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)

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 and canonically over the arithmetic symmetry shells of finite fields introduced and explored in the finite ring continuum (FRC) framework [7,8]. In FRC, the shells ${\mathbb{F}}_{\mathsf{p}}$ of information capacity $\kappa$ and prime cardinality $\mathsf{p}=4\kappa +1$ are presented with the framed datum ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$ of a chronon, an origin, a unit, and a chosen primitive generator. The frame determines the shell’s structural residues: the oriented quarter-turn $\mathrm{i}=-{\mathsf{g}}^{\kappa }={\mathsf{g}}^{-\kappa }$ with ${\mathrm{i}}^2=-1$, the half-period $\pi =2\kappa$ with ${\mathsf{g}}^{\pi }=-1$, and the exponential unit $e={\mathsf{g}}^{\mathrm{i}}$, carrying the roles of the imaginary unit, the half circle period constant, and Euler’s e; Section 3 supplies the formal layer, the orientation convention, and the generator covariance. The half-period $\pi$ is the shell’s own residue, the finite seat of the circle constant; the continuum $\pi$ appears below only inside labelled classical comparisons (§2). The operator constructions use only the generator–turn pair $(\mathsf{g},\mathrm{i})$.

The framed shell and its meridian–latitude geometry, generated by the additive and multiplicative actions over ${\mathbb{F}}_{\mathsf{p}}$, were introduced in the same FRC framework: the combinatorial 2-sphere complex ${\mathcal{S}}_{\mathsf{p}}$ of Figure 1 is the ensemble of vertices $v_{a,m}=a{\mathsf{g}}^m$, each meridian $M_m$ the additive half-orbit at fixed meridian index m, each latitude $L_a$ the multiplicative orbit at fixed additive coordinate a; the formal meridian definition, with its endpoint convention, is Equation (7) of Section 3. Figure 1 shows the specific case ${\mathbb{F}}_{13}(\tau ;0,1,2)$, with the prime meridian $M_0$ and the quarter-turn meridian $M_{\kappa }$ in blue and red, the latitudes $L_a$ as green orbits, and the observer’s origin at the vertex $v_{0,0}=0$.

The same FRC framework reconstructs the roles of e, $\pi$, and i from finite arithmetic within this finite Euclidean shell layer, and introduces the discrete shell Fourier transform W used below [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, using only the finite-field layer.

The main claim is that every shell meridian is a representation domain (Figure 2; formalised in Section 3). The prime meridian $M_0$ is the spatial domain. The meridian $M_{\kappa }$ — the quarter-turn meridian, whose great circle carries the imaginary axis $\pm \mathrm{i}$ — is the spectral domain. The meridian $M_{2\kappa }$ is the parity domain, and $M_{3\kappa }$ is the inverse-spectral domain. The finite FrFT is then the exact transform that rotates the representation from $M_0$ to $M_s$.

• Information-theoretic reading.

The quarter-turn between conjugate representations is, in information terms, the generator of complementarity. For a mutually unbiased pair the discrete entropic uncertainty relation [9,10,11,12] bounds the summed Shannon entropy of the two readouts below by $log(\mathsf{p}-1)$, the finite-dimensional sharpening of the Heisenberg principle. The finite shell makes this relation exact and parameter-free: the spatial and spectral meridians are a Fourier-conjugate (mutually unbiased) pair, the bound is saturated, and the fractional family interpolates the readout entropy along the meridian cycle (Proposition 7). This places the construction in the entropy layer of the FRC programme, where the shell dynamics is exactly reversible, and entropy is produced only by the observer’s finite-resolution readout [13].

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 3 (faithfulness on the full meridian cycle). The second is the finite-field Weil dictionary of Theorem 5, which matches the FRC-native family to the Weil realization on the cardinal Fourier skeleton. The third is the meridian-coordinate zoom of Theorem 4, which reads the same meridian cycle on the shell-coordinate side: the multiplicative map $S_r:x\mapsto {\mathsf{g}}^{r}x$ on ${\mathbb{F}}_{\mathsf{p}}$ sends the meridian $M_m$ to $M_{m+r}$, and in a local comparison chart where $\mathsf{g}$ 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.

The construction itself is compact. On the module $V={\mathbb{F}}_{\mathsf{p}}^{\Phi }$ of vectors over the meridian cycle $\Phi ={\mathbb{Z}}_{4\kappa }$, the shell Fourier matrix W, with entries ${\mathsf{g}}^{jk}$, squares to $-J$, where J is parity; the frame’s quarter-turn normalizes it, ${\mathcal{F}}_{\mathsf{p}}:=\mathrm{i}W$, closing the exact four-cycle ${\mathcal{F}}_{\mathsf{p}}^2=J$, ${\mathcal{F}}_{\mathsf{p}}^4=I$ (Section 4). The fractional family ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ of Equation (14) refines the four eigenvalues of this quarter-turn on the base ${\mathsf{g}}^{-1}$, whose quarter-cycle power is the oriented turn, ${\left({\mathsf{g}}^{-1}\right)}^{\kappa }=\mathrm{i}$ (Section 5).

The second layer connects this FRC-native construction to the finite-field Weil interpretation. The classical Weil representation realizes symplectic transformations of a finite phase plane as operators on a finite state space, with the Fourier transform the operator associated with the Weyl quarter-turn element [14,15]; recent work on arithmetic FrFTs similarly defines finite fractional Fourier transforms from finite arithmetic rotation groups [16]. In the present setting the frame residues supply the exact transfer: the phase value $z_s={\mathsf{g}}^{-s}$ maps the meridian cycle isomorphically onto the finite rotation group $SO(2,{\mathbb{F}}_{\mathsf{p}})$ of the form $x^2+y^2$ (Section 8), so the FRC meridian index and the finite-field Weil rotation index are the same finite datum written in two coordinate languages.

Fractional Fourier transforms over finite fields have a substantial prior literature: the GFrFT line constructs fractional powers of finite-field Fourier and trigonometric transforms through explicit eigenbasis choices [17,18], with applications to image encryption [19], extended to eigenvector-free matrix-function fractional number-theoretic transforms [20], and Floratos–Pavlidis build finite FrFTs from arithmetic rotation groups [16]. The present construction differs in three respects: the projectors are canonical polynomials in the operator itself, inside its rank-four algebra, so no eigenbasis choice and none of its multiplicity ambiguity enters (Remark 4); the normalization ${\mathcal{F}}_{\mathsf{p}}=\mathrm{i}W$ is internal to the shell, available exactly at the cycle length $n=\mathsf{p}-1\equiv -1$ (Remark 3); and the family is the frame’s own, carrying the canonicity, faithfulness, and dictionary statements of Section 5, Section 6, Section 7 and Section 8. What is claimed as new is this canonicity, shell-exactness, and the FRC dictionary, not finite-field fractionalization as such.

The construction is deliberately finite. It does not approximate a real angle. It replaces the continuum angle $\alpha$ by the shell meridian index $s\in {\mathbb{Z}}_{4\kappa }$. It also avoids the ambiguity of arbitrary fractional powers of a discrete Fourier matrix, whose repeated eigenvalues admit many inequivalent fractional definitions [5,21]. The projectors in (14) 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)

and

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

(3)

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 $D_s=(V,{\mathcal{B}}_s)$ on the same finite module V, indexed by a meridian $s\in {\mathbb{Z}}_{4\kappa }$, where the meridional basis ${\mathcal{B}}_s$ is obtained from the standard basis by the finite FrFT ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ 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

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

(4)

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

$$s\in {\mathbb{Z}}_{4\kappa }.$$

(5)

Thus, the values $s=0,\kappa ,2\kappa ,3\kappa$ correspond to the four cardinal representation domains.

3. Symmetry-Complete Shell Data

A symmetry-complete shell, in the sense used here, is built from a positive integer $\kappa$, the shell’s capacity (Remark 1): a symmetry-complete shell exists at $\kappa$ exactly when $\mathsf{p}:=4\kappa +1$ is prime, presented together with the framed datum ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$, the chronon $\tau$ its temporal datum (Remark 1) and $\mathsf{g}\in {\mathbb{F}}_{\mathsf{p}}^{\times }$ 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.

