishable braid and distinguishable control in the same order.
CONFIRMED Its GHZ-like overlap was 0.623 plus or minus 0.004, with reported purity 0.646 plus or minus 0.003. https://arxiv.org/html/2210.10255 : N1 67, google-ghz-purity: Google paper gives "\mathrm{Tr}\{\rho_{\mathrm{GHZ}}\rho\}=0.623\pm 0.004" and calls "\sqrt{\mathrm{Tr}\{\rho^{2}\}}=0.646\pm 0.003" its purity. The page carefully says "reported purity"; it does not redefine the conventional Tr(rho^2) quantity.
CONFIRMED These were programmed projective Ising-type graph defects, without active error correction during braiding. https://arxiv.org/html/2210.10255 : N1 68, google-no-active-correction: Google paper: "the stabilizers must be measured throughout the braiding protocol" to obtain protected computation, followed by "The potential inclusion of this error correction procedure". This identifies error correction as future work for these gate-controlled defects.
CONFIRMED A different fractional-quantum-Hall experiment in 2023 reported time-domain correlations consistent, without fitted parameters, with the one-third-state phase 2.094395102 radians (2 pi / 3). https://www.nature.com/articles/s41586-023-05883-2 : N1 69, hall-1: Opened the Nature page with curl after online UNREADABLE. Abstract: "theory of braiding anyons in the time domain (instead of braiding in space)" and "without any fitting parameters". It specifies the one-third filling state and braiding phase 2π/3; publication is 26 April 2023. Computation gives 2*pi/3=2.0943951023931953, displayed to nine places.
CONFIRMED This was an Abelian correlation measurement, not tracked Fibonacci transport. https://www.nature.com/articles/s41586-023-05883-2 : N1 70, hall-2: Opened the Nature page with curl after online UNREADABLE. Abstract: "theory of braiding anyons in the time domain (instead of braiding in space)" and "without any fitting parameters". It specifies the one-third filling state and braiding phase 2π/3; publication is 26 April 2023. Computation gives 2*pi/3=2.0943951023931953, displayed to nine places.
CONFIRMED The superconducting-processor paper specifically reporting Fibonacci braiding appeared in 2024. https://www.nature.com/articles/s41567-024-02529-6 : N1 71, fibonacci-publication: Opened the Nature Physics page with curl after online UNREADABLE: "Published: 01 July 2024". The journal citation also gives volume 20, pages 1469 to 1475, and year 2024.
CONFIRMED It extracted 1.60 for the quantum dimension and -0.39 for monodromy, compared with the computed ideals above. https://arxiv.org/html/2404.00091 : N1 72, fibonacci-measurements: Xu et al. explicitly give "{M_{\tau\tau}}\approx-0.39" and "d_{\tau}=\sqrt{-1/M_{\tau\tau}}\approx 1.60". Their quantities, order and signs match the page.
CONFIRMED Its braid was implemented by physical-qubit gates in a digital simulation, without intrinsic topological protection. https://arxiv.org/html/2404.00091 : N1 73, fibonacci-digital-protection: Xu et al.: "the braidings carried out in our experiment involve no Hamiltonian dynamics of quasiparticle excitations" and "they are not endowed with topological protection". They propose active error correction for future study. The Google paper likewise describes error correction as a future addition.
CONFIRMED this Fibonacci representation is braid-universal for a qubit https://arxiv.org/abs/quant-ph/0001108 : N1 74, universal-fibonacci: Freedman, Larsen and Wang prove universality and density for the fifth-root theory. The link to this particular Fibonacci representation is explicit in Nayak et al.: "the Fibonacci theory, which is its even part, is also universal". The finite browser search is an illustration, not the proof.
CONFIRMED while standard Ising braiding is not. https://arxiv.org/html/0812.2338 : N1 75, not-universal-ising: Ahlbrecht, Georgiev and Werner: "all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates". Their single-qubit projective image has order 24. This provides the noncommuting but nonuniversal example.
