Protocol Optimisation

The protocol-optimisation layer turns the GUM forward- propagation pipeline into a design tool: answer the precision-requirement question and the budget-allocation question in closed form.

Two operations:

• required_precision(m, σᵢ, target_uc) — the analytical "precision condition" on σᵢ to meet a target combined standard uncertainty (EARS REQ-060).
• budget_allocation(m, variables, sigmas, total_budget) — the Lagrange-multiplier optimal allocation of a fixed total budget across inputs (REQ-061).

required_precision — inequality framing

using Symbolics, SymbolicUncertainties

@variables V I σV σI
R = (V ± σV) / (I ± σI)

cond = required_precision(R, σV, 0.01)

\[ \begin{equation} \sqrt{I^{2} ~ \left( 0.0001 + \frac{ - \mathtt{{\sigma}I}^{2} ~ V^{2}}{I^{4}} \right)} \end{equation} \]

Read the returned Num as the boundary value: σV values strictly below satisfy m.err < target_uc; values equal saturate; values above violate.

required_precision is implemented as a direct delegate to infer_precision — see specs/007-protocol-and-inference/research.md R6 for the rationale. Both functions return the same Num expression; the distinct names preserve the two EARS framings (precision condition REQ-060 vs. solved equation REQ-090).

budget_allocation — Lagrange-optimal distribution

Given a fixed total uncertainty budget the measurement can afford, budget_allocation returns the per-input σᵢ* that minimises m.err subject to Σᵢ σᵢ = total_budget. The closed form is the inverse-sensitivity-squared weighting:

σᵢ* = total_budget · (1 / cᵢ²) / Σⱼ (1 / cⱼ²)

where cᵢ = sensitivity_coefficient(m, xᵢ).

Symmetric example

For a pure sum m = a + b + c with total budget B, all three sensitivity coefficients are 1, so every input gets B/3 — the symmetric optimum:

@variables a σa b σb c σc
m = propagate((x, y, z) -> x + y + z, [a ± σa, b ± σb, c ± σc])

alloc = budget_allocation(m, [a, b, c], [σa, σb, σc], 1.0)
# alloc[σa] = alloc[σb] = alloc[σc] = 1/3 after numeric
# evaluation.

Dict{Symbolics.Num, Symbolics.Num} with 3 entries:
  σb => 1//3
  σc => 1//3
  σa => 1//3

Asymmetric example

For m = 2a + 3b the sensitivity coefficients are c_a = 2, c_b = 3, so the inverse-squared weighting is 1/4 : 1/9 ⇒ the σ allocation ratio is 9 : 4. This is the stationarity condition cᵢ²·σᵢ = cⱼ²·σⱼ.

alloc = budget_allocation(
    propagate((x, y) -> 2x + 3y, [a ± σa, b ± σb]),
    [a, b], [σa, σb], 1.0,
)
# alloc[σa] ≈ 9/13,  alloc[σb] ≈ 4/13.

Dict{Symbolics.Num, Symbolics.Num} with 2 entries:
  σb => 0.307692
  σa => 0.692308

Error paths

• length(variables) != length(sigmas) → DimensionMismatch.
• Numeric total_budget < 0 → ArgumentError.
• Every sensitivity coefficient zero (measurand independent of every supplied input) → ArgumentError per REQ-091.

Per-variable sensitivity_coefficient failures (e.g. the REQ-021 unresolved-differential path) propagate unchanged.

API reference

required_precision(m::SymbolicMeasurement, σᵢ, target_uc) -> Num

budget_allocation(m, variables, sigmas, total_budget) -> Dict{Num, Num}

σᵢ* = total_budget · (1 / cᵢ²) / Σⱼ (1 / cⱼ²)