Tags
- I
- Import. A standard result, a corpus result, or a convention used here without reproof.
- D
- Definition. A naming, a declared dictionary, or a set-up move, carrying no empirical content.
- T
- Theorem. Derived within this paper from the rows above it, machine-verified in exact arithmetic by the named witness where a witness is named.
- R
- Realisation. States what a physical term denotes on the substrate: constitutive, not interpretive, and carrying its falsifier.
- O
- Open. Genuinely unresolved.
| # | Description | Status | Source |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p08001 | The framed substrate: the symmetry-complete shell $\p=4\kap+1$ with frame $\Fp(t;0,1,g)$, the derived quarter-turn $\It=g^{-\kap}$, $\pi=2\kap$, $e=g^{\It}$, the residue web with $c^{2}=2^{-1}$; the orbital shell with its meridians $M_m$ and latitude circles $L_a$; the relational stance under which the continuum is a projection. | I | |
| A2p08002 | Finite-field algebra: the quadratic extension $K=\Fp[\w]/(\w^{2}-\nu)$ with Frobenius conjugation and norm, Hilbert’s Theorem 90, the square classes by Euler’s criterion, the unitary group of a nondegenerate Hermitian form over a finite field. | I | |
| A3p08003 | The continuum antecedents: the Cayley transform and the Crank–Nicolson step, Clifford algebras and the Dirac matrices, the $1{+}1$ Lorentz boost and its spin lift, the Schwinger unitary basis and the finite-field quantum mechanics it is compared with. | I | [Crank & Nicolson 1947; Chevalley 1996; Porteous 1995; Schwinger 1960; Chang et al. 2013; Chang et al. 2013; Lev 2010] |
| A4p08004 | The Hartley (max-)entropy reading of image counts, and reversibility as zero information loss. | I | |
| A5p08005 | The programme’s admissibility congruences ($\kap$ even, $\kap\equiv1\pmod3$, $\p\equiv5\pmod{12}$), the laboratory Carrier $\Omega=2\,408\,561$ with $S=602\,140$, and the constants web ($c^{2}=2^{-1}$, $\hbar=\It$). | I | |
| A6p08006 | The programme position: the primary object a finite symmetry space of algebraic possibilities, every description an incomplete projection; the theorems of the paper stand independently of it. | I | [Akhtman 2025], § 7 |
| B. The shell, the coefficient field and the signature | |||
| B1p08007 | The shells in play: the worked Subject shells $\F_{13}$, $\F_{17}$, the coordinate shell $Y=\Fp^{4}$, the coefficient field $K$; the two conjugation registers — $\It\in\Fp$ Frobenius-fixed, the internal Euclidean phase, and $\w$ the conjugation of the extension; Carrier-scale statements are transfer targets, never sites of computation. | D | Rems. 2.2, 2.3 |
| B2p08008 | The class is the datum: $\Fpx/(\Fpx)^{2}$ has order two, any two nonsquares differ by a square factor, and the Lorentzian form depends on $\nu$ only through its class. | T | Lem. 2.4; o2.C2 1/1 |
| B3p08009 | The drive is canonical: $g$ is a nonsquare on every shell and the only named residue whose class is nonsquare on every symmetry-complete shell; $\It$, $2$, $2^{-1}$, $-2$ are squares exactly when $\kap$ is even; $e=g^{\It}$ has no stable class; $[g^{-1}]=[g]$. Exhaustive on $\p<2000$. | T | 3/3 |