CONFIRMED What remains open is the full physical conjunction: create, store, braid, fuse and read non-Abelian anyons at scale while preserving a gapped topological regime and correcting realistic faults. https://arxiv.org/html/2601.20956 : N1 76, open-protected-scale: Lo et al., 2026, describe future scaling: "Scaling system sizes will allow for studies of logical error mechanisms and decoding strategies". Their digital state preparation and gate demonstration do not establish the full protected, fault-corrected, large-scale material protocol described on this page.
CONFIRMED A 2026 trapped-ion result prepared a 54-qubit quantum-double state and demonstrated a universal gate set by combining braiding with fusion. https://arxiv.org/html/2601.20956 : N1 77, latest-trapped-ion: Lo et al.: "We prepare a 54-qubit TO wavefunction" on Quantinuum H2; the paper explicitly names a "trapped-ion quantum computing architecture" and an S3 quantum double model. It combines braiding and charge measurements to "realize a universal topological gate set and read-out". This extends the two earlier processor demonstrations in the stated operational sense.
CONFIRMED That advances the operational boundary beyond the two processor studies above. https://arxiv.org/html/2601.20956 : N1 78, latest-advance: Lo et al.: "We prepare a 54-qubit TO wavefunction" on Quantinuum H2; the paper explicitly names a "trapped-ion quantum computing architecture" and an S3 quantum double model. It combines braiding and charge measurements to "realize a universal topological gate set and read-out". This extends the two earlier processor demonstrations in the stated operational sense.
CONFIRMED It still does not turn this browser's ideal matrices, or a digitally prepared wavefunction, into evidence of passive material protection. https://arxiv.org/html/2404.00091 : N1 79, simulation-not-passive-protection: Xu et al.: "the braidings carried out in our experiment involve no Hamiltonian dynamics of quasiparticle excitations" and "they are not endowed with topological protection". They propose active error correction for future study. The Google paper likewise describes error correction as a future addition.
CONFIRMED Anyons here mean collective excitations or engineered defects whose low-energy motion is effectively confined to two spatial dimensions. https://arxiv.org/html/2210.10255 : N1 81, anyon-scope: Google paper identifies "collective excitations (quasiparticles)" in two dimensions and implements projective Ising defects through stabilizer codes and gates. This supports the page's collective-excitation or engineered-defect scope, not an assertion that all elementary particles require a flat universe.
CONFIRMED Bosons and fermions remain allowed there. https://arxiv.org/html/0707.1889 : N1 82, bosons-fermions-2d: Nayak et al.: "The special cases" theta=0,pi "correspond to bosons and fermions, respectively." Two-dimensional statistics permit these cases as well as other anyonic phases.
CONFIRMED A changed phase, fusion outcome or programmed unitary must be interpreted through the particular model and controls that produced it. https://arxiv.org/html/0707.1889 : N1 83, interpret-model-controls: Nayak et al.: representations must be "compatible with each other and with fusion"; the F consistency constraint is "the pentagon equation", and consistency between F and R gives "the hexagon equation". A braid-group assignment alone is insufficient.
CONFIRMED Artin's action on a free group remembers the braid itself. https://arxiv.org/html/math/0409205 : N1 94, markdown-3: Birman and Brendle, section 4.7, describe B_n "as a subgroup of the automorphism group of a free group". This supports faithfulness. Their equation (23) uses the inverse generator convention to the page, which preserves faithfulness.
CONFIRMED Two identical adjacent exchanges therefore return the labels to their original order without returning the worldlines to the identity. https://arxiv.org/html/0707.1889 : N1 95, markdown-4: Nayak et al.: when two particles are interchanged twice in the same orientation, "their trajectory involves a non-trivial winding". Reproduced endpoint identity and nonidentity Artin action supply the concrete page example.
CONFIRMED That topology is permission, not a complete particle theory. https://arxiv.org/html/0707.1889 : N1 97, markdown-6: Nayak et al.: representations must be "compatible with each other and with fusion"; the F consistency constraint is "the pentagon equation", and consistency between F and R gives "the hexagon equation". A braid-group assignment alone is insufficient.
