12-EFT.WP.Methods.Repro v1.0 | Appendix A Unified Terms and Symbols

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Appendix A Unified Terms and Symbols

I. Scope and Objectives

• This appendix unifies terminology, symbols, and expression gauges across the volume, spanning time and arrival time, randomness and reproducibility metrics, spectra and windows, environment locking and signatures, concurrency and run-time observability, benchmarks and scoring, and change management and release—so cross-chapter references remain consistent and forensically verifiable.
• All inline quantities must be wrapped in backticks. Any expression involving division, integrals, or composite operators must be parenthesized with an explicit measure declaration; dimensions and units follow the gauges in this appendix.

II. Notation and Basic Conventions

• Sets, sequences, intervals: {•} denotes a set; [a,b] a closed interval; x[0:N) a half-open sequence.
• Norms and inner products: norm( x ) is the Euclidean norm; <x,y> the inner product; real field is assumed unless stated.
• Probability and expectation: Pr(•), E[•], variance var( • ).
• Integration paths and measures: curve gamma(ell) with arc-length measure d ell; surface and volume measures dS, dV.
• Units and checks: check_dim(expr) denotes a dimensional audit. Dimensionless is 1. Frequency Hz; time s; length m; probability and similarity 1.

III. Unified Symbol Roster (by domain)

1. Time and time base
   • tau_mono: internal monotonic time base (s).
   • ts: external published time stamp (s), linearly mapped from tau_mono.
   • alpha, beta: time-base mapping coefficients, ts = alpha + beta * tau_mono.
   • r_tb: normalized time-base residual, r_tb = ( norm( ts - ( alpha + beta * tau_mono ) ) / norm( ts ) ) (1).
   • T_arr: time of arrival, reported in parallel via two gauges (s).
   • c_ref: reference phase speed (m/s).
   • n_eff(x,t): effective refractive index (1).
   • gamma(ell), d ell: propagation path and measure.
2. Randomness and determinism
   • seed: RNG seed (integer or byte sequence).
   • gate.rep: reproducibility pass gate (Boolean), derived from a set of thresholds.
   • E_NONDETERMINISM: enumerated nondeterminism exception.
   • RNG family and library versions must be recorded under EnvLock.
3. Reproducibility and metrics
   • delta_rep = ( norm( y_new - y_ref ) / norm( y_ref ) ) (1).
   • R_coef = 1 - delta_rep (1).
   • delta_psd: spectral discrepancy (1), see Section V.
   • eps_norm: normalization error (1).
   • eps_mass: physical-conservation residual (1).
   • score: benchmark aggregate score (1), weight-based per Chapter 8.
   • CI_score: lower confidence bound of score (1).
4. Spectra and windows
   • S_xx(f): one-sided power spectral density of signal x (units match normalization, default 1/Hz).
   • U_w: window energy normalization factor (1).
   • ENBW: equivalent noise bandwidth (Hz).
   • f: frequency (Hz); integration domain must be explicit.
5. Environment and signatures
   • EnvLock: environment-lock object.
   • hash(•): cryptographic hash.
   • fingerprint: artifact fingerprint—hash of content plus critical metadata.
   • anchor: release/rollback anchor, based on joint environment and artifact fingerprints.
   • H_k, sig_k: audit-chain hash and signature at step k.
6. Concurrency and run-time observability
   • TS.*: run-time observation namespace (e.g., TS.latency.p50, TS.thrpt, TS.error_rate).
   • hb: heartbeat event.
   • bp: back-pressure flag.
   • makespan, critical path: completion time and critical path of the job DAG.
7. Cards and report objects
   • PipelineCard: structured process/versions descriptor.
   • ParamCard: structured parameter/constraint descriptor.
   • BenchReport, RepReport: benchmark and reproducibility reports.
   • MigPlan, DiffMatrix, CutoverReport, RollbackReport: change-management artifacts.
   • ScoreSummary(old,new): score comparison summary.
8. Change and channels
   • version = MAJOR.MINOR.PATCH: semantic version.
   • channel ∈ {canary, stable, LTS}: release channel.
   • compat flag: set of compatibility toggles.
   • W_mig, T_dep, T_grace: migration window, deprecation activation time, and grace period.
   • budget = { b_rep , b_psd , b_tb , b_mass }: differential budgets.