| B4p08010 | The two speed-of-light seats: $\nu=g=c^{2}\cdot(2g)$ exactly on every shell; $[c^{2}]$ is even iff $\kap$ is even, $[2g]$ odd iff $\kap$ is even, the product odd always; the temporal coefficient is not $c^{2}$; anchors $\F_{13}$ ($2^{-1}=7$ nonsquare), $\F_{17}$ ($2,\It,e$ squares), and the laboratory Carrier ($c=171\,106$ base-rational, $1\,204\,281\cdot12=6=g$). | T | 5/5 master: 00:C3 |
| B5p08011 | The square class is chronon parity: the squares are $\langle g^{2}\rangle$, a residue’s class is the parity of its drive-step count; on $\kap$-even shells every element of $Q_4=\{1,\It,-1,-\It\}$ is a square, so the order-four chart grading carries no signature. Exhaustive over every primitive root of $\F_{13}$, $\F_{17}$; Euler form on $\p<2000$. | T | 2/2 |
| B6p08012 | Norm growth and parity: $N(gz)=g^{2}N(z)$ with $g^{2}$ a square, the single-chronon multiplier $g$ odd, $x^{2}=\nu$ unsolvable in $\Fp$; the flip $g\mapsto g^{-1}$ preserves the parity class. | T | 2/2 master: 00:C8 |
| B7p08013 | Signature lives on $\Fp$: the Euclidean form has $\p^{3}+\p^{2}-\p$ zeros (hyperbolic, Witt index two), $Q_\nu$ has $\p^{3}-\p^{2}+\p$ (elliptic, Witt index one) on every symmetry-complete $\p<60$; $2353$ and $2041$ at $\p=13$; over $K$ every element of $\Fp$ is a square and the dichotomy collapses. | T | 4/4 |
| B8p08014 | One-way and two-way transport: comparison (two-way) readings are even-parity, the one-way single-chronon multiplier odd, and the null-cone slope $\sqrt\nu=g^{1/2}$ is no base-field ratio — light-like transport carries no registered velocity, only its square class. Falsifier: a registered one-way velocity, or a two-way constant that changes under the synchronisation gauge $g\mapsto g^{-1}$ (B6). | R | Rem. 2.10; Thm. 2.9 |
| C. Hermitian spaces and Cayley dynamics | |||
| C1p08015 | The Hermitian form $\langle\psi,\phi\rangle_X=\sum\overline{\psi(x)}\phi(x)$ on $\calH(X)$, the adjoint, the finite unitary group; the Euclidean configuration space $X_d=\Fp^{d}$ with translations $T_u$, forward and backward differences and the Laplacian $\Delta_E$; the Hamiltonian $H_{E,V,\vkap}=-\vkap\Delta_E+M_V$ with the stiffness $\vkap$ distinct from the capacity $\kap$. | D | Defs. 3.1, 3.4, 3.7, 3.10 |
| C2p08016 | The form is Hermitian and nondegenerate; $T_u$ is unitary with $T_u^{\dagger}=T_{-u}$; $\Delta_E$ and $H_{E,V,\vkap}$ are self-adjoint; under $A\in\mathrm{GL}_d(\Fp)$ the pullback $A_*$ intertwines translations, differences and Laplacians of the two frames. | T | Lem. 3.2, Props. 3.5, 3.6, Cor. 3.8, Lem. 3.11 |
| C3p08017 | The Cayley step $U_{\alpha,H}=(I-\alpha H)^{-1}(I+\alpha H)$ for $\alpha\in K^{-}$ (the recursion $(I-\alpha H)\psi_{n+1}=(I+\alpha H)\psi_n$); the admissible set $\calA(H)=\{\alpha\in K^{-}:\det(I-\alpha H)\ne0\}$. | D | Defs. 4.1, 4.2, Rem. 4.3 |
| C4p08018 | Exact Hermitian preservation: for self-adjoint $H$ and admissible $\alpha$ the Cayley step is unitary, and every orbit is periodic. | T | 1/1 |