Remark 1

(Shell capacity). The parameter κ denotes the shell’s capacity, its primary structural datum: the count of representation states the meridian cycle carries, $n=\mathsf{p}-1=4\kappa$, matching the entropy bound of Proposition 7; the cardinality $\mathsf{p}=4\kappa +1$ is derived from it. The tuple’s first slot is the chronon τ: the frame’s one free temporal datum, assigned by the observer, the temporal analogue of the fixed second slot 0, its spatial origin. This paper’s constructions use only the structural side (§2). Composed shells combine capacity multiplicatively; a shared clock instead composes by least common multiple of the component cycles [22].

Let $\mathsf{p}=4\kappa +1$ be prime. The group ${\mathbb{F}}_{\mathsf{p}}^{\times }$ is cyclic of order $4\kappa$ [23, Ch. 2]. Choose a primitive generator

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

Define the oriented quarter-turn by

$$\mathrm{i}:=-{\mathsf{g}}^{\kappa }={\mathsf{g}}^{-\kappa }.$$

(6)

The opposite primitive fourth root of unity is $-\mathrm{i}={\mathsf{g}}^{\kappa }$. The orientation is fixed by the chart convention of the programme: the imaginary axis points up from the unit 1, while the phase rotation $m\mapsto {\mathsf{g}}^m$ advances clockwise, so the quarter-turn that carries 1 up is $\mathrm{i}=-{\mathsf{g}}^{\kappa }={\mathsf{g}}^{-\kappa }$ (see Figure 1). Then

$${\mathrm{i}}^2={\mathsf{g}}^{-2\kappa }={\mathsf{g}}^{2\kappa }={\mathsf{g}}^{(\mathsf{p}-1)/2}=-1$$

by Euler’s criterion, since $\mathsf{g}$ is primitive and ${\mathsf{g}}^{4\kappa }=1$. The quarter-turn subgroup is

$$Q=\{1,\mathrm{i},-1,-\mathrm{i}\}.$$

The half-period is

$$\pi :={\displaystyle \frac{\mathsf{p}-1}{2}}=2\kappa ,$$

so that ${\mathsf{g}}^{\pi }=-1$, the finite Euler identity of the frame. The exponential unit is the quarter-turn transform of the generator,

$$e:={\mathsf{g}}^{\mathrm{i}},$$

carrying the structural role of Euler’s e [8]. Identities of ${\mathbb{F}}_{\mathsf{p}}$ do not transfer to the exponent cycle: ${\mathrm{i}}^2=-1$ holds among residues, while exponents of $\mathsf{g}$ compose in ${\mathbb{Z}}_{4\kappa }$. The continuum reading $e^{\mathrm{i}\pi }=-1$ is accordingly a property of the chart class rather than of the frame: $e^{\mathrm{i}\pi }={\mathsf{g}}^{{\mathrm{i}}^2\pi \mathrm{mod}4\kappa }={\mathsf{g}}^{2\kappa \left(\mathrm{i}\mathrm{mod}2\right)}$, equal to $-1$ exactly when the quarter-turn residue $\mathrm{i}$ is odd, and the conjugate reframing $(\mathsf{g},\mathrm{i})\mapsto ({\mathsf{g}}^{-1},-\mathrm{i})$ toggles this parity, so exactly one member of each conjugate pair carries the identity. On the anchor shell ${\mathbb{F}}_{13}(\tau ;0,1,2)$ it holds: $\mathrm{i}=5$, $e^{\mathrm{i}\pi }=6^6=-1$. The exponential unit enters no operator construction of this paper, which uses only the generator–turn pair $(\mathsf{g},\mathrm{i})$.

Within the framed field the quarter-turn $\mathrm{i}$ acts as the imaginary unit of a finite complex plane [7,8]. Figure 1 illustrates this framed complex plane ${}^{\mathsf{f}}\mathbb{C}$ over ${\mathbb{F}}_{13}$: the f-real axis ${}^{\mathsf{f}}\mathbb{R}$ lies along the prime meridian $M_0$, and multiplication by $\mathrm{i}$ carries it to the orthogonal imaginary axis on the quarter-turn meridian $M_{\kappa }$, exhibiting $\mathrm{i}$ as the finite quarter-turn internal to ${\mathbb{F}}_{\mathsf{p}}$.

Remark 2

(The generator is the drive; reframings are relabellings). $\mathsf{g}$ is not a choice: it is the observer’s own drive, one step per chronon, the finite seat of the one-way speed. No predicate of the construction below depends on its residue value; only external labels do. An external comparer writes ${\mathsf{g}}^{\prime }={\mathsf{g}}^u$, $u\in {\mathbb{Z}}_{\mathsf{p}-1}^{\times }$, and sees the meridian relabelling $m\mapsto um$ [Propositions 1–3 [8]; every registered object (operator relations, cardinal values, faithfulness, entropy under the frame’s own readout) is invariant, so the family ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ below is the frame’s own, written in the labels of ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$. Two labels move. The quarter-turn name flips only on the conjugate (matter–antimatter) chart $u\equiv 3\left(\mathrm{mod}4\right)$, factually selected, not relabelled; the exponential-unit name moves already at $u\equiv 1\left(\mathrm{mod}4\right)$: $e^{\prime }={\mathsf{g}}^{u\mathrm{i}}\ne e$ whenever $\mathrm{i}(u-1) /\equiv 0\left(\mathrm{mod}4\kappa \right)$, as at $\mathsf{p}=13$, $\mathsf{g}=2$, $u=5$, where $e^{\prime }=6^5=2\ne 6=e$. The drive is real; e is the chart’s name for it and carries no invariant relation.

The meridian index group is

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

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

$$M_s=\{a{\mathsf{g}}^s:a\in I_{\mathsf{p}}\},$$

(7)

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 $I_{\mathsf{p}}=\{0,1,\dots ,\pi \}$.

Definition 1

(Cardinal meridians). The four cardinal meridians are

$$\begin{array}{cc}\hfill M_0& \mathit{spatial}\mathit{domain},\hfill \\ \hfill M_{\kappa }& \mathit{spectral}\mathit{domain},\hfill \\ \hfill M_{2\kappa }& \mathit{parity}\mathit{domain},\hfill \\ \hfill M_{3\kappa }& \mathit{\text{inverse-spectral}}\mathit{domain}.\hfill \end{array}$$

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

Recall the meridian index group $\Phi ={\mathbb{Z}}_n$, $n:=\mathsf{p}-1=4\kappa$, of Section 3, and let $V:={\mathbb{F}}_{\mathsf{p}}^{\Phi }$ be the module of vectors $v:\Phi \to {\mathbb{F}}_{\mathsf{p}}$. Define the reversal operator $J:V\to V$ by

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

(8)

Clearly $J^2=I$.

Definition 2

(Shell Fourier matrix). The shell Fourier matrix is the linear operator $W:V\to V$ with entries

$${\left(W\right)}_{k,j}={\mathsf{g}}^{jk},j,k\in \Phi .$$

(9)

Equivalently,

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

This is a finite-field Fourier transform over the cyclic phase group $\Phi$. It is internal to ${\mathbb{F}}_{\mathsf{p}}$ because $\mathsf{g}$ 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

$$W^2=-J.$$

Proof. 

For $k,l\in \Phi$,

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

If $k+l\equiv 0\left(\mathrm{mod}n\right)$, then every summand is 1 and the sum is $n=\mathsf{p}-1=-1$ in ${\mathbb{F}}_{\mathsf{p}}$. If $k+l /\equiv 0\left(\mathrm{mod}n\right)$, then ${\mathsf{g}}^{k+l}\ne 1$ and the finite geometric sum is

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

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

Definition 3

(Normalized shell Fourier quarter-turn). Define

$${\mathcal{F}}_{\mathsf{p}}:=\mathrm{i}W.$$

(10)

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

Remark 3

(The normalization is the unitary constant read in the field). Over ${\mathbb{F}}_{\mathsf{p}}$ the cycle length is $n=\mathsf{p}-1\equiv -1$, so the continuum unitary normalization $n^{-1/2}$ of the Fourier matrix becomes a square root of $1/n=-1$: the two unitary normalizations of W available inside ${\mathbb{F}}_{\mathsf{p}}$ are exactly $\pm \mathrm{i}$. The shell quarter-turn is the unitary normalization constant of the discrete Fourier transform, read inside the field.

