FRC Lab: Schrödinger—Dirac dynamics of the total state over the (13, 233) holographic substrate: iħ ∂ψ/∂t = Ĥψ ⇌ ψ_τ+1 = g·ψ_τ

The context: finitist mathematics, finite physics, it from bit. The lab stands in three lines of work. The finitist line: Zeilberger's programme reading continuum analysis as the degenerate case of discrete mathematics [11], and Lev's finite quantum theory over Galois fields, the continuum an approximation to a large finite characteristic [10]. The information line: Wheeler's it from bit, every physical quantity a registered answer to a yes–no question [8], and Susskind's holographic principle, the world's content carried on a boundary at finite density [9]. The structural line: Noether's identity of symmetry and conservation [12], and Henkin's model existence, a consistent theory building its own model from its own terms [13]. Against this background the lab's claim is that the three lines are one construction. The pair (13, 233) is a finite Henkin model; its symmetries are its conserved content; its sky is its holographic chart; its registered data are its bits. Nothing here discretises a continuum theory; the continuum enters only as a labelled chart. The current implementation is intended as a numerical laboratory, and a model existence proof, not currently a pedagogical exhibit.

Schrödinger dynamics on the finite substrate [3, 4]. The drive x ↦ gx is the shell's exact unitary evolution: one chronon multiplies every registered value by the generator, turning every phase arrow one quarter and stepping every tint one rung — the global phase and the diagonal law. The quantum paper identifies this chronon-indexed zonal evolution as the Schrödinger evolution of the windings [4], and the Dirac paper carries its exact unitary and spinor structure [3]. No dynamical quantity is approximated — every state, phase, and count is exact, with the screen projection the one labelled chart approximation: the three panels are three complementary readings of this one evolution — the frame's symmetry on the shell, the phase and the fold on the chart, the observed space on the sky — locked to one state pair and one view toggle.

The shell is the arithmetic symmetry sphere [3]. The node at longitude m, depth a carries the residue a·g^m. The meridians are the additive rays: the unit's ray reads 1…6 down one side and 12…7 up the other; the quarter-turn ray reads 8, 3, 11, 6, 1, 9. The latitudes are the drive orbits, all twelve residues in drive order. The Observer origin 0 sits at the pole. Every mark is frame data.

Schrödinger dynamics is rotation [3]. The drive x ↦ 2x sends node (m, a) to slot (m+1, a): a rigid turn of the sphere about its polar axis, 30° per chronon, latitudes preserved, origin fixed. The free evolution is the drive, complete in one move. The step is indivisible: the registered data are exact at every tick; the Smooth chart interpolates only the drawn geometry between them [approx].

The class quotient and the octant bridge [1:B5] [3]. The two cycles share the C₄ class quotient, and the registered state is the pair of cycle positions, each a residue of its own shell. The bridge between the shells is the quadratic extension: the coefficient plane K = F₁₆₉ of the zonal theorem, counted by the Carrier. The octant is the homomorphic bridge between the plane's units and the Carrier's, gcd(168, 232) = 8: an octant-equivariant counting transports the flip exactly, the image and the remaining 64 cells each closed under the flip, the remainder eight full octant fibres. In general the remainder is 8(S/2 − κ(2κ+1)) cells, whole octant fibres always.

