14-EFT.WP.Methods.Inference v1.0 | Appendix A: Unified Terms & Symbols

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Appendix A: Unified Terms & Symbols

I. Scope & Objectives

• Establish a unified baseline of terms, symbols, and dimensional conventions used across chapters, companion volumes, and implementation bindings. Coverage spans probability & inference, uncertainty & calibration, time & arrival time, spectra & windows, environment & signatures, performance & SLO, online/offline consistency, and canary/change management.
• All inline symbols must be wrapped in backticks; any fractional / composite operator must be parenthesized. For conditional probabilities, expectations, integrals, and path functionals, always declare the measure and support explicitly.

II. Notation & Typesetting Conventions

• Variables & operators: x, y, theta, D, y_hat, L(•), score, E_{p(•)}[•], p(•|•), grad, div, curl.
• Time & paths: tau_mono, ts, alpha, beta, gamma(ell), d ell, dS, dV; arrival-time dual formulations are presented in parallel (constant-factored and general).
• Dimensions vs. dimensionless: run check_dim(expr). Arguments to log/exp/ln must be dimensionless. Probabilities, calibration errors, quantiles, and relative discrepancies are dimensionless.

III. Unified Symbol Catalog (by domain)

1. Inference & probability
   • x: input (vector or tensor).
   • y: target / label.
   • theta: model parameters.
   • D: dataset or sample set.
   • p(y | x, theta): likelihood.
   • p(theta), p(theta | D): prior and posterior.
   • y_hat: prediction.
   • L(y, y_hat): per-sample loss; dimensionless.
   • R_emp = ( 1/N ) * Σ L( y_i, y_hat_i ).
   • R_exp = E_{p(x,y)}[ L( y, y_hat ) ].
   • NLL = - ( 1/N ) * Σ log p( y_i | x_i, theta ).
   • mean, var, quantile_q: distributional outputs.
   • CI_{1-delta}: confidence interval at level 1 - delta.
2. Calibration & uncertainty
   • ECE = Σ_{k=1..K} ( | acc(B_k) - conf(B_k) | * ( |B_k| / N ) ).
   • MCE = max_{k} | acc(B_k) - conf(B_k) |.
   • ECE_T: ECE after temperature scaling.
   • T_temp: temperature parameter; dimensionless.
   • NLL: as above; used jointly with ECE/MCE for calibration.
3. Time & arrival time
   • tau_mono: internal monotonic time base.
   • ts = alpha + beta * tau_mono: mapping to the published time base.
   • T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) (constant-factored).
   • T_arr = ( ∫ ( n_eff / c_ref ) d ell ) (general).
   • c_ref: reference phase speed; units m/s.
   • n_eff(x,t): effective refractive index; dimensionless.
   • gamma(ell): path parameterization; d ell is the arclength measure.
4. Spectra & windows
   • S_xx(f): power spectral density.
   • U_w: window energy-normalization constant.
   • ENBW: equivalent noise bandwidth.
5. Environment & signatures
   • hash(•): de-identified hash.
   • fingerprint: artifact fingerprint (includes hash aggregates).
   • EnvLock: environment lock.
   • anchor: artifact anchor (joint reference to model/features/dependencies).
   • rng.seed, rng_family: RNG identifiers.
6. Performance & observability
   • TS.latency_{p50,p95,p99}: latency percentiles; units ms.
   • TS.thrpt: throughput; units req/s.
   • TS.error: error rate; dimensionless.
   • budget.cpu/gpu/mem: resource budgets.
   • budget.power: power budget; units W.
   • cost.per.req: unit request cost; currency / request.
7. Consistency & discrepancies
   • delta_offon = ( norm( y_hat_off - y_hat_on ) / norm( y_hat_off ) ).
   • R_infer = 1 - delta_offon.
   • delta_AB = E_{D}[ L( y, y_hat_B ) - L( y, y_hat_A ) ].
   • p_fail = P( delta_AB > tau_noninf ) (estimated via bootstrap).
   • delta_dev: cross-device discrepancy.
   • CertEq: cross-device equivalence certificate.
8. SLO & composite scoring
   • score = Σ_k w_k * z_k (with z_k standardized sub-scores, Σ w_k = 1).
   • tau_error, tau_p99, tau_cost, tau_cal, tau_cons: threshold set.
   • score_LCB: lower-confidence bound of the composite score; compared with score_base.
9. Pipeline & configuration cards
   • IPC: InferPipelineCard.
   • PC: ParamCard.
   • PC.meta.parent_fingerprint: parent fingerprint reference.
   • Key fields: version, schema, artifacts, signature, resources, metrics.
10. Canary & change management
    • ver = major.minor.patch: semantic version.
    • channel ∈ {canary, stable, LTS}.
    • q: traffic share.
    • R = {R0, R1, R2, R3}: ring set.
    • W: observation window.
    • alpha_gate, tau_noninf: advancement significance and non-inferiority gate.
    • Gate: promotion gate.
    • g, s_max, q_cap: advancement-law parameters.
    • budget.blast: blast-radius budget.
    • kill_switch, holdback: emergency & holdback mechanisms.
    • PlanId, GateDecision, reasons: rollout decision keys.

