Scale-Shift and Fractional Fourier Transform as Rotations in Representation Space over Finite Fields

1. Introduction

The classical Fourier transform is usually described as a change of the basis of representation for both continuous and discrete signals. In its conventional analytic setting it exchanges a spatial variable with its spectral variable and satisfies the four-cycle

$${\mathcal{F}}^0=I,{\mathcal{F}}^1=\mathcal{F},{\mathcal{F}}^2=P,{\mathcal{F}}^4=I,$$

(1.1)

where P is parity. The fractional Fourier transform (FrFT) extends this four-cycle by interpreting the Fourier transform as a quarter-turn in representation space and by introducing intermediate rotated representations. This point of view appears in the work of Namias and in later optical and signal-processing treatments of the FrFT [1,2,3,4,5,6]. In the usual language these rotations are described as time-frequency rotations. In the present paper the corresponding shell terminology is spatial-spectral: the temporal/frequency vocabulary belongs to latitudinal phase analysis, while the meridional construction developed below rotates between spatial and spectral representation domains.

The aim of this paper is to define the FrFT exactly over a symmetry-complete finite fields that are defined in [7,8]. The specific setting is the finite shell

$$\mathsf{p}=4\mathsf{t}+1,$$

with $\mathsf{p}$ prime. The multiplicative group ${\mathbb{F}}_{\mathsf{p}}^{\times }$ then has order $4\mathsf{t}$ and contains a canonical quarter-turn subgroup once a primitive generator $e_{\mathsf{t}}$ is chosen. Setting

$$i_{\mathsf{t}}:=e_{\mathsf{t}}^{\mathsf{t}}$$

gives $i_{\mathsf{t}}^2=-1$. This is the finite shell analogue of the complex quarter-turn. The same shell datum also gives the half-period

$${\pi }_{\mathsf{t}}:={\displaystyle \frac{\mathsf{p}-1}{2}}=2\mathsf{t}.$$

The framed shell ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$ and its meridian-latitude geometry were introduced in earlier FRC work as a finite Euclidean shell layer in which the roles of e, $\pi$, and i are reconstructed from finite arithmetic data [7,8]. In particular, the discrete shell Fourier transform $W_{\mathsf{t}}$ used below is introduced in ([8] §6, Remark 13), where the representational reading of one Fourier step as a quarter-turn between primal and dual shell domains is also previewed and deferred to subsequent work ([8] end of §6); the present paper realizes that preview. Here we use only the finite-field layer: the cyclic group ${\mathbb{F}}_{\mathsf{p}}^{\times }$, its generator $e_{\mathsf{t}}$, the induced quarter-turn $i_{\mathsf{t}}$, and the meridian cycle ${\mathbb{Z}}_{4\mathsf{t}}$.

Figure 1 shows a concrete shell visualization in the case $\mathsf{p}=13$. The meridians and latitudes correspond to the additive and the multiplicative actions of the orbital complex, while “Observer” marks the observer’s origin. The red meridian is the prime meridian $M_0$ and the blue meridian is the spectral meridian $M_{\mathsf{t}}$.

The main claim is that every shell meridian is a representation domain. The prime meridian $M_0$ is the spatial domain. The meridian $M_{\mathsf{t}}$, obtained by multiplying the meridian direction by $i_{\mathsf{t}}$, is the spectral domain. The meridian $M_{2\mathsf{t}}$ is the parity domain, and $M_{3\mathsf{t}}$ is the inverse-spectral domain. The finite FrFT is then the exact transform that rotates the representation from $M_0$ to $M_s$.

The paper has three layers. The first is the FRC-native finite-field FrFT on the representation side, constructed by Theorem 1 (additivity and cardinal values) and completed by Theorem 2 (faithfulness on the full meridian cycle for $\mathsf{t}\ge 2$). The second is the finite-field Weil dictionary of Theorem 4, which matches the FRC-native family to the Weil realization on the cardinal Fourier skeleton. The third is the meridian-coordinate zoom of Theorem 3, which reads the same meridian cycle on the shell-coordinate side: the multiplicative map $S_r:x\mapsto e_{\mathsf{t}}^{r}x$ on ${\mathbb{F}}_{\mathsf{p}}$ sends the meridian $M_m$ to $M_{m+r}$, and in a local comparison chart where $e_{\mathsf{t}}$ is read as a real scale factor $\lambda >1$ this is the algebraic form of reversible zoom-out by factor ${\lambda }^r$, while the inverse shift $m\mapsto m-r$ is the corresponding zoom-in. The finite-field map is a bijection of ${\mathbb{F}}_{\mathsf{p}}$; lossy coarse-graining appears only after an observer readout is added.

To begin with the first layer, let

$$n:=\mathsf{p}-1=4\mathsf{t},{\Phi }_{\mathsf{t}}:={\mathbb{Z}}_n,$$

and let

$$V_{\mathsf{t}}:={\mathbb{F}}_{\mathsf{p}}^{{\Phi }_{\mathsf{t}}}.$$

Define the shell Fourier matrix $W_{\mathsf{t}}$ by

$${\left(W_{\mathsf{t}}\right)}_{k,j}=e_{\mathsf{t}}^{jk},j,k\in {\Phi }_{\mathsf{t}}.$$

Since $n=\mathsf{p}-1=-1$ in ${\mathbb{F}}_{\mathsf{p}}$, one obtains $W_{\mathsf{t}}^2=-J$, where J is parity. The normalized operator

$${\mathcal{F}}_{\mathsf{t}}:=i_{\mathsf{t}}{W}_{\mathsf{t}}$$

(1.2)

satisfies

$${\mathcal{F}}_{\mathsf{t}}^2=J,{\mathcal{F}}_{\mathsf{t}}^4=I.$$

(1.3)

The projectors onto the four eigenspaces of ${\mathcal{F}}_{\mathsf{t}}$ are polynomials in ${\mathcal{F}}_{\mathsf{t}}$:

$${\Pi }_{\ell }={\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=0}^3}{i}_{\mathsf{t}}^{-\ell r}{\mathcal{F}}_{\mathsf{t}}^r,\ell =0,1,2,3.$$

They define the finite fractional Fourier family

$${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}:={\displaystyle \sum _{\ell =0}^3}{e}_{\mathsf{t}}^{\ell s}{\Pi }_{\ell },s\in {\mathbb{Z}}_{4\mathsf{t}}.$$

This family is additive in the meridian index:

$${\mathcal{F}}_{\mathsf{t}}^{[s+r]}={\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}{\mathcal{F}}_{\mathsf{t}}^{\left[r\right]}.$$

It also has the required cardinal values

$${\mathcal{F}}_{\mathsf{t}}^{\left[0\right]}=I,{\mathcal{F}}_{\mathsf{t}}^{\left[\mathsf{t}\right]}={\mathcal{F}}_{\mathsf{t}},{\mathcal{F}}_{\mathsf{t}}^{\left[2\mathsf{t}\right]}=J,{\mathcal{F}}_{\mathsf{t}}^{\left[3\mathsf{t}\right]}={\mathcal{F}}_{\mathsf{t}}^{-1}.$$

The Weil dictionary relates this FRC-native construction to the finite-field Weil interpretation on the cardinal Fourier skeleton. The classical Weil representation realizes symplectic transformations of a finite phase plane as operators on a finite state space; the Fourier transform is the operator associated with the Weyl quarter-turn element [9,10]. Recent work on arithmetic FrFTs similarly defines finite fractional Fourier transforms from finite arithmetic rotation groups [11]. In the present setting the FRC constants give an exact dictionary. For $z_s=e_{\mathsf{t}}^s$, set

$$c_s={\displaystyle \frac{z_s+z_s^{-1}}{2}},d_s={\displaystyle \frac{z_s-z_s^{-1}}{2i_{\mathsf{t}}}}.$$

Then

$$R_s=\left(\begin{array}{cc}{c}_s& -d_s\\ d_s& c_s\end{array}\right)\in SO(2,{\mathbb{F}}_{\mathsf{p}}),$$

and $s\mapsto R_s$ is a group isomorphism from the meridian cycle to the finite-field rotation group $SO(2,{\mathbb{F}}_{\mathsf{p}})$ for the form $x^2+y^2$. Thus, the FRC meridian index and the finite-field Weil rotation index are the same finite datum written in two coordinate languages. This is the exact transfer supplied by $e_{\mathsf{t}}$, $i_{\mathsf{t}}$, and ${\pi }_{\mathsf{t}}$.