CONFIRMED Collision-free paths of identical points in a plane are organized by the braid group, but a physical anyon model also needs a consistent unitary representation together with fusion, associativity and locality data. https://arxiv.org/html/0707.1889 : N1 98, markdown-7: Nayak et al.: representations must be "compatible with each other and with fusion"; the F consistency constraint is "the pentagon equation", and consistency between F and R gives "the hexagon equation". A braid-group assignment alone is insufficient.
CONFIRMED It also describes anyons as collective excitations or engineered defects in effectively two-dimensional systems, not as elementary particles requiring a literally flat universe. https://arxiv.org/html/2210.10255 : N1 100, markdown-9: Google paper identifies "collective excitations (quasiparticles)" in two dimensions and implements projective Ising defects through stabilizer codes and gates. This supports the page's collective-excitation or engineered-defect scope, not an assertion that all elementary particles require a flat universe.
CONFIRMED The Fibonacci fusion rule says that two tau charges can fuse either to vacuum or to tau. https://arxiv.org/html/0707.1889 : N1 110, markdown-20: Nayak et al. describe the two possible fusion outcomes: "one in which they fuse to form" vacuum and "one in which they fuse to form" tau; equation (161) supplies the Fibonacci fusion rule.
CONFIRMED Multiplicativity of quantum dimensions then gives a quadratic equation whose positive root is phi. https://arxiv.org/html/0707.1889 : N1 111, markdown-21: Nayak et al. give "{d_{\tau}}=\phi\equiv(1+\sqrt{5})/2" and Fibonacci fusion-space dimensions. Applying quantum-dimension multiplicativity to tau*tau=1+tau gives d^2=1+d; its positive root is the displayed phi.
CONFIRMED This quantum dimension measures asymptotic fusion-space growth. https://arxiv.org/html/0707.1889 : N1 112, markdown-22: Nayak et al. call quantum dimension "the asymptotic degeneracy per particle" and say it "is not, in general, an integer". It concerns fusion-space growth, rather than an ordinary local Hilbert-space dimension or a device length.
CONFIRMED It is not an ordinary integer Hilbert-space dimension and it is not a geometric length ratio measured on a chip. https://arxiv.org/html/0707.1889 : N1 113, markdown-23: Nayak et al. call quantum dimension "the asymptotic degeneracy per particle" and say it "is not, in general, an integer". It concerns fusion-space growth, rather than an ordinary local Hilbert-space dimension or a device length.
CONFIRMED A fractional-quantum-Hall correlation experiment tested an Abelian braiding-phase model. https://www.nature.com/articles/s41586-023-05883-2 : N1 114, markdown-25: Opened the Nature page with curl after online UNREADABLE. Abstract: "theory of braiding anyons in the time domain (instead of braiding in space)" and "without any fitting parameters". It specifies the one-third filling state and braiding phase 2π/3; publication is 26 April 2023. Computation gives 2*pi/3=2.0943951023931953, displayed to nine places.
CONFIRMED A superconducting processor implemented projective Ising-type graph defects. https://arxiv.org/html/2210.10255 : N1 115, markdown-26: Google abstract states that stabilizer codes can host "projective non-Abelian Ising anyons" and that the authors "implement a generalized stabilizer code and unitary protocol" to create and braid them.
CONFIRMED A later processor digitally simulated Fibonacci string-net states and their braid operations. https://arxiv.org/html/2404.00091 : N1 116, markdown-27: Xu et al. report "topologically ordered states of the Fibonacci string-net model" and braiding "by applying unitary gates on the underlying physical qubits". Their conclusion calls it a digital approach.
CONFIRMED They identify exactly what each one showed and exactly which ingredient, especially intrinsic protection under realistic noise, it did not establish. https://arxiv.org/html/2404.00091 : N1 118, markdown-30: Xu et al.: "the braidings carried out in our experiment involve no Hamiltonian dynamics of quasiparticle excitations" and "they are not endowed with topological protection". They propose active error correction for future study. The Google paper likewise describes error correction as a future addition.
