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 and machine-verified in exact arithmetic by the named witness, or observed numerically and reproduced by it to the stated tolerance.
- O
- Open. Genuinely unresolved.
| # | Description | Status | Source |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p06001 | The framed substrate: the Subject $\Fp(\chron;0,1,\gen)$ with $\p=4\kap+1$, its register $\pi=2\kap$, $\im=-\gen^{\kap}=\gen^{-\kap}$, $e=\gen^{\im}$; the frame-relative labels and the atlas of presentations $\gen\mapsto\gen^{u}$; time as scale-dilation, the generator as the observer’s own advance. | I | |
| A2p06002 | The Weil representation $\omega_\p$ of $\SL(2,\Fp)$ and its Schrödinger model, the finite Fourier transform representing the rotation $w$, the finite harmonic oscillator; the scalar convention $\omega_\p(w)^{2}=\omega_\p(-I)$, $\omega_\p(w)^{4}=I$. | I | |
| A3p06003 | The classical evaluation of the twisted quadratic Gauss sum for $4\mid n$, $\sum_k\zeta_n^{uk^{2}}=\bigl(\tfrac{\kap}{u}\bigr)(1\pm i)\sqrt n$; the eigenvalue quartet of prime-modulus number-theoretic transforms; the multiplicity counts of the discrete Fourier eigenvalues. | I | |
| A4p06004 | The entropic uncertainty relation for a pair of bases, $H_{B}+H_{B'}\ge-2\log c$, $c$ the maximal overlap, $c=n^{-1/2}$ for mutually unbiased bases, saturated by basis-localized states. | I | |
| A5p06005 | The framed-rational zoom: the grids $G_n$ and their $(\p-1)$-periodicity $G_{n+(\p-1)}=G_n$, the meridian shift $m\mapsto m-1$ as the refinement step. | I | [Akhtman 2025, §4.3, Lemma 2] |
| A6p06006 | The measurement layer: probabilities are structural weights of the cyclotomic stratum, the finite-field module carries no Born magnitude, and the readout is an observer-side declaration; the shell dynamics a bijection, entropy attaching to the readout. | I | [Akhtman & Voether 2026; Akhtman 2025] |
| A7p06007 | The continuum fractional Fourier transform and the finite constructions the family is separated from: the eigenbasis and matrix-function transforms, the harmonic-oscillator propagator reading. | I | [Namias 1980; Ozaktas et al. 2001; Lima & Campello de Souza 2012; Pei et al. 2011; Lima et al. 2017] |
| B. The shell: the frame datum and the Fourier operator | |||
| B1p06008 | Shell data, capacity-first: $\kap$ primary, $\p=4\kap+1$, the meridian cycle $\Phit=\Z_{4\kap}$, the oriented quarter-turn $\im=-\gen^{\kap}$ (clockwise phase convention), $\pi=2\kap$, $e=\gen^{\im}$; the meridians $M_s=\{a\gen^{s}:a\in I_\p\}$; the six shells of Table 1 with their generators and quarter-turns. | D | 1/1 |
| B2p06009 | The Euler identity is a property of the chart class: $e^{\im\pi}=\gen^{2\kap(\im\bmod2)}\equiv-1$ exactly when the quarter-turn residue is odd; the conjugate reframing $(\gen,\im)\mapsto(\gen^{-1},-\im)$ toggles the parity, so one member of each conjugate pair carries it; on $\F_{13}(\chron;0,1,2)$, $\im=5$ and $6^{6}=-1$. Every primitive frame of the six shells. | T | § 3; A2 1/1 master: 00:C14 |