IV. Conflict Names & Legal Combinations

1. Strict distinctions
   • T_fil (tension) vs T_trans (transmission coefficient): must not be mixed.
   • n (number density) vs n_eff (effective refractive index): must not be substituted.
   • T_arr (arrival time) vs T_temp (calibration temperature): context must be explicit.
2. Legal combinations (examples)
   • With ts = alpha + beta * tau_mono, timestamps of any streaming observability TS.* must be mapped to ts before aggregation.
   • Within a single analysis, choose only one arrival-time convention (constant-factored or general) and state it in the title.
   • Post-calibration metrics carry the _T suffix, e.g., ECE_T, NLL_T.

V. Example Expressions & Dimensions

• Probability & expectation
  E_{p(z)}[ f(z) ] = ( ∫ f(z) p(z) dz ); dimensionless.
• Conditional risk & non-inferiority
  delta_AB = ( 1/N ) * Σ ( L_B - L_A ); if p_fail <= alpha_gate then non-inferiority passes.
• Consistency
  Require delta_offon <= tau_cons as a deployment gate; R_infer = 1 - delta_offon contributes to the composite score.
• Resources & cost
  cost.per.req = ( Σ resource_i * price_i ) / N_req; units follow price_i.
• Calibration
  y_hat = softmax( logit / T_temp ) with T_temp > 0; evaluate ECE_T, MCE post-update.

VI. Measurement Units & Abbreviations

• Time: ms, s. Frequency: Hz. Length: m. Speed: m/s. Power: W. Energy: Wh.
• Statistics: pXX denotes the XX-th percentile (e.g., p99); CI_{1-delta} denotes the confidence interval; LCB/UCB are lower/upper confidence bounds.
• Resources: cpu (vCPU.s), gpu (GPU.s), mem (GiB), power (W).

VII. Structured Objects & Key Names (summary)

• InferPipelineCard (IPC)
  Key keys: version, graph.sig, features.spec, artifacts, EnvLock, anchor, TS.targets, metrics.plan.
• ParamCard (PC)
  Key keys: version, ranges, defaults, units, constraints, meta.parent_fingerprint.
• Reports & evidence
  ScoreReport, CalibReport, DriftReport, ShadowReport, ConsistencyReport, RingMetrics, GatePolicy.

VIII. Thresholds & Defaults (placeholder conventions)

• alpha_gate ∈ [0.01, 0.10]; tau_noninf calibrated to business loss.
• tau_error, tau_p99, tau_cal, tau_cons are derived from Chapter 8 score weights; if no business customization, default to equal-weight parallel gating.

IX. Cross-References & Dependencies

• Calibration & uncertainty: Chapter 7.
• Performance metrics & SLO: Chapter 8.
• Online/offline consistency & time base: Chapter 6 and Chapter 3 P41-2.
• Pipeline & configuration cards: Chapter 9.
• Operations & alerting: Chapter 10.
• Cross-domain/device & equivalence: Chapter 11.
• Acceptance & announcement bundle: Chapter 12.
• Change management & canary: Chapter 13.

X. Numbering Reserve & Exception Enumeration (summary)

• Reserved numbering: postulates P41-*; minimal equations S42-*; flows Mx-4*; bindings I40-*.
• Exceptions: E_MODEL_MISMATCH, E_SCHEMA_MISMATCH, E_TIMEBASE_SKEW, E_NONDETERMINISM, E_CALIBRATION_FAIL, E_DRIFT_DETECTED, E_RESOURCE_EXCEEDED, E_PRECISION_LOSS.