{"version":"mathematics-knowledge/1.0","releasedOn":"2026-08-18","schemaPurpose":"A connective mathematical grammar for calculations, models, formalized traditions, and empirical tests. It does not transfer scientific validity between domains.","concepts":[{"id":"mathematics-angle-normalization","slug":"angle-normalization","name":"Angle normalization","category":"Geometry and coordinates","proofStatus":"method","description":"Represent cyclic angles in one declared interval without losing the original frame or direction of travel.","definition":"Angle normalization maps equivalent directions that differ by integer turns into a chosen half-open interval, commonly [0°, 360°). The modulo operation makes comparison deterministic, while metadata must preserve units, reference axis, orientation, and any unwrapped value needed to distinguish repeated crossings.","notation":["θ̄ = ((θ mod 360°) + 360°) mod 360°","θ ∈ ℝ; θ̄ ∈ [0°, 360°)"],"assumptions":["The angular unit is declared.","The zero direction and positive orientation are fixed.","A half-open output interval is chosen."],"invariants":["θ and θ + 360°k represent the same direction.","Normalization is idempotent.","The output remains inside the declared interval."],"procedure":["Convert the input to the canonical angular unit.","Apply floor-modulo rather than language remainder for negative inputs.","Store unwrapped angle or crossing index when chronology matters."],"errorBounds":["Modulo does not reduce upstream coordinate uncertainty.","Values near 0° require boundary-aware tolerances.","Rounded display values must not replace full-precision inputs."],"sourceIds":["iau-sofa","nist-dlmf-numerical"],"relatedSlugs":["modular-arithmetic","spherical-coordinates","root-finding"],"doesNotEstablish":"A normalized longitude is geometry. A sign, mansion, aspect, or interpretive label derived from it remains convention-dependent and does not gain predictive validity from the normalization."},{"id":"mathematics-spherical-coordinates","slug":"spherical-coordinates","name":"Spherical coordinates","category":"Geometry and coordinates","proofStatus":"definition","description":"Represent directions on a sphere with radius, longitude-like angle, and latitude-like angle under an explicit convention.","definition":"Spherical coordinates represent a point by radial distance and two angular coordinates. Multiple conventions swap angle names or origins, so a reproducible record must declare axis orientation, longitude range, latitude or colatitude, frame, origin, epoch, and whether the position is geometric, astrometric, or apparent.","notation":["x = r cos φ cos λ","y = r cos φ sin λ","z = r sin φ"],"assumptions":["The coordinate origin and frame are fixed.","Angular conventions are explicit.","Singularities at the poles are handled."],"invariants":["Rotation preserves radial distance.","Cartesian conversion preserves the represented point within numerical error.","Longitude is undefined at a coordinate pole."],"procedure":["Declare the spherical convention and frame.","Convert to Cartesian form for robust transformations.","Convert back with quadrant-aware inverse trigonometric functions."],"errorBounds":["Longitude becomes ill-conditioned near the poles.","Parallax matters when the observer origin changes.","Rounding near classification boundaries can change labels."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["angle-normalization","reference-frame-transformations"],"doesNotEstablish":"Spherical coordinates locate a direction. They do not supply physical explanation or astrological interpretation."},{"id":"mathematics-reference-frame-transformations","slug":"reference-frame-transformations","name":"Reference-frame transformations","category":"Geometry and coordinates","proofStatus":"method","description":"Transform the same physical direction between declared origins, axes, epochs, and correction states.","definition":"A reference-frame transformation maps coordinates between explicitly defined systems while preserving the represented physical direction to the accuracy of the transformation model. In astronomy this may include rotations, precession-nutation, Earth rotation, aberration, parallax, and origin changes in a specified sequence.","notation":["x_target = Rₙ ··· R₂R₁ x_source","RᵀR = I for an ideal rotation"],"assumptions":["Source and target frames are identified.","Epoch and timescale inputs are available.","Required physical corrections are declared."],"invariants":["An exact rotation preserves vector norm.","Round-trip transforms agree within tolerance.","Transformation order is part of the method."],"procedure":["Construct the source state and metadata.","Apply authoritative transformations in prescribed order.","Record target frame and verify a round trip."],"errorBounds":["Earth-orientation uncertainty can dominate topocentric work.","Approximate precession models have validity intervals.","Omitted corrections create systematic, not random, error."