The construction is deliberately finite. It does not approximate a real angle. It replaces the continuum angle $\alpha$ by the shell meridian index $s\in {\mathbb{Z}}_{4\mathsf{t}}$. It also avoids the ambiguity of arbitrary fractional powers of a discrete Fourier matrix. The projectors in () are canonical polynomials in the normalized finite Fourier operator.

2. Classical Fractional Fourier Transform

The classical FrFT is often introduced as a family of transforms ${\mathcal{F}}_{\alpha }$ such that

$${\mathcal{F}}_0=I,{\mathcal{F}}_{\pi /2}=\mathcal{F},{\mathcal{F}}_{\pi }=P,{\mathcal{F}}_{2\pi }=I,$$

(2.1)

and

$${\mathcal{F}}_{\alpha +\beta }={\mathcal{F}}_{\alpha }{\mathcal{F}}_{\beta }.$$

(2.2)

The angle-indexed convention ${\mathcal{F}}_{\alpha }$ is related to the iterate-indexed convention ${\mathcal{F}}^k$ of equation (1) by ${\mathcal{F}}^k={\mathcal{F}}_{k\pi /2}$; the cardinal values match in the two readings. The analytic kernel contains functions such as $cot\alpha$ and $csc\alpha$, but those functions are not the structural core. The structural core is the representation-space rotation. This observation is important because a finite-field construction has no need for a real angle or a trigonometric kernel. It needs only a finite cyclic rotation law with a Fourier quarter-turn.

The terminology used in this paper is as follows. A meridian of the shell is a representation axis. A latitude of the shell carries phase-cycle data. Hence, the meridional FrFT constructed here moves between spatial and spectral representation domains. The names temporal and frequency are not used for the meridional domains. They belong to a different layer of shell analysis, namely the latitudinal phase-cycle reading.

By a representation domain of the shell, in this paper, we shall always mean a basis-change pair $D_s=(V_{\mathsf{t}},{\mathcal{B}}_s)$ on the same finite module $V_{\mathsf{t}}$, indexed by a meridian $s\in {\mathbb{Z}}_{4\mathsf{t}}$, where the meridional basis ${\mathcal{B}}_s$ is obtained from the standard basis by the finite FrFT ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ constructed below (see Definition 5). This is an internal definition: it refers only to the finite-field FrFT and its meridian index, with no auxiliary chart-level interpretation.

The intended finite replacement of the continuum angle is

$${\alpha }_s={\displaystyle \frac{s}{\mathsf{t}}}{\displaystyle \frac{\pi }{2}},$$

(2.3)

only as an external comparison. Internally, the index is simply

$$s\in {\mathbb{Z}}_{4\mathsf{t}}.$$

(2.4)

Thus, the values $s=0,\mathsf{t},2\mathsf{t},3\mathsf{t}$ correspond to the four cardinal representation domains.

3. Symmetry-Complete Shell Data

A symmetry-complete shell, in the sense used here, is a prime $\mathsf{p}$ with $\mathsf{p}\equiv 1(mod4)$, presented together with the framed datum ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$ where $\mathsf{p}=4\mathsf{t}+1$ and $e_{\mathsf{t}}\in {\mathbb{F}}_{\mathsf{p}}^{\times }$ is a chosen primitive generator. This is the FRC programme shell setting introduced in [7] and developed in [8]; the present paper uses only its finite-field layer.

Let $\mathsf{p}=4\mathsf{t}+1$ be prime. The group ${\mathbb{F}}_{\mathsf{p}}^{\times }$ is cyclic of order $4\mathsf{t}$. Choose a primitive generator

$$e_{\mathsf{t}}\in {\mathbb{F}}_{\mathsf{p}}^{\times }.$$

Define the oriented quarter-turn by

$$i_{\mathsf{t}}:=e_{\mathsf{t}}^{\mathsf{t}}.$$

(3.1)

The opposite primitive fourth root of unity is $-i_{\mathsf{t}}=e_{\mathsf{t}}^{3\mathsf{t}}$; the choice of $e_{\mathsf{t}}$ fixes the orientation, and we take $i_{\mathsf{t}}=e_{\mathsf{t}}^{\mathsf{t}}$ throughout, consistent with ([8] §4). Then

$$i_{\mathsf{t}}^2=e_{\mathsf{t}}^{2\mathsf{t}}=e_{\mathsf{t}}^{(\mathsf{p}-1)/2}=-1$$

by Euler’s criterion, since $e_{\mathsf{t}}$ is primitive. The quarter-turn subgroup is

$$Q_{\mathsf{t}}=\{1,i_{\mathsf{t}},-1,-i_{\mathsf{t}}\}.$$

The half-period is

$${\pi }_{\mathsf{t}}:={\displaystyle \frac{\mathsf{p}-1}{2}}=2\mathsf{t}.$$

Remark 1 

(Dependence on the choice of $e_{\mathsf{t}}$). The construction below depends on the choice of primitive generator $e_{\mathsf{t}}$. Any other admissible choice has the form $e_{\mathsf{t}}^{\prime }=e_{\mathsf{t}}^u$ with $u\in {\mathbb{Z}}_{\mathsf{p}-1}^{\times }$. Under this change, the meridian relabelling $m\mapsto um$ realises the corresponding generator covariance of the framed shell established in ([8] Propositions 1–3); the FrFT family ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ constructed below is therefore canonical relative to the framed datum ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$ rather than relative to ${\mathbb{F}}_{\mathsf{p}}$ alone.

The meridian index group is

$${\Phi }_{\mathsf{t}}:={\mathbb{Z}}_{\mathsf{p}-1}={\mathbb{Z}}_{4\mathsf{t}}.$$

For each $s\in {\Phi }_{\mathsf{t}}$, the meridian direction is $e_{\mathsf{t}}^s$. In an observer-framed shell one may write the meridian as

$$M_s=\{a{e}_{\mathsf{t}}^s:a\in I_{\mathsf{p}}\},$$

(3.2)

where $I_{\mathsf{p}}$ is the finite meridian-step interval. For the operator construction of §§ 4–6 the precise choice of endpoint convention for $I_{\mathsf{p}}$ is irrelevant; the meridian index s is the essential representation-domain label. For the meridian-coordinate zoom of § 7 we fix the convention $I_{\mathsf{p}}=\{0,1,\dots ,{\pi }_{\mathsf{t}}\}$.

Definition 1 

(Cardinal meridians). The four cardinal meridians are

$$\begin{array}{cc}\hfill M_0& \mathrm{spatial}\mathrm{domain},\hfill \\ \hfill M_{\mathsf{t}}& \mathrm{spectral}\mathrm{domain},\hfill \\ \hfill M_{2\mathsf{t}}& \mathrm{parity}\mathrm{domain},\hfill \\ \hfill M_{3\mathsf{t}}& \mathrm{inverse}-\mathrm{spectral}\mathrm{domain}.\hfill \end{array}$$

This definition records the intended shell reading. The rest of the paper proves that the finite Fourier and fractional Fourier operators realize exactly this meridional cycle at the representation level.

4. The Normalized Shell Fourier Operator

Let

$$n:=\mathsf{p}-1=4\mathsf{t},{\Phi }_{\mathsf{t}}:={\mathbb{Z}}_n,V_{\mathsf{t}}:={\mathbb{F}}_{\mathsf{p}}^{{\Phi }_{\mathsf{t}}}.$$

Vectors in $V_{\mathsf{t}}$ are functions $v:{\Phi }_{\mathsf{t}}\to {\mathbb{F}}_{\mathsf{p}}$. Define the reversal operator $J:V_{\mathsf{t}}\to V_{\mathsf{t}}$ by

$${\left(Jv\right)}_k=v_{-k}.$$

(4.1)

Clearly $J^2=I$.

Definition 2 

(Shell Fourier matrix). The shell Fourier matrix is the linear operator $W_{\mathsf{t}}:V_{\mathsf{t}}\to V_{\mathsf{t}}$ with entries

$${\left(W_{\mathsf{t}}\right)}_{k,j}=e_{\mathsf{t}}^{jk},j,k\in {\Phi }_{\mathsf{t}}.$$

(4.2)

Equivalently,

$${\left(W_{\mathsf{t}}v\right)}_k=\sum _{j\in {\Phi }_{\mathsf{t}}}{e}_{\mathsf{t}}^{jk}{v}_j.$$

This is a finite-field Fourier transform over the cyclic phase group ${\Phi }_{\mathsf{t}}$. It is internal to ${\mathbb{F}}_{\mathsf{p}}$ because $e_{\mathsf{t}}$ is a root of unity of order $n=\mathsf{p}-1$ in ${\mathbb{F}}_{\mathsf{p}}^{\times }$. The same matrix is recorded as the shell DFT in ([8] §6, Remark 13); the present section adds only the shell quarter-turn normalization and the resulting four-cycle.

