Tags
- I
- Import. A standard result or measured datum used here without reproof.
- R
- Realisation. The row states what a physical term denotes in the substrate: constitutive, not interpretive, defended by the totality of its registered consequences.
- D
- Definition. A naming or set-up move, carrying no empirical content.
- 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.
- E
- Prediction. A falsifiable consequence of the rows above, stated with its falsifier; marked forced, sector, or structural in its row.
- Ω
- $\Omega$-hard. Decided by the totality; not closeable by a bounded observer.
- O
- Open (conjecture). Stated ahead of derivation, carrying its exposure; a derivation re-homes the row.
| # | Description | Status | Source |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p27001 | The finite substrate $\F_\Omega$, cardinality $\Omega\sim10^{122}$ (Planck units) fixed by the de Sitter entropy, with the phase cycle $\F_\Omega^\times=C_{\Omega-1}$ and the quarter-turn $Q_4$ (the residue $\Omega\equiv5\!\pmod{12}$ is derived, C11). | I | |
| A2p27002 | Lattice gauge theory and the Wilson plaquette action; lattice potential theory and the lattice Green’s function. | I | [wilson1974; duffin1953; lawler1991] |
| A3p27003 | The continuum uniqueness of Maxwell, non-abelian Yang–Mills with its asymptotic freedom, and the renormalisation-group running of couplings. | I | [yangmills1954; grosswilczek1973; politzer1973] |
| A4p27004 | The electroweak template ($\SU(2)\times\U(1)$ with the Higgs mechanism) and the $\SO(10)$ grand-unified template the generation falls into. | I | [glashow1961; weinberg1967; georgiglashow1974; fritzschminkowski1975] |
| A5p27005 | The Deligne–Lusztig / Weil character equidistribution invoked for the high-curvature extension of the correspondence. | I | |
| A6p27006 | The measured quantities used for comparison only: the three couplings and $\sin^2\sW=0.231$, the fermion masses and the CKM/PMNS mixings, the proton-decay bounds. | I | measured [superk2017] |
| B. Realisations: mathematics as physics (the constitutive moves) | |||
| B1p27007 | A cell-local change of an internal frame datum is a gauge symmetry; the connection that compensates it is the gauge field. | R | § 3, 9 master: 00:D6 |
| B2p27008 | The multiplicative phase frame gauged is electromagnetism: charge is the winding index, masslessness is drive-invariance. | R | § 5 |
| B3p27009 | The spinorial frame gauged is the weak interaction: chirality is the drive-coherent versus drive-rotated split of the spinor winding. | R | § 6 |
| B4p27010 | The colour frame gauged is the strong interaction: confinement is the saturation that closes the channel at the ladder’s $5\!\to\!1$ return. | R | § 7 |
| B5p27011 | One fermion generation is the spinor $\mathbf{16}$ of the rank-five internal frame $\C^3\oplus\C^2$; mass is winding rate. | R | § 8 master: 00:D2 |
| B6p27012 | The four primitive arithmetic roles are the frame data of the four interactions; the non-generative counting role is the bosonic carrier and the three generative roles the three generations. | R | § 9 |
| C. Derived: theorems and consequences within this paper | |||
| C1p27013 | The finite gauge correspondence: the abelian inclusion $\Z/M\hookrightarrow\U(1)$, the positive cyclotomic action, the exact $q{=}3$ embedding $\SU(2,\F_3)\cong2T$, the low-curvature Yang–Mills kinetic operator, and the uniform window expectation bound $O(H/\Omega)$ (Theorem 4.11, proof at sketch level); the cross-scale running is $\Omega$-hard (Z1). | T | Thm. 4.1; fld.correspondence 1/1 |
