STAR-PHASER GALAXY

§1 · Hero Positioning

First clean field architecture in Region I

From ℤ₁₀ decimal structure to algebraically closed GF(2⁴).

★

STAR-PHASER lineup

GEMINI → SOLAR → GALAXY → TETRIS → NOVA

GEMToday

GEMINI — d=2, GF(2) dual-rail

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v1 shipping product. Path-encoded qubit on Si₃N₄; measurement-based fusion; single-node OPC hygiene. Product is dual-rail path encoding — not frequency-bin field ISA.

d=2 · Region I · TODAY

SOLℤ₁₀

SOLAR — d=10, structured composite

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Human-readable decimal: 2×5 tensor-product subspace with ℤ₁₀ addition — convenient for symbolic encoding, not a field (5 has no inverse mod 10). Not algebraically closed.

d=10 · ring, not field · Region I

GALGF(16)

GALAXY — d=16, first true GF system

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Sixteen mutually coherent frequency bins labeled as GF(2⁴) elements. Addition = XOR on 4-bit polynomial coefficients; multiplication = polynomial multiply mod primitive irreducible p(x) = x⁴ + x + 1. Every nonzero element has an inverse.

Hero claim: GFADD/GFMUL kernels, RS(15,11) syndrome arithmetic, and X₁₆/Z₁₆ stabilizer Paulis become first-class ISA primitives — not transcoder hacks on decimal.

4 bits/photon · 750 GHz · Q ≈ 0.08–0.15 · roadmap

GALAXY highlighted in STAR-PHASER lineup

NOVGF(64)

NOVA — d=64, Region II endpoint

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Optimal balance chip; 8×8 octet tiling at full scale; GALAXY's dual-octet layout is the genesis step. GALAXY QUASAR-transition prototype rehearses NOVA materials at d=16.

d=64 · Region II · WS12 synthesis

STAR-PHASER lineup with GALAXY gold highlight — **Figure 1. Lineup hero.** GALAXY is the first frequency-bin GF(2^n), n>1 field architecture — the deliberate bridge from SOLAR's ℤ₁₀ to NOVA's GF(64).

Q-metric gauge showing GALAXY in Region I — **Figure 2. Q-metric gauge.** Target Q ≈ 0.08–0.15 — linear-dominant, well inside Region I (Q < 0.2). 750 GHz ≪ 12 THz OPC band.

Hilbert

d = 16 = 2⁴

4 bits per photon in Hilbert space; 4 GF(2) coefficient bits per field element in polynomial basis.

Carrier

Δf = 50 GHz

B_comb = 750 GHz = 15·Δf. C-band telecom-native; AWG-precedented spacing.

Footprint

~5×7 mm² die

Three co-integrated tiles — source, QFP/FAU, readout — scaled from B07 decit tile.

§2 · Encoding Physics

16-bin lattice & GF(16) spectral map

One photon, sixteen colors — Hilbert dimension and finite-field arithmetic coincide by design.

16

Spectral commitments

Lattice · labeling · orthogonality · OPC margin

StateQudit

|ψ⟩ = Σ c_n |f_n⟩, n = 0…15

→

f_n = f_ref + n·Δf ,   Δf = 50 GHz ,   Σ|c_n|² = 1

Coefficients are amplitudes over GF(16)-labeled bins. Quantum superposition is distinct from classical GF(16) addition implemented by FAU interference.

How we do it: All sixteen teeth descend from one pump via SFWM — mutual phase coherence at birth [demonstrated, B02].

16-bin frequency lattice at 50 GHz spacing

MapGF(16)

|f_k⟩ ↔ α^k, k = 0…14

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Primitive polynomial p(x) = x⁴ + x + 1. Computational basis = exponential/log basis (native for RS roots). Polynomial basis for XOR-add via F₁₆ QFP rotation.

Guard bin f₁₅: reserved — α¹⁵ = α⁰ degeneracy isolation; monitor or RS zero [designed/target]. Active compute window: bins 0–14.

PurityG33

Orthogonality & linewidth targets

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Adjacent-bin overlap F_orth ≥ 0.999 after shaper windowing. Per-bin linewidth δν = 0.19–1.9 GHz (Q=10⁵–10⁶); Δf/δν ≥ 26. Crosstalk budget ε ≤ 1% per neighbour.

OPC compatibility: 750 GHz ≪ 12 THz claimed FWM band → one OPC pass conjugates all sixteen bins coherently.

GALAXY 16-bin spectral lattice over 750 GHz — **Figure 3. Sixteen-bin GF(16) lattice.** Uniform 50 GHz spacing over 750 GHz; bin 15 hatched as guard; pump placed ≥2Δf outside compute window.

