| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p28001 | Substrate arithmetic: $\F_\Om$, the multiplicative cycle, the Frobenius $C_3$ Galois orbit, the quarter-turn $Q_4$, the admissible residue $\Om\equiv5\,(\mathrm{mod}\,12)$. | I | |
| A2p28002 | One generation as the $\mathbf{16}$ of $\SO(10)$: field content, hypercharges, charges, anomaly freedom, $\sin^2\theta_W=3/8$, and the massless-photon / confining-gluon residues. | I | master: 00:G12 |
| A3p28003 | The Froggatt–Nielsen flavour mechanism and the Wolfenstein parametrisation. | I | [froggattnielsen1979; wolfenstein1983] |
| A4p28004 | The Gatto–Sartori–Tonin, Georgi–Jarlskog, and lopsided-$\SU(5)$ textures. | I | [gattosartoritonin1968; georgijarlskog1979; albrightbarr1998] |
| A5p28005 | The seesaw mechanism and the renormalisation-group running. | I | [minkowski1977] |
| A6p28006 | Measured anchors: the two $\Delta m^2$, $M_R$, the charged-lepton and quark masses, $V_{us}$ and the mixing angles; the Koide identity is P1’s prediction, not an input. | I | [koide1983; nufit60] |
| B. Realisations: mathematics $\to$ physics | |||
| B1p28007 | Matter is the circulant amplitude matrix $\sqrt M$; a mass is the squared modulus of a generation amplitude (the Born square). Falsifier: a charged-lepton spectrum departing from the circulant Koide form at improved precision (present agreement $6\times10^{-6}$). | R | master: 00:G16 |
| B2p28008 | The three generations are the Frobenius $C_3$ Galois orbit: identical in every gauge number, differing only in mass. Falsifier: a fourth generation, or two generations differing in any gauge number. | R | master: 00:G14 |
| B3p28009 | The role depths $(0,1,2)$ of the closed ladder are the Froggatt–Nielsen flavour charges. Falsifier: a confirmed departure of the Wolfenstein hierarchy from the depth ladder $(0,1,2)$, e.g. $V_{us}$ outside the band of the Gatto relation for every phase $\varphi$. | R | |
| B4p28010 | The colour excess of the quark amplitudes is the Georgi–Jarlskog Clebsch $\Nc=3$; the squared amplitude is the cyclotomic norm $N(1-\zeta_n)$. Falsifier: a quark-to-lepton amplitude ratio departing from the $\Nc=3$ Clebsch at improved precision. | R | |
| B5p28011 | A colourless sector (charged leptons, neutrinos) carries the undressed quarter-turn amplitude $r=\sqrt2$, i.e. $Q=2/3$. Falsifier: $Q_\ell\neq2/3$ at improved charged-lepton precision; for neutrinos, inverted ordering, a degenerate spectrum, or $\sum m_\nu\gtrsim120$ meV. | R | |
| B6p28051 | Drive-invariant boundary branch: the lightest colourless neutrino sits at the quarter-turn zero (as the electron), selecting the normal near-$m_1{=}0$ branch. Falsifier: inverted ordering. | R | § 5 |
| C. Derived: theorems within this paper | |||
| C1p28012 | Generation universality $\Rightarrow$ the amplitude matrix is a $C_3$-circulant $\Rightarrow$ the Koide functional form. | T | |
| C2p28013 | $Q=2/3\Leftrightarrow r=\sqrt2\Leftrightarrow\rho^2=\tfrac12$ (the equivalence an identity); the value $\rho^2=\tfrac12$ is the realisation B5. | T | master: 00:G16 |
| C3p28014 | The leading boundary phase is the framed-rational winding $\delta_{\mathrm{LO}}=\tfrac38-\tfrac13=\tfrac1{24}$ cycle (split-torus boundary $\tfrac38$ minus generation $\tfrac13$; continuum $\pi/12$ [approx]); massless lightest generation. | T | |
| C4p28015 | The squared amplitudes are the cyclotomic norms $r_\ell^2=N(1-i)=2$, $r_u^2=N(1-\omega)=3$ (on B5), hence $Q_\ell=\tfrac23$, $Q_u=\tfrac56$ in framed arithmetic ($Q_u=0.849$ in the working scheme, band $0.83$–$0.89$); the down sector is the GJ-dressed intermediate. | T | master: 00:G17 |
