The Fine-Structure Constant on Entangled Pairs: the 2×3 Cell's Spectrum and Wissner-Gross's 172-Record Count as Two Readings of One Maximally Entangled Pair, with the Count as the Tower's Boundary Term
Year: 2026
Type: article
Abstract: Description The four blocks of the 2×3 cell, ℂ ⊕ M₂ ⊕ M₃ ⊕ End(ℂ²⊗ℂ³), carry the eigenvalues λ = 1, 1/2, 1/3, 1/6: the normalised trace 1/N on a block of size N = 1, 2, 3, 6. The normalised trace is exactly the expectation value in the maximally entangled pair of dimension N, tr A/N = ⟨ω_N|A⊗1|ω_N⟩. The cell’s spectrum is that pair read one strand at a time and squared, (1/√N)² = λ. Read over all N strands and summed, the same pair gives N/√N = λ^(−1/2): the weights 1, √2, √3, √6 of A. D. Wissner-Gross’s 172-record count (2026), whose inverse 31104/(79 + 48√2 + 42√3 + 3√6) = 137.035999190 sits 0.60σ from CODATA 2022. The product of pairs is the pair of the product, ω₂⊗ω₃ = ω₆, which is the cell’s rule 1/6 = 1/2 · 1/3; the cell’s thermal block, entropy ln 6 per tick, is ω₆ seen from one end. Every other number of the count is an object of the cell. The readout dimension is d₄χ⁴ = 24 · 36² = 31104. The letters are the traceless parts of the colour and weak blocks. The 169 product records are d₃² + d₂² + 4d₄ = 13², an equality that among integer seeds holds only at (2, 3). The four sector sizes 79, 48, 42, 3 are the symmetric and antisymmetric squares of the two adjoints. In the cell’s terms α = (d₄χ⁴)⁻¹ Σ n_s λ_s^(−1/2). Unweighted, the records fill 172/31104 = 43/6⁵ of the readout space, the tower’s height over χ⁵, and both factors of that identity are the seed’s alone. The exchange that sorts the records is also the cell’s. Exchanging two letters at one end is conjugation by the swap, and the swap is the partial transpose of the unnormalised pair, (N|ω_N⟩⟨ω_N|)^T₂ = F_N: the same partial transpose that selects the seed, since a read fixes both ends of a pair only where every positive map is decomposable, d₁d₂ ≤ 6 (Størmer, Woronowicz, Choi). So is the end at which each exchange collects its factor. The same F_N and ω_N give each end’s Frobenius–Schur indicator: the weak end has a pair invariant under the same unitary at both ends (ν = −1), the colour end has none (ν = 0). Only the weak end can be a far end, so the reads are taken at the colour end, which is also the larger. And so is the way the strands add. The squared weight is a d-path interference intensity, (1 + 2V_X)(1 + V_Y), with coherence V = 1 for a record the exchange returns with its own sign and V = 0 otherwise: amplitudes add for alternatives that cannot be told apart, probabilities for those that can. The flipped sign is a label, not a phase, because a readout invariant under the exchange cannot see a phase between its two eigenspaces; read as a phase, every flipped record cancels. And so are the alternatives: carried across the joining pair by the transpose, the exchange becomes the contraction Σᵢⱼ|ii⟩⟨jj|, full coherence on exactly the pair’s d strands, so the squared weight at an end is ⟨ω_d|R^T₂|ω_d⟩ with R the exchange for a survivor and the identity otherwise. Six substitutes fail: the cell’s eigenvalues as weights (69.2), the pair projection, the factor where the letters sit (139.03), the Werner twirl (135.89, the two ends exchanged by an identity unique to (2, 3)), the sign as a phase (4232.7), and the projector in place of the exchange (153.69). Three clauses of the source’s definitions are taken as given; with them the count is a computation on the cell. The count’s charge is hypercharge: Q = 6Y acts on the pair as d₂ − d₁ = 1, the smallest hypercharge the ℤ₆ quotient of the Standard Model group allows. The passage to electric charge needs no factor. The pair reads the normalised trace, so its mean electric charge is its hypercharge (Gell-Mann–Nishijima); and at the zero-momentum pole the hypercharge current couples only to the photon, with e per unit Y — the factor cos²θ_W belongs to the hypercharge field’s own coupling g′, not to that residue. The source reads the count as that residue; the reading has the shape of a Källén–Lehmann pole weight against a sum rule — the bare insertion is the pair on the readout space, the records hold 0.007297353 of it and the directions no read fixes the rest — and its place at q² = 0 is the tower’s infrared cap. What remains is the first clause of the source’s conjecture: that the qutrit–qubit link is the photon’s elementary link. Read through the compositeness criterion, the physical photon is its bare link with probability 1/137.036. Up to one basis change at the qutrit, the link is a massless vector’s polarisation map: the qubit is its helicity (±1 = 2T₃), the matched plane its transverse polarisations, the unread direction its longitudinal, gauge direction; applied to polarisation maps, the read cap admits massive spin ½ and the massless vector and excludes the massive vector and the graviton. The reading holds in radiation gauge. With Poincaré symmetry, two polarisation states force p² = 0, so the link propagates at E = |k| along the direction no read fixes; the tower’s log-periodic texture