| C5p08019 | The finite Cayley transform: $\varphi(\lambda)=(1+\alpha\lambda)/(1-\alpha\lambda)$ is defined on all of $\Fp$, takes values in the norm-one torus $N^{1}$, and with $\varphi(\infty)=-1$ bijects $\mathbb P^{1}(\Fp)$ onto $N^{1}$, $|N^{1}|=\p+1$; the unitary group of one channel is $N^{1}$; the orientation gauge $\alpha\mapsto-\alpha$ inverts every $\varphi(\lambda)$ and imports the derived orientation of the shell. | T | 2/2 |
| C6p08020 | The $\F_{13}$ Schrödinger example: $H=-\Delta_E$ on $\F_{13}$ is nilpotent, so $\calA(H)=K^{-}$; at $\alpha=c$, $U^{13}=I$; every $\alpha=kc$, $k\ne0$, gives order $13$, and $U$ is unitary for the form. | T | 2/2 |
| D. The Dirac operator and the boost transport | |||
| D1p08021 | Spinor fields $\calS(Y)=K^{4}$-valued on the coordinate shell, the forward differences $\nabla_\mu$, the gamma matrices (21) with $\eta=\diag(-\nu,1,1,1)$, the standard finite Dirac operator $\mathcal D_{E_0}=\sum_\mu\gamma^{\mu}\nabla_\mu$ and the finite Dirac equation $(\mathcal D_{E_0}-m)\psi=0$. | D | (21), Def. 5.2 |
| D2p08022 | The Clifford relations $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta_{\mu\nu}I_4$. | T | Prop. 5.1; fin.S3 1/1 |
| D3p08023 | Dirac-to-Klein–Gordon factorization: $(\mathcal D_{E_0}-m)(\mathcal D_{E_0}+m)=\operatorname{KG}^{\mathrm{spin}}_{E_0}-m^{2}$. | T | Thm. 5.3 |
| D4p08024 | The $1{+}1$ boost $\Lambda(x,y)$ with entries $A,B$ from $x^{2}-\nu y^{2}=\delta\ne0$; the spin lift $S(x,y)=xI_4+y\gamma^{0}\gamma^{1}$; the spinor transport $\mathcal T_{\Lambda,S}=S\Lambda_*$ and the transported operator $\mathcal D_{\Lambda,S}$. | D | Def. 5.5, (38) |
| D5p08025 | The boost group is the non-split torus: $\Lambda(x,y)\in O(Q_\nu,\Fp)$, $G_\nu$ a subgroup, $z\mapsto\Lambda$ a surjection $K^{\times}\to G_\nu$ with kernel $\Fpx$ and $A-B\w=z/\bar z$; $G_\nu\simeq N^{1}\simeq C_{\p+1}$, cyclic; an order-three boost exists iff $3\mid\p+1$ ($|G_3|=18$ at $17$, $|G_2|=14$ at $13$). | T | 6/6 |
| D6p08026 | Spin conjugation: $S^{-1}\gamma^{0}S=A\gamma^{0}+\nu B\gamma^{1}$, $S^{-1}\gamma^{1}S=B\gamma^{0}+A\gamma^{1}$, $\gamma^{2},\gamma^{3}$ fixed. | T | Prop. 5.4; fin.S3 1/1 |
| D7p08027 | The transported family: $\nabla^{E_\Lambda}_\mu\Lambda_*=\Lambda_*\nabla^{E_0}_\mu$; $\mathcal D_{\Lambda,S}=\sum\widehat\gamma^{\mu}\nabla^{E_\Lambda}_\mu$ with $\widehat\gamma^{0}=A\gamma^{0}-\nu B\gamma^{1}$, $\widehat\gamma^{1}=-B\gamma^{0}+A\gamma^{1}$; the transported Clifford relations and factorization. | T | 1/1 |
| D8p08028 | Exact covariance: $\mathcal D_{\Lambda,S}\mathcal T_{\Lambda,S}\psi=\mathcal T_{\Lambda,S}\mathcal D_{E_0}\psi$, solutions transported to solutions; verified on a sample field at $\p=13$. | T | Cor. 5.13; fin.S3 1/1 |
| D9p08029 | The spinor form: under the plain conjugate transpose the gammas carry mixed signs ($\gamma^{2}$ the obstruction) and the $\gamma^{0}$-twist fails; $X=\gamma^{0}\gamma^{1}\gamma^{3}$ is Hermitian with $X^{2}=\nu$, commutes with $\gamma^{0},\gamma^{1},\gamma^{3}$, anticommutes with $\gamma^{2}$, and $X^{-1}(\gamma^{\mu})^{\dagger}X=-\gamma^{\mu}$ for all $\mu$; $\langle\cdot,\cdot\rangle_X$ is nondegenerate Hermitian, $T-T^{-1}$ nilpotent and anti-self-adjoint, $\mathcal D^{s}$ $X$-self-adjoint and nilpotent ($\p=5,13,17$). | T | 4/4 |