IV. Collision Names and Gauges (mandatory constraints)

• T_fil denotes tension only; T_trans denotes transmission coefficient only; neither may be used for time or latency.
• n denotes number density; n_eff denotes effective refractive index. Expressions involving T_arr may only use n_eff.
• score is a multi-metric aggregate; R_coef is a single relative-consistency metric. They are not interchangeable.
• seed is not equivalent to nonce; this volume uses seed exclusively for replayable RNGs.
• ts is the external time stamp; internal timing uses tau_mono. Any cross-device/node comparison must first map to the common ts.

V. Example Expressions and Dimensions

1. Dual arrival-time gauges
   • Constant-out: T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) (s).
   • General: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) (s).
   • Discrepancy: delta_form = ( | T_arr(const) - T_arr(general) | / max( |T_arr(const)| , |T_arr(general)| ) ) (1).
2. Time-base mapping and residual
   • ts = alpha + beta * tau_mono (s).
   • r_tb = ( norm( ts - ( alpha + beta * tau_mono ) ) / norm( ts ) ) (1).
3. Reproducibility metrics
   • delta_rep = ( norm( y_new - y_ref ) / norm( y_ref ) ), R_coef = 1 - delta_rep (1).
   • var( x ) ≈ ( ∫ S_xx(f) df ) holds when S_xx(f) and x share the same normalization.
4. Spectral discrepancy and windowing
   • delta_psd = ( ∫ | S_xx(f) - S_yy(f) | df ) / ( ∫ S_xx(f) df ) (1), with matched bands and windows.
   • ENBW = ( ( Σ w[n]^2 ) / ( Σ w[n] )^2 ) * Fs (Hz), where Fs is sampling rate (Hz), w[n] the window.
   • U_w = 1 / ( Σ w[n] / N ) or per-chapter normalization (1).
5. Environment and signatures
   • EnvLock = hash( OS || kernel || drivers || libs || compiler || cpu_id || gpu_id || flags || rng_family ).
   • fingerprint = hash( bytes(artifact) || meta ).
   • anchor = hash( EnvLock || PipelineCard || ParamCard || fingerprint ).

VI. Composite Gauges and Minimal Lock Fields

1. Environment-lock minimal fields
   OS, kernel, driver, libc, numerical libraries and accelerator-stack versions, cpu_id/gpu_id, compiler and build flags, rng_family and seed policy, parallel controls (omp/threads), and TS.* sampling frequency.
2. Card minimal fields
   • PipelineCard: step DAG, I/O anchors, EnvLock binding, version & channel, observation points and alert rules.
   • ParamCard: parameter name, value, unit, dimension, range, default, constraints, and reference path.
3. Audit-chain minimal fields
   Event time ts, actor & role, change summary, H_k/sig_k, impact surface & delta summary, rollback anchor.

VII. Gates and Pass Conditions (retrospective definitions)

1. Repro gate gate.rep
   • Hard gates: delta_rep <= b_rep ∧ delta_psd <= b_psd ∧ r_tb <= b_tb ∧ eps_mass <= b_mass.
   • Soft gates: lower( CI_score ) >= tau_score ∧ score_new - score_old >= -tau_reg.
2. Score synthesis (outline)
   score = w1*( 1 - Delta_agg ) + w2*( 1 - PSD_agg ) + w3*( 1 - TB_agg ) + ..., with weights summing to 1.

VIII. Cross-Volume and Cross-Chapter Anchors

• Time base and sea reference: see Core.Sea for time-base gauges and references.
• Spectral gauges and metrology: see Core.Metrology for S_xx(f), U_w, ENBW, and experimental procedures.
• Concurrency and run-time observability: see Core.Threads for TS.*, hb, bp, and critical-path semantics.
• Conservation and minimal equations: see Core.Equations for the unified conservation framework and audits.

IX. Symbol Index and Naming Rules

• Use lower-case English with underscores or camelCase for symbols; constants and enums in ALL-CAPS or CamelCase enums.
• Spectral auto-correlations carry subscript xx; cross-spectra use xy; window-related quantities append _w.
• Use noun phrases for events and cards; suffix *Report for reports, *Plan for plans, *Summary for summaries.

X. Versioning and Change Policy (this appendix)

• Any gauge change in this appendix is a MAJOR change; adding symbols without breaking existing gauges is MINOR; errata are PATCH.
• Every change must update the CHANGELOG, cross-chapter references, and anchor, and be registered on the Chapter 9 audit chain with H_k/sig_k.