The Carrier's quarter-turn [1:A4a], and the precession [1:C6]. The Carrier itself is the timeless torsor; its chart, in the observer's reading, runs the complete period: each quarter, 58 ticks, multiplies the chart by 78⁵⁸ = h with h² = −1, the Fourier flip of the plane, space and momentum exchanging roles. The four cardinals of the ring are the quarter operators {1, h, −1, ħ}; h and ħ share class 2, and the flip's chirality is the C14 bit, Subject-selected. Against this the Subject's orbit precesses: 12 ticks per orbit against the 58-tick quarter advances the q/p frame by 6/29 of a quarter per orbit: the apsidal fraction per revolution is the winding ratio κ/S [1:C6], the mass-ratio reading of D2, and at κ = 3 it is the Schwarzschild leading form for an orbit of semi-latus rectum S gravitational radii. The frame returns to minus itself at the pair event (home, 116), the chart reading dlog(−1): the Carrier's own −1 carries the spinor sign; and it closes only at (home, home), the pair space exhausted between closures [1:B5]: the double cover as event structure within the totality, the spinor structure in the time domain. The quarter takes exactly S chronons: the register and the Carrier's quarters wind one tick-angle per chronon — no Subject is an ideal of the Carrier, so the frame leaks, the q/p axes riding the register with their Carrier addresses flipping ×h at each cardinal. In the Subject view the Carrier chart turns ↺ under the standing register; in the Carrier view the register drifts ↻ past the standing quarters; q aligns with the h cardinal at τ = S, and the chart closes at Ω − 1 = 232. The frame ray therefore moves at the composed rate 1/12 + 1/232, the coprime numerator κ + S = 61: its absolute direction returns exactly at the pair event (home, home) and stands antipodal at (home, 116), the double cover carried by the dynamics itself, with the prograde sign of the astronomical advance. The chirality is Subject-selected: C14's odd-member rule gives ħ = 144 the i-orientation, exactly as it gives i = 5 on the shell, the gauge bit k_Bc = h concurs, and the transported flip lands on h; the orientation is orbit-relative, on both sides. The gravitational realisation [1:C8]: the registration dictionary of this precession is derived from the two leak faces and the dilation reading, and the realisation stands with its strong-field falsifier — gravitational apsidal advance [1:C8], Mercury to the S2 star.

Two views, one event [4]. In the Carrier view the residues stand, because the Carrier's chart is timeless, and it is the Subject's frame that turns: its prime meridian sweeps clockwise around the axis, one step per chronon, lying on the ray of g^τ. In the Subject's view the frame is static by definition and the residues evolve, x ↦ gx: the depth-a slot reads a·g^τ, the unit slot walking 1, 2, 4, 8, … forward with the drive, the same readings the Carrier view shows under the swept frame. The pullback lives in the naming: the Subject names the fixed unit cell g^−τ, the ringed node of the ring panel, and this opposite-running name, with count positivity, derives the orientation i = −g^κ = 5.

The Subject and the Carrier [3]. This episode holds the Subject and the Carrier: the evolution is x ↦ gx, with the fixed shell winding numbers κ = 3 and S = 58. One identification is exact and belongs here: read as a K-valued function on the cycle, the register comb is g^τ·χ₁, the shell's own fundamental character, because the identity register is χ₁: the frame register is the fundamental winding, and Object modes are k ≠ 1 selections read against it. The winding ladder, the plane waves, and the dispersion are the spectrum of this same drive, and they enter the series when an Object contributes such a selection.

Two finite unit circles [3]. The split phase cycle C₁₂ carries the free evolution; the norm-one torus N¹ ≅ C₁₄ of K = F₁₆₉ carries the Cayley interactions and the boosts; they meet only at ±1. The three consecutive tori of F₁₃ hold the three dynamical roles: 12, 13, 14. Kinetic Cayley orbits on the translation shell return in exactly 13 steps, machine-verified over F₁₆₉.

The exact claims [3]. The episode's exact claims are unitarity, phases, and periods. Probability enters the series later, through registration counting.