| C5p28016 | The cross-sector phase lock $\delta_\ell:\delta_d:\delta_u=1:\tfrac12:\tfrac13$ in the $\overline{\mathrm{MS}}$ working scheme of Table 2 ($0.388$ for the up ratio with the pole charm mass). | T | § 4.5; flv.delta |
| C6p28017 | The Gatto Cabibbo relation $V_{us}=|\sqrt{m_d/m_s}-e^{i\varphi}\sqrt{m_u/m_c}|$, the phase $\varphi$ a kernel datum ($85^\circ\pm10^\circ$ from the data). | T | master: 00:G15 |
| C7p28018 | Up-sector doubling $m^u_i\sim(m^d_i)^2$ from the $\mathbf{10}\cdot\mathbf{10}$ Yukawa. | T | Prop. 3; flv.up_doubling |
| C8p28019 | Lopsided $M_e=M_d^{\mathsf T}$ splits CKM (small) from PMNS (large); the CKM angle values reduce to the winding kernel and the shared $\sigma$. | T | § 4.4; flv.tier_b master: 00:G15 |
| C9p28020 | The reactor angle at leading order, $\sin\theta_{13}=\sin\theta_C/\sqrt2\simeq\delta_0/\sqrt2$ (quark–lepton complementarity): $0.159$ against the measured $0.149$, $7\%$, no correction derived. | T | master: 00:G15 |
| C10p28021 | TM2 selected: the trivial $C_3$ character $(1,1,1)/\sqrt3$ is the protected column, so $\cos\delta_{\mathrm{CP}}$ is locked to $\theta_{23}$ by (8); the magic Jarlskog $J=1/(6\sqrt3)$ is maximal CP, the sign fixed by the drive. | T | |
| C11p28022 | Seesaw of circulants $\Rightarrow$ Koide-form (theorem); with $M_R\propto m_D$ (Y1) the seesaw gives $m_\nu=m_D^{\mathsf T}$, $r_\nu=\sqrt2$ (transpose fixed point). | T | master: 00:G15 |
| C11bp28050 | Both orderings solved: the signed $r_\nu=\sqrt2$ circulant solves both neutrino orderings exactly, so the boundary branch (B6) is the sole ordering assumption. | T | |
| C12p28024 | The generation $\mathbf{16}$ is the four-chart reflection algebra $\Lambda^{\bullet}(\C^4)\cong\Lambda^{\mathrm{even}}(\C^5)$, the quarter-turn the fifth direction; projection anchored at $\nu^c=\Lambda^0$. | T | Prop. 1 |
| X. What the construction explains: the surplus over the flavour data | |||
| X1p28025 | The Koide ratio is exactly $2/3$ given the quarter-turn realisation B5, the cube-root generation orbit bisected at the quarter-turn ($\rho^2=2^{-1}$), an exact value rather than an unexplained empirical coincidence. | T | Proposition 2; flv.framed_koide |
| X2p28026 | The up-quark Koide value is $5/6$ in framed arithmetic ($0.849$ in the working scheme, $0.832$ with pole masses), the cube-root colour norm $r_u^2=N(1-\omega)=3$ against the colourless $N(1-i)=2$, a ratio where the Standard Model carries no relation at all. | T | Remark 4.5; flv.quark_amp |
| X3p28027 | The flavour charges are the role depths $(0,1,2)$ of the closed primitive-role ladder, derived as the three generative roles rather than assigned to fit the hierarchy. | T | § 4.2; flv.tier_b |
| X4p28028 | The Cabibbo angle is the Gatto relation $V_{us}=|\sqrt{m_d/m_s}-e^{i\varphi}\sqrt{m_u/m_c}|$ following from the role-depth texture, the leading term at $0.3\%$ and the phase $\varphi$ a kernel datum, so two otherwise-independent observables are tied rather than measured apart. | T | Proposition 4; flv.spurion |
| X5p28029 | The Georgi–Jarlskog factor is the colour count $N_c=3$, the $B\!-\!L$ Clebsch of the $\overline{\mathbf{126}}$, not a texture coefficient inserted by hand; its double ratio $9$ against the measured $10.4\pm0.5$ is the open $15\%$ excess of the minimal Clebsch. | T | § 4.3; flv.tier_b |
| X6p28030 | The up-sector doubling $m^u_i\sim(m^d_i)^2$ is forced by the $\mathbf{10}\cdot\mathbf{10}$ Yukawa, so the steeper up hierarchy is a consequence rather than a separate assumption. | T | Proposition 3; flv.up_doubling |
| X7p28031 | The electron’s extreme lightness is the quarter-turn boundary: the lightest generation sits where the winding amplitude vanishes, massless at leading order, rather than an anomalously small Yukawa tuned by hand. | T | Proposition 6; flv.m10 |