§3 · Gate Set

SU(16) QFP & GF(16) named primitives

First chip where GF arithmetic and SU(d) quantum gates share one QFP stack.

Generator set

{X₁₆, Z₁₆, F₁₆, BS₂}

X₁₆	\|n⟩ → \|n+1 mod 16⟩	1 stage
Z₁₆	\|n⟩ → ω^n\|n⟩	1 stage
F₁₆	DFT₁₆ / field FFT	6–10 stages
BS₂	2-bin Hadamard	1 stage

FAU kernels (Tier A)

GFADD · GFMUL · GFINV · GFMAC

GFADD	XOR on 4-bit poly	1–2 depth
GFMUL	Log/antilog or F-basis conv	2–4 depth
GFINV	α^k → α^{15−k}	1 (table)
GFMAC	Parity accumulate	3–6 depth

Tiering: Tier A (production) = named generators + FAU only. Tier B (calibration) = offline Reck SU(16). No runtime arbitrary unitaries in QEC hot loop. Harmonic RF bus: tones m·Δf, m=1…8 on 0.8–1.2 cm TFLN traveling-wave segment.

F16 QFP cascade with OPC inserts — **Figure 4. F₁₆ QFP cascade.** 6–10 EOM→16-ring shaper sandwiches; distributed AlGaAs OPC every 6–8 stages; target F_ρ ≥ 0.95.

Tier-A GF(16) gate ISA wheel — **Figure 5. Tier-A ISA wheel.** Production hot path: X₁₆, Z₁₆, F₁₆, GFADD, GFMUL, X₁₆(α), Z₁₆(β). Tier B offline tomography only.

§4 · Source & Fab

16-line comb & dual-octet tiling begins

Scaled decit tile — not a new materials story.

FAB

Three-tile floorplan

SOURCE · QFP/FAU · READOUT · ~5×7 mm²

SRCComb

16-line SiN microcomb @ 50 GHz

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Route A (primary): SFWM ring R ≈ 480 µm (FSR ≈ 50 GHz), Q_loaded 10⁵–10⁶, 16 signal teeth + heralding partner set. Route B (fallback): EO sideband comb — R≈50 µm seed + TFLN harmonics.

How we do it: Pump rejection ≥110 dB; f_p placed ≥2Δf outside nearest compute bin.

OCT2×8

Dual-octet shaper tiling

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OCT0: bins 0–7  ── local trim ──► bus
OCT1: bins 8–15 ── local trim ──► bus

First appearance of octet tiling — scales to G46 eight-octet NOVA. Trade: +1.5 dB router loss vs monolithic; gain: parallel bring-up, yield per octet.

ΔB07

Scale-up from decit tile

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+6 shaper rings, +6 WDM channels, +2 octet trim loops vs B07. Heater DAC ~32–40 channels. AlGaAs BEOL overlay for 2–3 distributed FWM-OPC cells (QUASAR-transition).

How we do it: Retarget B07 GDS — duplicate 8-ring octet cell; AlGaAs windows post-TFLN, pre-cap.

GALAXY die floorplan 5x7 mm2 — **Figure 6. Die floorplan.** SOURCE (16-tooth comb) → QFP/FAU (dual-octet shapers, TFLN EOM, AlGaAs OPC mesh) → READOUT (16-ch AWG fan-out). Octet tiling begins at d=16.

§5 · Routing & Readout

16-channel WDM — first GF readout

Every detector channel maps to a field element, not a decimal digit.

Topology A

16-ch AWG baseline

50 GHz spacing; ≤−20 dB adjacent crosstalk; ≤4 dB IL. v1 baseline [designed/target].

Topology B

16-ring cascade

Serial bus + add-drop rings; tunable grid fallback when AWG yield fails.

Topology C

2×8 octet mini-AWG

OCT0/OCT1 each 8-ch; matches S24 tiling; +1.5 dB router.

Readout fidelity [model]: single-stage AWG ε=1% → F_read ≈ 0.981; cascaded −30 dB → ≈0.998. Product mode: 16 SPAD single-shot GF(16) computational read. Bring-up: 1–2 SNSPD serial scan with project-then-detect [B09]. Non-computational basis (MUB/F₁₆): QFP rotation before demux.

16-channel readout tree with octet alignment — **Figure 7. 16-ch readout tree.** Octet-aligned fan-out; 16×16 confusion matrix as fidelity object; detector i → bin i → α^i.

Project-then-detect measurement flow — **Figure 8. Project-then-detect.** Non-computational GF(16) readout requires QFP unitary U before demux; syndrome s ∈ GF(16) to FPGA.