One step, two exact readings [3]. The exact operator content of one step is the basis-label permutation P₂|a⟩ = |2a⟩: unitary, P₂¹² = I on the twelve units, the origin fixed. In component form it is the pullback (P₂ψ)(a) = ψ(2⁻¹a), character for character the drive D of the zonal theorem: the free evolution is the drive, its sector meeting the Cayley kinetic sector only at ±1. The comb animates the active reading: register labels advance, R_τ(a) = a·g^τ, one comb step composed with the pullback giving the identity, so the animation runs by D⁻¹ while the coefficient wavefunction, Ψ_τ(a) = Ψ₀(g^−τa) = D^τΨ₀(a), runs by D: on one cell the comb sends 1 to 2 while (Dχ₁)(1) = 2⁻¹ = 7. The two are duals through the inverse table, one dynamics read from two frames; wavefunction names the coefficient side. The register vector is isotropic, Σa² ≡ 0 (mod 13), the zonal theorem's own isotropy of the winding lines; unitarity is carried by the permutation, exactly as the theorem's proof reindexes. The Cayley kinetic sector, the step U = (I − wH)⁻¹(I + wH) with H = 2I − T − T⁻¹ on the translation shell, holds exact unitarity at order 13 and enters the display with the Object.

The state is the register section [1:D11] [2]. The register section is an instantaneous, space-like object: the residue vector of the frame's prime great circle, the additive line through the Observer. Thirteen cells: the origin fixed at 0 at the pole and twelve components R_τ(a) = a·g^τ; the full space splits as Kδ₀ ⊕ K^F₁₃×, the step acting as the identity on the origin summand and as D on the units. The comb on the shell displays it: each cell carries the 4-state phase vector i^r of its registered residue, r = dlog(x) mod 4 in the fold x = i^r·3^s, tinted by its rung s. The coefficient wavefunction of the fold [1:A2] is this section's passive dual, read through the inverse table.

The phase arrow is orthogonal to the meridian fiber [3]. The phase is a fiber datum: the arrow lies in the transverse plane at its cell, spanned by the two directions orthogonal to the fiber, the surface radial and the latitude tangent. The imaginary axis is zonal, because the class is a shell operation: i = g⁹, nine drive steps, one counterclockwise quarter along the latitudes. The real axis is the one direction orthogonal to both zonal and meridional: the radial. So the four states are radial out (1), zonal counterclockwise (i), radial in (−1), zonal clockwise (−i), and one drive step, R_τ+1 = g·R_τ, turns every arrow one quarter about its meridian fiber: the comb corkscrews around the space-like circle, the transverse-wave picture of the continuum recovered exactly, with Re radial and Im zonal. The tint completes the registration: each arrow is coloured by its rung s = dlog(x) mod 3, in the three fibre colours of the phase panel, and since (r, s) determines the residue, direction and tint together display the complete value of every cell. The unit cell's tint is the lit fibre α = τ mod 3. Per chronon the tint advances one rung as the direction advances one quarter: the diagonal law, now visible on the comb itself. Per shell period the comb makes κ = 3 fiber turns against the frame's one axis turn: the winding rate made visible. The classes equipartition: exactly three cells per class; the rungs equipartition: exactly four cells per rung. The antipode is the fiber half-turn: −x = i²x, so opposite cells sit two quarters apart in their fiber planes, zonal states landing exactly parallel and radial states mirrored through the horizontal plane. The origin at the pole has no fiber plane: the meridians converge there and the latitude shrinks to a point; its fixedness (0·g = 0) and its phaselessness are one fact. The latitude reading of the same step is time-like: the phase panel's middle ring is this L₁ latitude itself, its cells re-registering one drive step per chronon and carrying the same phase arrows read in the time domain, the ring's radial the real axis and the zonal tangent the imaginary, i one quarter counterclockwise. This vector is residue-valued, the registration layer: K-valued amplitudes arrive with the Object.

The Carrier is a timeless torsor [2]. Its determined content is the register, fixed generator-free by square roots and the gauge bit, and the canonical subgroup chain: the core Q₄, the octant, the four crossing classes. The tick coordinates of the ring are the Subject's declared chart on the torsor (78 = 3⁻¹, the reframing gauge, ħ at the top cardinal), and the one-tick-per-chronon motion on the ring is the Subject's registration sweeping the torsor.