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["angle-normalization","spherical-coordinates","uncertainty-propagation"],"doesNotEstablish":"Frame conversion can explain why two systems assign different labels to one direction; it cannot decide which symbolic tradition is true."},{"id":"mathematics-modular-arithmetic","slug":"modular-arithmetic","name":"Modular arithmetic","category":"Time and periodicity","proofStatus":"framework","description":"Reason about equivalence classes and repeating cycles without mistaking a wrapped label for elapsed distance.","definition":"Modular arithmetic identifies numbers that differ by integer multiples of a modulus. It is the natural grammar for clock time, angular cycles, weekday indices, lunar-limb boundaries, and ring buffers, but calculations that need elapsed turns must retain an unwrapped counter alongside the residue.","notation":["a ≡ b (mod n)","[a]ₙ = {a + kn : k ∈ ℤ}"],"assumptions":["The modulus is positive and fixed.","Residue convention is declared.","Wrapped and unwrapped quantities are distinguished."],"invariants":["Congruence is preserved by addition and multiplication.","Every integer has one canonical residue in a chosen complete system.","A residue alone does not reveal cycle count."],"procedure":["Choose the modulus from the actual cycle.","Compute a canonical residue.","Retain epoch or cycle index for ordering across wraps."],"errorBounds":["Boundary rounding can select the adjacent residue class.","A variable physical period cannot be modeled as fixed modulo without residual error.","Modulo comparisons need circular rather than linear distance."],"sourceIds":["nist-dads","iau-sofa"],"relatedSlugs":["angle-normalization","periodic-functions-and-phase","calendar-and-timescale-mappings"],"doesNotEstablish":"A repeating mathematical index does not establish that historical outcomes repeat or that a symbolic cycle causes events."},{"id":"mathematics-periodic-functions-and-phase","slug":"periodic-functions-and-phase","name":"Periodic functions and phase","category":"Time and periodicity","proofStatus":"framework","description":"Describe repeating variation through period, frequency, amplitude, and phase while testing whether periodicity is actually stable.","definition":"A periodic function repeats after a period T, while phase identifies position within that cycle. Fourier representations decompose suitable signals into sinusoidal components, but finite, noisy, drifting, or irregularly sampled observations require uncertainty estimates and tests against non-periodic alternatives.","notation":["f(t + T) = f(t)","φ(t) = 2πt/T + φ₀"],"assumptions":["The proposed period is stable over the analyzed interval.","Sampling can resolve the frequency.","Trend and seasonality are not conflated."],"invariants":["Phase is equivalent modulo 2π.","Frequency is the reciprocal of period.","A time shift produces a phase shift."],"procedure":["Inspect sampling and detrend only under a declared model.","Estimate candidate frequency and phase.","Validate out of sample and test aliases."],"errorBounds":["Aliasing can create false periods.","Short windows yield broad frequency uncertainty.","Phase drift invalidates fixed-period extrapolation."],"sourceIds":["nist-statistical-handbook","nist-dlmf-numerical"],"relatedSlugs":["modular-arithmetic","time-series-models","change-point-detection"],"doesNotEstablish":"Detecting periodic structure does not identify a celestial cause, and a visually aligned cycle is not evidence of forecast skill."},{"id":"mathematics-calendar-and-timescale-mappings","slug":"calendar-and-timescale-mappings","name":"Calendar and timescale mappings","category":"Time and periodicity","proofStatus":"method","description":"Map civil labels, atomic scales, rotational time, and ephemeris arguments without treating them as interchangeable.","definition":"A calendar date and clock reading are labels that map to an instant only with a calendar, timezone history, offset policy, and fold or gap resolution. Astronomical work then converts among UTC, TAI, TT, UT1, and other scales using leap-second and Earth-orientation data.","notation":["instant = resolve(calendar, local time, zone, fold)","TT = TAI + 32.184 s"],"assumptions":["Calendar system and timezone identifier are known.","Leap-second and Earth-orientation tables are versioned.","Ambiguous or nonexistent local times are resolved explicitly."],"invariants":["A resolved instant can be represented in multiple scales.","Timezone display changes do not change the instant.","Scale offsets follow their defining standards."],"procedure":["Resolve the civil timestamp to an instant.","Convert with authoritative scale tables.","Store input label, resolution decision, instant, scale, and data version."],"errorBounds":["Historical timezone records can be uncertain.","UT1 prediction degrades beyond measured Earth orientation.","Date-only records imply an interval, not an exact instant."],"sourceIds":["iau-sofa","jpl-horizons"],"relatedSlugs":["modular-arithmetic","uncertainty-propagation","cryptographic-commitments"],"doesNotEstablish":"A precise event time improves reproducibility but does not make a natal, corporate, or electional interpretation empirically valid."