| B3p06010 | Chart covariance: under $\gen'=\gen^{u}$, $u\in\Zp^{\times}$, the quarter-turn name flips exactly on $u\equiv3\pmod4$ and the exponential-unit name moves already at $u\equiv1\pmod4$ whenever $\im(u-1)\not\equiv0\pmod{4\kap}$ ($\p=13$, $u=5$: $e'=2\neq6$); the registered objects (C3, C4, C9, F3) are invariant. | T | Rem. 3.2; A3 1/1 |
| B4p06011 | The shell Fourier matrix $\Wt_{kj}=\gen^{jk}$ on $\Vt=\Fp^{\Phit}$, the reversal $J$, the normalized shell Fourier operator $\Ft=\im\Wt$. | D | (4.2), (4.1), (4.3) |
| B5p06012 | $\Wt^{2}=-J$, $\Ft^{2}=J$, $\Ft^{4}=I$: the normalized operator generates the four-cycle, on the six shells of Table 1. | T | 1/1 master: 00:C2 |
| B6p06013 | The normalization is the unitary constant read in the field: $1/n\equiv-1$, its square roots in $\Fp$ are exactly $\pm\im$, and $(c\Wt)^{2}=J$ if and only if $c=\pm\im$. | T | Rem. 4.4; A5 1/1 |
| B7p06014 | $\Wt J=J\Wt$, hence $\Ft J=J\Ft$; $\Vt=V^{+}\oplus V^{-}$ with $\dim V^{+}=2\kap+1$, $\dim V^{-}=2\kap-1$. | T | Lem. 5.5; A6 1/1 |
| C. The fractional family | |||
| C1p06015 | The projectors $\Pi_\ell=\tfrac14\sum_{r=0}^{3}\im^{-\ell r}\Ft^{r}$ and the principal framed character lift $\Ft^{[s]}=\sum_\ell\gen^{-\ell s}\Pi_\ell$, $s\in\Phit$: the refinement base is the inverse generator, $(\gen^{-1})^{\kap}=\im$, each projector’s character the $\ell$-th power of the meridian phase $z_s=\gen^{-s}$. | D | (5.2), (5.3), Rem. 5.3 |
| C2p06016 | $\Pi_\ell^{2}=\Pi_\ell$, $\Pi_\ell\Pi_m=0$ for $\ell\neq m$, $\sum_\ell\Pi_\ell=I$, $\Ft\Pi_\ell=\im^{\ell}\Pi_\ell$. | T | Lem. 5.1; B1 1/1 |
| C3p06017 | The exact finite-field FrFT: $s\mapsto\Ft^{[s]}$ is a representation of $\Phit$, $\Ft^{[s+r]}=\Ft^{[s]}\Ft^{[r]}$ on every pair of the six shells, with the cardinal values $\Ft^{[0]}=I$, $\Ft^{[\kap]}=\Ft$, $\Ft^{[2\kap]}=J$, $\Ft^{[3\kap]}=\Ft^{-1}$, and $(\Ft^{[1]})^{\kap}=\Ft$. | T | 2/2 master: 00:C2 |
| C4p06018 | Faithfulness: $s\mapsto\Ft^{[s]}$ is injective on $\Z_{4\kap}$ for every $\kap\ge1$; at $\p=5$ the surviving odd projector carries the faithful character. | T | Thm. 5.9; B4 1/1 |
| C5p06019 | The multiplicities $m_\ell=\operatorname{rank}\Pi_\ell=\dim\ker(\Ft-\im^{\ell}I)$: $m_0+m_2=2\kap+1$, $m_1+m_3=2\kap-1$; $m_0,m_2\ge1$; $m_1,m_3\ge1$ for $\kap\ge2$; at $\p=5$ exactly one of $\Pi_1,\Pi_3$ vanishes. | T | Lem. 5.6; B5 1/1 |
| C6p06020 | Multiplicities are chart data: the vertex relabelling $m\mapsto um$ carries $\Ft(\gen)$ to $\Ft(\gen^{u^{2}})$ by a coordinate permutation; at $\p=13$ the frames $\gen=2$ and $\gen=6$ give $(3,3,4,2)$ and $(4,2,3,3)$, traces $4$ and $9$. | T | Rem. 5.7; B6 1/1 |
