--- slug: quantum-hall-chern passed: 2026-10-18 by: the assay line (research/assay-line/claude-1018b/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5) checked: 90 confirmed: 81 # 76 CONFIRMED lines + 5 MINOR lines counted here wrong: 5 unverifiable: 0 observed: 4 stale: 0 --- This was a directed repair pass over the 10 triage items in research/assay-line/claude-1018b/quantum-hall-chern/triage.json; the claim lines below are the checker's (N2-claude.md, 90 claims against the N1 source record plus added claims). Of its five WRONG items, four were re-read and fixed on the page: the 2024 Nature Communications paper was credited to "A. Kumar and collaborators" though its first author is Simrandeep Kaur and no author is named Kumar (Crossref and arXiv re-read); the verifier summary listed "projector idempotence", which the verifier does not test; and at every even q (95 of 95 fractions with q up to 31) the spectrum offered the interval between the two touching central bands as a gap and printed a predicted Hall integer C = +q/2 for it, now fixed in the page engine (gapLabel() searches 2|C| < q only, spectrum() never offers r = q/2 at even q), which also answers two MINOR items resting on the same touching. The fifth WRONG, the placard's "runs from 2 files this site serves", reproduces (a stranger's run of the served page exits 1 on check 111 because the edge injects the Cloudflare beacon) and is held by the shared placard and beacon notes, not fixed here. Three other MINOR items were listed under the tier B rule and the page left as it was. ## Claims - CONFIRMED | K. von Klitzing, G. Dorda and M. Pepper, "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance", Physical Review Letters 45 (1980), 494–497 | https://api.crossref.org/works/10.1103/PhysRevLett.45.494 | Crossref: authors K. v. Klitzing; G. Dorda; M. Pepper; same title; PRL 45(6) 494-497; issued 1980-08-11; the page's DOI link points here - CONFIRMED | D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, "Quantized Hall Conductance in a Two-Dimensional Periodic Potential", PRL 49 (1982), 405–408 | https://api.crossref.org/works/10.1103/PhysRevLett.49.405 | Crossref: same four authors, title, PRL 49(6) 405-408, issued 1982-08-09 - CONFIRMED | Qian Niu, D. J. Thouless and Yong-Shi Wu, "Quantized Hall conductance as a topological invariant", PRB 31 (1985), 3372–3377 | https://api.crossref.org/works/10.1103/PhysRevB.31.3372 | Crossref: Qian Niu; D. J. Thouless; Yong-Shi Wu; PRB 31(6) 3372-3377; issued 1985-03-15 - CONFIRMED | Douglas R. Hofstadter, "Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields", PRB 14 (1976), 2239–2249 | https://api.crossref.org/works/10.1103/PhysRevB.14.2239 | Crossref: given "Douglas R.", family "Hofstadter"; PRB 14(6) 2239-2249, 1976-09-15. The sources tool's "DOI MISMATCH (gives Hofstadter the initial R.)" is a false alarm of the tool: the page prints "Douglas R. Hofstadter", exactly Crossref's name - CONFIRMED | Takahiro Fukui, Yasuhiro Hatsugai and Hiroshi Suzuki, "Chern Numbers in Discretized Brillouin Zone", J. Phys. Soc. Jpn. 74 (2005), 1674–1677 | https://api.crossref.org/works/10.1143/JPSJ.74.1674 | Crossref: same authors, JPSJ 74(6) 1674-1677, 2005-06; full title adds ": Efficient Method of Computing (Spin) Hall Conductances" (the page prints the main title only, a normal short form) - CONFIRMED | Raffaello Bianco and Raffaele Resta, "Mapping topological order in coordinate space", PRB 84 (2011), 241106(R) | https://arxiv.org/abs/1111.5697 | arXiv journal reference "Phys. Rev. B 84, 241106(R) (2011)"; Crossref same title, PRB 84(24) article 241106, 2011-12-13 - CONFIRMED | Keith Slevin and Tomi Ohtsuki, "Critical exponent for the quantum