Formal specifications

Progressive formal specification for research and implementation

This page is the evolving execution contract for PiP research and build programs. It now uses explicit mathematical notation, typed formal statements, and LaTeX-compatible canonical forms so specifications can be compiled directly into formal technical documents.

Scope and governance

This specification governs research assumptions, implementation contracts, and release readiness for PiP systems.

Any architecture, dataset, tooling, or fabrication change that can alter measured behavior must be reflected here before deployment decisions.

Normative levels: MUST (hard gate), SHOULD (strong recommendation), MAY (optional extension).

Mechanism

Specification to implementation to measurement to revision.

Boundary

No claims are considered official unless mapped to explicit benchmark and validation identifiers.

Adopted formal language

The specification adopts a typed, set-theoretic first-order form with operator semantics and temporal progression constraints for runtime stability claims.

Core form: ⟨𝒮, 𝒯, 𝒱, 𝒢⟩ where 𝒮 is signature, 𝒯 is theory (axioms/theorems), 𝒱 is verification obligations, and 𝒢 is gate thresholds.

Normative operators: MUST (hard validity requirement), SHOULD (strong recommendation), MAY (optional extension).

Canonical statement pattern

Definition 𝒟ᵢ:
  ∀x ∈ 𝒳, ∃y ∈ 𝒴 such that y = Fᵢ(x, θᵢ).

Obligation 𝒪ⱼ:
  If preconditions Pⱼ hold, then metric Mⱼ ≥ τⱼ MUST hold.

Gate 𝒢ₖ:
  Promote revision r only if 𝒪ⱼ is satisfied for all required j ∈ 𝒥ₖ.

Notation and symbols

x ∈ 𝒳 ⊂ ℝⁿ: input tensor space.

uₜ: perturbation input at step t.

P_θ: PiP physical operator parameterized by geometry/material vector θ.

sₜ: latent physical state at step t.

R_ψ: readout operator with calibration parameters ψ.

D_φ: decode map to output space 𝒴.

Primary equations (reference form)

Forward transform:
  ŷₜ = D_φ( R_ψ( P_θ( E_α(xₜ), uₜ ), sₜ ) )

State evolution:
  sₜ₊₁ = F_η(sₜ, uₜ) + εₜ

Quality constraint:
  Q(ŷₜ, yₜ) ≥ τ_Q

Continuity constraint:
  C(ŷₜ, ŷₜ₋₁) ≤ τ_C

Energy-per-update objective:
  E_update(t) ∝ ‖uₜ - uₜ₋₁‖₂

Revision protocol

Step 1: Proposal intake

Open a revision proposal with affected modules, expected behavior delta, and risk class (low, medium, high).

Step 2: Simulation impact scan

Run controlled simulation suite and compare output manifold drift against current approved baseline.

Step 3: Fabrication and metrology check

Verify the revision is manufacturable within process windows and does not violate tolerance bands.

Step 4: Runtime and benchmark rerun

Re-execute benchmark matrix, sign off only when quality and stability gates remain valid.

Core formal specification

The core specification is organized as a contract across input encoding, physical transform, readout, and decode layers. Every layer must expose measurable acceptance criteria.

Input representation

A signal is encoded as optical boundary conditions derived from model input.

Physical operator

Engineered photonic geometry performs transformation through propagation.

Measured state

Output state is sampled at designated readout points with calibrated sensing.

Output

Measured response is decoded into task-specific prediction space.

Focused stage

A signal is encoded as optical boundary conditions derived from model input.

FlowStateMeasurementConstraint

Axiomatic contract layer

Axiom A1 (Determinism under calibration):
  Given fixed (θ, ψ, φ) and bounded perturbation class 𝒰,
  Var[ŷ | x, 𝒰] ≤ τ_det.

Axiom A2 (Operator rank sufficiency):
  rank(J_θ) ≥ r_min, where J_θ = ∂ŷ/∂x.

Axiom A3 (Temporal stability):
  supₜ ‖sₜ₊₁ - sₜ‖₂ ≤ τ_state under envelope Ω.

Axiom A4 (Benchmark parity floor):
  Q_PiP ≥ Q_baseline - Δq_max for every required benchmark b.

Input encoding contract (MUST)

Input tensor normalization range, boundary condition mapping, and perturbation envelope are explicitly versioned.

Physical operator contract (MUST)

Geometry, material class, and simulation fidelity must map to fabrication files with traceable identifiers.

Readout contract (SHOULD)

Sensor placement, sampling cadence, and calibration model should remain stable within approved drift bands.

Decode contract (MUST)

Decode mapping to task outputs must preserve quality target thresholds for all benchmark suites.

Interface contracts

compile: 𝒢 ⟶ (θ, 𝑀, 𝒦) where 𝒢 is model graph, 𝑀 mask artifacts, 𝒦 calibration keys.

simulate: (θ, x, u) ⟶ (ŷ, ∇, σ) with uncertainty estimate σ.

fabricate: (𝑀, process) ⟶ (lot_id, metrology_report).

run: (xₜ, uₜ, lot_id) ⟶ (ŷₜ, telemetryₜ).

Validation and gate criteria

Gate A (Determinism): Var[ŷ | x] ≤ τ_det over repeated trials.

Gate B (Rank/separability): rank(J_θ) ≥ r_min and probe accuracy Acc ≥ τ_acc.

Gate C (Stability): supₜ ‖sₜ₊₁-sₜ‖₂ ≤ τ_state under envelope Ω.

Gate D (Benchmark parity): quality floor and promised deltas versus GPU/NPU baselines are satisfied.

LaTeX compatibility and formal publishing profile

This specification is authored to be LaTeX-native. All equations and formal statements follow notation compatible with amsmath, amssymb, mathtools, and theorem environments.

Engine support: pdflatex, xelatex, and lualatex.

Bibliography support: bibtex and biber profiles.

Math profile: numbered equations, theorem/proof blocks, Unicode symbol support (∀, ∃, ∈, ≤, ≥, ∇, ∂).

Export profile: PDF-ready and archive-ready builds with deterministic references.

Minimal preamble (reference)

\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{bm}
\usepackage{siunitx}
\usepackage{hyperref}
\usepackage{cleveref}

\newtheorem{definition}{Definition}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}

Compilation commands

pdflatex spec.tex && bibtex spec && pdflatex spec.tex && pdflatex spec.tex
xelatex spec.tex && biber spec && xelatex spec.tex && xelatex spec.tex
lualatex spec.tex && biber spec && lualatex spec.tex && lualatex spec.tex

Tooling and artifact stack

Model and compiler

PyTorch or JAX training pipelines, custom graph-lowering passes, and physical-constraint compiler outputs.

Simulation and optimization

Differentiable photonic simulation stack (FDTD/FEM classes), gradient-based geometry optimization, and sweep orchestration.

Fabrication and metrology

Mask layout artifacts, process control logs, interferometric measurements, and drift tracking reports.

Runtime and observability

Readout calibration services, deployment runtime adapters, and benchmark telemetry with reproducibility metadata.

Publication cadence and update instructions

Cadence: biweekly specification review, monthly baseline freeze, quarterly formal revision release.

Versioning: semantic pattern spec-major.minor.patch with change rationale and affected benchmarks.

Required attachments: simulation diff report, metrology summary, benchmark delta table, and deployment risk note.

For implementation sequencing, open the dedicated roadmap page: Implementation Roadmap.