§6 · Calibration

240 phase relationships on 50 GHz grid

Calibration complexity scales O(d²); GALAXY is still feasible on Kintex-7 BRAM.

N_pairs = d·(d−1) = 16·15 = 240 directed pairwise phase relationships φ_{ij}

Each relationship is relative phase required for F₁₆, GFMUL convolution, and stabilizer QFP rotations. Errors map to field-element phase slips, not anonymous bin drift. Target residual: σ_φ ≤ 0.05 rad rms → F₁₆ fidelity ≥ 0.95.

Servos

Comb lock & octet trims

f_ref lock — heterodyne vs GPS RF, 10 kHz
16 per-tooth trims — power monitor array, 1 Hz
OCT0/OCT1 global heaters — cross-octet φ drift guard
2–3 OPC pump phase biases per AlGaAs cell
6–10 EOM IQ tables × 8 harmonics — frozen offline

Ladder comparison

Phase relationship count

GEMINI d=2	2
SOLAR d=10	90
GALAXY d=16	240
TETRIS d=32	992
NOVA d=64	4032

240 pairwise phase relationship heatmap — **Figure 9. 240-pair phase tensor.** 16×16 directed matrix (diagonal blank); octet blocks 0–7 and 8–15 outlined; self-calibration state machine: PAIR_SWEEP → UPDATE → F₁₆_PROBE → OPC_ECHO → SIGNOFF.

§7 · OPC & CV · QUASAR-Transition

Distributed OPC & the canonical prototype recipe

GALAXY rehearses Region II hardware while staying Region I in the Q-metric.

Required · QUASAR-transition prototype recipe

Canonical build card — GALAXY → NOVA handoff

The GALAXY QUASAR-transition prototype is the reference stack for rehearsing Region II fab before NOVA. It deliberately trials III-V heterogeneous nonlinear and optional CV assist without crossing Q≈0.2 in the v1 GALAXY SKU.

1	Backbone	Stoichiometric Si₃N₄ — 200 mm damascene, 0.1 dB/cm class routing
2	Source	SiN microcomb — 16 lines, Δf=50 GHz, B_comb=750 GHz
3	Mixer	Heterobond TFLN TW-EOM, >110 GHz BW, 8-harmonic RF bus
4	Shaper	16 microring SPU — dual-octet OCT0/OCT1
5	Nonlinear	AlGaAs overlay — distributed FWM-OPC cells ×2–3 (2–4 mm each, η_pulse 0.5–5%)
6	CV assist	3–6 dB squeezing optional at encoding boundary — calibration/syndrome SNR, not gate requirement
7	Readout	16-ch AWG/ring demux + SPAD array — octet-aligned
8	Region	I — target Q ≈ 0.08–0.15 (< 0.2)
9	Signoff	T-GALAXY matrix before NOVA G46 tape-out

Q = (H_NL + H_OPC + H_CV) / H_linear ≈ (0.06 + 0.04 + 0.02) / 0.88 ≈ 0.14  ✓ Region I

What it previews: distributed OPC mesh (2→3 cells → 8+ at NOVA), octet tiling (2×8 → 8×8), optional squeeze (3–6 dB → mandatory higher at NOVA). What it does not claim: field computing (Region III–IV), single-photon OPC demonstrated, production AlGaAs yield.

OPC

Distributed FWM-OPC

AlGaAs cells · comb-derived pumps · phase pre-layer

CellAlGaAs

ω_idler = 2ω_pump − ω_signal

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2–3 distributed cells along QFP bus; γ_eff 10–40 W⁻¹m⁻¹; pump detuning 7.4 THz Raman-null. Pumps derived from same 16-line microcomb — phase-locked conjugation reference [G45].

Insert every 6–8 QFP sandwiches. Steady-state visibility V_∞ ≈ 0.94 with M=6 stages, σ=0.08 rad, η=2% [model].

CV3–6 dB

Optional squeeze at encoding boundary

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3–6 dB squeezing (10–25% variance reduction) improves homodyne SNR for phase-syndrome checks and squeeze-assisted 240-pair calibration — not required for computational-basis SPAD readout.

Mechanism: AlGaAs OPO shunt or off-chip SFG v1. Low duty preserves Q < 0.2. Preview of H_CV bucket for QUASAR without activating field dynamics.

vsAs₂S₃

AlGaAs chosen for QUASAR rehearsal

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GEMINI/NOVA docs emphasize As₂S₃ chalcogenide overlay. GALAXY prototype deliberately trials AlGaAs for III-V heterogeneous nonlinear — foundry-accessible, strong χ⁽³⁾. Production NOVA may reunify on As₂S₃; both paths [designed/target] until T9.