Proposition 1.

The normalized shell Fourier operator satisfies

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

(11)

Proof. 

Using Lemma 1 and ${\mathrm{i}}^2=-1$,

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

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

Thus, ${\mathcal{F}}_{\mathsf{p}}$ 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 [21]. The construction below avoids this issue by using projectors that are polynomials in ${\mathcal{F}}_{\mathsf{p}}$ itself.

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

$$x^4-1=(x-1)(x-\mathrm{i})(x+1)(x+\mathrm{i}),$$

(12)

with four distinct roots. Define

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

(13)

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

$${\mathcal{F}}_{\mathsf{p}}{\Pi }_{\ell }={\mathrm{i}}^{\ell }{\Pi }_{\ell }.$$

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 ([23], Ch. 5)

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

The relation ${\mathcal{F}}_{\mathsf{p}}{\Pi }_{\ell }={\mathrm{i}}^{\ell }{\Pi }_{\ell }$ follows by multiplying (13) by ${\mathcal{F}}_{\mathsf{p}}$ and reindexing powers modulo 4.    □

Definition 4

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

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

(14)

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 ${\mathrm{i}}^{\ell }$ to ${\mathsf{g}}^{-\ell s}$: the refinement base is the inverse generator ${\mathsf{g}}^{-1}$, whose quarter-cycle power is the oriented turn, ${\left({\mathsf{g}}^{-1}\right)}^{\kappa }=\mathrm{i}$, in accordance with the clockwise phase convention of (6). The projector route separates this family from the eigenbasis constructions of [17,18], which inherit exactly the multiplicity ambiguity the projectors avoid; the matrix-function transforms of [20] share the polynomial-in-the-operator genus, and the separation there is the frame-tied resolution of the branch ambiguity (Remark 4), the shell-internal normalization at $n=\mathsf{p}-1\equiv -1$, full-cycle faithfulness, and the scale-action intertwiner of Proposition 5.

Remark 4

(Classification: the principal framed character lift). Additivity and the cardinal values do not by themselves single out (14): every exponent choice $a_{\ell }\equiv \ell \left(\mathrm{mod}4\right)$ yields an additive family $U_s^{\left(a\right)}={\sum }_{\ell }{\mathsf{g}}^{-a_{\ell }s}{\Pi }_{\ell }$ with the same group law and the same cardinal Fourier skeleton (at $s=\kappa$, ${\mathsf{g}}^{-a_{\ell }\kappa }={\mathrm{i}}^{a_{\ell }}={\mathrm{i}}^{\ell }$), and the faithful members include the power-tower lifts ${\chi }_{\ell }\left(s\right)={\left({\mathsf{g}}^{-us}\right)}^{\ell }$ on the $\mathsf{g}$-frame projectors, $u\equiv 1\left(\mathrm{mod}4\right)$ coprime to $4\kappa$ (Remark 2); a family canonical in another chart uses that chart’s projectors and, when $u^2 /\equiv 1\left(\mathrm{mod}4\kappa \right)$, does not even commute with ${\mathcal{F}}_{\mathsf{p}}$ (machine-verified at $\mathsf{p}=29$, $u=5$), hence lies outside the classified set. Equation (14) is the principal framed character lift: the unique member whose eigenvalue system is the power tower of the meridian phase, ${\chi }_{\ell }\left(s\right)=z_s^{\ell }$ with $z_s={\mathsf{g}}^{-s}$ — each projector’s character the ℓ-th power of one frame datum. Canonicity is therefore frame-internal: the principal lift is the frame’s own, written in the labels of ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$.

Theorem 1

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

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

(15)

for all $s,r\in \Phi$. Moreover,

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

(16)

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

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

Proof. 

Using Lemma 2,

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

This proves additivity. For the cardinal values, use ${\mathsf{g}}^{-\kappa }=\mathrm{i}$. Then

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

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

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

and

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

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

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

   □

The cardinal values of Theorem 1 are correct without faithfulness; they only require the four projectors and the additivity. The full $4\kappa$-domain reading, however — the statement that every shell meridian $s\in \Phi$ gives a representation domain distinct from the others — requires the family $s\mapsto {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ to be injective on $\Phi$. Injectivity in turn requires an odd-index eigenprojector to be nonzero, and the full four-fold refinement requires all four. 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:

$$WJ=JW,$$

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

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

with

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

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

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

(17)

Proof. 

For $k,j\in \Phi$,

$${\left(WJ\right)}_{k,j}={\left(W\right)}_{k,-j}={\mathsf{g}}^{-kj}={\left(W\right)}_{-k,j}={\left(JW\right)}_{k,j},$$

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

Lemma 4

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

$$m_0+m_2=2\kappa +1,m_1+m_3=2\kappa -1.$$

For every $\kappa \ge 1$, $m_0\ge 1$ and $m_2\ge 1$. For every $\kappa \ge 2$, $m_1\ge 1$ and $m_3\ge 1$. At $\kappa =1$ ($\mathsf{p}=5$) 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 $\{0,1,2,3\}=\{0,\kappa ,2\kappa ,3\kappa \}$ 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 ${\mathcal{F}}_{\mathsf{p}}$ (eigenvalues $\pm 1$, ${\mathrm{i}}^0=1$, ${\mathrm{i}}^2=-1$), while $V^{-}$ is the sum of the $\ell =1,3$ eigenspaces (eigenvalues $\pm \mathrm{i}$).

For the eigenvalue $\pm 1$ pair on $V^{+}$, consider the symmetric vector $e_0$. Then ${\left(W{e}_0\right)}_k={\mathsf{g}}^{0·k}=1$ for every k, so $W{e}_0=\mathbf{1}$, the all-ones vector. Thus ${\mathcal{F}}_{\mathsf{p}}{e}_0=\mathrm{i}\mathbf{1}$ has nonzero entries at every index $k\in \Phi$, whereas $e_0$ has support $\left\{0\right\}$. Consequently ${\mathcal{F}}_{\mathsf{p}}{e}_0\notin {\mathbb{F}}_{\mathsf{p}}·e_0$, so ${\mathcal{F}}_{\mathsf{p}}{|}_{V^{+}}$ is not a scalar operator. Since ${\left({\mathcal{F}}_{\mathsf{p}}{|}_{V^{+}}\right)}^2=I$, the minimal polynomial of ${\mathcal{F}}_{\mathsf{p}}{|}_{V^{+}}$ 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 $\pm \mathrm{i}$ pair on $V^{-}$ with $\kappa \ge 2$, consider the antisymmetric vector $v=e_1-e_{-1}$. Then

$${\left(Wv\right)}_k={\mathsf{g}}^k-{\mathsf{g}}^{-k},$$

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

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

Remark 5

(Multiplicities are chart data). The individual multiplicities are labels, not registered structure: the Fourier kernel ${\mathsf{g}}^{jk}$ is quadratic in the frame’s time-labels, so the vertex relabelling $m\mapsto um$ carries ${\mathcal{F}}_{\mathsf{p}}\left(\mathsf{g}\right)$ to ${\mathcal{F}}_{\mathsf{p}}\left({\mathsf{g}}^{u^2}\right)$, and presentations outside one square-class orbit are genuinely non-conjugate — at $\mathsf{p}=13$ the frame $\mathsf{g}=2$ gives $(m_0,m_1,m_2,m_3)=(3,3,4,2)$ while $\mathsf{g}=6$, the same shell and orientation, gives $(4,2,3,3)$ (traces 4 and 9, exact). The results above use only the invariant coarse facts: $m_0,m_2\ge 1$, an odd projector nonzero, and the sums $m_0+m_2=2\kappa +1$, $m_1+m_3=2\kappa -1$. The chart values themselves are classified exactly.

