53-Model Card Template v1.0 | Chapter 4 — Architecture & Control Equations

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Chapter 4 — Architecture & Control Equations

I. Purpose & Scope

• Provide normative descriptions of the model architecture and control equations, including pre/post-processing, core network, losses & constraints, inference path, and boundary conditions.
• For path quantities (arrival time/phase), the text must explicitly show gamma(ell) and the measure d ell, and the data side must record delta_form ∈ {general, factored}; arrival/phase equations use the parenthesized unified forms; publication requires p_dim = 1.0.

II. Architecture Overview

• Composition: preprocess → f_θ(core) → postprocess, where f_θ is the core network/operator stack.
• Dataflow: x ∈ R^{B×T×D} (example) is aligned/normalized, fed to f_θ, producing ŷ and optional uncertainty elements.
• Interface contract: I/O fields and units/dimensions align with Dataset Card Ch. 4; constraints & assumptions are specified here and stored in manifests.

III. Control Equations

1. Preprocessing (examples)
   • Normalization: x' = ( x − μ ) / σ; masking: x'' = x' ⊙ m.
   • Path resampling (if applicable): (gamma_ell, d_ell, n_eff) → align(gamma_ell, d_ell, n_eff).
2. Core mapping
   • Form: ŷ = f_θ(x''), with trainable θ; f_θ composed of operator sequences/attention/convolution/kernel regression, etc.
   • Linearization (for Delta method): f_θ(x) ≈ f_θ(x̂) + J_x (x − x̂), J_x = ∂f/∂x |_{x̂}.
3. Loss & regularization
   • Total objective: L(θ) = E[ ℓ(f_θ(x), y) ] + λ R(θ); examples ℓ ∈ {CE, MSE, Huber}, R(θ) ∈ {‖θ‖_2^2, TV, KL}.
   • Multi-task weighting: L = ∑_k w_k L_k (freeze w_k in manifests).
4. Unified path forms
   • Arrival (two equivalent):
     T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
     T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Phase: Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )
   • Evaluation alignment order: time → path (gamma_ell/d_ell/n_eff sync) → phase (reference window).
5. Post-processing
   Rescaling: ŷ_phys = a ⊗ ŷ + b; interval conversion: interval = convert(coverage_mode, params).

IV. Assumptions & Boundaries

• Domain: paraxial/small-angle, slowly varying medium, within coherence windows; |∇ n_eff|·L_coh ≪ 1.
• Sampling & sync: f_s ≥ 2 f_max; clock_state = "locked"; |ts_start − calib.timestamp| ≤ τ_calib.
• Path consistency: len(gamma_ell) = len(d_ell) = len(n_eff) ≥ 2; Δell ≤ ( c_ref / f_s ) / max(n_eff).
• Disabled conditions: strong dispersion/unlock/missing path blocks/invalid units → [Restricted].

V. Units & Dimensions

• All expressions must specify units/dimensions; any division/integral/composite expression is parenthesized.
• Before release run I70-dim_check, output check_dim_report.json, require p_dim = 1.0.

VI. Train/Inference Paths

• Training: x → preprocess → f_θ → loss/regularizer → update(θ); supports incremental/micro-batch; RNG & seed pinned.
• Inference: x → preprocess → f_θ → postprocess → ŷ; idempotency & state-independence: same idempotency_key → invariant outputs.

VII. Gate Mapping

• G1 Schema completeness (I/O vs contract).
• G2 Citation compliance (anchor coverage ≥ 90%).
• G3 Path conventions (gamma/measure/delta_form complete; step & alignment compliant).
• G4 Dimensional closure (p_dim = 1.0).
• G5 Freshness (clock_state="locked", τ_calib valid).
• G6 Coverage consistency (k/alpha/quantile unified with Error Budget).
• G7 Covariance consistency (Σ PD, aligned with cov_group).
• G8 Uniqueness & acyclicity (record_id/checksum, lineage DAG).
• Trigger S1–S5 to block; tag [Restricted] when needed.

VIII. Machine-Readable Artifacts
A. model_arch.svg: core structure graph (nodes/edges/shapes/units).
B. equations.md (excerpt)

# Control Equations

- Pre: x' = (x − μ) / σ

- Core: ŷ = f_θ(x'')

- Loss: L = E[ℓ(f_θ(x), y)] + λ R(θ)

- Arrival: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

- Phase: Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )

C. control_spec.yaml

version: "1.0.0"

preprocess:

normalize: { mean: "μ", std: "σ" }

path_align: { require: true, delta_form: "general" }

core:

f_theta: { type: "hybrid", ops: ["conv","attn","mlp"] }

loss:

total: "L = E[ℓ] + λ R"

tasks: ["task_main","task_aux"]

path_forms:

T_arr: "( ∫ ( n_eff / c_ref ) d ell )"

Phi: "( (2π / lambda_ref) * ∫ n_eff d ell )"

postprocess:

scale: { a: "learned", b: "learned" }

coverage: { mode: "k", k: 2 }

IX. Anti-Patterns & Fixes

• Anti: T_arr = ∫ n_eff / c_ref d ell (missing parentheses) → Fix: parenthesize to the unified form.
• Anti: only gamma(ell) without d ell/delta_form → Fix: complete and align with n_eff.
• Anti: missing units/dimensions → Fix: complete and pass I70-dim_check.
• Anti: inference dependent on training state (non-idempotent) → Fix: remove state coupling or isolate via idempotency key.

X. Cross-References

• Dataset: Ch. 2/4 (I/O contracts), Ch. 6 (splits).
• Error Budget: Ch. 5/6/8 (covariance/coverage).
• Pipeline: Ch. 6/11 (stages/interfaces), Ch. 12 (release).
• Parameter Registration: Ch. 4/6/8 (units/freshness/covariance groups).

XI. Checklist

• Architecture diagram & control equations stored; units/dimensions complete; all division/integral/composite expressions parenthesized.
• For path quantities, explicit gamma(ell)/d ell, delta_form recorded; Δell & alignment constraints satisfied.
• I70-dim_check passed, p_dim = 1.0; coverage mode k/alpha/quantile aligned with Error Budget.
• Gates G1–G8 passed; non-compliances tagged [Restricted].
• Machine-readable set (model_arch.svg/equations.md/control_spec.yaml/check_dim_report.json) and signatures complete.