The octant completes the general case [2]. On the Carrier S = 58 is even: the crossing classes of the core are even, 1 and −1 in class 0, ħ and h together in class 2, and the octant ζ₈ = 97, with ζ₈² = h, carries the odd classes, sitting at chart exponent S/2 = 29, odd by the predicate clause S ≡ 2 (mod 4).

Transport to the Carrier [4]. The two phase charts register through the common C₄ class quotient, gcd(12, 232) = 4. The lifts are shell-specific: the Subject lifts all four classes in its Q₄; the Carrier core lifts the even classes, and the octant supplies the odd pair. The transport invariant is S ≡ 2 (mod 4). The chronon is a cycle position: the shell returns every 12, the Carrier every 232, and the registered state is the pair of cycle positions, each a residue of its own shell. The joint recurrence is the pair space [1:B5]: every class-compatible pair occurs exactly once between consecutive (home, home) events.

The grid is the nodes [4, 5]. The Subject's space-time grid is its twelve phase-carrying nodes, the register itself; nothing else is observed. Distance is phase decoherence, counted in relational steps along the meridian: node a sits |a| quanta from the Observer and is registered as it was |a| chronons ago, reading a·g^τ−|a|: the unit cone slope. The 3D chart of the nodes is derived, the polar coordinates read off one coherence budget: a cell is one phase with two readings, temporal along the scale cycle and spatial along the meridian, and the sky is the dyad of that pair.

The half-phase clock: the extension. The two angles exist only through the quadratic extension K = F₁₃[η] with η = √g, the one further frame datum the observer supplies. The spinor clock ⟨η⟩ has order 24, the double cover of the drive cycle: even steps are the drive orbit, odd steps are K-proper, and η¹² = −1. Every chart angle lands on the sky halved: the drive step 30° becomes 15°, the chart quarter becomes 45°, the Carrier tick half a tick. A head-turn is a congruence, moving every node rigidly with the fibers fixed; a Fourier flip is a chart quarter, 45° on the azimuths with every fiber turned: the two operations differ in both coordinates, so no rotation of the observer is the Fourier dual.

The sky map: two covers. The two angles are read off two double covers, both of this universe. The temporal cover is the spinor clock C₂₄, η = √g, doubling the drive cycle; the spatial cover is the walk C₂₆ = 2p, doubling the space circle — the loop itself. Colatitude is the walk position: θ = k·180/13°, one C₂₆ half-step per route quantum; the terminus k = 13 is the antipodal pole, and the horizon falls strictly between rows 6 and 7 (1080 < 1170 < 1260): the direct/echo boundary, with nothing registered on it. Azimuth is the sum of the two cover readings: α = (τ+3k)·15° on the spinor clock — the transport, 45° per decoherence quantum, direction-blind since decoherence is a count — plus β = c·180/13°, the cell's own half-angle: the space-circle position halved, a datum of the cell alone, the same from either route — the sign of −1 distributed, one half-step per cell, half the cover per circuit, since −1 is the half-turn of the space circle and position accumulates it. The fine curl adds half a tick-angle per lookback chronon [1:C1]. The rung never enters the position: tint and arrow ride the fiber, the energy reading.