Theorem 2

(Multiplicity dichotomy). Let $G={\sum }_{k\in {\mathbb{Z}}_{4\kappa }}{\mathsf{g}}^{k^2}\in {\mathbb{F}}_{\mathsf{p}}$ be the frame’s exponent-cycle Gauss sum. Then $G=\epsilon (1+\mathrm{i})$ for a sign $\epsilon \in \{\pm 1\}$, and the multiplicity tuple takes exactly two values across the frames of a shell:

$$(m_0,m_1,m_2,m_3)=\left\{\begin{array}{cc}(\kappa ,\kappa ,\kappa +1,\kappa -1),\hfill & \epsilon =+1,\hfill \\ (\kappa +1,\kappa -1,\kappa ,\kappa ),\hfill & \epsilon =-1.\hfill \end{array}\right.$$

Conjugate frames carry opposite patterns, $\epsilon \left({\mathsf{g}}^{-1}\right)=-\epsilon \left(\mathsf{g}\right)$, and on the orientation-preserving class $\epsilon \left({\mathsf{g}}^u\right)=\left(\frac{\kappa }{u}\right)\epsilon \left(\mathsf{g}\right)$ for $u\equiv 1\left(\mathrm{mod}4\right)$ (Jacobi symbol).

Proof. 

Write $G^{*}={\sum }_k{\mathsf{g}}^{-k^2}=G\left({\mathsf{g}}^{-1}\right)$. First, $G{G}^{*}=-2$: substituting $k=l+d$ gives ${\sum }_{k,l}{\mathsf{g}}^{k^2-l^2}={\sum }_d{\mathsf{g}}^{d^2}{\sum }_l{\mathsf{g}}^{2ld}$, the inner geometric sum is $n\equiv -1$ exactly at $2d\equiv 0\left(\mathrm{mod}n\right)$, i.e. $d\in \{0,2\kappa \}$, and ${\mathsf{g}}^{d^2}=1$ at both. Second, $G^2=2\mathrm{i}$: in the readout algebra of Definition 10 the element $\widehat{G}={\sum }_k{X}^{k^2}$ satisfies ${\widehat{G}}^2=2n{X}^{\kappa }$ — the classical evaluation of the twisted quadratic Gauss sum for $4\mid n$ [24] gives ${\sum }_k{\zeta }_n^{u{k}^2}=\left(\frac{\kappa }{u}\right)(1+i)\sqrt{n}$ for $u\equiv 1\left(\mathrm{mod}4\right)$ and $\left(\frac{\kappa }{u}\right)(1-i)\sqrt{n}$ for $u\equiv 3\left(\mathrm{mod}4\right)$, so ${\widehat{G}}^2-2n{X}^{\kappa }$ vanishes at every primitive embedding and is divisible by ${\Phi }_n\left(X\right)$. Reducing, $G^2=2n{\mathsf{g}}^{\kappa }=2(-1)(-\mathrm{i})=2\mathrm{i}$, whence $G=\epsilon (1+\mathrm{i})$ and $G^{*}=-2/G=-\epsilon (1-\mathrm{i})$. Third, the traces: $Tr{\mathcal{F}}_{\mathsf{p}}=\mathrm{i}G$, $Tr{\mathcal{F}}_{\mathsf{p}}^2=TrJ=2$, and ${\mathcal{F}}_{\mathsf{p}}^3=\mathrm{i}WJ$ gives $Tr{\mathcal{F}}_{\mathsf{p}}^3=\mathrm{i}{G}^{*}$. Since the trace of an idempotent is its rank, $m_{\ell }\equiv \frac{1}{4}{\sum }_{r=0}^3{\mathrm{i}}^{-\ell r}Tr{\mathcal{F}}_{\mathsf{p}}^r\left(\mathrm{mod}\mathsf{p}\right)$, which evaluates, using $n\equiv -1$ and $\kappa \equiv -\frac{1}{4}$, to the two displayed patterns according to $\epsilon$; since $0\le m_{\ell }\le n<\mathsf{p}$, the congruences pin the integers. The conjugate law is $G\left({\mathsf{g}}^{-1}\right)=G^{*}=-\epsilon (1-\mathrm{i})=-\epsilon (1+{\mathrm{i}}^{\prime })$ with ${\mathrm{i}}^{\prime }=-\mathrm{i}$ the conjugate frame’s quarter-turn; the transformation law is the framed reduction of the twisted evaluation, $G\left({\mathsf{g}}^u\right)=\left(\frac{\kappa }{u}\right)G\left(\mathsf{g}\right)$ for $u\equiv 1\left(\mathrm{mod}4\right)$.    □

Machine verification: exact on all 38 primitive frames of $\mathsf{p}\in \{5,13,17,29,37\}$, the two classes equally populated at every shell; the Table 1 frames with $\kappa \ge 2$ all lie in class $\epsilon =+1$, and at $\mathsf{p}=5$ the two frames realize the two patterns, $(2,0,1,1)$ at $\mathsf{g}=2$ ($\epsilon =-1$) and $(1,1,2,0)$ at $\mathsf{g}=3$ ($\epsilon =+1$). Under the continuum chart the dichotomy is the Y-bifurcation of Definition 10: the in-field Gauss root reduces to $\epsilon \mathrm{i}$, so the frame’s decree $Y\mapsto -\mathrm{i}$ occupies the $\mp S$ component according to $\epsilon$, and the two patterns are the classical DFT multiplicity counts of $\mp S$ [21] transported through the label flip $\rho \left(X^{\kappa }\right)=-\mathrm{i}$.

Theorem 3

(Faithfulness on the meridian cycle). For every $\kappa \ge 1$, the map

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

is injective. In particular, the $4\kappa$ meridional images are pairwise distinct.

Proof. 

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

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

By Lemma 4, ${\Pi }_1\ne 0$ for $\kappa \ge 2$; take $m=1$. For $\kappa =1$ exactly one of ${\Pi }_1,{\Pi }_3$ is nonzero; take m to be its index, so that $gcd(m,4\kappa )=gcd(m,4)=1$. In either case ${\mathsf{g}}^{-m(s-r)}=1$ forces $m(s-r)\equiv 0\left(\mathrm{mod}4\kappa \right)$, since $\mathsf{g}$ has order $4\kappa$ in ${\mathbb{F}}_{\mathsf{p}}^{\times }$, and coprimality gives $s\equiv r\left(\mathrm{mod}4\kappa \right)$.    □

6. Meridional Representation Domains

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

Definition 5

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

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

The meridional representation domain $D_s$ is the pair

$$D_s:=(V,{\mathcal{B}}_s).$$

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

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

(18)

Corollary 1

(Cardinal domains). The four cardinal domains are

$$\begin{array}{cc}\hfill D_0& =\mathit{spatial}\mathit{domain},\hfill \\ \hfill D_{\kappa }& =\mathit{spectral}\mathit{domain},\hfill \\ \hfill D_{2\kappa }=D_{\pi }& =\mathit{parity}\mathit{domain},\hfill \\ \hfill D_{3\kappa }& =\mathit{\text{inverse-spectral}}\mathit{domain}.\hfill \end{array}$$

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

Proof. 

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

Corollary 2

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

Proof. 

By Theorem 3, the map $s\mapsto {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ is injective, so the meridional bases ${\mathcal{B}}_s={\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{\mathcal{B}}_0$ are pairwise distinct, and hence so are the pairs $D_s=(V,{\mathcal{B}}_s)$.    □

Remark 6

(Framed domains and measurement bases). The corollary counts framed (ordered) bases: the meridional basis ${\mathcal{B}}_s$ is an ordered tuple, and the framed object is the primitive one — the orientation of the cycle is derived frame data. Read as unordered measurement bases, $B_{s+2\kappa }=B_s$, because parity J permutes the standard basis; the cycle therefore carries $4\kappa$ oriented representation domains and at most $2\kappa$ measurement bases, the factor two the parity involution.