Hall transition", PRB 80 (2009), 041304(R) | https://arxiv.org/abs/0905.1163 | arXiv journal reference "Phys. Rev. B 80, 041304(R) (2009)"; Crossref same authors and title, issued 2009-07-09 - WRONG | A. Kumar and collaborators, "Universality of quantum phase transitions in the integer and fractional quantum Hall regimes", Nature Communications 15 (2024), 8535 | https://api.crossref.org/works/10.1038/s41467-024-52927-w | title, journal, volume, article 8535 and year are right, but no author is named Kumar: Crossref lists Simrandeep Kaur; Tanima Chanda; Kazi Rafsanjani Amin; Divya Sahani; Kenji Watanabe; Takashi Taniguchi; Unmesh Ghorai; Yuval Gefen; G. J. Sreejith; Aveek Bid. Should read "S. Kaur and collaborators" - WRONG | (same citation, author entry) A. Kumar and collaborators | https://arxiv.org/abs/2312.06194 | arXiv 2312.06194 (journal ref "Nat Commun 15, 8535 (2024)") gives the same ten authors, first "Simrandeep Kaur"; positive control: the same author lists name Kaur, Chanda, Bid, so the search would have shown a Kumar - CONFIRMED | C. T. Tai and collaborators, "Engineering quantum criticality in the integer quantum Hall regime through a screening layer", accepted in Physical Review Letters on 27 July 2026 | https://api.crossref.org/works/10.1103/7vkh-q7rd | Crossref: first author C. T. Tai (then P. T. Madathil, A. Gupta, L. N. Pfeiffer, K. W. Baldwin, M. Shayegan); same title; PRL. The tool's "does not name P. T. Madathil" is expected for "and collaborators". Since published: PRL 137(11) 116501, issued 2026-09-09 (the citation could now carry it) - CONFIRMED | accepted in Physical Review Letters on 27 July 2026 | https://eutils.ncbi.nlm.nih.gov/entrez/eutils/efetch.fcgi?db=pubmed&id=42789839&retmode=xml | PubMed history: received 2026-03-12, revised 2026-07-20, accepted 2026-07-27 - MINOR | The SI values and practical decimal are from the BIPM Mise en pratique for electrical units | https://www.bipm.org/documents/20126/41489676/SI-App2-ampere.pdf/0987a90e-051b-dd7f-827d-3f7b32751a61 | read with pdftotext: the MeP gives e ("fixed numerical value of the elementary charge e to be 1.602 176 634 × 10–19") and "RK = 25 812.807 459 3045 Ω", but not the value of h: it refers to "the values of h and e given in [1]" (the SI Brochure). h's value is at https://www.bipm.org/en/measurement-units/si-defining-constants - CONFIRMED | h = 6.62607015 × 10⁻³⁴ J s | https://www.bipm.org/en/measurement-units/si-defining-constants | table: "h 6.626 070 15 x 10 –34 J s" - CONFIRMED | e = 1.602176634 × 10⁻¹⁹ C | https://www.bipm.org/en/measurement-units/si-defining-constants | table: "e 1.602 176 634 x 10 –19 C" - CONFIRMED | both exact in the SI | https://www.bipm.org/en/measurement-units/si-defining-constants | "The numerical values of the seven defining constants have no uncertainty." - CONFIRMED | both exact in the SI since 20 May 2019 | https://www.bipm.org/en/measurement-units | "Since 20 May 2019, all SI units have been defined by fixing the numerical values of seven defining constants." - CONFIRMED | The quotient is exact as a definition | https://physics.nist.gov/cgi-bin/cuu/Value?rk | "von Klitzing constant Numerical value 25 812.807 45... Standard uncertainty (exact)" - CONFIRMED | The displayed decimal is rounded, and a laboratory realization still has experimental uncertainty | https://www.bipm.org/documents/20126/41489676/SI-App2-ampere.pdf/0987a90e-051b-dd7f-827d-3f7b32751a61 | MeP: "this truncated recommended value is in error by less than 1 part in 10^15"; the page displays 25812.8074593045 Ω (toFixed(10)), digit for digit the MeP Eq. (2); the realization half is general metrology, consistent with the MeP's realization sections - CONFIRMED | Under ideal integer-plateau conditions, abs(Rxy) = RK/abs(C), not always RK (page prints the abs bars as pipes) | https://en.wikipedia.org/w/index.php?title=Quantum_Hall_effect&action=raw | "R_{xy} = \frac{V_\text{Hall}}{I_\text{channel}} = \frac{h}{e^2\nu}" - CONFIRMED | A clean isolated band already carries an integer Hall conductance | https://web.archive.org/web/20240228003834/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405 | TKNN abstract: "a uniform magnetic field and a periodic substrate potential U. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap." - CONFIRMED | In this noninteracting model, σxy = C e²/h | https://en.wikipedia.org/w/index.php?title=Quantum_Hall_effect&action=raw | "\rho_{xy}=\frac{h}{\nu e^2}\Rightarrow \sigma=\nu \frac{e^2}{h}" - CONFIRMED | In this noninteracting model, σxy = C e²/h (the integer as a topological invariant) | https://web.archive.org/web/20231028154636/https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.3372 | NTW abstract: "Whenever the Fermi level lies in a gap (or mobility gap) the bulk Hall conductance can be expressed in a topologically invariant form showing the quantization explicitly." - CONFIRMED | Moderate disorder can then widen the interval over which transport keeps that integer, because many of the new bulk states are localized | https://en.wikipedia.org/w/index.php?title=Quantum_Hall_effect&action=raw | "These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels." - CONFIRMED | the integer exists in a clean gap, while moderate disorder can give a plateau width by localizing states | https://web.archive.org/web/20231028154636/https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.3372 | NTW: "gap (or mobility gap)" and "generalizes the earlier result by Thouless, Kohmoto, Nightingale, and den Nijs to the situation where many-body interaction and substrate disorder are also present" - CONFIRMED | Strong disorder can close the mobility gap and destroy the plateau | https://api.openalex.org/works/doi:10.1103/physrevlett.78.318 | the record had no source. Sheng and Weng, PRL 78, 318 (1997), abstract: "The disappearance of integer quantum Hall effect (IQHE) at strong disorder and weak magnetic field is studied in a lattice model ... extended levels between those plateaus do not float up in energy but keep merging together after the destruction of plateaus" - CONFIRMED | the magnetic unit cell has q sites and therefore q magnetic bands | https://en.wikipedia.org/w/index.php?title=Hofstadter%27s_butterfly&action=raw | "if α=P/Q ... there are exactly Q energy bands" - CONFIRMED | If the reduced flux through each plaquette is (p/q), the magnetic unit cell contains (q) sites | https://en.wikipedia.org/w/index.php?title=Hofstadter%27s_butterfly&action=raw | same passage; the page engine's harper() builds a q by q matrix after reducing p/q - CONFIRMED | with possible touchings | public/strata/quantum-hall-chern/index.html | the record had no source. Recreated with the page's own engine (scratch sweep-code/touch.mjs, touch2.mjs): harper(1,2,π/2,π/2) has eigenvalues 0 and 0 (analytically E = ±2t sqrt(cos²kx + cos²ky)); at α = 1/4 the two central bands meet (direct gap 9.6e-17 at kx = 0, ky = −π/2). research/quantum-hall-chern/README.md states the same: "for even q the two central bands touch at E = 0 with linear (Dirac) dispersion" - CONFIRMED | Click a sampled spectral gap and the page solves its Diophantine gap label | https://en.wikipedia.org/w/index.php?title=Hofstadter%27s_butterfly&action=raw | "one can obtain a Diophantine equation that describes the system" and "each phase is characterized by an integral Hall conductance ... known as Chern numbers"; the