The closure. The sky map is injective everywhere: twenty-four images, twenty-four points, at every chronon — no folds. At each pole the two branches' azimuths differ by exactly half the cover: the branch flip of passing through a pole, so both poles are exact pass-throughs — one smooth closed loop, no seams, no cusps, no twists. The direct ring runs continuous through the horizon: space-neighbors 6 and 7 land one half-step apart on the sky. And the covers close the clock: positions return after 24 chronons, values after 12; one drive circuit turns every azimuth half-way around, the spinor sign of [3], S(−1) = −I — visible in the Carrier view, where the drive turns the sky and the register-anchored pattern drifts ↻ with the register past the standing quarters, half a tick-angle per chronon: the frame leak on the spinor cover, continuous through the Carrier wrap on the sheet σ. The leak's boost component splits by observability into two sections whose difference is the observable: a pure boost is unobservable at the quotient ([1:B6]: observation is the section), so the common fiber flow is gauge and appears only in the harness view — the Carrier-referenced chart realization at the dial's full rate, anchored at the exact C₄ stations, the frame dilation, the mass phase [1:C6], the boost torus being the Dirac layer's mass channel [3] and the mass sector's interaction face [6]. What the Subject observes is the relative fiber displacement between base points, and the retardation supplies it exactly: the image at lookback k lags the common flow by k tick-angles — the boost-curl [1:C1], static, integer-exact, gauge-free, drawn in the Subject view: the gravitational face in the Objectless lab, gravitation as phase synchronisation [5], the apsidal reading of [1:C8]; time-varying dilation is relational between systems of different rates and arrives with the first Object — rendered in the sheet-fair chart, where each point projects from the guard on its own side of S³ so that the antipodal map is the ball's exact point reflection. Stations, exact and gauge-free: at the Carrier quarter the sharp shell disperses into a radial band, the Fourier dual's spread — the dual has the units of the inverse, localized to dispersed; at the half the pattern reassembles as the congruent parity image, wF = −w exactly at full scale; home with the Carrier cycle, the pattern with the pair at 696 — the joint return a proxy-chart datum, S = 58 Subject-cycles out, registered by the declaring chart; the pair itself registers the monotone dephasing. The chart radius along a leaf is bounded below by the flow-invariant |z₁|: the ensemble breathes and never collapses, and only pole-crossing beads pass the center, along the observer's axis leaf. Flowed curves bend where they cross the mid-sphere, the seam where the atlas's two charts glue — the boundary between the home and parity halves of the totality. The Subject view holds the register: the drive is hidden by re-registration and the sky stands, the values walking through it; the views are locked across all panels.

The echo sky: the loop and the trace [1:A4b, D2]. Space is a closed circle, so every cell is observed by two routes: the direct way at lookback |a|, and around the world at lookback 13−|a|. One formula covers the whole sphere — arm ε, route k = 1..12, cell εk, value cell·g^τ−k. The direct images fill the near cap; the echo continues past the horizon, and since values are 12-periodic the wrapped past cone is the advanced cone: the echo shows the future value ×g⁻¹, the circumnavigation chronon. The two images of every cell share their half-angle and sit at colatitudes summing exactly 180°, their transports an odd multiple of 45° apart: the cover's parity signature. C7 lives on the values — retarded × conjugate = a², the matter–antimatted gauge as the norm-circle relation — and the instance's matter content selects the direct branch. The loop's anchors are the two null points of the walk: at k = 0 both arms sit on cell 0, the observer's own null registration, the north pole and the time-axis direction; at k = 13 both arms reach the antipodal pole — the observer's null self-echo, beyond the faithful window of 12, one drive cycle: the loop passes straight through, the azimuth branch flipping by half the cover, and closes. Per drive circuit the direct events number 12² + 12 = 156 = p(p−1), one per element of the affine group: the cone chart [1:B1] realized event by event; the two covers share exactly the one sign, gcd(24, 26) = 2: their central product, glued over the common −1, carries 24·26/2 = 2 × 156 cells, twice the Borel count — the closed circuit — while the base cycles are coprime, gcd(13, 24) = 1.