AlGaAs vs As2S3 nonlinear path comparison

QUASAR-transition prototype stack cross-section — **Figure 10. QUASAR-transition stack (required).** Si₃N₄ core + TFLN bond + AlGaAs OPC pads ×3 + optional 3–6 dB squeeze at input. Canonical recipe for fab and marketing handoff.

Distributed OPC along QFP bus preserving 16-bin coherence — **Figure 11. Distributed OPC bus.** Phase flip φ→−φ at each AlGaAs cell; 16-bin spectrum coherence unbroken across 750 GHz. OPC ≠ QEC — pre-layer beneath RS/stabilizer.

QUASAR-transition recipe to NOVA handoff diagram — **Figure 12. Recipe → NOVA handoff.** Scale octets 2→8, d 16→64, Q→~0.4. GALAXY prototype is the NOVA materials rehearsal — not a Region II claim.

§8 · QEC

First field-native classical + quantum ECC

RS/BCH and X₁₆/Z₁₆ stabilizers share the native GF(16) symbol field.

L7 classical

RS(15,11) default

n=15 symbols, k=11 message, t=2 errors or 4 erasures. Syndromes S_j ∈ GF(16). Photonic RSENC via GFMUL chain; FPGA Berlekamp–Massey decode; RSFIX via FAU XOR.

Erasure-first: heralded photon loss → known bin position → RS gains 2× vs error-only.

L8 quantum

X₁₆ / Z₁₆ stabilizers

15 X-type + 15 Z-type generators per qudit — X₁₆(α^ℓ), Z₁₆(β) with β, α^ℓ ∈ GF(16)*. Syndrome extraction: QFP U_k → 16-ch demux → s ∈ GF(16) → CORRECT within ~11.2 ns.

GF(16) dual QEC stack L7 RS and L8 stabilizer — **Figure 13. Dual QEC stack.** L7 RS(15,11) parity via GFADD/GFMUL; L8 stabilizer grid with X₁₆/Z₁₆; syndromes are field elements — not digits, not bin indices.

§9 · Applications

First true GF-based application stack

Field-native pilots for coding theory, crypto, and NOVA bring-up validation.

APP

Pilot workloads

RS · GFMUL · stabilizer · field-ISA validation

RSL7

RS(15,11) end-to-end round-trip

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Encode 11 symbols → transmit 15-bin photon block → induce erasures or phase slips → decode. First on-chip photonic RS round-trip milestone [roadmap]. Interop with Magma/Sage — same polynomial x⁴+x+1.

ISASDK

Field-ISA microcode validation

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GFMUL truth table (225 nonzero products); RS syndrome match FPGA golden; stabilizer commute tableau; OPC on/off F₁₆ delta < 5%. Validates microcode before GF(64) opcode explosion at NOVA.

Decimal I/O via G04 transcoder at API boundary only — SOLAR remains decimal-native; GALAXY is field-native.

Publishable framing: Photonic qudit system of dimension d=16 implemented in frequency-bin synthetic spectral space with OPC-assisted CV stabilization and GF(2⁴) algebraic control layers — first true GF-based system in STAR-PHASER lineup. Throughput sketch [model]: ~4 Mb/s symbolic per spatial mode at pilot rates — not a product claim.

GALAXY application stack from hardware to pilot apps — **Figure 14. Application stack.** SiN+TFLN+AlGaAs hardware → GF(16) FAU ISA → RS/BCH L7 → stabilizer L8 → pilot apps. Sidebar: SOLAR ℤ₁₀ | GALAXY GF(16).

Parts & System Integration

Bill of materials + control stack

Every component in the GALAXY d=16 system — first frequency-bin chip on the QUASAR-transition path — with verified vendor part numbers.