Lemma 1. 

The shell Fourier matrix satisfies

$$W_{\mathsf{t}}^2=-J.$$

Proof. 

For $k,l\in {\Phi }_{\mathsf{t}}$,

$${\left(W_{\mathsf{t}}^2\right)}_{k,l}=\sum _{j\in {\Phi }_{\mathsf{t}}}{e}_{\mathsf{t}}^{j(k+l)}.$$

If $k+l\equiv 0(modn)$, then every summand is 1 and the sum is $n=\mathsf{p}-1=-1$ in ${\mathbb{F}}_{\mathsf{p}}$. If $k+l\neg \equiv 0(modn)$, then $e_{\mathsf{t}}^{k+l}\ne 1$ and the finite geometric sum is

$${\displaystyle \sum _{j=0}^{n-1}}{e}_{\mathsf{t}}^{j(k+l)}={\displaystyle \frac{e_{\mathsf{t}}^{n(k+l)}-1}{e_{\mathsf{t}}^{k+l}-1}}=0.$$

Therefore, ${\left(W_{\mathsf{t}}^2\right)}_{k,l}=-1$ exactly when $l=-k$ and 0 otherwise. This is precisely $-J$. □

Definition 3 

(Normalized shell Fourier quarter-turn). Define

$${\mathcal{F}}_{\mathsf{t}}:=i_{\mathsf{t}}{W}_{\mathsf{t}}.$$

(4.3)

The normalization by $i_{\mathsf{t}}$ is forced by the shell constants: $i_{\mathsf{t}}$ is the quarter-turn already present in the symmetry-complete field.

Proposition 1. 

The normalized shell Fourier operator satisfies

$${\mathcal{F}}_{\mathsf{t}}^2=J,{\mathcal{F}}_{\mathsf{t}}^4=I.$$

(4.4)

Proof. 

Using Lemma 1 and $i_{\mathsf{t}}^2=-1$,

$${\mathcal{F}}_{\mathsf{t}}^2={\left(i_{\mathsf{t}}{W}_{\mathsf{t}}\right)}^2=i_{\mathsf{t}}^2{W}_{\mathsf{t}}^2=(-1)(-J)=J.$$

Since $J^2=I$, it follows that ${\mathcal{F}}_{\mathsf{t}}^4=I$. □

Thus, ${\mathcal{F}}_{\mathsf{t}}$ is an exact finite quarter-turn: one application gives the spectral representation, two applications give parity, and four applications return to the original representation.

5. Canonical Finite Fractional Powers

A direct fractional power of a Fourier matrix can be non-canonical if eigenspaces have multiplicity and an arbitrary eigenbasis is chosen. The construction below avoids this issue by using projectors that are polynomials in ${\mathcal{F}}_{\mathsf{t}}$ itself.

Since ${\mathcal{F}}_{\mathsf{t}}^4=I$ and $\mathsf{p}\ne 2$, the polynomial $x^4-1$ splits in ${\mathbb{F}}_{\mathsf{p}}$ as

$$x^4-1=(x-1)(x-i_{\mathsf{t}})(x+1)(x+i_{\mathsf{t}}),$$

(5.1)

with four distinct roots. Define

$${\Pi }_{\ell }={\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=0}^3}{i}_{\mathsf{t}}^{-\ell r}{\mathcal{F}}_{\mathsf{t}}^r,\ell =0,1,2,3.$$

(5.2)

Here $1/4$ denotes the inverse of 4 in ${\mathbb{F}}_{\mathsf{p}}$.

Lemma 2. 

The operators ${\Pi }_{\ell }$ are pairwise orthogonal idempotents satisfying

$${\Pi }_{\ell }^2={\Pi }_{\ell },{\Pi }_{\ell }{\Pi }_m=0(\ell \ne m),{\displaystyle \sum _{\ell =0}^3}{\Pi }_{\ell }=I,$$

and

$${\mathcal{F}}_{\mathsf{t}}{\Pi }_{\ell }=i_{\mathsf{t}}^{\ell }{\Pi }_{\ell }.$$

Proof. 

This is the standard idempotent decomposition of the cyclic group algebra ${\mathbb{F}}_{\mathsf{p}}\left[C_4\right]$ after adjoining all fourth roots of unity. A direct verification uses the finite character orthogonality identity

$${\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=0}^3}{i}_{\mathsf{t}}^{r(a-b)}=\left\{\begin{array}{cc}1,\hfill & a\equiv b(mod4),\hfill \\ 0,\hfill & a\neg \equiv b(mod4).\hfill \end{array}\right.$$

The relation ${\mathcal{F}}_{\mathsf{t}}{\Pi }_{\ell }=i_{\mathsf{t}}^{\ell }{\Pi }_{\ell }$ follows by multiplying (15) by ${\mathcal{F}}_{\mathsf{t}}$ and reindexing powers modulo 4. □

Definition 4 

(FRC-native finite FrFT). For each meridian index $s\in {\Phi }_{\mathsf{t}}={\mathbb{Z}}_{4\mathsf{t}}$, define

$${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}:={\displaystyle \sum _{\ell =0}^3}{e}_{\mathsf{t}}^{\ell s}{\Pi }_{\ell }.$$

(5.3)

This is the central definition of the paper. The continuum angle has been replaced by a finite meridian index, and the four eigenvalues of the Fourier quarter-turn have been refined from $i_{\mathsf{t}}^{\ell }$ to $e_{\mathsf{t}}^{\ell s}$.

Theorem 1 

(Exact finite-field FrFT). The family $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ is a cyclic representation of the meridian group ${\Phi }_{\mathsf{t}}$. In particular,

$${\mathcal{F}}_{\mathsf{t}}^{[s+r]}={\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}{\mathcal{F}}_{\mathsf{t}}^{\left[r\right]}$$

(5.4)

for all $s,r\in {\Phi }_{\mathsf{t}}$. Moreover,

$${\mathcal{F}}_{\mathsf{t}}^{\left[0\right]}=I,{\mathcal{F}}_{\mathsf{t}}^{\left[\mathsf{t}\right]}={\mathcal{F}}_{\mathsf{t}},{\mathcal{F}}_{\mathsf{t}}^{\left[2\mathsf{t}\right]}=J,{\mathcal{F}}_{\mathsf{t}}^{\left[3\mathsf{t}\right]}={\mathcal{F}}_{\mathsf{t}}^{-1},{\mathcal{F}}_{\mathsf{t}}^{\left[4\mathsf{t}\right]}=I.$$

(5.5)

Consequently, ${\mathcal{F}}_{\mathsf{t}}^{\left[1\right]}$ is an exact $\mathsf{t}$-th root of the normalized shell Fourier transform:

$${\left({\mathcal{F}}_{\mathsf{t}}^{\left[1\right]}\right)}^{\mathsf{t}}={\mathcal{F}}_{\mathsf{t}}.$$

Proof. 

Using Lemma 2,

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}{\mathcal{F}}_{\mathsf{t}}^{\left[r\right]}& =\left({\displaystyle \sum _{\ell =0}^3}{e}_{\mathsf{t}}^{\ell s}{\Pi }_{\ell }\right)\left({\displaystyle \sum _{m=0}^3}{e}_{\mathsf{t}}^{mr}{\Pi }_m\right)\hfill \\ \hfill & ={\displaystyle \sum _{\ell =0}^3}{e}_{\mathsf{t}}^{\ell (s+r)}{\Pi }_{\ell }={\mathcal{F}}_{\mathsf{t}}^{[s+r]}.\hfill \end{array}$$

This proves additivity. For the cardinal values, use $e_{\mathsf{t}}^{\mathsf{t}}=i_{\mathsf{t}}$. Then

$${\mathcal{F}}_{\mathsf{t}}^{\left[\mathsf{t}\right]}={\displaystyle \sum _{\ell =0}^3}{i}_{\mathsf{t}}^{\ell }{\Pi }_{\ell }={\mathcal{F}}_{\mathsf{t}},$$

because ${\Pi }_{\ell }$ is the $i_{\mathsf{t}}^{\ell }$ eigenspace projector for ${\mathcal{F}}_{\mathsf{t}}$. Similarly,

$${\mathcal{F}}_{\mathsf{t}}^{\left[2\mathsf{t}\right]}={\displaystyle \sum _{\ell =0}^3}{(-1)}^{\ell }{\Pi }_{\ell }={\mathcal{F}}_{\mathsf{t}}^2=J,$$

and

$${\mathcal{F}}_{\mathsf{t}}^{\left[3\mathsf{t}\right]}={\displaystyle \sum _{\ell =0}^3}{(-i_{\mathsf{t}})}^{\ell }{\Pi }_{\ell }={\mathcal{F}}_{\mathsf{t}}^3={\mathcal{F}}_{\mathsf{t}}^{-1}.$$