page's gapLabel() solves r + pC = sq (but see the added even-q claim below) - CONFIRMED | Chiral edge transport is the boundary consequence of the bulk invariant | https://web.archive.org/web/20240228003842/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.3697 | Hatsugai 1993 abstract: "One is the Thouless–Kohmoto–Nightingale–den Nijs (TKNN) integer in the infinite system and the other is a winding number of the edge state." (the page cites no source for this sentence) - CONFIRMED | Finite-size participation cannot by itself prove Anderson localization, which is an infinite-size statement | https://arxiv.org/abs/0707.4378 | the record had no source. Evers and Mirlin, Rev. Mod. Phys. 80, 1355 (2008), Sec. II.3.1 (read at ar5iv): IPR scaling with system size, "⟨P_q⟩ ... ∼ L^{−τ_q}" and "In a metal D_q = d, in an insulator D_q = 0": localization is classified by how the IPR scales with L, which one size cannot show - CONFIRMED | Bianco-Resta local Chern marker | https://arxiv.org/abs/1111.5697 | abstract: the topological order "can be mapped by means of a 'topological marker', defined in r-space" - CONFIRMED | Its marker cancels when averaged over the whole lattice, so this page averages only the central 6 by 6 sites and says so | https://ar5iv.labs.arxiv.org/html/1111.5697 | after Eq. (7): the whole-sample trace "guarantees a zero result, whenever P projects over a finite-dimensional manifold. This confirms that the global topology is trivial within open boundary conditions" - CONFIRMED | the full-lattice average is avoided because boundary cancellation is part of the real-space construction | https://ar5iv.labs.arxiv.org/html/1111.5697 | same passage: "traces must be avoided when addressing finite and/or inhomogeneous samples" - MINOR | The local marker convention is −4π Im diag(PXPYP) | https://ar5iv.labs.arxiv.org/html/1111.5697 | Bianco and Resta Eq. (6): "C = −(1/π)(2π)²/A_c Im tr_cell{PxQy} = (4π/A_c) Im tr_cell{PxPy}", i.e. +4π; the page's −4π is the opposite sign and the page does not say so. Their note 10: "The choice of the sign of C is not uniform in the literature." The page's sign is internally consistent (marker −0.788 beside clean C = −1) - CONFIRMED | The local marker is approximate because the lattice is small, open, disordered, and evaluated at one seed | https://ar5iv.labs.arxiv.org/html/1111.5697 | "The choice of boundary conditions becomes irrelevant in the limit of a large sample"; a statement about the page's own finite computation - CONFIRMED | The Chern sum is a discretized gauge-invariant computation, not symbolic integration | https://arxiv.org/abs/cond-mat/0503172 | "We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone." - CONFIRMED | takes normalized determinant overlaps between neighboring occupied subspaces | https://ar5iv.labs.arxiv.org/html/cond-mat/0503172 | FHS Eq. (16): "U_μ(k_ℓ) = (1/N_μ(k_ℓ)) det ψ†(k_ℓ)ψ(k_ℓ+μ̂)" with "N_μ(k_ℓ) ≡ abs(det ψ†(k_ℓ)ψ(k_ℓ+μ̂))"; engine detOverlap() and chernMesh() do this - CONFIRMED | A high-precision noninteracting Chalker-Coddington calculation reported a localization-length exponent | https://ar5iv.labs.arxiv.org/html/0905.1163 | abstract: "critical exponent of the Chalker-Coddington model"; text: "the critical exponent ν that describes the divergence of the localization length", "Such high precision is required", "models of non-interacting electrons cannot explain" - CONFIRMED | ν = 2.593 with interval [2.587, 2.598] in 2009 | https://arxiv.org/abs/0905.1163 | "We report an estimate ν = 2.593 [2.587,2.598]"; PRB 80, 041304(R) (2009) - CONFIRMED | A 2024 trilayer-graphene experiment instead reported γ = 2.4 ± 0.2 and κ = 0.41 ± 0.02 across integer and fractional