| C2p27014 | Electromagnetism: charge as a quantised winding index, the massless photon by drive-invariance, the repulsive sign, the unique Maxwell operator, and $\al_{\mathrm{bare}}=1/4\pi$ as the phase-channel capacity normalisation. | T | Prop. 5.2, Thm. 5.4, Prop. 5.5; fld.em1_prototype, fld.enumerate_maxwell, fld.audit_finitism, fld.o2_numbers, fld.p2 5/5 |
| C3p27015 | The Coulomb law and Maxwell dynamics, as corollaries of the correspondence on the resolvable window. | T R | 2/2 master: 00:E3 |
| C4p27016 | The weak sector: the cell-local $\SU(2)$ Wilson connection, electroweak breaking as drive-torus misalignment with the photon kept massless, and maximal $V\!-\!A$ parity violation with the Frobenius branch decoupling exactly. | T | 4/4 |
| C5p27017 | The propagating neutral spectrum and custodial $\rho=1$: the Hessian of $S_\rho$ at the broken vacuum gives massive $W^\pm,Z$ against a massless photon, $\rho=1$ as $\det\equiv0$, $M_W^2/M_Z^2=\cos^2\sW$, which is $5/8$ at the unification point (Prop. 8.3); the imported electroweak template (A4) evaluated on the realisation B3 (fld.weak_spectrum, fld.ew1). | T R | 2/2 master: 00:H1 |
| C6p27018 | The Weinberg trace formula $\sin^2\sW=\Tr(T_3^2)/\Tr(Q^2)=3/8$ for a complete generation, the unification-point value, and the four-fermion amplitude with $G_F=1/(\sqrt2\,v^2)$; the imported template (A4) evaluated on the realisations B3, B5 (fld.ew1, fld.weak_current). | T R | 4/4 |
| C7p27019 | The strong sector: the colour $\SU(3,\Omega)$ frame forced as the minimal triality-carrying rank, the exact $\Z_3$ centre, the gluon connection, and the confinement area law $\sigma>0$ from positivity, with the string tension $\sigma(\beta)=-\ln c_1(\beta)$ computed in finite units (closed form on $2T$; $c_1^{\SU(3)}=\tfrac\beta{18}+\tfrac{\beta^2}{216}+\cdots$). | T | Prop. 7.1, Prop. 7.2, Prop. 4.12; fld.correspondence, fld.qcd, fld.p5, fld.string_tension, fld.missing_rank 5/5 |
| C8p27020 | One complete Standard-Model generation as the spinor $\mathbf{16}$ of the rank-five frame: field content, hypercharges, charges, and full anomaly freedom; the imported $\SO(10)$ template (A4) evaluated on the realisation B5 (fld.generation). | T R | 2/2 master: 00:J1 |
| C9p27021 | Simple $\SO(10)$ unification forced: the colour–isospin off-block reframings are gauge automorphisms, the $3+2$ torsion sitting at carrier scale above the coherence horizon $\sqrt\Omega$. | T R | 2/2 master: 00:Z5 |
| C10p27022 | The closure of the four primitive roles ($5\!\to\!1$): four interactions and no fifth, three generations and no fourth, the boson/fermion split, and the generation ordering, given the role-to-matter realisation (B6). | T R | 3/3 master: 00:J4 |
| C11p27023 | The admissible substrate residue $\Omega\equiv5\!\pmod{12}$ forced by self-consistency. | T | Prop. 9.1; fld.p11 1/1 master: 00:B5 |
| X. What the construction explains: the surplus over the standard postulates | |||
| X1p27024 | The gauge group $\U(1)\times\SU(2)\times\SU(3)$ is the gauging of the substrate’s three frame data, the phase, spinor, and colour frames read by the one recipe, not a product of groups chosen to fit. | T | § 5, 6, 7 |
| X2p27025 | Charge quantisation is automatic: electric charge is an integer winding index, so the electron and proton charges match exactly and no fractional residue survives, with no quantisation condition imposed. | T | Prop. 5.2; fld.em1_prototype 1/1 |
| X3p27026 | The photon and its dynamics are forced: masslessness is drive-invariance and the exact rank-one kernel of the broken vacuum, and Maxwell is the unique adjacency-local gauge-invariant quadratic operator, neither put in by hand. | T | 2/2 |
| X4p27027 | The Weinberg angle is the exact rational $\sin^2\sW=3/8$ at the unification point, fixed by the charge spectrum of the realised generation (B5) with the group-theoretic step imported (A4), not a free parameter measured into the theory. | T | 1/1 |