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 $s\in \Phi$ on the representation side, as the basis-change pair $D_s=(V,{\mathcal{B}}_s)$ acted on by ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$. The present section reads the same meridian index $m\in \Phi$ on the coordinate side, as the subset $M_m\subset {\mathbb{F}}_{\mathsf{p}}$ acted on by multiplication by ${\mathsf{g}}^m$, and records the corresponding scale-shift law; on the shell geometry this is motion along the multiplicative, latitudinal direction of Figure 2. 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 7 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

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

(19)

of cardinality $\pi +1=2\kappa +1$. The meridian $M_m=\{a{\mathsf{g}}^m:a\in I_{\mathsf{p}}\}$ from (7) is then a labelled subset of ${\mathbb{F}}_{\mathsf{p}}$, traversed in the order $a=0,1,2,\dots ,\pi$.

Definition 6

(Meridian-scale map). For $r\in \Phi$, define

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

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

Proposition 2

(Meridian-scale covariance). For all $m,r\in \Phi$,

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

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

Proof. 

Using Definition 6 and the meridian definition,

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

   □

Remark 7

(Identification with the framed-rational zoom). Proposition 2 and the cyclic group $\{S_r:r\in \Phi \}$ 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 generator g is the same primitive generator in both settings; the remaining dictionary to the present section is

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

so that increasing the framed-rational scale level n by one — which divides the grid step by g, i.e. zooms in — corresponds to the meridian shift $m\mapsto m-1$, in agreement with the orientation convention of Theorem 4 below. The coordinate range $I_{\mathsf{p}}=\{0,1,\dots ,\pi \}$ 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 $\mathsf{g}=2$.

Corollary 3

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

$$0,{\mathsf{g}}^m,2{\mathsf{g}}^m,\dots ,\pi {\mathsf{g}}^m.$$

Proof. 

The elements of $M_m$, listed by the ordering of $I_{\mathsf{p}}$, are $a{\mathsf{g}}^m$ for $a=0,1,\dots ,\pi$. The difference of consecutive entries in this ordering is $(a+1){\mathsf{g}}^m-a{\mathsf{g}}^m={\mathsf{g}}^m$, 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 ${\alpha }_s=(s/\kappa )(\pi /2)$ introduced in Section 2.

Definition 7

(Local scale chart). A local scale chart for the meridian frame is a partial comparison map in which the shell unit $\mathsf{g}$ is represented by a positive real scale factor

$$\lambda >1.$$

Under this comparison, the effective shell step ${\mathsf{g}}^m$ 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 4

(Meridian zoom). Fix a local scale chart (Definition 7; an external comparison map, not additional finite-field structure) with $\mathsf{g}\mapsto \lambda >1$, and read each meridian index through its integer lift $\tilde{m}\in \{0,1,\dots ,4\kappa -1\}$: the chart cannot descend to the residue $m\in \Phi$, since ${\lambda }^{\tilde{m}+4\kappa }\ne {\lambda }^{\tilde{m}}$. Then:

(i) 

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

(ii) 

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

(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 ordered readings “finer” and “coarser” hold within the no-wrap window of Remark 9. On the canonical lifts the cardinal meridians are read as

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

Proof. 

Item (i) is the case $m=0$ of Corollary 3, where the effective step is ${\mathsf{g}}^0=1$ 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 ${\mathsf{g}}^m$ 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 ${\mathsf{g}}^{\pm 1}$, read as ${\lambda }^{\pm 1}$ in the chart. The cardinal values are the cases $m\in \{0,\kappa ,2\kappa ,3\kappa \}$.    □

Remark 8

(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. In information terms the bijection $S_r$ is entropy-conserving, so meridian zoom produces no entropy; the information convention is that of a deterministic readout $Y=C\left(X\right)$, for which $H\left(Y\right)\le H\left(X\right)$ by the data-processing inequality, the discarded information being the conditional entropy $H(X\mid Y)$.

Example 1

(Meridian-step ladder at $\mathsf{p}=13$). Let $\mathsf{p}=13$, $\kappa =3$, $\pi =6$, $\mathsf{g}=2$, $\mathrm{i}=5$ (from $\mathrm{i}=-{\mathsf{g}}^{\kappa }=-8=5$, Equation (6)). The meridian-step interval is $I_{\mathsf{p}}=\{0,1,\dots ,6\}$ and the first four meridians of the zoom ladder are

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

The effective step doubles at every meridian shift, in agreement with Theorem 4 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 9

(Scope of the unification). The meridian cycle Φ carries two coherent algebraic readings: the representation-side rotation

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

faithful on the full cycle by Theorem 3, 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 is $4\kappa$-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 Φ and the cardinal interpretations at $\{0,\kappa ,2\kappa ,3\kappa \}$, both built from the shell constants $\mathsf{g}$, $\mathrm{i}$, π. The phrasing “two facets of one rotation” is therefore to be read at the cyclic-group level and at the level of cardinal labeling. The operator-level relation is settled in SubSection 8.1: full conjugacy on V is excluded by spectrum (Proposition 4), while the two sides share an exact common character sector (Proposition 5) and the cardinal skeleton acts by exact Heisenberg covariance (Proposition 6).

The scale-shift interpretation of $S_r$ as multiplication by ${\lambda }^r$ in a local scale chart holds in the no-wrap range: for an observer window of w meridian steps, the listed shell-values $a{\mathsf{g}}^r$, $0\le a\le w$, remain unwrapped as long as $w{\mathsf{g}}^r<\mathsf{p}$ in the integer reading, i.e. for $r<{log}_{\lambda }(\mathsf{p}/w)$ clean zoom steps, with λ the chosen positive integer representative of $\mathsf{g}$; 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 wrap visible from $M_2$ onward in Example 1, where $w=\pi$). 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 Φ.

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 $\mathsf{p}=4\kappa +1$, the element $\mathrm{i}$ exists in ${\mathbb{F}}_{\mathsf{p}}$ and ${\mathrm{i}}^2=-1$. 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 $s\in \Phi$, set

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

(20)

the phase value of the meridian index on the inverse-generator base, so that $z_{\kappa }={\mathsf{g}}^{-\kappa }=\mathrm{i}$, and define

$$c_s:={\displaystyle \frac{z_s+z_s^{-1}}{2}},d_s:={\displaystyle \frac{z_s-z_s^{-1}}{2\mathrm{i}}}.$$

(21)

Finally set

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

(22)

Lemma 5.

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

Proof. 

Since ${\mathrm{i}}^2=-1$,

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

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

Proposition 3

(FRC constants parametrize finite rotations). The map

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

is a group isomorphism.

Proof. 

The formulas (21) are exactly the change of variables

$$z_s=c_s+\mathrm{i}{d}_s,z_s^{-1}=c_s-\mathrm{i}{d}_s.$$

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

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

Under the identification

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

multiplication of z corresponds to multiplication of matrices. Hence, $R_s{R}_r=R_{s+r}$. Since $\mathsf{g}$ is primitive, so is ${\mathsf{g}}^{-1}$, and $s\mapsto z_s$ is an isomorphism $\Phi \simeq {\mathbb{F}}_{\mathsf{p}}^{\times }$. 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:

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

Let ${\omega }_{\mathsf{p}}$ denote the finite Weil representation of $SL(2,{\mathbb{F}}_{\mathsf{p}})$ in a chosen Schrödinger 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

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

(23)

is represented, up to the usual scalar convention, by the finite Fourier transform [14,15]; the finite harmonic oscillator realizing this Schrödinger model is constructed in [25]. 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

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

(24)

Definition 9

(Cardinal-skeleton equivalence of FrFT realizations). Two FrFT families $A_s$ and $B_s$ are called cardinal-skeleton equivalent when there exists one cyclic index group Φ such that both are representations of Φ and the four cardinal elements $0,\kappa ,2\kappa ,3\kappa$ 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 $\Phi$ 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 $s\notin \{0,\kappa ,2\kappa ,3\kappa \}$.

Theorem 5

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

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

(25)

and on the cardinal indices

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

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

Proof. 