Finally, $e_{\mathsf{t}}^{4\mathsf{t}}=1$, hence ${\mathcal{F}}_{\mathsf{t}}^{\left[4\mathsf{t}\right]}=I$. The root identity follows from additivity:

$${\left({\mathcal{F}}_{\mathsf{t}}^{\left[1\right]}\right)}^{\mathsf{t}}={\mathcal{F}}_{\mathsf{t}}^{\left[\mathsf{t}\right]}={\mathcal{F}}_{\mathsf{t}}.$$

□

The cardinal values of Theorem 1 are correct without faithfulness; they only require the four projectors and the additivity. The full $4\mathsf{t}$-domain reading, however — the statement that every shell meridian $s\in {\Phi }_{\mathsf{t}}$ gives a representation domain distinct from the others — requires the family $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ to be injective on ${\Phi }_{\mathsf{t}}$. Injectivity in turn requires the four eigenprojectors ${\Pi }_{\ell }$ to be individually nonzero. In characteristic zero this would be automatic, but over ${\mathbb{F}}_{\mathsf{p}}$ a verification is needed. The remainder of this section supplies it.

Lemma 3 

(Symmetric/antisymmetric decomposition). The Fourier matrix commutes with the reversal:

$$W_{\mathsf{t}}J=J{W}_{\mathsf{t}},$$

and hence so does ${\mathcal{F}}_{\mathsf{t}}$. Decompose

$$V_{\mathsf{t}}=V^{+}\oplus V^{-},V^{+}=ker(J-I),V^{-}=ker(J+I),$$

with

$$dim{V}^{+}=2\mathsf{t}+1,dim{V}^{-}=2\mathsf{t}-1.$$

Then ${\mathcal{F}}_{\mathsf{t}}$ stabilizes $V^{\pm }$, with

$${\mathcal{F}}_{\mathsf{t}}^2{{|}_{V^{+}}=I,{\mathcal{F}}_{\mathsf{t}}^2|}_{V^{-}}=-I.$$

(5.6)

Proof. 

For $k,j\in {\Phi }_{\mathsf{t}}$,

$${\left(W_{\mathsf{t}}J\right)}_{k,j}={\left(W_{\mathsf{t}}\right)}_{k,-j}=e_{\mathsf{t}}^{-kj}={\left(W_{\mathsf{t}}\right)}_{-k,j}={\left(J{W}_{\mathsf{t}}\right)}_{k,j},$$

which gives the commutation. The involution J acts on ${\Phi }_{\mathsf{t}}={\mathbb{Z}}_{4\mathsf{t}}$ by $k\mapsto -k$ with fixed points $\{0,2\mathsf{t}\}$ and $(4\mathsf{t}-2)/2=2\mathsf{t}-1$ pairs. Hence $dim{V}^{+}=2+(2\mathsf{t}-1)=2\mathsf{t}+1$ and $dim{V}^{-}=2\mathsf{t}-1$. Stabilization follows from the commutation, and ${\mathcal{F}}_{\mathsf{t}}^2=J$ gives the eigenvalues on $V^{\pm }$. □

Lemma 4 

(Multiplicity). Write $m_{\ell }=rank{\Pi }_{\ell }$ for $\ell =0,1,2,3$. Then

$$m_0+m_2=2\mathsf{t}+1,m_1+m_3=2\mathsf{t}-1.$$

For every $\mathsf{t}\ge 1$, $m_0\ge 1$ and $m_2\ge 1$. For every $\mathsf{t}\ge 2$, $m_1\ge 1$ and $m_3\ge 1$. At $\mathsf{t}=1$ ($\mathsf{p}=5$) the antisymmetric subspace is one-dimensional, so exactly one of ${\Pi }_1,{\Pi }_3$ is zero; but ${\mathbb{Z}}_4$ is exhausted by the cardinal indices $\{0,1,2,3\}=\{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$ in that case, so no intermediate meridian is lost.

Proof. 

The first two identities follow from Lemma 3: $V^{+}$ is the sum of the $\ell =0,2$ eigenspaces of ${\mathcal{F}}_{\mathsf{t}}$ (eigenvalues $\pm 1$, $i_{\mathsf{t}}^0=1$, $i_{\mathsf{t}}^2=-1$), while $V^{-}$ is the sum of the $\ell =1,3$ eigenspaces (eigenvalues $\pm i_{\mathsf{t}}$).

For the eigenvalue $\pm 1$ pair on $V^{+}$, consider the symmetric vector $e_0$. Then ${\left(W_{\mathsf{t}}{e}_0\right)}_k=e_{\mathsf{t}}^{0·k}=1$ for every k, so $W_{\mathsf{t}}{e}_0=\mathbf{1}$, the all-ones vector. Thus ${\mathcal{F}}_{\mathsf{t}}{e}_0=i_{\mathsf{t}}\mathbf{1}$ has nonzero entries at every index $k\in {\Phi }_{\mathsf{t}}$, whereas $e_0$ has support $\left\{0\right\}$. Consequently ${\mathcal{F}}_{\mathsf{t}}{e}_0\notin {\mathbb{F}}_{\mathsf{p}}·e_0$, so ${\mathcal{F}}_{\mathsf{t}}{|}_{V^{+}}$ is not a scalar operator. Since ${\left({\mathcal{F}}_{\mathsf{t}}{|}_{V^{+}}\right)}^2=I$, the minimal polynomial of ${\mathcal{F}}_{\mathsf{t}}{|}_{V^{+}}$ divides $x^2-1=(x-1)(x+1)$; as it has degree at least 2, it equals $x^2-1$ and both eigenvalues $\pm 1$ occur. Hence $m_0\ge 1$ and $m_2\ge 1$.

For the eigenvalue $\pm i_{\mathsf{t}}$ pair on $V^{-}$ with $\mathsf{t}\ge 2$, consider the antisymmetric vector $v=e_1-e_{-1}$. Then

$${\left(W_{\mathsf{t}}v\right)}_k=e_{\mathsf{t}}^k-e_{\mathsf{t}}^{-k},$$

which vanishes iff $e_{\mathsf{t}}^{2k}=1$, iff $2\mathsf{t}\mid k$, iff $k\in \{0,2\mathsf{t}\}$. For $\mathsf{t}\ge 2$ the index $k=2$ lies outside $\{0,\pm 1,2\mathsf{t}\}$ and satisfies $e_{\mathsf{t}}^{2k}=e_{\mathsf{t}}^4\ne 1$ since $e_{\mathsf{t}}$ has order $4\mathsf{t}>4$. Hence ${\left(W_{\mathsf{t}}v\right)}_2=e_{\mathsf{t}}^2-e_{\mathsf{t}}^{-2}\ne 0$, while $v_2=0$, so ${\mathcal{F}}_{\mathsf{t}}v\notin {\mathbb{F}}_{\mathsf{p}}·v$ and ${\mathcal{F}}_{\mathsf{t}}{|}_{V^{-}}$ is not a scalar. Since ${\left({\mathcal{F}}_{\mathsf{t}}{|}_{V^{-}}\right)}^2=-I$, its minimal polynomial divides $x^2+1=(x-i_{\mathsf{t}})(x+i_{\mathsf{t}})$ and equals this product, so both eigenvalues $\pm i_{\mathsf{t}}$ occur. Hence $m_1\ge 1$ and $m_3\ge 1$.

For $\mathsf{t}=1$, $dim{V}^{-}=1$, so ${\mathcal{F}}_{\mathsf{t}}{|}_{V^{-}}$ is a scalar; that scalar squares to $-1$, hence equals either $i_{\mathsf{t}}$ or $-i_{\mathsf{t}}$, and exactly one of $m_1,m_3$ is zero. □

Theorem 2 

(Faithfulness on the meridian cycle). For $\mathsf{t}\ge 2$, the map

$$s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]},s\in {\mathbb{Z}}_{4\mathsf{t}},$$

is injective. In particular, the $4\mathsf{t}$ meridional images are pairwise distinct.

Proof. 