| X8p28032 | The large-lepton, small-quark mixing split is the lopsided $M_e=M_d^{\mathsf T}$: one off-diagonal feeds the invisible right-handed quark rotation and, transposed, the visible left-handed lepton rotation, so the two mixing matrices are one structure rather than two unrelated ones. | T | § 4.4 |
| X9p28033 | The reactor angle is, at leading order, $\sin\theta_{13}=\sin\theta_C/\sqrt2=0.159$, quark–lepton complementarity through the quarter-turn, $7\%$ above the observed $0.149$ with no correction derived: a leading-order estimate of the mixing parameter, not a prediction. | T | § 6; flv.theta13 |
| X10p28034 | Near-maximal leptonic CP is the cube-root phase: $\mathrm{Im}\,\omega=\sqrt3/2$ gives the maximal Jarlskog $J=1/(6\sqrt3)$, so the cubic reappears as CP violation rather than a free phase fitted to data. | T | § 6; flv.pmns_cp |
| X11p28035 | The neutrino spectrum (a near-massless lightest state, $\sum m_\nu\simeq59\,$meV on the normal branch) follows from colourlessness fixing $Q_\nu=2/3$, the transpose fixed point under the alignment $M_R\propto m_D$ (Y1) and the boundary branch that selects the ordering (B6), rather than three independent neutrino masses. | T | |
| P. Falsifiable predictions | |||
| P1p28036 | Charged-lepton Koide $Q_\ell=2/3$ exact on B5 (measured $0.666661\pm0.000007$); $m_e=0.5106$ vs $0.5110$ MeV. Falsifier: a lepton-mass shift of $Q_\ell$ off $2/3$ beyond the $\tau$-mass tolerance. | T | master: 00:G16 |
| P2p28037 | Up-quark Koide $Q_u=5/6$ in framed arithmetic, $r_u^2=N(1-\omega)=3$; continuum $0.849$ in the working scheme, $0.832$ with pole masses, band $0.83$–$0.89$. Falsifier: up/charm/top masses moving $Q_u$ outside the band in every standard scheme. | T | master: 00:G17 |
| P3p28038 | Cabibbo $V_{us}=|\sqrt{m_d/m_s}-e^{i\varphi}\sqrt{m_u/m_c}|$: leading term $0.224^{+0.011}_{-0.004}$ vs $0.2243$ ($0.3\%$), the up term $\pm0.04$ with $\varphi$ open. Falsifier: no $\varphi$ satisfying the relation within the input band. | T | |
| P5p28039 | Normal neutrino ordering, from $Q_\nu=2/3$, the two $\Delta m^2$ and the boundary branch (B6); the signed circulant solves both orderings (C11b). Falsifier: inverted ordering. | T | |
| P6p28040 | Mass-sum floor $\sum m_\nu=58.7$ meV ($m_1{\simeq}0$, $m_2{\simeq}8.6$, $m_3{\simeq}50$ meV; $58.8$ at NuFIT 6.0), the $m_1=0$ normal-ordering floor. Falsifier: a cosmological $\sum m_\nu<0.058$ eV. | T | master: 00:P3 |
| P7p28041 | Effective Majorana mass $m_{\beta\beta}{\simeq}1.5$–$3.7$ meV (NO, $m_1{\simeq}0$, circulant phase). Falsifier: a $0\nu\beta\beta$ signal at the inverted scale $15$–$50$ meV. | T | |
| P8p28042 | Leptonic CP locked to the octant by (8): $\delta_{\mathrm{CP}}\in(204^\circ,310^\circ)$ over the $3\sigma$ $\theta_{23}$ range, $\pm124^\circ$/$\pm74^\circ$ at the NuFIT 6.0 best fits, $\sin\delta_{\mathrm{CP}}{<}0$, $J=1/(6\sqrt3)$. Falsifier: CP-conserving $\delta_{\mathrm{CP}}$, $\sin\delta_{\mathrm{CP}}{>}0$, or $(\theta_{23},\delta_{\mathrm{CP}})$ off the relation. | T | |
| V. Machine verification | |||
| V1p28043 | The validation package finite-ring-space/src/28-flavour: the sixteen scripts as written — flv.exact_core, flv.framed_koide, flv.tier_b, flv.m10, flv.delta, flv.revision_checks, flv.quark_amp, flv.up_doubling, flv.spurion, flv.theta13, flv.neutrino, flv.pmns_cp, flv.tm2_jointfit, flv.scale_a, flv.coupling_anchor, flv.alpha_probe — run through one registry, one family per script, its micro-checks the script’s own verdict lines together with the registry’s predicates on the script’s namespace and output (four scripts print without asserting; every headline numeral is pinned by a labelled check in flvcommon.py), each family naming the rows it witnesses; the notebook 28-flavour-main.ipynb executes them in the browser and the driver run_all.py writes results.json. | T | Sec. 10; run_all |