By Theorem 1, $s\mapsto {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ is a representation of $\Phi$ with the cardinal values listed in (16). By Proposition 3, $s\mapsto R_s$ is an isomorphism $\Phi \simeq SO(2,{\mathbb{F}}_{\mathsf{p}})$. 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 $\Phi$. The cardinal values follow from the explicit matrices $R_0$, $R_{\kappa }$, $R_{2\kappa }$, and $R_{3\kappa }$ together with the chosen Weil normalization. Therefore both families satisfy Definition 9.    □

Remark 10

(Scope of the dictionary). Theorem 5 pins down agreement of the two FrFT families only on the four cardinal indices $\{0,\kappa ,2\kappa ,3\kappa \}$, which generate the order-4 Fourier sub-skeleton of Φ. Two cardinal-skeleton equivalent families may therefore differ freely at intermediate meridian indices $s\notin \{0,\kappa ,2\kappa ,3\kappa \}$. Faithfulness of $s\mapsto {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ on the full meridian cycle Φ is established separately by Theorem 3, but it is a statement about the FRC-native family in isolation rather than a comparison with the Weil family.

The FRC-native family ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ acts on ${\mathbb{F}}_{\mathsf{p}}^{\Phi }$ and is internal to the shell Fourier cycle. The standard Schrödinger 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. Subsection 8.1 shows the obstruction is intrinsic rather than an artifact of the dimension mismatch: even inside V, the fractional family is Weyl-covariant exactly on the cardinal skeleton.

8.1. Operator-Level Comparison of the Three Readings

The dictionary of Theorem 5 and the coordinate-side zoom of the previous section share the index group $\Phi$; this subsection settles in what sense they share operators. Lift the scale shift to V as the exponent-shift permutation ${\left(\sigma v\right)}_k=v_{k-1}$, and let $D_1=diag{\left({\mathsf{g}}^k\right)}_{k\in \Phi }$ be the modulation operator.

Proposition 4

(Spectral obstruction). σ is diagonalizable over ${\mathbb{F}}_{\mathsf{p}}$ with n distinct eigenvalues: its characteristic polynomial is $x^n-1={\prod }_{a\in {\mathbb{F}}_{\mathsf{p}}^{\times }}(x-a)$. Every member of the family ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ has at most four eigenvalues. Hence for $\kappa \ge 2$ no invertible T on V satisfies $T\sigma T^{-1}={\mathcal{F}}_{\mathsf{p}}^{\left[1\right]}$: the cyclic groups $\langle \sigma \rangle$ and $\langle {\mathcal{F}}_{\mathsf{p}}^{\left[1\right]}\rangle$, of the same order $4\kappa$, are not conjugate.

Proof. 

$\sigma$ is the cyclic shift, with characteristic polynomial $x^n-1$; over ${\mathbb{F}}_{\mathsf{p}}$ with $n=\mathsf{p}-1$ every nonzero residue is a root, so the polynomial splits with n distinct roots and $\sigma$ is diagonalizable with simple spectrum. ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}={\sum }_{\ell }{\mathsf{g}}^{-\ell s}{\Pi }_{\ell }$ takes at most four eigenvalues. Conjugation preserves the spectrum, and $4<4\kappa$ for $\kappa \ge 2$.    □

Proposition 5

(Common character sector). Let $E_1=im{\Pi }_1$, nonzero for $\kappa \ge 2$ (Lemma 4). On $E_1$ the fractional family acts by the meridian character, ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{|}_{E_1}={\mathsf{g}}^{-s}I$. For any nonzero $v\in E_1$, the map $T_v:{\mathbb{F}}_{\mathsf{p}}\to E_1$, $T_v\left(x\right)=xv$, intertwines exactly:

$${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{T}_v=T_v{S}_{-s}\mathit{for}\mathit{all}s\in \Phi ,$$

and the rotation $R_s$ of (22) carries the same character on its eigenline: $R_s{(1,-\mathrm{i})}^{\mathsf{T}}={\mathsf{g}}^{-s}{(1,-\mathrm{i})}^{\mathsf{T}}$. The representation rotation, the coordinate-side scale shift, and the defining $SO(2,{\mathbb{F}}_{\mathsf{p}})$ rotation underlying the Weil dictionary thus share one exact common character sector.

Proof. 

${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{T}_v\left(x\right)={\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}\left(xv\right)=x{\mathsf{g}}^{-s}v=T_v\left({\mathsf{g}}^{-s}x\right)=T_v{S}_{-s}\left(x\right)$, by linearity and the eigenvalue of ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$ on $E_1$. For the eigenline, $R_s{(1,-\mathrm{i})}^{\mathsf{T}}=(c_s+\mathrm{i}{d}_s){(1,-\mathrm{i})}^{\mathsf{T}}$ by direct multiplication, and $c_s+\mathrm{i}{d}_s=z_s={\mathsf{g}}^{-s}$ by (21).    □

Proposition 6

(Cardinal Heisenberg covariance). The normalized shell Fourier operator intertwines shift and modulation exactly:

$${\mathcal{F}}_{\mathsf{p}}\sigma {\mathcal{F}}_{\mathsf{p}}^{-1}=D_1,{\mathcal{F}}_{\mathsf{p}}{D}_1{\mathcal{F}}_{\mathsf{p}}^{-1}={\sigma }^{-1}:$$

the cardinal skeleton acts as an exact quarter-rotation on the multiplicative Heisenberg pair $(\sigma ,D_1)$.

Proof.${\left(W\sigma v\right)}_k={\sum }_j{\mathsf{g}}^{jk}{v}_{j-1}={\sum }_j{\mathsf{g}}^{(j+1)k}{v}_j={\mathsf{g}}^k{\left(Wv\right)}_k$, so $W\sigma =D_{1}W$; and ${\left(W{D}_{1}v\right)}_k={\sum }_j{\mathsf{g}}^{j(k+1)}{v}_j={\left(Wv\right)}_{k+1}={\left({\sigma }^{-1}Wv\right)}_k$, so $W{D}_1={\sigma }^{-1}W$. The scalar $\mathrm{i}$ cancels in conjugation, giving both identities for ${\mathcal{F}}_{\mathsf{p}}=\mathrm{i}W$.    □

The covariance does not extend beyond the cardinal skeleton: machine verification across the shells of Table 1 ($\mathsf{p}\in \{13,17,29,37,41\}$) shows ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}\sigma {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]-1}$ monomial exactly at the four cardinal indices and non-monomial at every one of the 112 intermediate indices; a general proof is open — the obstruction is the non-vanishing of a sparse six-term Laurent polynomial off subgroup cosets.

The three propositions convert the scope limitation of Remark 10 into structure: the FRC-native family is a phase-space rotation exactly on the cardinal skeleton, the coordinate and representation readings are never conjugate as full cycles, and what all three families share is one exact character sector — the operator-level content of “two facets of one rotation”.

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

Let

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

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

$$\mathrm{i}=-{\mathsf{g}}^{\kappa }=-2^3=-8=5,$$

and

$${\mathrm{i}}^2=5^2=25=12=-1\left(\mathrm{mod}13\right).$$

The remaining frame constants are the half-period $\pi =6$ and the exponential unit $e={\mathsf{g}}^{\mathrm{i}}=2^5=6$; neither enters the computations below. The meridian cycle has twelve representation domains:

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

The cardinal domains are

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

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

$$c_{\kappa }={\displaystyle \frac{5+8}{2}}=0,d_{\kappa }={\displaystyle \frac{5-8}{2·5}}={\displaystyle \frac{-3}{10}}=(-3)·4=-12=1.$$

Thus,

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

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 4 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.

Table 1. Exact modular validation of the finite Fourier and rotation identities for several symmetry-complete prime shells. The row $\mathsf{p}=5$ is the boundary case $\kappa =1$, in which the meridian cycle ${\mathbb{Z}}_4$ is exhausted by the four cardinal indices; the multiplicity Lemma 4 leaves exactly one of ${\Pi }_1,{\Pi }_3$ nonzero, and the surviving odd projector still carries a faithful character, so the faithfulness of Theorem 3 holds there as well. The rows with $\kappa \ge 2$ carry intermediate meridians and realize the full $4\kappa$-domain reading.