},{"id":"mathematics-interpolation","slug":"interpolation","name":"Interpolation","category":"Numerical methods","proofStatus":"method","description":"Estimate values between computed or measured samples under a declared local model and bounded domain.","definition":"Interpolation constructs an approximating function that agrees with known samples and estimates values between them. Polynomial, spline, rational, and trigonometric methods make different smoothness and stability assumptions; interpolation should not be silently extended into extrapolation.","notation":["p(xᵢ) = yᵢ","f(x) = p(x) + R(x)"],"assumptions":["The target lies inside the supported sample range.","Sampling resolves relevant variation.","The selected interpolant matches local smoothness."],"invariants":["The interpolant reproduces declared nodes within tolerance.","Units are preserved.","Method and node set determine the result."],"procedure":["Select bracketing samples.","Choose a stable interpolation family.","Estimate or test residual error against denser references."],"errorBounds":["High-degree equispaced polynomials can oscillate.","Sparse nodes miss sharp changes.","Interpolation error is separate from source-data error."],"sourceIds":["nist-dlmf-numerical","jpl-horizons"],"relatedSlugs":["root-finding","convergence-precision-and-error","time-series-models"],"doesNotEstablish":"Interpolation fills a numerical gap under a model; it does not create observations or justify extrapolated predictions."},{"id":"mathematics-root-finding","slug":"root-finding","name":"Root finding and event location","category":"Numerical methods","proofStatus":"method","description":"Locate times or states where a continuous residual crosses a target, with bracketing and convergence evidence.","definition":"Root finding solves f(x)=0 numerically. Bracketing methods preserve an interval containing a sign-changing root under continuity; open methods can converge faster but require stronger local conditions and may converge to an unintended root.","notation":["find x*: f(x*) = 0","aₖ ≤ x* ≤ bₖ"],"assumptions":["The residual is defined in the search region.","Continuity or differentiability matches the algorithm.","Repeated and tangent roots are considered."],"invariants":["A valid sign-change bracket retains at least one root for a continuous function.","Stopping criteria are declared in input and residual units.","Multiple crossings require separate brackets."],"procedure":["Define a continuous residual and scan for candidate brackets.","Refine each candidate with a safeguarded solver.","Verify residual, bracket width, direction, and duplicate handling."],"errorBounds":["Sampling can miss tangent or rapid roots.","Time error depends on local slope.","Ephemeris and timescale errors remain in the located event."],"sourceIds":["nist-dlmf-numerical","jpl-horizons"],"relatedSlugs":["interpolation","uncertainty-propagation","angle-normalization"],"doesNotEstablish":"A precisely located ingress, station, phase, or boundary is an event calculation; it does not prove an interpretation attached to that event."},{"id":"mathematics-numerical-integration","slug":"numerical-integration","name":"Numerical integration","category":"Numerical methods","proofStatus":"method","description":"Approximate accumulated quantity over an interval with a stated quadrature rule and convergence check.","definition":"Numerical integration approximates a definite integral from sampled function values. Accuracy depends on smoothness, interval partition, singularities, oscillation, and the chosen quadrature rule; adaptive methods allocate evaluations according to estimated local error.","notation":["I = ∫ₐᵇ f(x) dx","I ≈ Σ wᵢf(xᵢ)"],"assumptions":["The integral exists under the chosen definition.","The integrand can be evaluated accurately.","Discontinuities and singularities are isolated."],"invariants":["Additivity holds across exact subinterval partitions.","Units equal integrand units times integration-variable units.","Convergence should stabilize under refinement."],"procedure":["Partition at known discontinuities.","Apply an appropriate fixed or adaptive rule.","Repeat with tighter tolerance or an independent rule."],"errorBounds":["Quadrature estimates can fail on unresolved spikes.","Cancellation can hide large local errors.","Model and measurement uncertainty remain separate."],"sourceIds":["nist-dlmf-numerical","nist-gum"],"relatedSlugs":["convergence-precision-and-error","uncertainty-propagation","dynamical-systems"],"doesNotEstablish":"An accurately accumulated exposure or signal does not by itself identify causation or predictive usefulness."