| D10p08030 | Exact Dirac evolution: the Cayley step of $H=\mathcal D^{s}-mI$ preserves $\langle\cdot,\cdot\rangle_X$; for $m=0$ admissibility is automatic, $U$ unipotent with order a power of $\p$ ($5$ in the $1{+}1$ computation at $\p=5$); for $m\ne0$, $U=\varphi(-m)V$ and $\operatorname{ord}(U)=\operatorname{lcm}(\p^{a},\operatorname{ord}_{N^{1}}\varphi(-m))$ ($75$, $150$ at $m=1,2$). | T | 2/2 |
| D11p08031 | The two Dirac operators are one covariant family: the pullback identity holds for the symmetric differences, $\mathcal T_{\Lambda,S}$ carries $\mathcal D^{s}$ to its transported form, and the spinor form transports by the norm, $S(x,y)^{\#}=S(x,-y)=\delta S^{-1}$: norm-one lifts preserve $\langle\cdot,\cdot\rangle_X$ exactly, the odd coset scales it by the nonsquare class. | T | Prop. 5.16; o8.X8 1/1 |
| D12p08032 | The worked shells: $\F_{13}$ with $\nu=g=2$, $x=y=1$: $A=10$, $B=2$, $S=I+\gamma^{0}\gamma^{1}$, $\widehat\gamma^{0}=10\gamma^{0}-4\gamma^{1}$, $\widehat\gamma^{1}=-2\gamma^{0}+10\gamma^{1}$; $\F_{17}$ the minimal admissible shell ($17\equiv5\bmod12$, no smaller symmetry-complete prime), $\It=4$, $2,\It,e$ squares, $\nu=3$, $A=15$, $B=1$, $|G_3|=18$, triality present. | T | 4/4 |
| D13p08033 | Mass is winding rate: the finite Dirac equation equates the meridional development rate of the spinor field to $m$ times the field, the finite seat of the energy–momentum–mass relation, $E=hf$ an identity. Falsifier: that of the corpus row it instantiates — a rest mass with no cycle winding, or a stationary state whose registered frequency is not its winding index. | R | Rem. 5.17 |
| E. Reversibility and the information counts | |||
| E1p08034 | The image count $I_X(f)=|f(X)|$ and the loss factor $L_X(f)=|X|/|f(X)|$; the Hartley reading, a map losing exactly $\log L_X(f)$ of distinguishability. | D | Def. 6.1, Rem. 6.6 |
| E2p08035 | A self-map is bijective iff $L_X(f)=1$; the power map $x\mapsto x^{\varepsilon}$ on $\Fpx$ has image $(\p-1)/d$ and loss factor $d=\gcd(\varepsilon,\p-1)$, every nonempty fibre of size $d$ ($\varepsilon=1,2,3,4,6,12$ on $\F_{13}$). | T | 1/1 |
| E3p08036 | Cayley propagators and boost transports have loss factor one: the wave dynamics is isentropic, compression confined to the power-map layer. | T | 1/1 |
| E4p08037 | Period dichotomy: kinetic Hamiltonians are nilpotent, fully admissible, with unipotent propagators of order $\p$; potential Hamiltonians give diagonal propagators with phases in $N^{1}$ and order dividing $\p+1$ ($14$ at $13$, $18$ at $17$); the mixed datum $\operatorname{ord}(U)=1563$ at $\p=5$; composites take the lcm ($13,14\mapsto182$); the three tori $C_{\p-1}$, $C_\p$, $C_{\p+1}$ each govern one dynamical role. | T | 4/4 |
| F. The shell reading | |||
| F1p08038 | The four domains of one framed shell: the additive Fourier pair on the meridians $M_0$ (space) and $M_\kap$ (momentum), the multiplicative pair on the latitudes $L_1$ (time) and $L_{\kap+1}$ (energy); zonal and meridional motion; both dualities the index advance by the capacity. | D | § 7, Fig. 2 |
