Data and code for: One quarter, three families: quantized flavor, grand unification, and a magnetic origin in minimal-Yukawa SO(10)

The best-fit Yukawa matrices of the minimal $10_H + \overline{126}_H$ Yukawa sector of SO(10), obtained in the sector's benchmark global fits with no texture assumption, admit a decomposition $|Y_{ij}| = c_{ij}\,|Y_{33}|\,\varepsilon^{\,q_i+q_j}$ with a single expansion parameter and two integer charge vectors, $q_f=(2,1,0)$ in the $\overline{126}$ channel and $q_h=(3,2,0)$ in the $10$ channel, all constrained coefficients at $\varepsilon = 1/4$ inside the factor-of-three band adopted here as the criterion for order one. That band is a convention rather than a measurement, and at matched parameter count the 18-observable menu does not separate this two-vector reading from a single-vector one, which it does not exclude; the two-vector assignment is a model-building hypothesis supported by coefficient economy, not a decomposition the data establish. Re-fits with this structure imposed and $\varepsilon = 1/4$ frozen reproduce the full 18-observable GUT-scale menu within $3\sigma$ in both the supersymmetric model ($\chi^2 = 22.1$) and the non-supersymmetric model ($\chi^2 = 2.4$ under the hardened optimization, $18.6$ in the earlier search); the best-found supersymmetric profile of $\chi^2$ over the grading scale has its minimum at $\varepsilon \simeq 0.27$, inside the dressed band of the bare quarter predicted by the mechanism below ($\varepsilon_{\rm eff} = 0.29 \pm 0.06$), with $1/4$ on the basin shoulder $2.5$ units above that minimum and the Cabibbo value $13$ units above the quarter, both on a best-found upper envelope that has not been re-hardened. Hardening the non-supersymmetric profile flattens its basin: $\varepsilon$ is not identified between $0.222$ and $0.286$ on this menu, an interval of hop ratios $a/b \in [0.86, 1.20]$ that contains the equal-weight value, and the feature it does resolve, again on a best-found upper envelope rather than a certified profile, is the wall closing the basin from above near $\varepsilon \simeq 0.30$ rather than a preferred value inside. A density-matrix treatment of the same seesaw sector, with the family-basis conjugation fixed by the model's own seesaw term, with the $SU(2)_L$ triplet that the fit's type-I/type-II hybrid structure requires treated as a dynamical species (decays, inverse decays, gauge annihilations, its own $CP$ asymmetry, and consistent pole handling), and with epoch-correct spectator effects and a first-principles $\Delta L = 2$ washout under a derived diagonal chemical-potential dressing, does not predict the baryon asymmetry at the non-supersymmetric point: $|\eta_B|/\eta_B^{\rm obs}$ spans a convention envelope from $0$ (the signed asymmetry crosses zero inside the model's own reheating window, a cancellation between the $N$-sector and triplet-sector sources) to $11.1$ at the ceiling under the adopted compact $CP$ normalization, $6.9$ under the literal finite-mass alternative, the spread running over the reheating window, the heavy-state initial abundances, and three convention choices that current theory input does not fix: the triplet $CP$ normalization (a factor $3.8$, since the fitted spectrum has $M_\Delta/M_3 = 9.0$ and neither the hierarchical limit nor a compact form is controlled), the $\Delta L = 1$ washout dressing (a 21 percent ceiling shift), and the triplet initial condition (a factor $8.3$ with a sign flip). The observed value lies inside the envelope in every variant; the sign and mechanism are those of hybrid $N$-plus-triplet leptogenesis, and no numerical value is claimed. The kinetic corrections act on the uncorrected $0.11\,\eta_B^{\rm obs}$ by damping a near-cancellation between the $N_2$- and $N_3$-sourced terms rather than by adding a source. Under a vanishing abundance at reheating the same fixed texture contains an 18/18 point at $\chi^2 = 5.33$, below the $\chi^2 = 7.31$ of the reference point, whose quoted $1.02\,\eta_B^{\rm obs}$ is conditional on the superseded static engine and the superseded narrower