| V2p28044 | The discipline, recorded per family as its class: EXACT — integer, $\F_p$, $\F_{p^2}$ or cyclotomic arithmetic throughout, no float in any asserted claim (flv.exact_core, flv.framed_koide, the load-bearing identities of the mass sector); MIXED — an own exact check for at least one claim, the continuum confined to labelled [approx] comparisons (seven families); APPROX — continuum or numerical by construction, a numerical reconfirmation of an identity proven exactly in flv.exact_core, a comparison with measured data, or an $\Om$-hard reading (seven families). | T | Sec. 10, Sec. 7 |
| V3p28045 | What the run does not decide: the $\Om$-hard residues (Z1–Z4) enter only as labelled readings (the transmutation exponent, the coupling anchor, the one-loop fermion sum, $\delta_0=Q/3$ as a confrontation); the joint mixing fit of flv.tm2_jointfit is status, not a test (five parameters for four observables); the reactor angle is a leading-order estimate at $7\,\%$ (C9), not a prediction; the Georgi–Jarlskog double ratio is carried as the open $15\,\%$ excess; five of the scripts’ own checks are statements, not decisions, and are recorded as such in the package README. | T | Sec. 10 |
| Z. Residues: $\Om$-hard | |||
| Z1p28046 | The lepton winding phase $\delta_0$: the one $\Om$-hard drive-orientation residue. Neither a geometric nor a profinite derivation fixes it, the drive-orientation Gauss sum’s phase taking no fixed value across the shells of the totality, equidistributed in its large-shell reading. Sub-horizon reading $\delta_0=Q/3=2/9$ ($3\delta_0=Q$); leading $\delta_{\mathrm{LO}}=\tfrac38-\tfrac13$ framed-rational (C3); phase residue carried by $\mathrm{Re}(b^{3})$. | Ω | master: 00:G16 |
| Z2p28047 | With the quark phases locked to $\delta_0$ by C5 ($\delta_d=\delta_0/2$, $\delta_u=\delta_0/3$), the neutrino phase $\delta_\nu\simeq0.48$, which the seesaw resets, and the quark CP phase are independent $\Om$-hard residues. | Ω | |
| Z3p28048 | The absolute masses and the electroweak scale $v$: $\Om$-hard, the matter face of the scale operator (master ledger Z6). | Ω | master: 00:Z6 |
| Z4p28049 | $\alpha^{-1}(0)$: the electromagnetic face of the scale operator, $\Om$-hard (master ledger Z8). | Ω | master: 00:Z8 |
| Y. Hypotheses: stated ahead of their test | |||
| Y1p28052 | The $\overline{\mathbf{126}}$ alignment, $M_R\propto m_D$, on which the seesaw’s transpose fixed point rests (C11): a hypothesis stated ahead of its test; no falsifier that tests it alone is yet stated. | O | Prop. 8 |
Nothing matches that filter.
Status
- I
- Import. A standard result or measured datum used here without reproof.
- R
- Realisation. A forced identification of a physical term with the substrate object it denotes: constitutive, not interpretive; defended by the totality of consequences registered under it; killed by the falsifier stated in its row.
- D
- Definition. A naming or set-up move.
- T
- Theorem. Derived within this paper from the rows above it; where a family identifier is named, its content is verified by the validation package.
- O
- Open. A hypothesis stated ahead of its test; a row that rests on it names it.
- Ω
- $\Om$-hard. Decided by the totality; not closeable by a bounded observer.
Sources
- section
- The statement in the paper's body that carries the predicate: a proposition, a remark or a section.
- rests on
- The predicates it rests on, by label: realisations, definitions and imports of this ledger, predicates of other ledgers (
32:B3) and of the master (00:D15).