Suppose ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}={\mathcal{F}}_{\mathsf{t}}^{\left[r\right]}$. Multiplying ${\sum }_{\ell }(e_{\mathsf{t}}^{\ell s}-e_{\mathsf{t}}^{\ell r}){\Pi }_{\ell }=0$ by ${\Pi }_m$ and using ${\Pi }_m^2={\Pi }_m$ together with ${\Pi }_m{\Pi }_{\ell }=0$ for $\ell \ne m$ gives

$$(e_{\mathsf{t}}^{ms}-e_{\mathsf{t}}^{mr}){\Pi }_m=0.$$

By Lemma 4, ${\Pi }_1\ne 0$ for $\mathsf{t}\ge 2$. Taking $m=1$ therefore gives $e_{\mathsf{t}}^{s-r}=1$, and since $e_{\mathsf{t}}$ has order $4\mathsf{t}$ in ${\mathbb{F}}_{\mathsf{p}}^{\times }$, this forces $s\equiv r(mod4\mathsf{t})$. □

6. Meridional Representation Domains

We now state the representation-domain reading. Let ${\mathcal{B}}_0={\left({\delta }_j\right)}_{j\in {\Phi }_{\mathsf{t}}}$ be the standard basis of $V_{\mathsf{t}}$, interpreted as the coordinate basis of the prime spatial domain.

Definition 5 

(Meridional representation domain). For $s\in {\Phi }_{\mathsf{t}}$, the operator ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ is invertible: its eigenvalues $e_{\mathsf{t}}^{\ell s}$ ($\ell =0,1,2,3$) are nonzero powers of $e_{\mathsf{t}}$ in ${\mathbb{F}}_{\mathsf{p}}$, so ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ has nonzero determinant. Define the meridional basis

$${\mathcal{B}}_s:={\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}{\mathcal{B}}_0.$$

The meridional representation domain $D_s$ is the pair

$$D_s:=(V_{\mathsf{t}},{\mathcal{B}}_s).$$

Thus all domains use the same finite module $V_{\mathsf{t}}$, but each domain writes its coordinates in a different meridional basis. The transform from $D_r$ to $D_s$ is

$${\mathcal{F}}_{\mathsf{t}}^{[s-r]}:D_r\to D_s.$$

(6.1)

Corollary 1 

(Cardinal domains). The four cardinal domains are

$$\begin{array}{cc}\hfill D_0& =\mathrm{spatial}\mathrm{domain},\hfill \\ \hfill D_{\mathsf{t}}& =\mathrm{spectral}\mathrm{domain},\hfill \\ \hfill D_{2\mathsf{t}}=D_{{\pi }_{\mathsf{t}}}& =\mathrm{parity}\mathrm{domain},\hfill \\ \hfill D_{3\mathsf{t}}& =\mathrm{inverse}-\mathrm{spectral}\mathrm{domain}.\hfill \end{array}$$

The parity domain sits at meridian index $2\mathsf{t}={\pi }_{\mathsf{t}}$, identifying the role of the half-period ${\pi }_{\mathsf{t}}=(\mathsf{p}-1)/2$ in the dictionary.

Proof. 

The statement follows immediately from Theorem 1. At $s=0$, the basis is unchanged. At $s=\mathsf{t}$, the basis is transformed by the normalized Fourier quarter-turn. At $s=2\mathsf{t}={\pi }_{\mathsf{t}}$, the basis is transformed by parity J. At $s=3\mathsf{t}$, the basis is transformed by the inverse Fourier quarter-turn. □

Corollary 2 

(Distinct representation domains). For $\mathsf{t}\ge 2$, the $4\mathsf{t}$ meridional representation domains $D_0,D_1,\dots ,D_{4\mathsf{t}-1}$ are pairwise distinct.

Proof. 

By Theorem 2, the map $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ is injective for $\mathsf{t}\ge 2$, so the meridional bases ${\mathcal{B}}_s={\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}{\mathcal{B}}_0$ are pairwise distinct, and hence so are the pairs $D_s=(V_{\mathsf{t}},{\mathcal{B}}_s)$. □

This proves the first main thesis: shell meridian rotation is represented algebraically as rotation in representation space.

7. Meridian-Coordinate Zoom

The previous section read each meridian index $s\in {\Phi }_{\mathsf{t}}$ on the representation side, as the basis-change pair $D_s=(V_{\mathsf{t}},{\mathcal{B}}_s)$ acted on by ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$. The present section reads the same meridian index $m\in {\Phi }_{\mathsf{t}}$ on the coordinate side, as the subset $M_m\subset {\mathbb{F}}_{\mathsf{p}}$ acted on by multiplication by $e_{\mathsf{t}}^m$, and records the corresponding scale-shift law. The two readings live on opposite sides of the shell Fourier correspondence and are exact at the finite-field level. The scale-shift law recorded below is the FrFT-meridian repackaging of the framed-rational zoom and scale-periodicity established in ([7] §4.3, Lemma 2); the dictionary between the two formulations is given in Remark 2 below, and a visual depiction of the same $(\mathsf{p}-1)$-step zoom cycle at $\mathsf{p}=13$ appears as Figure 7 of that paper.

For the zoom interpretation we fix the meridian-step interval

$$I_{\mathsf{p}}:=\{0,1,\dots ,{\pi }_{\mathsf{t}}\},$$

(7.1)

of cardinality ${\pi }_{\mathsf{t}}+1=2\mathsf{t}+1$. The meridian $M_m=\{a{e}_{\mathsf{t}}^m:a\in I_{\mathsf{p}}\}$ from (9) is then a labelled subset of ${\mathbb{F}}_{\mathsf{p}}$, traversed in the order $a=0,1,2,\dots ,{\pi }_{\mathsf{t}}$.

Definition 6 

(Meridian-scale map). For $r\in {\Phi }_{\mathsf{t}}$, define

$$S_r:{\mathbb{F}}_{\mathsf{p}}\to {\mathbb{F}}_{\mathsf{p}},S_r\left(x\right):=e_{\mathsf{t}}^{r}x.$$

Since $e_{\mathsf{t}}^r\in {\mathbb{F}}_{\mathsf{p}}^{\times }$ is a unit, each $S_r$ is a bijection of ${\mathbb{F}}_{\mathsf{p}}$, with inverse $S_{-r}$. The collection $\{S_r:r\in {\Phi }_{\mathsf{t}}\}$ is a cyclic group under composition.

Proposition 2 

(Meridian-scale covariance). For all $m,r\in {\Phi }_{\mathsf{t}}$,

$$S_r\left(M_m\right)=M_{m+r}.$$

In particular, the meridian shift $m\mapsto m+1$ multiplies the meridian direction by $e_{\mathsf{t}}$, and the meridian shift $m\mapsto m-1$ multiplies it by $e_{\mathsf{t}}^{-1}$.

Proof. 

Using Definition 6 and the meridian definition,

$$S_r\left(M_m\right)=\{e_{\mathsf{t}}^r\left(a{e}_{\mathsf{t}}^m\right):a\in I_{\mathsf{p}}\}=\{a{e}_{\mathsf{t}}^{m+r}:a\in I_{\mathsf{p}}\}=M_{m+r}.$$

□

Remark 2 

(Identification with the framed-rational zoom). Proposition 2 and the cyclic group $\{S_r:r\in {\Phi }_{\mathsf{t}}\}$ are the FrFT-meridian repackaging of the framed-rational scale-periodicity developed in ([7] §4.3). In that setting, ${\mathbb{F}}_{\mathsf{p}}$ is presented as the coordinate ring of framed rationals

$$[x,n]:={\displaystyle \frac{x}{g^n}},0\le x<\mathsf{p},n\in \mathbb{N},$$

with $g\in {\mathbb{F}}_{\mathsf{p}}^{\times }$ a fixed primitive generator, and the zoom map is

$$Z:[x,n]⟼[x,n+1].$$

The dictionary to the present section is

$$g⟷e_{\mathsf{t}},[x,n]⟷x·e_{\mathsf{t}}^{-n},Z⟷S_{-1},$$

so that increasing the framed-rational scale level n by one — which divides the grid step by g, i.e. zoomsin— corresponds to the meridian shift $m\mapsto m-1$, in agreement with the orientation convention of Theorem 3 below. The coordinate range $I_{\mathsf{p}}=\{0,1,\dots ,{\pi }_{\mathsf{t}}\}$ used here is the meridian half-range; the algebra paper uses the full range $\{0,1,\dots ,\mathsf{p}-1\}$. With these identifications, the $(\mathsf{p}-1)$-periodicity $S_{r+(\mathsf{p}-1)}=S_r$ is exactly Lemma 2 of [7], which states $G_{n+(\mathsf{p}-1)}=G_n$ on the framed-rational grids $G_n=\{[x,n]:0\le x<\mathsf{p}\}$. A visual rendering of the resulting cyclic zoom ladder, drawn at $\mathsf{p}=13$, is Figure 7 of that paper; the textual zoom ladder of Example 1 below gives the same information for $\mathsf{p}=13$ with primitive generator $e_{\mathsf{t}}=2$.