Bill of materials — GALAXY components

Category	Part	Vendor / P/N	Spec	Qty	Source
Photonics	Si₃N₄ waveguide backbone	LIGENTEC AN800	200 mm damascene, 0.1 dB/cm	Backbone	FEOL
	16-line SFWM microcomb	LIGENTEC AN800	R≈480 µm, FSR≈50 GHz, Q 10⁵–10⁶	1 ring	FEOL
	EO sideband comb (Route B)	QLT + LIGENTEC	R≈50 µm seed + TFLN harmonics, fallback	1	Hybrid
	16-ch AWG demux	LIGENTEC AN800	Baseline; alt: 2×8 octet mini-AWG or 16-ring cascade	1	FEOL
	16 microring SPU (shaper)	LIGENTEC AN800	Dual-octet OCT0/OCT1 tiling	16 rings	FEOL
	Harmonic RF bus	QLT bond	0.8–1.2 cm TFLN traveling-wave segment	1	Bond
Active	Heterobond TFLN TW-EOM	QLT + X-Celeprint MTP	>110 GHz BW, 8-harmonic RF bus	1	Bond
	Heater DAC channels	Analog Devices AD5372	16 per-tooth + OCT globals + OPC bias	~32–40 ch	Purchased
	AlGaAs FWM-OPC cells	MBE + X-Celeprint MTP	γ_eff 10–40 W⁻¹m⁻¹, 2–4 mm each	2–3 cells	MTP
Detectors	InGaAs SPAD module	Aurea Technology	18-ch (16+2 herald), PDE ≥28%, free-running + gated	1 module	OEM
Detectors	SNSPD (bring-up)	—	1–2 ch serial scan, lab only	1–2	Lab
Electronics	FPGA	AMD/Xilinx Kintex-7	BRAM phase tables, GF(16) decode	1	Dev board
Electronics	Supervisor MCU	STMicro STM32H7	Housekeeping, DFU, calibration	1	Purchased
Lasers	CW pump laser	Santec TSL-570	C-band SFWM pump, 150 mW	1	Purchased
Lasers	Mode-locked clock	Menlo Systems C-fiber	100 MHz, 150 fs	1	Purchased
Materials	AlGaAs (Al₀.₂Ga₀.₈As) coupons	MBE-grown	2–4 mm rib, MTP into SiN recess	2–3	MTP
	As₂S₃ (alternative OPC)	QLT BEOL	6N purity, <210 °C thermal evap	Alt path	BEOL
	TFLN thin film	QLT bond	300–600 nm, ion-cut	EOM layer	Bond

Split-fab architecture

GALAXY adds AlGaAs OPC + TW-EOM to the GEMINI baseline

Phase A — FEOL: LIGENTEC AN800. SiN microcomb ring, AWG, shaper banks, routing. Same 200 mm X-FAB Erfurt line.

Phase B — TFLN + AlGaAs bond: QLT. TW-EOM for 8-harmonic RF bus. X-Celeprint MTP for AlGaAs OPC coupons.

Phase C — BEOL: As₂S₃ alternative OPC path. Arsenic-isolated cleanroom.

Phase D — Assembly: 18-ch Aurea SPAD module, fiber array, hermetic package.

Key delta from GEMINI

QUASAR-transition prototype

Comb: 16-line microcomb @ 50 GHz (750 GHz span) vs GEMINI's path encoding.

OPC material: AlGaAs (γ_eff 10–40 W⁻¹m⁻¹) primary, As₂S₃ fallback — vs GEMINI As₂S₃ only.

Readout: 18-ch SPAD for single-shot GF(16) read vs GEMINI's 12-ch.

Squeezing: 3–6 dB optional at encoding boundary (new for GALAXY).

System integration

LUV → QIL → FPGA → GALAXY photonic chip

LUV Language v0.3 .luv source → luvc build → QIL IR 0.1.0 · run_freqbin_shot (gate-only path)

QIL Middleware — Three Artifact Contract dac.voltages[40] → 16 per-tooth trims + OCT globals + OPC pump bias
qfp_schedule[] → GF(16) frequency-bin unitary programs
shot_contract → 18-ch SPAD pattern + comb_lock gate

rf_schedule[] → 8-harmonic EOM IQ tables (new for GALAXY)

FPGA Firmware — Kintex-7 GF(16) arithmetic decode · BRAM phase tables · DAC SPI · TDC · RF scheduling

Kintex-7 BRAM for expanded phase table storage

Supervisor MCU — STM32H7 Cortex-M7 Housekeeping · DFU · calibration · comb lock monitoring

Control Electronics AD5372 DAC (~40 ch) · TDC ASIC · Santec TSL-570 pump · Menlo C-fiber clock

Aurea 18-ch InGaAs SPAD (16+2 herald) · R&S SMA100B RF gen (50 GHz)

Photonic Chip — GALAXY (~5×7 mm²) LIGENTEC AN800 SiN · 16-line microcomb · 16-ch AWG · 16 microring SPU · TW-EOM
AlGaAs OPC cells ×2–3 · Raman-null pump detuning 7.4 THz

Δf=50 GHz · B_comb=750 GHz · d=16 · GF(2⁴) · OPC margin ~16×

GALAXY is Rung 2 on the encoding progression. See the GF(64) roadmap, today's dual-rail platform, and manufacturing path.

GALAXY manufacturing traveler Base 2 → GF(256)How We Do It Manufacturing