window; the clean-origin intermediate-scale ordering fails at that point's own $v_R$ in all nine threshold cases, so it is an existence statement about the family under a stated cosmology and not a replacement reference point. The earlier values ($0.58$--$1.14$ from the static pole-subtracted engine, $0.11\times$ with its $0.004$--$0.6$ band, $0.34\times$ without the triplet term, a flavored-Boltzmann $0.48\times$ at the supersymmetric point that carried convention defects, and an analytic $1.04\times$ with a conjugation error) are superseded. The neutrino-sector predictions are set by the flavor fit and move by less than $0.3$~meV across the corrected asymmetry re-fit. A mechanism fixes the grading scale by counting rather than fitting: a clockwork chain of vector-like $16$-plet triplets whose hop is the equal-weight sum of the two $S_3$-invariant structures has subleading eigenvalue $1/(N_f+1)$, its stationary democratic mode is the third generation, and, completed by a single Georgi--Jarlskog-type operator, it reproduces all eighteen observables end to end. That counting reading carries one condition the construction imposes rather than derives, that the chain's flavon-to-cutoff ratio lie below the same eigenvalue; above it a direct operator with the same charge powers dominates and the expansion parameter is a vacuum-expectation-value ratio again. In the non-supersymmetric model the same texture supports a candidate completion: a $210_H$-driven Pati--Salam chain in which two-loop gauge unification, with intermediate-scale scalars drawn from the model's own Higgs content, selects in its compressed realization $M_I = (1.01\text{--}1.05)\,v_R$, the seesaw scale the flavor fit requires, with $\tau(p \to e^+\pi^0)$ within Hyper-Kamiokande reach and targeted tree-level checks for the first two breaking stages. A speculative magnetic-first model gives the counting a candidate origin: the hop is the uniform average over the four reversible flux moves of a three-plaquette torus, unique at $N_f = 3$ two ways, with the same complex certifying internal dimension $N_f + 1$ through a flux witness. A survey of the six nearest cells of the resulting $(N_f, m)$ lattice, the four- and five-family cells recomputed on their own exact textures over charges derived up to a minimality choice, finds them dead in two epoch classes rather than six modes, a light-quark detuning at the QCD crossover in two cells and an inversion of the nucleon ordering at weak freeze-out in the other four; the record axis on its own is not restrictive, holding at four of the seven computed cells and at $68$ of the $121$ cells of the swept grid, and the home cell $(3,4)$ is singled out by the conjunction rather than by any one axis, being the only computed cell that holds stable matter, a flavor-sector arrow, and retained records together, a uniqueness that is convention-conditional: a charge-conjugated hydrogen branch of the $(4,5)$ cell survives every gate the programme can compute at a probability the deuteron-binding literature brackets as $[0.00, 0.14]$, $0.131$ under the favored response form, with the branch's split pion multiplet the open structural channel, so the claim holds under the stated charge-sign reading of the baryonic criterion and the stated anchoring convention. Predictions include $\delta_{PMNS}$ ($294^\circ$ at the representative good-fit point NS-P), $m_{\beta\beta} \simeq 0.3$~meV (SUSY point) to $0.6$--$0.8$~meV (non-supersymmetric), $\Sigma m_\nu \simeq 63$--$68$~meV at the current DESI+CMB bound, a two-sided tensor window $r \in [5\times10^{-16},\, 1.0\times10^{-13}]$ whose floor is independent of the axion decay constant and whose ceiling, with the axion dark-radiation component $\Delta N_{\rm eff} = 0.002$--$0.020$ (rising to $0.026$ only if the light doublet mixes near $\tan\beta \simeq 1$, where the top-Yukawa production channel thermalizes the axion), is conditional on the inflationary branch and on the Peccei--Quinn completion adopted in the paper, the proton-decay branching pattern, and the $\theta_{23}$ octant as the sharpest near-term test.