Corollary 3 

(Effective shell step). The meridian $M_m$ realizes the prime-meridian coordinate vector $a=0,1,2,\dots ,{\pi }_{\mathsf{t}}$ at effective shell step $e_{\mathsf{t}}^m$:

$$0,e_{\mathsf{t}}^m,2e_{\mathsf{t}}^m,\dots ,{\pi }_{\mathsf{t}}{e}_{\mathsf{t}}^m.$$

Proof. 

The elements of $M_m$, listed by the ordering of $I_{\mathsf{p}}$, are $a{e}_{\mathsf{t}}^m$ for $a=0,1,\dots ,{\pi }_{\mathsf{t}}$. The difference of consecutive entries in this ordering is $(a+1)e_{\mathsf{t}}^m-a{e}_{\mathsf{t}}^m=e_{\mathsf{t}}^m$, which is the effective step. □

To compare different meridians as scale-shifted copies of one another, we record an observer-side comparison chart, in the same spirit as the angle comparison ${\alpha }_s=(s/\mathsf{t})(\pi /2)$ introduced in Section 2.

Definition 7 

(Local scale chart). Alocal scale chartfor the meridian frame is a partial comparison map in which the shell unit $e_{\mathsf{t}}$ is represented by a positive real scale factor

$$\lambda >1.$$

Under this comparison, the effective shell step $e_{\mathsf{t}}^m$ is read as the real scale step ${\lambda }^m$.

The chart adds no new finite-field structure. It supplies the external observer vocabulary in which the words “finer” and “coarser” acquire a meaning; ${\mathbb{F}}_{\mathsf{p}}$ itself carries no compatible global order.

Theorem 3 

(Meridian zoom). Fix a local scale chart with $e_{\mathsf{t}}\mapsto \lambda >1$. Then:

(i)

$M_0$ represents the meridian-coordinate vector at step 1.

(ii)

$M_m$ represents the same meridian-coordinate vector at step ${\lambda }^m$ for every $m\in {\Phi }_{\mathsf{t}}$.

(iii)

The forward shift $m\mapsto m+1$ multiplies the local scale step by λ and is the algebraic form of zoom-out.

(iv)

The inverse shift $m\mapsto m-1$ multiplies the local scale step by ${\lambda }^{-1}$ and is the algebraic form of zoom-in.

The cardinal meridians are read as

$$M_0\leftrightarrow 1,M_{\mathsf{t}}\leftrightarrow {\lambda }^{\mathsf{t}},M_{2\mathsf{t}}=M_{{\pi }_{\mathsf{t}}}\leftrightarrow {\lambda }^{{\pi }_{\mathsf{t}}},M_{3\mathsf{t}}\leftrightarrow {\lambda }^{3\mathsf{t}}.$$

Proof. 

Item (i) is the case $m=0$ of Corollary 3, where the effective step is $e_{\mathsf{t}}^0=1$ and the local-scale image is ${\lambda }^0=1$. Item (ii) is the general case of the same corollary together with Definition 7, which sends the effective shell step $e_{\mathsf{t}}^m$ to ${\lambda }^m$. Items (iii) and (iv) follow because Proposition 2 gives $S_{\pm 1}\left(M_m\right)=M_{m\pm 1}$, so the effective step is multiplied by $e_{\mathsf{t}}^{\pm 1}$, read as ${\lambda }^{\pm 1}$ in the chart. The cardinal values are the cases $m\in \{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$. □

Remark 3 

(Reversible zoom versus lossy coarse-graining). Each $S_r$ is a bijection of ${\mathbb{F}}_{\mathsf{p}}$, so meridian zoom is reversible at the finite-field level: $S_{-r}\circ S_r={\mathrm{id}}_{{\mathbb{F}}_{\mathsf{p}}}$. A genuinely lossy coarse-graining requires an additional observer map: a bounded comparison window that discards points outside it, a resolution map that identifies nearby comparison-chart points, a projection from ${\mathbb{F}}_{\mathsf{p}}$ onto a smaller measured alphabet, or a truncation of the meridian-coordinate vector. The precise statement is therefore

meridian shift is exact reversible zoom; coarse-graining is zoom followed by observer readout.

This keeps the proposition algebraically exact while preserving the intended observational interpretation.

Example 1 

(Meridian-step ladder at $\mathsf{p}=13$). Let $\mathsf{p}=13$, $\mathsf{t}=3$, ${\pi }_{\mathsf{t}}=6$, $e_{\mathsf{t}}=2$, $i_{\mathsf{t}}=8$. The meridian-step interval is $I_{\mathsf{p}}=\{0,1,\dots ,6\}$ and the first four meridians of the zoom ladder are

$$\begin{array}{ccc}{M}_0\hfill & \mathrm{step}1\hfill & \{0,1,2,3,4,5,6\},\hfill \\ M_1\hfill & \mathrm{step}2\hfill & \{0,2,4,6,8,10,12\},\hfill \\ M_2\hfill & \mathrm{step}4\hfill & \{0,4,8,12,3,7,11\},\hfill \\ M_3\hfill & \mathrm{step}8\hfill & \{0,8,3,11,6,1,9\}.\hfill \end{array}$$

The effective step doubles at every meridian shift, in agreement with Theorem 3 for $\lambda =2$. Starting from $M_2$ the listing wraps around ${\mathbb{F}}_{13}$; the corresponding “coarser” reading is observer-local rather than a global order statement on ${\mathbb{F}}_{13}$.

Remark 4 

(Scope of the unification). The meridian cycle ${\Phi }_{\mathsf{t}}$ carries two coherent algebraic readings: the representation-side rotation

$$s⟼{\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}\in End\left(V_{\mathsf{t}}\right),$$

faithful on the full cycle for $\mathsf{t}\ge 2$ by Theorem 2, and the coordinate-side scale shift

$$r⟼S_r:{\mathbb{F}}_{\mathsf{p}}\to {\mathbb{F}}_{\mathsf{p}},$$

an exact reversible bijection for every r. The two families act on carriers of different dimension over ${\mathbb{F}}_{\mathsf{p}}$ — $V_{\mathsf{t}}$ is $4\mathsf{t}$-dimensional, while ${\mathbb{F}}_{\mathsf{p}}$ is one-dimensional over itself — and are therefore not equal as operators. What they share is the cyclic index group ${\Phi }_{\mathsf{t}}$ and the cardinal interpretations at $\{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$, both built from the shell constants $e_{\mathsf{t}}$, $i_{\mathsf{t}}$, ${\pi }_{\mathsf{t}}$. The phrasing “two facets of one rotation” is therefore to be read at the cyclic-group level and at the level of cardinal labeling, with the operator-level unification — an explicit intertwining map relating $S_r$ on ${\mathbb{F}}_{\mathsf{p}}$ to ${\mathcal{F}}_{\mathsf{t}}^{\left[r\right]}$ on $V_{\mathsf{t}}$ — recorded as the natural next target of the programme and left to future work.

The scale-shift interpretation of $S_r$ as multiplication by ${\lambda }^r$ in a local scale chart holds in the zoom-like range $e_{\mathsf{t}}^r<\mathsf{t}$, where the shell-value of the cumulative dilation remains below the radius parameter; outside that range, the same algebraic action wraps around ${\mathbb{F}}_{\mathsf{p}}^{\times }$ and manifests as phase intermixing across the cycle rather than as a clean external scale shift. The coordinate-side reading was developed at the framed-rational level in ([7] §4.3, Lemma 2 and Figure 7); the present section recasts it in FrFT-meridian language so that the two layers share the same index group ${\Phi }_{\mathsf{t}}$.

8. Finite-Field Weil Dictionary

The previous construction is internal to the cyclic shell Fourier structure over ${\mathbb{F}}_{\mathsf{p}}$. We now connect it to the finite-field Weil reading.