| C7p06021 | The multiplicity dichotomy: $G=\sum_{k}\gen^{k^{2}}=\varepsilon(1+\im)$; the tuple is $(\kap,\kap,\kap+1,\kap-1)$ for $\varepsilon=+1$ and $(\kap+1,\kap-1,\kap,\kap)$ for $\varepsilon=-1$; $\varepsilon(\gen^{-1})=-\varepsilon(\gen)$, $\varepsilon(\gen^{u})=\bigl(\tfrac{\kap}{u}\bigr)\varepsilon(\gen)$ for $u\equiv1\pmod4$, the two classes equally populated; the proof’s identities $GG^{*}=-2$, $G^{2}=2\im$, $\operatorname{Tr}\Ft=\im G$, $\operatorname{Tr}\Ft^{2}=2$, $\operatorname{Tr}\Ft^{3}=\im G^{*}$. Exact on the $38$ primitive frames of $\p\in\{5,13,17,29,37\}$ and the $16$ of $\p=41$. | T | 2/2 |
| C8p06022 | Classification: every exponent lift $a_\ell\equiv\ell\pmod4$ is additive with the same cardinal skeleton; a family canonical in the chart $\gen^{u}$ with $u^{2}\not\equiv1\pmod{4\kap}$ does not commute with $\Ft$ ($\p=29$, $u=5$, and every such chart of the six shells); the principal lift is the unique power-tower member. | T | Rem. 5.3; B9 1/1 |
| C9p06023 | The conjugate reframing $(\gen,\im)\mapsto(\gen^{-1},-\im)$: exactly $\Ft'=-\Ft^{-1}$ and $\Pi'_\ell=\Pi_{\ell+2}$; the operator relations, cardinal values, additivity and faithfulness hold on the conjugate frame, and its multiplicity tuple is the other pattern of C7. | T | 1/1 master: 00:C7 |
| D. Representation domains and the coordinate-side zoom | |||
| D1p06024 | The meridional representation domain $D_s=(\Vt,\mathcal B_s)$, $\mathcal B_s=\Ft^{[s]}\mathcal B_0$, every $\Ft^{[s]}$ invertible; the cardinal domains spatial, spectral, parity, inverse-spectral. | D | 1/1 |
| D2p06025 | The $4\kap$ framed domains are pairwise distinct; read as unordered measurement bases $B_{s+2\kap}=B_s$, since $\Ft^{[s+2\kap]}=\Ft^{[s]}J$ and $J$ permutes the standard basis, and the cycle carries exactly $2\kap$ of them. | T | 2/2 |
| D3p06026 | The meridian-step interval $I_\p=\{0,1,\dots,\pi\}$, the meridian-scale map $S_r(x)=\gen^{r}x$, and the local scale chart $\gen\mapsto\lambda>1$ reading the effective step $\gen^{m}$ as $\lambda^{m}$ [chart]. | D | (7.1), Def. 7.1, Def. 7.5 |
| D4p06027 | Meridian-scale covariance: $S_r(M_m)=M_{m+r}$ for every $(m,r)$, as ordered lists; consecutive entries of $M_m$ differ by the effective step $\gen^{m}$; $S_{r+(\p-1)}=S_r$, the periodicity of A5 in meridian form. | T | 2/2 master: 00:C2 |
| D5p06028 | Meridian zoom [chart]: $M_m$ reads the meridian-coordinate vector at step $\lambda^{\tilde m}$, the forward shift zoom-out and the inverse shift zoom-in, the seam an aliasing return; at $\p=13$, $\gen=2$ the ladder $M_0,\dots,M_3$ at steps $1,2,4,8$, unwrapped within $w\gen^{r}<\p$ ($w=\pi=6$: $r\le1$) and wrapping from $M_2$. | T | 1/1 |
| D6p06029 | Scope of the unification: the representation-side rotation and the coordinate-side scale shift act on carriers of different dimension and are not equal as operators; they share the index group $\Phit$ and the cardinal labels, and the operator-level relation is E5–E7. | D | Rem. 7.9 |