cannot ride on that propagation (burst timing excludes it by thirteen orders) and belongs to the running, where analyticity puts its test in the timelike spectral density. The count has no scale and cannot run; the tower runs π per layer. Entered as the tower’s boundary term, α⁻¹(d) = π(d + 1) + ln 7 + δ with δ = 0.0016049 = π × 0.000510867 layers, it equals the count at the cap and moves no mass of the series’ master table by more than 8.1 ppm, against a best residual of 300 ppm. Spread over the layers without the boundary term, it breaks the leptons (m_μ/m_e from −0.11 to +0.51 percent). The layer count 43π + ln 7 is bare. The 11.71 ppm between it and the count is one-loop QED’s dressing between zero momentum and Q* = 141.10 keV, a scale fixed to 6.6 ppm at which the tower supplies none; the two are tied by the running, not by a dressing. With the count as α(0), the layer count at the Thomson point is 43.000510867. Falsifiers: a resolution of the rubidium–caesium recoil disagreement that settles more than a few of the count’s standard deviations from 137.035999190 (the boundary term then falls back to the tower and no mass changes); a count that is not the zero-momentum residue (read as the hypercharge field’s coupling g′ it would give 178.2); a link that is not the photon’s elementary link; a log-periodic analysis of the timelike spectral density R(s) without the tower’s texture of period ln 6 (the running carries only about 1/141 of a depth in R(s), so percent-level e⁺e⁻ data already hold the spacelike texture below 3×10⁻⁷). Fifty-six exact checks are reproduced by the deposited script fsc_entangled_receipts.py. The finite and symbolic statements — the pair identity for every N, the Brauer relations, the Frobenius–Schur indicators, the weight law, every substitute’s numerator, the charge and polarisation statements — and integer certificates for the value and its offsets are kernel-checked in Lean 4 (FscEntangled.lean) and Agda (FscEntangled.agda). The count, its construction and its value are Wissner-Gross’s and are cited throughout. Crystal Topos algebra series; builds on The Masses and The Layer Count of the Fine Structure Constant. Keywords: fine-structure constant, maximally entangled state, entangled qudits, normalised trace, purification, Choi-Jamiolkowski isomorphism, positive maps, decomposable maps, partial transpose, Peres-Horodecki criterion, swap operator, Frobenius-Schur indicator, interference, indistinguishability, superselection, symmetric square, antisymmetric square, Plancherel weights, Wedderburn decomposition, Standard Model gauge group, hypercharge, running coupling, vacuum polarisation, bare and dressed coupling, boundary term, layer count, CODATA 2022, electron g-2, atom recoil, 2×3 cell, Crystal Topos, Wissner-Gross I am targetig December 1, 2026 for the first explainer video in 3brown1blue format using manim. You can follow here to get notified when the first video drops https://www.youtube.com/@CrystalToe The mathematics of this paper sits on top of the six foundational Crystal Topos papers. Load them into the model before this paper, as separate uploads in separate turns: a single bundle is compressed into summaries and the algebraic detail is lost. Confirm each — "please confirm you have read this paper and can state its main result" — before sending the next. Load these six first, in this order, one upload per turn: Discrete Core — the algebra , the seed , and its dual. https://doi.org/10.5281/zenodo.20232800 MERA Scaling Spectrum — the two rescalings, the eigenvalues and the multiplicities . https://doi.org/10.5281/zenodo.20236511 Type II Core — the von Neumann factor type and the semi-finite trace. https://doi.org/10.5281/zenodo.20257603 Triangular Duality — the three structures as one categorical object. https://doi.org/10.5281/zenodo.20257953 The Eight Towers and the Foliated Grid — the grid, the eightfold cluster and the 43-layer tower from the spectral data. https://doi.org/10.5281/zenodo.21475952 The Spacetime Constants , , from Substrate Adjacency — the units; the two speeds of the circuit. https://doi.org/10.5281/zenodo.20485003 Then, for the vacuum, the anchor, the lattice and the trees: The Higgs Boson as the Singlet Projector — the vacuum. https://doi.org/10.5281/zenodo.20422007 The Cube Mass Generator — the Yukawa anchor. https://doi.org/10.5281/zenodo.21470949 The Design Tower — the lattice. https://doi.org/10.5281/zenodo.22079045 Pronic Growth — the lattice. . https://doi.org/10.5281/zenodo.23539196 The Masses https://doi.org/10.5281/zenodo.23541978 Then this paper. Questions about why the rings, the signs, the floors and the one column tier take the form they do will then be answered against the construction rather than by pattern-matching. Copyright © 2026 Daland Montgomery. All Rights Reserved. Creative Commons Attribution Non Commercial No Derivatives 4.0 International For a private, proprietary license, contact: [email protected] Software Implementation: The formulas and constants derived in this work
Author: Montgomery, Daland
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225464923