Let ${\mathbb{F}}_{\mathsf{p}}^2$ carry the quadratic form

$$q(x,y)=x^2+y^2.$$

Because $\mathsf{p}=4\mathsf{t}+1$, the element $i_{\mathsf{t}}$ exists in ${\mathbb{F}}_{\mathsf{p}}$ and $i_{\mathsf{t}}^2=-1$. The corresponding rotation group is

$$SO(2,{\mathbb{F}}_{\mathsf{p}})=\left\{\left(\begin{array}{cc}c& -d\\ d& c\end{array}\right):c^2+d^2=1\right\}.$$

For each meridian index $s\in {\Phi }_{\mathsf{t}}$, set

$$z_s:=e_{\mathsf{t}}^s\in {\mathbb{F}}_{\mathsf{p}}^{\times },$$

(8.1)

and define

$$c_s:={\displaystyle \frac{z_s+z_s^{-1}}{2}},d_s:={\displaystyle \frac{z_s-z_s^{-1}}{2i_{\mathsf{t}}}}.$$

(8.2)

Finally set

$$R_s:=\left(\begin{array}{cc}{c}_s& -d_s\\ d_s& c_s\end{array}\right).$$

(8.3)

Lemma 5. 

For every $s\in {\Phi }_{\mathsf{t}}$, $R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}})$.

Proof. 

Since $i_{\mathsf{t}}^2=-1$,

$$\begin{array}{cc}\hfill c_s^2+d_s^2& ={\displaystyle \frac{{(z_s+z_s^{-1})}^2}{4}}+{\displaystyle \frac{{(z_s-z_s^{-1})}^2}{4i_{\mathsf{t}}^2}}\hfill \\ \hfill & ={\displaystyle \frac{{(z_s+z_s^{-1})}^2-{(z_s-z_s^{-1})}^2}{4}}={\displaystyle \frac{4}{4}}=1.\hfill \end{array}$$

Hence, $R_s$ preserves $x^2+y^2$ and has determinant 1. □

Proposition 3 

(FRC constants parametrize finite rotations). The map

$${\rho }_{\mathrm{rot}}:{\Phi }_{\mathsf{t}}\to SO(2,{\mathbb{F}}_{\mathsf{p}}),s\mapsto R_s,$$

is a group isomorphism.

Proof. 

The formulas (23) are exactly the change of variables

$$z_s=c_s+i_{\mathsf{t}}{d}_s,z_s^{-1}=c_s-i_{\mathsf{t}}{d}_s.$$

Multiplication of the elements $z_s$ and $z_r$ gives

$$z_s{z}_r=z_{s+r}.$$

Under the identification

$$z=c+i_{\mathsf{t}}d\mapsto \left(\begin{array}{cc}c& -d\\ d& c\end{array}\right),$$

multiplication of z corresponds to multiplication of matrices. Hence, $R_s{R}_r=R_{s+r}$. Since $e_{\mathsf{t}}$ is primitive, $s\mapsto z_s$ is an isomorphism ${\Phi }_{\mathsf{t}}\simeq {\mathbb{F}}_{\mathsf{p}}^{\times }$. The above identification gives ${\mathbb{F}}_{\mathsf{p}}^{\times }\simeq SO(2,{\mathbb{F}}_{\mathsf{p}})$, so ${\rho }_{\mathrm{rot}}$ is an isomorphism. □

The cardinal meridians become the usual finite quarter-turns:

$$\begin{array}{cc}\hfill R_0& =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),R_{\mathsf{t}}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\hfill \\ \hfill R_{2\mathsf{t}}& =\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right),R_{3\mathsf{t}}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).\hfill \end{array}$$

Let ${\omega }_{\mathsf{p}}$ denote the finite Weil representation of $SL(2,{\mathbb{F}}_{\mathsf{p}})$ in a chosen Schrodinger model. We only need its restriction to $SO(2,{\mathbb{F}}_{\mathsf{p}})\subset SL(2,{\mathbb{F}}_{\mathsf{p}})$ and the standard fact that the Weyl element

$$w=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

(8.4)

is represented, up to the usual scalar convention, by the finite Fourier transform [9,10]. Fix the scalar convention so that ${\omega }_{\mathsf{p}}{\left(w\right)}^2={\omega }_{\mathsf{p}}(-I)$ and ${\omega }_{\mathsf{p}}{\left(w\right)}^4=I$.

Definition 8 

(Weil FrFT pulled back by FRC constants). Define

$${FrFT}_{\mathsf{t}}^{\mathrm{Weil}}\left(s\right):={\omega }_{\mathsf{p}}\left(R_s\right),s\in {\Phi }_{\mathsf{t}}.$$

(8.5)

Definition 9 

(Cardinal-skeleton equivalence of FrFT realizations). Two FrFT families $A_s$ and $B_s$ are calledcardinal-skeleton equivalentwhen there exists one cyclic index group ${\Phi }_{\mathsf{t}}$ such that both are representations of ${\Phi }_{\mathsf{t}}$ and the four cardinal elements $0,\mathsf{t},2\mathsf{t},3\mathsf{t}$ are sent respectively to identity, Fourier quarter-turn, parity, and inverse Fourier quarter-turn in their corresponding state spaces.

This definition is intentionally representation-theoretic. It compares the order-4 Fourier sub-skeleton of ${\Phi }_{\mathsf{t}}$ and the cardinal interpretations attached to its four elements. It does not require the two realizations to act on vector spaces of the same dimension or over the same coefficient field. It also does not constrain the two families at intermediate (non-cardinal) meridian indices $s\notin \{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$.

Theorem 4 

(FRC–Weil cardinal-skeleton dictionary). The FRC-native FrFT family ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ and the finite-field Weil FrFT family ${FrFT}_{\mathsf{t}}^{\mathrm{Weil}}\left(s\right)$ are cardinal-skeleton equivalent in the sense of Definition 9. The exact transfer is the dictionary

$$M_s⟷s\in {\Phi }_{\mathsf{t}}⟷z_s=e_{\mathsf{t}}^s⟷R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}}),$$

(8.6)

and on the cardinal indices

$$\begin{array}{cc}\hfill M_0& ⟷R_0=I,\hfill \\ \hfill M_{\mathsf{t}}& ⟷R_{\mathsf{t}}=w,\hfill \\ \hfill M_{2\mathsf{t}}=M_{{\pi }_{\mathsf{t}}}& ⟷R_{2\mathsf{t}}=-I,\hfill \\ \hfill M_{3\mathsf{t}}& ⟷R_{3\mathsf{t}}=w^{-1},\hfill \end{array}$$

where the parity meridian $M_{2\mathsf{t}}=M_{{\pi }_{\mathsf{t}}}$ uses the half-period ${\pi }_{\mathsf{t}}=(\mathsf{p}-1)/2$. The dictionary identifies the order-4 Fourier sub-skeleton of ${\Phi }_{\mathsf{t}}$ inside $SO(2,{\mathbb{F}}_{\mathsf{p}})$.

Proof. 

By Theorem 1, $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ is a representation of ${\Phi }_{\mathsf{t}}$ with the cardinal values listed in (18). By Proposition 3, $s\mapsto R_s$ is an isomorphism ${\Phi }_{\mathsf{t}}\simeq SO(2,{\mathbb{F}}_{\mathsf{p}})$. Since ${\omega }_{\mathsf{p}}$ is a representation of $SL(2,{\mathbb{F}}_{\mathsf{p}})$, its restriction to $SO(2,{\mathbb{F}}_{\mathsf{p}})$ makes $s\mapsto {\omega }_{\mathsf{p}}\left(R_s\right)$ a representation of the same cyclic group ${\Phi }_{\mathsf{t}}$. The cardinal values follow from the explicit matrices $R_0$, $R_{\mathsf{t}}$, $R_{2\mathsf{t}}$, and $R_{3\mathsf{t}}$ together with the chosen Weil normalization. Therefore both families satisfy Definition 9. □

Remark 5 

(Scope of the dictionary). Theorem 4 pins down agreement of the two FrFT families only on the four cardinal indices $\{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$, which generate the order-4 Fourier sub-skeleton of ${\Phi }_{\mathsf{t}}$. Two cardinal-skeleton equivalent families may therefore differ freely at intermediate meridian indices $s\notin \{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$. Faithfulness of $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ on the full meridian cycle ${\Phi }_{\mathsf{t}}$ is established separately by Theorem 2, but it is a statement about the FRC-native family in isolation rather than a comparison with the Weil family.

The FRC-native family ${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ acts on ${\mathbb{F}}_{\mathsf{p}}^{{\Phi }_{\mathsf{t}}}$ and is internal to the shell Fourier cycle. The standard Schrodinger model of the Weil representation usually acts on functions on the additive group ${\mathbb{F}}_{\mathsf{p}}$ with complex or cyclotomic coefficients. Genuine isomorphism of the two families is therefore unavailable: they act on spaces of different dimension over different coefficient rings. Equality of a chosen operator model at intermediate s is obtained only after selecting a common realization and a scalar convention.