The radial ladder: three fibers [1:B2]. The law is scale-generic: a Subject of capacity κ (p = 4κ+1) resolves exactly κ radial decoherence steps within its horizon — 4a < p ⇔ a ≤ κ, identically — at the axial radii R·sin(2aπ/p) — the latitude circles' own radii, the view down the main axis, the outer shell nearly at the horizon rim — the chart flattening; the indices and counts are exact, the instance numerals enter only the instance. The drive cycle factorizes uniquely, C₁₂ = C₄ × C₃: each fiber realizes the quarter factor internally (the arrows, the T = 4 wave), and the rung factor is realized externally as the ladder — the fold x = iʳ·3ˢ drawn, r inside each fiber, s as the radius. Consecutive fibers are one rung apart, ×3 = g⁴, and the third-of-a-turn appears on both faces: on the values as the rung shift, and on the mounting as the orientation — each shell rigidly rotated 120° about the main axis, at full angle, since a congruence is not a phase; the quarter is blind to both, so the arrows repeat identically across the shells while the tints cycle one rung. Each shell carries the complete aligned fiber, 48 = 4×12 cover-nodes; with the origin stabilizer — the pure drive, the Observer's own clock, 12 null events — the count closes: 3×48 + 12 = 156 = p(p−1), the affine group: the totality of the observable [1:B1], decomposed with nothing left over. Equal weight per shell is the uniform counting measure on the three C₃ fibers, resolved through coherent sums; the Born readout of [4] lives at the registration layer, and consumes this measure. The instance is Objectless: in the generic (q, p, Ω) case, with p and Ω enormous, the ladder densifies into the radial coordinate, and radial probability densities with quantum numbers — an Object's orbital structure on the κ_O ladder [1:§9] — live inside that chart as Object physics; the toy previews the geometry, and contains no orbitals to scale. Visibility is depth alone.

The light cone: the meridian is the scale-tower circuit [1:A3, D3] [3, 4]. The meridian law runs colatitude aπ/13 at radius R·|sin(2aπ/13)| — |sin| of the tower angle, one full period per cycle, period points at a = 0, 13/2, 13. The radius is the scale coordinate: the approach to the origin is the decoherence descent L₃ → L₂ → L₁ into registration, absorption the completed cascade, the continuation down the scale-periodic tower; between the quarter folds the circuit runs below resolution — unobservable, undrawn. The cycle closes because the tower is periodic: κ scale steps of four heights each, 4κ = p − 1, one drive revolution — and the drive step factors exactly, g = g⁴·g⁻³ = 3·i on F₁₃: one chronon = one scale step × one quarter turn, ψ_τ+1 = i·(3ψ_τ). The capacity rule 4d < 13 selects the twelve exact stations per meridian; the winding family m = 1…12 under the one law covers all 72 nodes, 24 per shell — each drawn meridian one label of the covering family, isotropy at the resolved mesh, refined down the tower within capacity; every node is collected once on odd rows and thrice on even — the sheet parity of the fold decides each slot, and the completeness is a corollary of that parity law. One Dirac flow in every representation; the Subject's own locus is never dressed (ownership): every ray anchors at the drawn Observer at every clock state, the gauge's half-turn acting as the chart's exact point reflection — fixing the Observer — and the quarter sweep kissing the tangency rim, the drawn spinorial quarter [1:D5]. M₀ and M₃ (winding 5 = g⁹ = −g³, the ±i line) are the field pair F = E + iB: co-propagating at the unregistered origin, quarter-separated through the shells, fused pairwise at the four horizon fold points — the duality exchange with route and time both flipped; all four M₃ rays pass the half-cell point 13/2 exactly, the ramification of the double cover. The clock bead displays the frame phase.

The sequel (E2): the frame chart is projective. The observer's chart is the projective line, so the observable frame group is PGL₂(F₁₃), order 12 × 13 × 14 = 2184: its Borel — the point stabilizer, the affine group AGL(1, 13), our cone chart of 156 cells — and its nonsplit C₁₄, acting regularly on the fourteen points of P¹, factorize it exactly, 2184 = 156 × 14, stabilizer times transversal. SL₂ → PSL₂ with kernel {±I}, PSL₂ the index-two rotation half of PGL₂: the Clifford layer, carrying η, the lifts, and the spinor sign S(−1) = −I. The boosts change the observer; their study is the rank-3 sequel, the declared horizon.