| X5p27028 | The custodial relation $\rho=1$ is the rank-one identity $\det\equiv0$ forced by the spinor-doublet Higgs, not a tree-level accident. | T | 2/2 |
| X6p27029 | Maximal parity violation is the exact decoupling of the Frobenius branch from the drive, so the $V\!-\!A$ current is derived rather than the left-handed doublets assigned by hand. | T | 2/2 |
| X7p27030 | The colour sector is fixed: three colours is the minimal triality-carrying frame, confinement is the compact-group area law from positivity, and asymptotic freedom is the gluon self-coupling, not $N_c=3$ supplemented by an assumed confining phase. | T | 3/3 |
| X8p27031 | One complete generation is the spinor $\mathbf{16}$ of the rank-five internal frame, so its hypercharges, charges, and anomaly freedom follow and a right-handed neutrino is present, with $\SO(10)$ unification forced, rather than a hypercharge assignment whose anomaly cancellation is a coincidence; given B5, the anomaly cancellation is the imported $\SO(10)$ fact (A4) evaluated on the realised content. | T | 3/3 |
| X9p27032 | The interaction and generation counts are the $1+3$ split of the closed primitive-role ladder, so four forces and three families are fixed and a fifth force and a fourth generation are forbidden, not left unexplained. | T | 1/1 |
| X10p27033 | The Koide ratio is exactly $2/3$, the cube-root generation orbit at the self-dual quarter-turn $\rho^2=2^{-1}$, an exact value rather than an unexplained numerical coincidence. | T | Prop. 8.8; fld.koide 1/1 |
| X11p27034 | The strong-CP angle vanishes with no axion: no substrate-native topological term, and Hermitian quark masses with real determinant, give $\bar\theta=0$, so the problem does not arise rather than being tuned away. | T | Prop. 8.9; fld.strongcp 1/1 |
| P. Falsifiable predictions: the exposures | |||
| P1p27035 | No fourth generation and no fifth interaction [forced]. The role ladder closes, the fifth step returns to counting, and the Carrier is the entire substrate, so a fourth chiral family and a fifth fundamental force are forbidden, not merely absent (Tier 4). Falsifier: any fourth-generation fermion, or any new fundamental interaction. | E | Prop. 8.7; fld.p9b 1/1 master: 00:N1 |
| P2p27036 | A desert to the substrate scale [forced]. The gauge group is exactly $\SU(3)\times\SU(2)\times\U(1)$ until the substrate resolves the colour–isospin split; no new gauge boson, vector-like fermion, or extra scalar lies between the electroweak and substrate ($\sim10^{16}$–$10^{19}\,$GeV) scales. Falsifier: a $Z'$, $W'$, leptoquark, second Higgs doublet, or compositeness signature at the LHC or a future collider. This is the sharpest divergence from supersymmetry, extra dimensions, and composite-Higgs scenarios. | E | Prop. 8.2 master: 00:N1 |
| P3p27037 | No TeV supersymmetry [forced]. The ${\sim}13\%$ three-coupling near-miss is closed by the $X,Y$ completion at the substrate scale, with the running carried by purely Standard-Model content; there are no superpartners enforcing exact low-scale unification. Falsifier: discovery of superpartners, or evidence that unification demands TeV-scale thresholds. | E | Prop. 8.3; fld.p1 1/1 master: 00:N1 |
| P4p27038 | The proton is effectively stable, $\tau_p\gtrsim10^{45}\,$yr [forced]. The $X,Y$ leptoquarks acquire mass at the substrate scale, not at $\sim10^{16}\,$GeV, so $\tau_p\sim M_X^4/(\al_{\mathrm{GUT}}^2 m_p^5)$ reaches $10^{45}$–$10^{47}\,$yr for $M_X$ near the Planck scale, orders beyond the $\sim10^{36}\,$yr of supersymmetric $\SO(10)$ and the $\sim10^{35}\,$yr reach of Hyper-Kamiokande. Falsifier: observation of $p\to e^+\pi^0$ near $10^{34}$–$10^{35}\,$yr in Hyper-Kamiokande, DUNE, or JUNO, against the current bound [superk2017]. The programme predicts these searches find nothing. | E | Prop. 8.2; fld.p8 1/1 master: 00:N1 |