| F2p08039 | The latitude identities: $\kap+1$ is the first rung past the midpoint of the ladder $1,\dots,2\kap$; $m\mapsto m+\kap$ is multiplication by $g^{\kap}=\pm\It$ and $a\mapsto a+\kap$ sends $1$ to $\kap+1$; $\kap=-4^{-1}$ so the energy radius is $3\cdot4^{-1}$; $\pi=2\kap=-2^{-1}=-c^{2}$, on the Carrier $\pi_\Omega=2S$; the unit-norm circle has $\p-1$ points. On $\F_{13}$, $\F_{17}$ and the laboratory Carrier. | T | 5/5 |
| F3p08040 | The free evolution is the drive: the pullback $D\psi(x)=\psi(g^{-1}x)$ is unitary, the characters $\chi_k$ are its eigenvectors with $D\chi_k=g^{-k}\chi_k$, and every character line is isotropic except $k=0,(\p-1)/2$ with eigenvalues $\pm1=N^{1}\cap\mu_{\p-1}$. Exhaustive on $\F_{13}$, $\F_{17}$. | T | 2/2 |
| F4p08041 | Two evolution sectors, one space: the split-phase sector of the drive and the norm-one sector of the Cayley propagators meet in $\{\pm1\}$ alone ($\operatorname{ord}(g^{-1})=\p-1$); the cycle Laplacian $\Delta_\Phi=D+D^{-1}-2I$ is self-adjoint, its Cayley steps commute with $D$ and the composite is unitary ($\p=13$); the dispersion relation $-\Delta_\Phi\chi_k=(2-g^{k}-g^{-k})\chi_k$; no meridional plane waves ($\gcd(\p,|K^{\times}|)=1$); $\mathcal D^{s}-m$ invertible for $m\ne0$, so mass is an evolution phase, not a kernel condition. | T | 2/2 |
| F5p08042 | The shell reading: Schrödinger dynamics is the exact zonal evolution (the drive on $L_1$, its spectrum the winding ladder at $L_{\kap+1}$), Dirac dynamics the exact meridional development, the three tori the three dynamical seats ($C_{\p-1}$ free evolution, $C_\p$ translations and kinetic periods, $C_{\p+1}$ interactions and boosts), the light cone a square-class boundary. Falsifier: a registered dynamics of the shell requiring a phase outside the two tori, or a mass without winding (D13). | R | Rem. 7.3 |
V. Machine verification (the validation package finite-ring-space/src/8-dirac) | |||
| V1p08043 | The worked examples and the shell scans: worked_checks (the $\F_{13}$ examples, $3$ checks over finite_checks), shell_checks ($4$ families, $68$ micro-checks), o2_checks ($9$ families, $1048$), o7_checks ($6$ families, $1735$). | T | |
| V2p08044 | The tori, the spinor form and the signature: o134_checks ($11$ families, $907$ micro-checks), o8_checks ($8$ families, $55$), latitude_checks ($5$ families, $35$), o9_signature_counts ($4$ checks). | T | |
| V3p08045 | The driver: eight block scripts, one notebook run in the browser, $52$ family checks over $3848$ exact micro-checks, a record per check written to results.json keyed to these rows. | T | run_all; the notebook |
| O. The open front | |||
| O1p08046 | The classification of the admissible set $\calA(H)$ for a general self-adjoint Hamiltonian — the roots of $\det(I-\alpha H)$ in $K^{-}$ — beyond the nilpotent case ($\calA=K^{-}$, C6, E4) and the diagonal case (the Cayley transform of the spectrum, C5). | O | Rem. 4.3 |
Nothing matches that filter.
Ledger history
Ledger history. 2026-09-13: ledger added (the paper predates the corpus convention), formulated from the labelled statements of Sections 2–7 and the validation package of the same date. One open row, O1, the classification of the admissible set for a general self-adjoint Hamiltonian; when it closes its content is re-homed in block C and the row leaves. No row has been retired.