$\mathsf{p}$	$\mathit{\kappa }$	$\mathsf{g}$	$\mathrm{i}$	$W^2=-J$	${\left(\mathrm{i}W\right)}^2=J$	${\left(\mathrm{i}W\right)}^4=I$	$R_{\kappa }$
5	1	2	3	true	true	true	$\left(\begin{array}{cc}0& 4\\ 1& 0\end{array}\right)$
13	3	2	5	true	true	true	$\left(\begin{array}{cc}0& 12\\ 1& 0\end{array}\right)$
17	4	3	4	true	true	true	$\left(\begin{array}{cc}0& 16\\ 1& 0\end{array}\right)$
29	7	2	17	true	true	true	$\left(\begin{array}{cc}0& 28\\ 1& 0\end{array}\right)$
37	9	2	6	true	true	true	$\left(\begin{array}{cc}0& 36\\ 1& 0\end{array}\right)$
41	10	6	9	true	true	true	$\left(\begin{array}{cc}0& 40\\ 1& 0\end{array}\right)$

Table 1. Exact modular validation of the finite Fourier and rotation identities for several symmetry-complete prime shells. The row $\mathsf{p}=5$ is the boundary case $\kappa =1$, in which the meridian cycle ${\mathbb{Z}}_4$ is exhausted by the four cardinal indices; the multiplicity Lemma 4 leaves exactly one of ${\Pi }_1,{\Pi }_3$ nonzero, and the surviving odd projector still carries a faithful character, so the faithfulness of Theorem 3 holds there as well. The rows with $\kappa \ge 2$ carry intermediate meridians and realize the full $4\kappa$-domain reading.

$\mathsf{p}$	$\mathit{\kappa }$	$\mathsf{g}$	$\mathrm{i}$	$W^2=-J$	${\left(\mathrm{i}W\right)}^2=J$	${\left(\mathrm{i}W\right)}^4=I$	$R_{\kappa }$
5	1	2	3	true	true	true	$\left(\begin{array}{cc}0& 4\\ 1& 0\end{array}\right)$
13	3	2	5	true	true	true	$\left(\begin{array}{cc}0& 12\\ 1& 0\end{array}\right)$
17	4	3	4	true	true	true	$\left(\begin{array}{cc}0& 16\\ 1& 0\end{array}\right)$
29	7	2	17	true	true	true	$\left(\begin{array}{cc}0& 28\\ 1& 0\end{array}\right)$
37	9	2	6	true	true	true	$\left(\begin{array}{cc}0& 36\\ 1& 0\end{array}\right)$
41	10	6	9	true	true	true	$\left(\begin{array}{cc}0& 40\\ 1& 0\end{array}\right)$

10. Entropy on the Meridian Cycle

Definition 10

(Cyclotomic observer readout). Let $A=\mathbb{Z}\left[\frac{1}{4n},X,Y,Y^{-1}\right]/\left({\Phi }_n\left(X\right),Y^2-n\right)$ be the structural readout algebra: X a formal primitive n-th root of unity and Y a formal square root of n, adjoined rather than selected inside a field. Two specializations are declared separately: the observer’s continuum chart ${\sigma }_{\mathbb{C}}:X\mapsto e^{2\pi i/n}={\zeta }_n$, $Y\mapsto +\sqrt{n}$, and the framed reduction $\rho :X\mapsto \mathsf{g}$, $Y\mapsto -\mathrm{i}$. On A set $D_{jk}=Y^{-1}{X}^{jk}$, $\iota =X^{-\kappa }$, ${\widehat{\Pi }}_{\ell }=\frac{1}{4}{\sum }_{r=0}^3{\iota }^{-\ell r}{D}^r$, and ${\widehat{F}}^{\left[s\right]}={\sum }_{\ell }{X}^{-\ell s}{\widehat{\Pi }}_{\ell }$; the framed reduction carries these termwise to the shell objects, $\rho \left(D\right)=\mathrm{i}W={\mathcal{F}}_{\mathsf{p}}$, $\rho \left({\widehat{\Pi }}_{\ell }\right)={\Pi }_{\ell }$, $\rho \left({\widehat{F}}^{\left[s\right]}\right)={\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$. The formal adjunction of Y is what keeps the frame’s decree consistent: the Gauss root inside the selected complex field carries its own sign under reduction, which need not match $-\mathrm{i}$ (the bifurcation classified by Theorem 2). The cyclotomic observer readout of a shell state ψ in a meridian basis is the Born vector of a selected lift under ${\sigma }_{\mathbb{C}}$, $p_j=|{\psi }_j{|}^2/{\sum }_k{\left|{\psi }_k\right|}^2$, with Shannon entropy $H\left(p\right)=-{\sum }_j{p}_{j}log{p}_j$; the reduction is many-to-one, so the lift is part of the observer datum, and the localized inputs ${\delta }_j$ used below have canonical lifts. The chart also fixes an embedding: the Galois twist $X\mapsto e^{2\pi iu/n}$ relabels the intermediate curve by $s\mapsto us$ while leaving the cardinal values invariant. The readout is the observer-side declaration the FRC measurement layer supplies [22]: probabilities are structural weights of the cyclotomic stratum, and the finite-field module itself carries no Born magnitude.

This entropy is the Shannon (informational) entropy of a measurement-outcome distribution in the stated meridian basis: dimensionless, reported in nats, and not thermodynamic entropy — no Boltzmann constant, temperature, or physical microstate count is invoked or computed anywhere below. That the entropy attaches to the declared readout is the paper’s point rather than an anomaly: the shell dynamics is a bijection and conserves entropy, and entropy is produced only at the readout [13]; the basis dependence is the ordinary content of the entropic uncertainty relation [9,10,11,12], and no claim about physical randomness (e.g. thermal noise) is made anywhere in this paper. The representation entropy is then an exact function of the meridian index.

Proposition 7

(Entropic cardinal values). Under the readout of Definition 10, for a state localized at a single site of the spatial meridian the spatial-readout entropy of ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}\psi$ takes the exact cardinal values

$$H\left(0\right)=0,H\left(\kappa \right)=logn,H\left(2\kappa \right)=0,H\left(3\kappa \right)=logn,$$

minimal at the spatial and parity meridians and maximal at the two Fourier meridians. The spatial and spectral readouts obey the discrete entropic uncertainty relation

$$H_{B_0}+H_{B_{\kappa }}\ge logn,$$

because the bases $B_0$ and $B_{\kappa }$ are mutually unbiased: the shell Fourier matrix W has entries of equal readout magnitude. The bound is saturated in particular by the basis-localized states; subgroup-coset combs saturate it as well — at $n=12$, $({\delta }_0+{\delta }_6)/\sqrt{2}$ gives $log2+log6=logn$, machine-verified.

Proof. 

The cardinal values ${\mathcal{F}}_{\mathsf{p}}^{\left[0\right]}=I$ and ${\mathcal{F}}_{\mathsf{p}}^{\left[2\kappa \right]}=J$ are coordinate permutations, so they preserve the localized readout and give $H=0$. The quarter-turn ${\mathcal{F}}_{\mathsf{p}}^{\left[\kappa \right]}={\mathcal{F}}_{\mathsf{p}}$ and its inverse ${\mathcal{F}}_{\mathsf{p}}^{\left[3\kappa \right]}={\mathcal{F}}_{\mathsf{p}}^{-1}$ are the shell Fourier operator (up to the unit $\mathrm{i}$), whose rows have equal magnitude, so a localized state maps to the uniform readout with $H=logn$. Equal magnitude of every overlap between the $B_0$ and $B_{\kappa }$ bases is the mutually unbiased condition, for which the Maassen–Uffink bound equals $logn$ and is attained by the basis-localized states [9,10,11].    □

Proposition 8

(Closed-form interpolation for the localized input). Under the readout of Definition 10, the spatial readout of ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{\delta }_0$ is two-valued:

$$p_0=1-{\displaystyle \frac{n-1}{n}}{t}_s,p_j={\displaystyle \frac{t_s}{n}}(j\ne 0),t_s={\displaystyle \frac{2-{\zeta }_n^{2s}-{\zeta }_n^{-2s}}{4}},$$

and the entropy along the cycle is the exact function

$$H\left(s\right)=-\left(1-\frac{n-1}{n}{t}_s\right)log\left(1-\frac{n-1}{n}{t}_s\right)-\frac{n-1}{n}{t}_{s}log{\displaystyle \frac{t_s}{n}}.$$

The weight $t_s$ is a totally real cyclotomic quantity, exact in $\mathbb{Z}[{\zeta }_n,\frac{1}{4}]$; its labelled continuum reading is $t_s={sin}^2(\pi s/2\kappa )$. As s runs from 0 to κ, $t_s$ increases strictly from 0 to 1 and H increases strictly from 0 to $logn$; the full cycle carries two entropic oscillations.