transitions | https://arxiv.org/abs/2312.06194 | "the same critical scaling exponent κ = 0.41 ± 0.02 and localization length exponent γ = 2.4 ± 0.2 for fractional and integer quantum Hall transitions ... ultra-high mobility trilayer graphene devices"; Nat Commun 15, 8535, issued 2024-10-02 - CONFIRMED | while documenting a much wider historical experimental range for κ | https://ar5iv.labs.arxiv.org/html/2312.06194 | "Experimental investigations of scaling in the IQH regime have reported κ varying between 0.16 ≤ κ ≤ 0.81" - CONFIRMED | The critical theory connecting adjacent plateaux, especially how interactions, screening, dephasing, and disorder range select measured exponents, remains unsettled | https://ar5iv.labs.arxiv.org/html/2312.06194 | "This wide variation has been attributed to varying disorder correlation lengths ... This lack of a tight constraint on κ has hindered any claims of their universality"; dephasing enters there as "The phase coherence length L_φ ... (L_φ = T^{−p/2})"; interactions and screening from Slevin-Ohtsuki and Tai et al. - CONFIRMED | An accepted 27 July 2026 experiment reports that a nearby screening layer changes the dynamical critical behavior, direct evidence that electron interaction range matters | https://eutils.ncbi.nlm.nih.gov/entrez/eutils/efetch.fcgi?db=pubmed&id=42789839&retmode=xml | abstract: "z=1, consistent with a theoretical picture where electrons have a long-range (1/r) interaction. Theory also predicts that z=2 for short-range electron interaction ... when screening is introduced, κ is reduced to ≃0.22, implying z=2. Our results provide direct experimental evidence for the role of electron-electron interaction" - CONFIRMED | As of 1 August 2026, the existence of integer quantization is not the open problem | https://en.wikipedia.org/w/index.php?title=Quantum_Hall_effect&action=raw | "The quantization of the Hall conductance ... has the important property of being exceedingly precise." - CONFIRMED | The immersive page constructs that (q) by (q) Harper Hamiltonian at 256 magnetic momenta and diagonalizes every one | public/strata/quantum-hall-chern/index.html | updateSpectrum(): "spectrum(f.p,f.q,16)", 16 by 16 loop; verifier PASS 66 - CONFIRMED | The denominator ... is the dimension of the magnetic Bloch Hamiltonian being diagonalized at every sampled momentum | public/strata/quantum-hall-chern/index.html | harper() builds matrix(q) after reduceFraction - CONFIRMED | Moving the flux control therefore changes the matrix being solved, not merely the label on a stored image | public/strata/quantum-hall-chern/index.html | "flux.addEventListener('input',updateSpectrum,{passive:true})" - CONFIRMED | Click a sampled spectral gap and the page solves its Diophantine gap label (engine) | public/strata/quantum-hall-chern/index.html | canvas click handler -> chooseGap() -> gapLabel(p,q,r) - CONFIRMED | 21 by 21 momentum mesh, normalized determinant overlaps, sums the plaquette phases | public/strata/quantum-hall-chern/index.html | chernRefined(): chernMesh(p,q,occ,15) and chernMesh(p,q,occ,21) - CONFIRMED | A 15 by 15 calculation supplies a visible refinement check | public/strata/quantum-hall-chern/index.html | "'15 mesh: '+fmt(R.coarse.chern,5)+' · 21 mesh: '+fmt(F.chern,5)" - MINOR | The page reports the minimum direct gap and the smallest determinant overlap beside the Chern sum, and refuses to call the result resolved when the occupied subspace is not numerically isolated | public/strata/quantum-hall-chern/index.html | the gap is measured only on the mesh points ("resolved:fine.minGap>1e-6&&..."), so it misses touchings between mesh points. Recreated (sweep-code/evendepth.mjs): at the even-q