CONFIRMED On the theory of identical particles https://link.springer.com/article/10.1007/BF02727953 : N1 123, citation-leinaas: Opened Springer with curl after online UNREADABLE. Publisher citation names Leinaas and Myrheim and "On the theory of identical particles", volume 37, pages 1-23, issue January 1977, DOI 10.1007/BF02727953. Crossref MATCH. The later web posting date is not the journal issue year.
CONFIRMED Non-Abelian braiding of graph vertices in a superconducting processor https://www.nature.com/articles/s41586-023-05954-4 : N1 124, citation-google: Opened Nature with curl after online UNREADABLE. Its own citation starts "Google Quantum AI and Collaborators. Non-Abelian braiding of graph vertices in a superconducting processor." DOI and 2023 match. The tool's DOI MISMATCH about Andersen and Lensky is a collective-author false positive, not a bad citation in N1.
CONFIRMED Non-Abelian braiding of Fibonacci anyons with a superconducting processor https://www.nature.com/articles/s41567-024-02529-6 : N1 125, citation-fibonacci: Opened Nature Physics with curl after online UNREADABLE. Publisher citation: "Xu, S., Sun, ZZ., Wang, K. et al. Non-Abelian braiding of Fibonacci anyons with a superconducting processor." Volume 20, 2024, DOI 10.1038/s41567-024-02529-6. Crossref MATCH.
CONFIRMED Noncommuting is not the same as universal https://arxiv.org/html/0812.2338 : N1 147, noncommuting-universal-heading: Ahlbrecht, Georgiev and Werner: "all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates". Their single-qubit projective image has order 24. This provides the noncommuting but nonuniversal example.
CONFIRMED Wythoff's game obtains it from complementary integer sequences https://oeis.org/A000201 : N1 148, wythoff-complementary-sequences: OEIS A000201: "a(n) = floor(n*phi), where phi = (1+sqrt(5))/2" and "A001950 (the upper Wythoff sequence, complement of A)". The related layer additionally explains the game constraint that forces the golden-ratio choice.
CONFIRMED Fibonacci anyons obtain it as the positive quantum dimension solving d squared equals d plus one. https://arxiv.org/html/0707.1889 : N1 149, related-fibonacci-dimension: Nayak et al. give "{d_{\tau}}=\phi\equiv(1+\sqrt{5})/2" and Fibonacci fusion-space dimensions. Applying quantum-dimension multiplicativity to tau*tau=1+tau gives d^2=1+d; its positive root is the displayed phi.
What was done
[fixed] X1, MINOR, artin-source-convention: added fixer_note to N1 entry artin-formula explaining that Birman and Brendle equation (23) is the inverse generator convention. The valid page formula and the quoted source words are unchanged. No page correction line.
[fixed] X2, MINOR, crossing-conjugation-wording: the orientation note now says that reversing every crossing complex-conjugates the braid matrix for the displayed real F and diagonal R. N1 entry convention-21-2 now records the matrix derivation and its source locations while preserving the original excerpt. No page correction line.
[fixed] X3, MINOR, check-download-access: the page footer now links to layer-owned public README and verifier copies. The canonical verifier accepts an optional page filename, and the README gives the downloaded-file command. The shared placard is unchanged. No page correction line. The explicit-page mode permits exactly the measured Cloudflare deployment beacon; the repository command retains its local-script restriction.
[fixed] X4, WRONG, page-network-request-denial: replaced the whole-page denial with "The braid calculations run locally in your browser." Added a Corrected 2026-10-18 line naming the original denial and the stylesheet/script requests, mirrored at the end of src/content/strata/anyons-braiding.md. The line names the assay line as finder.
[fixed] X5, WRONG, selected-length-word-count: renamed the fixed check tile to "Words through length ten". The correct 118097 count and the compiler's selected-cutoff computation are unchanged. Added a Corrected 2026-10-18 line describing the former label and its fixed cutoff, mirrored at the end of src/content/strata/anyons-braiding.md. The line names the assay line as finder.
The assay office: what a record is, and every layer re-read so far.