Proof. 

$J{\delta }_0={\delta }_0$, and $V^{+}$ is the sum of the $\ell \in \{0,2\}$ eigenspaces (Lemma 3), so ${\Pi }_1{\delta }_0={\Pi }_3{\delta }_0=0$ and, under the chart ${\sigma }_{\mathbb{C}}$, ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{\delta }_0={\Pi }_0{\delta }_0+{\zeta }_n^{-2s}{\Pi }_2{\delta }_0$. There $D{\delta }_0=D^{-1}{\delta }_0=n^{-1/2}\mathbf{1}$, whence ${\Pi }_0{\delta }_0=\frac{1}{2}({\delta }_0+n^{-1/2}\mathbf{1})$ and ${\Pi }_2{\delta }_0=\frac{1}{2}({\delta }_0-n^{-1/2}\mathbf{1})$. The off-site components of ${\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}{\delta }_0$ are therefore $\frac{1}{2}{n}^{-1/2}(1-{\zeta }_n^{-2s})$, of squared magnitude $(2-{\zeta }_n^{2s}-{\zeta }_n^{-2s})/4n=t_s/n$, and the family is unitary in this realization, so the weights sum to one. Monotonicity: the distribution is the mixture between the point mass ($t=0$) and the uniform vector ($t=1$) and is majorization-decreasing in t, so H is strictly increasing in $t\in [0,1]$; in the continuum reading $t_s={sin}^2(\pi s/2\kappa )$ is strictly increasing on $0\le s\le \kappa$. The cardinal values of Proposition 7 are the evaluations $t_0=t_{2\kappa }=0$, $t_{\kappa }=t_{3\kappa }=1$.    □

Remark 11

(Input dependence of the intermediate curve). The cardinal values of Proposition 7 hold for every site-localized input ${\delta }_j$. The intermediate curve does not: for $j\notin \{0,2\kappa \}$ the input meets all four projectors and the interpolation differs — at $\mathsf{p}=13$, $H\left(1\right)/logn=0.55$ for ${\delta }_1$ against $0.44$ for ${\delta }_0$. Figure 3 and Proposition 8 are stated for ${\delta }_0$.

Thus, one meridian step is a discrete step of entropic change between complementary representations (the space–momentum pair of Figure 2), and the FrFT index s is an exact discrete entropic angle. For $\kappa \ge 2$ the intermediate meridians give strictly intermediate entropies for the localized input — by Proposition 8, since $t_s\in (0,1)$ off the cardinal indices — the entropic counterpart of the faithfulness of Theorem 3.

• Illustration at $\mathsf{p}=13$.

For the shell ${\mathbb{F}}_{13}(\tau ;0,1,2)$ ($\kappa =3$, $n=12$), under the readout of Definition 10 (continuum realization ${\zeta }_n=e^{2\pi i/n}$), the spatial-readout entropy along the full meridian cycle is

s	0	1	2	3	4	5	6	7	8	9	10	11
$H\left(s\right)/logn$	0	$.44$	$.91$	1	$.91$	$.44$	0	$.44$	$.91$	1	$.91$	$.44$

exactly zero at the spatial and parity meridians $M_0,M_{2\kappa }=M_6$, exactly $logn$ at the spectral and inverse-spectral meridians $M_{\kappa }=M_3,M_{3\kappa }=M_9$, and strictly intermediate and symmetric elsewhere (Figure 3) — the values are exact evaluations of Proposition 8. The cycle carries two full entropic oscillations over one representation turn; the computation is reproducible in code/entropy_meridian_cycle.py.

11. Discussion and Conclusion

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

First, the Fourier transform becomes a literal algebraic quarter-turn in representation space. The normalization ${\mathcal{F}}_{\mathsf{p}}=\mathrm{i}W$ is essential: the unnormalized ring Fourier matrix satisfies $W^2=-J$, while the normalized shell operator satisfies ${\mathcal{F}}_{\mathsf{p}}^2=J$ 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 ${\mathcal{F}}_{\mathsf{p}}$, and the replacement ${\mathrm{i}}^{\ell }\mapsto {\mathsf{g}}^{-\ell s}$ is fixed by the shell generator $\mathsf{g}$ and the orientation convention $\mathrm{i}={\mathsf{g}}^{-\kappa }$. Thus the one-step transform ${\mathcal{F}}_{\mathsf{p}}^{\left[1\right]}$ is an exact $\kappa$-th root of the shell Fourier quarter-turn.

Third, the finite-field Weil interpretation is not external to the shell. The constants $\mathsf{g}$, $\mathrm{i}$, and $\pi$ give the exact passage from a meridian index to a finite-field rotation matrix. The same element $s\in {\mathbb{Z}}_{4\kappa }$ can be read as a shell meridian, as the nonzero field element $z_s={\mathsf{g}}^{-s}$, 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 5 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 ${\mathsf{g}}^r$ sends $M_m$ to $M_{m+r}$ exactly, and in any local scale chart $\mathsf{g}\mapsto \lambda >1$ 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 algebraic readings of one common index: a representation-side rotation $s\mapsto {\mathcal{F}}_{\mathsf{p}}^{\left[s\right]}$, faithful on the full cycle; 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; the first and third share the exact common character sector of Proposition 5, full operator conjugacy being excluded by Proposition 4.

The rotation family constructed here sits inside the finite metaplectic/Weil representation literature: the finite-field Weil representation and its associated finite harmonic oscillator [14,15,25] realize symplectic group actions, including the Fourier quarter-turn, on finite state spaces, and Theorem 5 supplies the cardinal-skeleton dictionary from the present FRC-native family to that literature rather than a new abstract Weil-representation result. The entropy layer developed above similarly sits inside the mutually-unbiased-bases and entropic-uncertainty-relation literature [9,10,11,12], where the present shell supplies an exact, finite, parameter-free instance rather than a new abstract inequality.

In conclusion, for every symmetry-complete prime shell $\mathsf{p}=4\kappa +1$, the framed datum ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$ 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 $M_{2\kappa }=M_{\pi }$ is the parity domain. For $\kappa \ge 2$, every intermediate meridian is a genuinely distinct intermediate representation domain.

The main algebraic construction is

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

(26)

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

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

$$M_s⟷z_s={\mathsf{g}}^{-s}⟷R_s\in SO(2,{\mathbb{F}}_{\mathsf{p}})⟷{\omega }_{\mathsf{p}}\left(R_s\right),$$

(27)

and Theorem 5 matches the FRC-native and Weil families on the four cardinal meridians $\{0,\kappa ,2\kappa ,3\kappa \}$. 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 4: the multiplicative map $S_r:{\mathbb{F}}_{\mathsf{p}}\to {\mathbb{F}}_{\mathsf{p}}$, $S_r\left(x\right)={\mathsf{g}}^{r}x$, sends $M_m$ to $M_{m+r}$, and in a local scale chart $\mathsf{g}\mapsto \lambda >1$ this becomes reversible meridian-step zoom-out (or zoom-in, for the inverse shift). The same meridian index $s\in {\mathbb{Z}}_{4\kappa }$ therefore admits a representation-side reading, a Weil rotation reading, and a coordinate-side scale-shift reading, all derived from the framed shell datum ${\mathbb{F}}_{\mathsf{p}}(\tau ;0,1,\mathsf{g})$. In entropy terms, the meridian index also dials the representation entropy between the localized spatial reading and the maximally spread spectral reading, so the meridian step is a discrete step of entropic change as well as of time evolution and scale dilation; the arrow it defines is the coarse-graining direction of the observer readout, within the finite budget set by the substrate capacity S, the de Sitter entropy fixing the FRC substrate cardinality $\Omega =4S+1$ [22]: a Carrier-register datum, expressible by no embedded observer, which the Subject reads only as the framed bit-chart $\sim log\Omega$.