| P5p27039 | Majorana neutrinos, heavy singlets, no light sterile neutrino [sector]. The $\nu^c$ is the gauge-singlet $\Lam^0$ of the generation, so it takes a Majorana mass at unification and drives the seesaw, the three right-handed states sitting at $\sim10^{14}$–$10^{15}\,$GeV. Falsifier: a confirmed eV-scale sterile neutrino, or evidence that the light neutrinos are Dirac. This diverges from Dirac-neutrino and sterile-neutrino interpretations and is consistent with $N_{\mathrm{eff}}=3$. | E | 1/1 |
| P6p27040 | No new particle of dark matter [sector]. The matter content is exactly one Standard-Model generation, triplicated, with heavy $\nu^c$; the construction contains no weakly-interacting massive particle and no axion. Falsifier: a positive signal in a direct-detection (LZ, XENONnT) or axion (ADMX) experiment. Dark matter, if not a Standard-Model state, must then reside in the gravitational/substrate sector, a residual burden of the programme. | E | Thm. 8.1 master: 00:N1 |
| P7p27041 | No new-physics contribution to the muon anomaly [sector]. With no states beyond the Standard Model, $(g-2)_\mu$ must be accounted for by Standard-Model (notably hadronic) contributions. Falsifier: a robustly established beyond-Standard-Model excess; the current lattice and experimental trend toward the Standard-Model value favours this prediction. | E | Thm. 8.1 master: 00:N1 |
| P8p27042 | The charged-lepton Koide relation is exactly $2/3$ [structural, $T\mid\Omega$]. The empirical identity $(m_e+m_\mu+m_\tau)/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2=2/3$, which the Standard Model does not explain and which holds to one part in $10^{5}$, is the cube-root generation orbit at the self-dual value $\rho^2=2^{-1}$: $Q=\tfrac13+\tfrac23\rho^2$ is exact and $\rho^2=2^{-1}\in\F_\Omega$ is the quarter-turn ($4\mid\Omega-1$) self-dual amplitude on the cubic ($3\mid\Omega+1$) plane. The functional form and the self-dual value are derived; the selection of the charged-lepton mass vector onto that orientation is the $\Omega$-hard flavour residual (the carrier-scale winding phase, Z1), not derived here. Falsifier: a charged-lepton mass measurement displacing $Q$ from $2/3$ beyond the per-mille winding-phase corrections. | E | Prop. 8.8; fld.koide 1/1 master: 00:K2 |
| P9p27043 | Vanishing strong-CP angle without an axion [structural, $T\mid R$]. The QCD vacuum angle vanishes on the substrate, $\bar\theta=0$ with no axion: the colour action carries no topological term and colour is drive-invariant, so the bare $\theta=0$, and the Hermitian quark mass matrices have real determinant, so $\arg\det M_q=0$, while the Cabibbo–Kobayashi–Maskawa phase sits in the up–down misalignment. Falsifier: a neutron electric dipole moment above the QCD floor, or a discovered axion. | E | Prop. 8.9; fld.strongcp 1/1 master: 00:I4 |
| V. Machine verification | |||
| V1p27044 | The validation package finite-ring-space/src/27-fields: the twenty-eight scripts as written — fld.em1_prototype, fld.enumerate_maxwell, fld.audit_finitism, fld.o2_numbers, fld.p2, fld.correspondence, fld.p3, fld.ew1, fld.weak_spectrum, fld.weak_current, fld.p4, fld.p10, fld.v_scale, fld.qcd, fld.p5, fld.string_tension, fld.p6, fld.missing_rank, fld.generation, fld.p8, fld.p8b, fld.p9, fld.p9b, fld.p11, fld.p1, fld.p7, fld.koide, fld.strongcp — 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 (most scripts print their booleans and numbers without asserting them; the decision is made in fldcommon.py, one labelled predicate per stated value), each family naming the rows it witnesses (Appendix G the per-script map); the notebook 27-fields-main.ipynb executes them in the browser and the driver run_all.py writes results.json. | T | Sec. 12; run_all |