9. Worked Example: $\mathsf{p}=\mathbf{13}$

Let

$$\mathsf{p}=13,\mathsf{t}=3,n=\mathsf{p}-1=12.$$

The element $e_{\mathsf{t}}=2$ is primitive in ${\mathbb{F}}_{13}^{\times }$. Therefore,

$$i_{\mathsf{t}}=e_{\mathsf{t}}^{\mathsf{t}}=2^3=8,$$

and

$$i_{\mathsf{t}}^2=8^2=64=12=-1(mod13).$$

The meridian cycle has twelve representation domains:

$$D_0,D_1,\dots ,D_{11}.$$

The cardinal domains are

$$D_0=\mathrm{spatial},D_3=\mathrm{spectral},D_6=\mathrm{parity},D_9=\mathrm{inverse}-\mathrm{spectral}.$$

The quarter-turn rotation matrix is obtained from $z_{\mathsf{t}}=e_{\mathsf{t}}^{\mathsf{t}}=8$. Since $8^{-1}=5$ in ${\mathbb{F}}_{13}$,

$$c_{\mathsf{t}}={\displaystyle \frac{8+5}{2}}=0,d_{\mathsf{t}}={\displaystyle \frac{8-5}{2·8}}=1.$$

Thus,

$$R_{\mathsf{t}}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}0& 12\\ 1& 0\end{array}\right)\mathrm{over}{\mathbb{F}}_{13}.$$

On the coordinate side, the same shell gives the zoom ladder $M_0=\{0,1,2,3,4,5,6\}$ (step 1), $M_1=\{0,2,4,6,8,10,12\}$ (step 2), $M_2=\{0,4,8,12,3,7,11\}$ (step 4), $M_3=\{0,8,3,11,6,1,9\}$ (step 8), illustrating Theorem 3 at $\lambda =2$. The wrap-around visible from $M_2$ onward shows that “coarser” is an observer-local statement rather than a global order claim on ${\mathbb{F}}_{13}$.

The exact arithmetic checks used for this paper are shown in Table 1. The entries report exact modular computations; no floating-point arithmetic is involved.

10. Discussion and Conclusion

The construction gives a finite replacement for the continuum FrFT based on shell meridians rather than real angles. It has three immediate consequences.

First, the Fourier transform becomes a literal algebraic quarter-turn in representation space. The normalization ${\mathcal{F}}_{\mathsf{t}}=i_{\mathsf{t}}{W}_{\mathsf{t}}$ is essential: the unnormalized ring Fourier matrix satisfies $W_{\mathsf{t}}^2=-J$, while the normalized shell operator satisfies ${\mathcal{F}}_{\mathsf{t}}^2=J$ and hence realizes the standard Fourier-parity four-cycle exactly.

Second, the fractional transforms are canonical. They are not arbitrary choices of fractional matrix powers. The projectors ${\Pi }_{\ell }$ are polynomials in ${\mathcal{F}}_{\mathsf{t}}$, and the replacement $i_{\mathsf{t}}^{\ell }\mapsto e_{\mathsf{t}}^{\ell s}$ is fixed by the shell generator $e_{\mathsf{t}}$. Thus the one-step transform ${\mathcal{F}}_{\mathsf{t}}^{\left[1\right]}$ is an exact $\mathsf{t}$-th root of the shell Fourier quarter-turn.

Third, the finite-field Weil interpretation is not external to the shell. The constants $e_{\mathsf{t}}$, $i_{\mathsf{t}}$, and ${\pi }_{\mathsf{t}}$ give the exact passage from a meridian index to a finite-field rotation matrix. The same element $s\in {\mathbb{Z}}_{4\mathsf{t}}$ can be read as a shell meridian, as the nonzero field element $e_{\mathsf{t}}^s$, or as the rotation $R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}})$; on the four cardinal indices it can also be read as the Weil operator ${\omega }_{\mathsf{p}}\left(R_s\right)$, and Theorem 4 establishes the FRC-native and Weil families as cardinal-skeleton equivalent. At intermediate meridian indices the two readings act on spaces of different dimension over different coefficient rings, and are not required to coincide.

Fourth, the meridian cycle has a coordinate-side counterpart to the representation-side rotation. Multiplication by $e_{\mathsf{t}}^r$ sends $M_m$ to $M_{m+r}$ exactly, and in any local scale chart $e_{\mathsf{t}}\mapsto \lambda >1$ this realizes the meridian step as a reversible zoom: forward shift coarsens the local step, inverse shift refines it. The finite-field map $S_r$ is a bijection of ${\mathbb{F}}_{\mathsf{p}}$; lossy coarse-graining requires an additional observer readout. The meridian cycle of a symmetry-complete shell therefore admits three coherent readings of one common index: a representation-side rotation $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$, faithful on the full cycle for $\mathsf{t}\ge 2$; a finite-field Weil rotation $s\mapsto R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}})$, cardinal-skeleton equivalent to the FRC-native family; and a coordinate-side reversible zoom $r\mapsto S_r$, exact at the finite-field level. The first two are linked by the cardinal-skeleton dictionary; connecting the first and third in a single intertwining theorem is a natural target for future work.

Physical interpretation The continuum FrFT is established as the evolution operator of the quantum harmonic oscillator, ${\mathcal{F}}_{\alpha }$ acting as the propagator at time $\alpha$ [1]. The finite meridian rotation $s\mapsto {\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}$ described by Theorem 1 therefore admits a natural reading as a finite, chronon-indexed Schrödinger evolution on the symmetry-complete shell, with the four-cycle ${\mathcal{F}}_{\mathsf{t}}^4=I$ as its closure. A finite Schrödinger–Dirac formalism built on exactly this shell rotation is developed in [12]. Furthermore, Theorem 3 invites a parallel cosmological reading: in any local scale chart $e_{\mathsf{t}}\mapsto \lambda >1$, the meridian shift realizes a uniform exponential dilation by factor $\lambda$ per chronon step, so the meridian cycle ${\Phi }_{\mathsf{t}}$ supplies an exact finite analogue of a Hubble-like expansion of the scale frame. Establishing this reading as a viable physical theory will require extensive dedicated programmatic work and is left to future research on both wave-function-level, and the cosmological extremes of the epistemological scale.

In conclusion, for every symmetry-complete prime shell $\mathsf{p}=4\mathsf{t}+1$, the framed datum ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$ determines an exact finite-field fractional Fourier transform. The prime meridian is the spatial domain. The quarter-turn meridian is the spectral domain. The half-turn meridian $M_{2\mathsf{t}}=M_{{\pi }_{\mathsf{t}}}$ is the parity domain. For $\mathsf{t}\ge 2$, every intermediate meridian is a genuinely distinct intermediate representation domain.

The main algebraic construction is

$${\mathcal{F}}_{\mathsf{t}}^{\left[s\right]}={\displaystyle \sum _{\ell =0}^3}{e}_{\mathsf{t}}^{\ell s}{\Pi }_{\ell },s\in {\mathbb{Z}}_{4\mathsf{t}},$$

(10.1)

where the projectors ${\Pi }_{\ell }$ are canonical polynomials in the normalized finite Fourier quarter-turn ${\mathcal{F}}_{\mathsf{t}}=i_{\mathsf{t}}{W}_{\mathsf{t}}$. This family satisfies exact additivity, exact cardinal values, and exact periodicity, and is faithful on the meridian cycle whenever $\mathsf{t}\ge 2$.

The finite-field Weil interpretation is equivalent on the cardinal Fourier sub-skeleton. The shell constants provide the transfer

$$M_s⟷e_{\mathsf{t}}^s⟷R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}})⟷{\omega }_{\mathsf{p}}\left(R_s\right),$$

(10.2)

and Theorem 4 matches the FRC-native and Weil families on the four cardinal meridians $\{0,\mathsf{t},2\mathsf{t},3\mathsf{t}\}$. The two readings are not literally equal on the full cycle, because they act on spaces of different dimension over different coefficient rings; what they share is the order-4 Fourier sub-skeleton and its cardinal interpretation.

The coordinate-side reading of the meridian cycle is supplied by Theorem 3: the multiplicative map $S_r:{\mathbb{F}}_{\mathsf{p}}\to {\mathbb{F}}_{\mathsf{p}}$, $S_r\left(x\right)=e_{\mathsf{t}}^{r}x$, sends $M_m$ to $M_{m+r}$, and in a local scale chart $e_{\mathsf{t}}\mapsto \lambda >1$ this becomes reversible meridian-step zoom-out (or zoom-in, for the inverse shift). The same meridian index $s\in {\mathbb{Z}}_{4\mathsf{t}}$ therefore admits a representation-side reading, a Weil rotation reading, and a coordinate-side scale-shift reading, all derived from the framed shell datum ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$.