| E. The Weil dictionary and the operator-level comparison | |||
| E1p06030 | The Weil-side data: $z_s=\gen^{-s}$, $c_s=(z_s+z_s^{-1})/2$, $d_s=(z_s-z_s^{-1})/(2\im)$, the rotation $R_s$; the pulled-back family $\operatorname{FrFT}^{\rm Weil}_\kap(s)=\omega_\p(R_s)$; cardinal Weil correspondence of two families. | D | (8.1)–(8.3), (8.5), Def. 8.4 |
| E2p06031 | $R_s\in SO(2,\Fp)$, and $s\mapsto R_s$ is an isomorphism $\Phit\simeq SO(2,\Fp)$ with $|SO(2,\Fp)|=\p-1=4\kap$: the full rotation group of the $\p^{2}$-point plane is the meridian cycle. | T | 2/2 |
| E3p06032 | The cardinal-skeleton dictionary $M_s\leftrightarrow s\leftrightarrow z_s\leftrightarrow R_s$ with $R_0=I$, $R_\kap=w$, $R_{2\kap}=-I$, $R_{3\kap}=w^{-1}$ (the $R_\kap$ column of Table 1), $z_\kap=\im$; the FRC-native and Weil families stand in cardinal Weil correspondence, the Weil side by the normalization of A2. | T | Thm. 8.5; D3 1/1 |
| E4p06033 | Scope and reading of the dictionary: agreement on the four cardinal indices only, the two families acting on spaces of different dimension over different coefficient rings; the holographic reading, the rotation symmetry of the $\p^{2}$-point plane carried by a cycle of the size of its square root, the state side of dimension $\p$. | D | Rem. 8.6, Rem. 8.7 |
| E5p06034 | Spectral obstruction: the exponent shift $\sigma$ has the $4\kap$ simple eigenvalues $\Fpx$, every $\Ft^{[s]}$ at most four; for $\kap\ge2$ the cyclic groups $\langle\sigma\rangle$ and $\langle\Ft^{[1]}\rangle$ are not conjugate. | T | Prop. 8.8; D4 1/1 |
| E6p06035 | Common character sector: on $E_1=\operatorname{im}\Pi_1\neq0$ ($\kap\ge2$) $\Ft^{[s]}=\gen^{-s}I$, the intertwiner $\Ft^{[s]}T_v=T_vS_{-s}$ on every $x\in\Fp$ and $s$, and $R_s(1,-\im)^{\mathsf T}=\gen^{-s}(1,-\im)^{\mathsf T}$. | T | Prop. 8.9; D5 1/1 |
| E7p06036 | Cardinal Heisenberg covariance: $\Ft\sigma\Ft^{-1}=D_1$, $\Ft D_1\Ft^{-1}=\sigma^{-1}$; $\Ft^{r}\sigma=\sigma_r\Ft^{r}$ with $(\sigma,D_1,\sigma^{-1},D_1^{-1})$; the expansion (8.7); $\Ft^{[s]}\sigma\Ft^{[s]-1}$ monomial exactly at the four cardinal indices and non-monomial at every one of the $112$ intermediate indices of $\p\in\{13,17,29,37,41\}$. | T | 2/2 |
| F. The cyclotomic observer readout and the entropy cycle | |||
| F1p06037 | The readout algebra $A=\Z[\tfrac{1}{4n},X,Y,Y^{-1}]/(\Phi_n(X),Y^{2}-n)$ with the continuum chart $\sigma_\C$ ($X\mapsto\zeta_n$, $Y\mapsto+\sqrt n$) and the framed reduction $\rho$ ($X\mapsto\gen$, $Y\mapsto-\im$); the readout of a shell state as the Born vector of a selected lift, its Shannon entropy in nats, not thermodynamic. | D | Def. 9.1 |
| F2p06038 | $\widehat G^{2}=2n\,X^{\kap}$ in $A$, exact modulo $\Phi_n(X)$; under $\sigma_\C$, $(\sum_k\zeta_n^{k^{2}})^{2}=2n\,i$; under $\rho$, $G^{2}=2\im$, the reduction the proof of C7 uses. | T | 1/1 |