central gap the green "resolved" lamp is shown for 8 of 17 fractions with q ≤ 12 (1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 5/12, 7/12), each with FHS sum 0.000 and minimum direct gap 0.106t at 1/4, while the bands touch (gap 9.6e-17) - CONFIRMED | calls the invariant unresolved below a direct gap of 10⁻⁶t or a link determinant of 10⁻⁸, and requires the two meshes to agree within 0.2 | public/strata/quantum-hall-chern/index.html | "resolved:fine.minGap>1e-6&&fine.minDet>1e-8&&Math.abs(fine.chern-coarse.chern)<.2" - CONFIRMED | 12 by 12 open lattice with seeded random onsite energy | public/strata/quantum-hall-chern/index.html | disorderHamiltonian(12,...): bonds only inside the block (x+11e-7)"), but the midpoint mesh never lands on the even-q Dirac points, so a real touching IS counted: recreated (sweep-code/evenall.mjs), the central touching is listed as a gap for 95 of 95 even-q fractions (α = 1/2: "width 0.873t") - CONFIRMED | The disorder is uniform on [−W/2,W/2) from a named 32-bit seed | public/strata/quantum-hall-chern/index.html | "onsite[i]=W*(rng()-.5)", mulberry32 returns [0,1); seed 137, Reseed adds 2654435761 mod 2^32 - CONFIRMED | double-precision cyclic complex Jacobi rotations with tolerance 10⁻¹¹ and at most 40 sweeps | public/strata/quantum-hall-chern/index.html | "tol=tol===undefined?1e-11:tol;maxSweeps=maxSweeps" (double pipe) "40"; no call passes other values - CONFIRMED | The density-of-states Gaussian width is η = 0.12t | public/strata/quantum-hall-chern/index.html | "gaussianDOS(vals,state.EF,.12)" - CONFIRMED | diagonalizing 666 clean matrices | public/strata/quantum-hall-chern/index.html | 15² + 21² = 225 + 441 = 666 - CONFIRMED | Height is the participation fraction 1/(N·IPR) | public/strata/quantum-hall-chern/index.html | "pr[a]=1/(n*s)" - CONFIRMED | normalized determinant overlap links, four links per plaquette, sum divided by 2π | public/strata/quantum-hall-chern/index.html | "return {chern:sum/(2*PI),minGap:minGap,minDet:minDet,mesh:M}" - CONFIRMED | This page computes h/e² from h = 6.62607015 × 10⁻³⁴ J s and e = 1.602176634 × 10⁻¹⁹ C | public/strata/quantum-hall-chern/index.html | "H_SI = 6.62607015e-34", "E_SI = 1.602176634e-19", siValues() rk=H_SI/(E_SI*E_SI) - CONFIRMED | sign follows this page's gauge convention | public/strata/quantum-hall-chern/index.html | gapLabel() solves r + pC = sq, smallest abs(C), ties to positive C; verifier PASS 14 C = −1 at 1/3, r 1 - CONFIRMED | choose q ≤ 13 for the depth calculation | public/strata/quantum-hall-chern/index.html | "document.getElementById('run-chern').disabled=!g" (double pipe) "f.q>13" - WRONG | More than forty independent assertions cover ... projector idempotence ... | research/quantum-hall-chern/verify-quantum-hall-chern.mjs | 126 assertions, but none tests P·P = P: the projector checks are PASS 93 "occupied projector trace equals occupied count" and PASS 94 "occupied projector is Hermitian". Searched the file for "idempot" and "P*P": no hit; positive control: "Hermitian" finds lines 86, 93, 206 - CONFIRMED | The verifier ... independently checks exact SI arithmetic, solvable Harper spectra, gap labels, Hermiticity, eigenvector residuals, Chern values, deterministic disorder, sum rules, localization bounds, projector properties, and the finite marker | research/quantum-hall-chern/verify-quantum-hall-chern.mjs | PASS 01-08, 23-31, 14-18, 20/33/80, 62-64, 71-73, 35-40/78-79, 26-28/83-84, 85-87, 93-94, 89-90 - CONFIRMED | It also extracts and executes the shipped page engine, then perturbs constants and algorithmic choices to show that its checks detect wrong answers | research/quantum-hall-chern/verify-quantum-hall-chern.mjs | html.match(/