| V2p27045 | The discipline, recorded per family as its class (Appendix G): EXACT — integer, $\F_p$, $\F_{p^2}$ or cyclotomic arithmetic throughout, a pass a proof on the tested instances; MIXED — an exact core with a labelled continuum-comparison layer, the exact claim never resting on the float part; APPROX — a continuum-comparison or dimensional-transmutation reading by construction. Fifteen families EXACT, eleven MIXED, two APPROX; audit_finitism.py cross-checks that the load-bearing EM/O1 claims rest on integer and cyclotomic arithmetic alone. | T | Sec. 12, Sec. 10 |
| V3p27046 | What the run does not decide: the $\Omega$-hard scales (Z1) enter only as labelled readings ($\Lambda_{\mathrm{QCD}}$ by one-loop transmutation, the electroweak scale against the transmutation forms, the proton lifetime’s $M_X^4$ scaling); the correspondence’s high-curvature extension and the cross-scale running (Remark 4.2) are not computed; the narrative scripts (fld.p7, fld.p8, fld.p9b, fld.p10, fld.o2_numbers) are decided on their arithmetic identities and stated bands only. | T | Sec. 12 |
| Z. Horizon: the residue beyond the bounded observer | |||
| Z1p27047 | The cross-scale beta-function running and the absolute scales — $\al(1/137)$, $\Lambda_{\mathrm{QCD}}$, the electroweak scale $v$ — are decided by the totality but lie beyond the coherence horizon; the same residue as the number-theoretic horizon clause. | Ω | 3/3 |
| O. Open: the conjecture stated ahead of derivation | |||
| O1p27048 | The electroweak scale as the minimal winding weight of the gauged spinor frame at channel unity: $\langle\phi^0\rangle=v/\sqrt2=m_P\big(e^{-S_{+1/2}}+e^{-S_{-1/2}}\big)$ with $S_{\pm1/2}=4\pi^2/g^2$ the action of the 't Hooft half-charge sector of the frame group $\SU(2)/\Z_2$ at $g=1$ (the frame closes in one circuit, the spinor in two), i.e. $v=2^{3/2}m_P\,e^{-(2\pi)^2}=247.2$ GeV against $246.2$ ($0.38\,\%$ high, a real residual at the precision of $m_P$). The exponent is the sector’s action, verified in the Wilson normalisation of Section 4 (fld.v_scale); the amplitude identification, the two-sector prefactor and the residual are underived. Exposure: the prefactor lands within $0.38\,\%$ with odds about $1$ in $45$; the strong channel obeys no such rule (its $\Z_3$ sector gives $5\times10^{7}$ GeV); the Standard-Model coupling at $m_P$ ($g_2^2=0.25$) is not the channel-unity coupling, so the running cannot test it. Falsifier: a derivation of the amplitude identification producing a residual other than $0.38\,\%$, or a framework closed form for the residual that fails. | O | Rem. 8.5; fld.v_scale 1/1 master: 00:H3 |
Nothing matches that filter.
Ledger history
Ledger history. 2026-06-25: the status section with three tables (what the construction explains X1–X11, the falsifiable predictions P1–P9, the predicate ledger A–C, Z, Y); the review rounds of July–August 2026 added the weak-spectrum, weak-current, string-tension, missing-rank, koide, strongcp and audit scripts and rows C5–C7, C11, X10–X11, P8–P9, Y1. 2026-09-14 (T26): the three tables merged into one ledger in the corpus format — X and P are blocks of it, V added, the conjecture row Y1 relabelled O1 (block O holds the open rows, as in the corpus), every machine-verified row citing the family identifiers of the validation package (finite-ring-space/src/27-fields), accession keys assigned. No row has been retired.