| F3p06039 | Entropic cardinal values $H(0)=H(2\kap)=0$, $H(\kap)=H(3\kap)=\log n$ for every site-localized input; $B_0$ and $B_\kap$ mutually unbiased, $H_{B_0}+H_{B_\kap}\ge\log n$ saturated by the localized states and by the comb $(\delta_0+\delta_6)/\sqrt2$ at $n=12$, $\log2+\log6=\log12$ [approx]. | T | 2/2 |
| F4p06040 | The closed form for $\delta_0$: $p_0=1-\tfrac{n-1}{n}t_s$, $p_{j\neq0}=t_s/n$, $t_s=(2-\zeta_n^{2s}-\zeta_n^{-2s})/4$, read as $\sin^{2}(\pi s/2\kap)$; $H(s)$ strictly increasing on $[0,\kap]$, two oscillations per cycle, strictly intermediate off the cardinal indices [approx]. | T | Prop. 9.3; E4 1/1 |
| F5p06041 | $\p=13$: $H(s)/\log n=0,\,.44,\,.91,\,1,\,.91,\,.44$ over each half-cycle; Figure 2 regenerated [approx]. | T | § 9.1; E5 1/1 |
| F6p06042 | Input dependence: $\delta_1$ gives $H(1)/\log n=0.55$ against $0.44$ for $\delta_0$; $\delta_j$ meets the odd projectors exactly for $j\notin\{0,2\kap\}$, and $\delta_{2\kap}$ reproduces the $\delta_0$ curve [approx]. | T | Rem. 9.4; E6 1/1 |
| F7p06043 | The Galois twist $X\mapsto\zeta_n^{u}$ relabels the $\delta_0$ curve by $s\mapsto us$ for every unit $u$ and leaves the cardinal values of every $\delta_j$ invariant; the conjugate twist $u=-1$ relabels the curve of every $\delta_j$, while for $j\notin\{0,2\kap\}$ a twist $u\not\equiv\pm1$ changes the intermediate curve ($n=12$, $\delta_1$, $u=5$: $0.55\mapsto0.44$ at $s=1$) [approx]. | T | Def. 9.1; E7 1/1 |
V. Machine verification (the validation package finite-ring-space/src/6-fourier) | |||
| V1p06044 | Blocks A–D of the package: $29$ integer-exact checks of B1–B3, B5–B7, C2–C9, D1–D2, D4–D5, E2–E3 and E5–E7 on the six shells of Table 1, every primitive frame where a statement ranges over frames ($54$), all $4096$ pairs for additivity and covariance, the $112$-index sweep. | T | |
| V2p06045 | Block E: the readout on $\C^{n}$, $n=4,12,16,28,36,40$: F2 exact in $\Z[X]$, F3–F7 to $10^{-9}$, Figure 2 regenerated. | T | |
| V3p06046 | The driver: five block scripts, one notebook run in the browser, $36$ checks ($30$ exact, $6$ [approx]), a record per check written to results.json keyed to these rows; the corpus master ledger’s rows C2, C14 and, on the transform layer, C7 are witnessed by B5, C3, D4, by B2, and by C9. | T | run_all; the notebook |
| O. The open front | |||
| O1p06047 | Cardinal exclusivity: for every $s\notin\{0,\kap,2\kap,3\kap\}$ the operator (8.7) is not monomial — sweep-verified on the $112$ intermediate indices of E7, the proof obstructed by the possible vanishing of sparse six-term Laurent polynomials on subgroup cosets. | O | Conj. 8.11 |
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 3–9 and the validation package of the same date. One open row, O1, the cardinal-exclusivity conjecture; when it closes its content is re-homed in block E and the row leaves. No row has been retired.