},{"id":"mathematics-convergence-precision-and-error","slug":"convergence-precision-and-error","name":"Convergence, precision, and error","category":"Numerical methods","proofStatus":"framework","description":"Separate approximation error, floating-point behavior, input uncertainty, and model discrepancy.","definition":"Convergence describes whether an approximation approaches a limiting value as resolution or iteration changes. Precision describes numerical representation or repeatability, while error is deviation from a reference quantity. Reproducible software must not collapse truncation, roundoff, measurement uncertainty, and model discrepancy into one number.","notation":["eₙ = xₙ − x*","|eₙ₊₁| ≤ C|eₙ|ᵖ"],"assumptions":["A target quantity and reference meaning are defined.","Stopping criteria are scale-aware.","Arithmetic and library versions are recorded."],"invariants":["More printed digits do not imply lower error.","Convergence to a value does not imply convergence to the correct model.","Tolerance is not identical to uncertainty."],"procedure":["Identify error sources before computation.","Run refinement and independent-reference checks.","Report precision, tolerance, residual, and uncertainty separately."],"errorBounds":["Catastrophic cancellation can dominate.","Ill-conditioned problems amplify tiny perturbations.","Unknown model error cannot be inferred from solver residual alone."],"sourceIds":["nist-dlmf-numerical","nist-gum"],"relatedSlugs":["interpolation","root-finding","uncertainty-propagation"],"doesNotEstablish":"Numerical agreement establishes implementation consistency only within tested conditions; it cannot validate a symbolic or causal claim."},{"id":"mathematics-uncertainty-propagation","slug":"uncertainty-propagation","name":"Uncertainty propagation","category":"Probability and statistics","proofStatus":"method","description":"Carry declared input uncertainty and covariance through a measurement or calculation model.","definition":"Uncertainty propagation estimates the distribution or standard uncertainty of an output produced by uncertain inputs. Linearized covariance propagation, interval methods, and Monte Carlo simulation answer different questions and require declared distributions, dependencies, and model equations.","notation":["Σᵧ ≈ JΣₓJᵀ","y = g(x)"],"assumptions":["Input uncertainty models are defensible.","Dependencies and covariance are represented.","The forward model covers material effects."],"invariants":["Units and covariance dimensions remain consistent.","Perfectly shared error must not be counted as independent.","Output uncertainty is conditional on the model."],"procedure":["Inventory uncertain inputs and correlations.","Select linearized, interval, or simulation propagation.","Check sensitivity and convergence, then report included components."],"errorBounds":["Linearization fails for strong nonlinearity or boundaries.","Unknown systematics remain outside the budget.","Monte Carlo sampling error must be quantified."],"sourceIds":["nist-gum","nist-statistical-handbook"],"relatedSlugs":["convergence-precision-and-error","reference-frame-transformations","calibration-and-reliability"],"doesNotEstablish":"A quantified uncertainty budget does not certify that omitted variables, interpretations, or causal assumptions are correct."},{"id":"mathematics-bayesian-updating","slug":"bayesian-updating","name":"Bayesian updating","category":"Probability and statistics","proofStatus":"framework","description":"Update a declared prior distribution with a likelihood generated by observed data.","definition":"Bayesian inference combines a prior distribution and likelihood to produce a posterior distribution. The result is conditional on the model, prior, data-generating assumptions, and observation process; posterior concentration does not protect against misspecification or leakage.","notation":["p(θ|D) ∝ p(D|θ)p(θ)","posterior ∝ likelihood × prior"],"assumptions":["Prior and likelihood are declared before evaluation.","The observation model reflects sampling and censoring.","Model comparison accounts for complexity."],"invariants":["The posterior normalizes to one.","Sequential updating is coherent for conditionally independent batches.","Changing the prior can change sparse-data conclusions."],"procedure":["Pre-register parameterization and priors.","Compute and diagnose the posterior.","Run prior sensitivity and posterior predictive checks."],"errorBounds":["Approximate inference adds computational error.","Misspecified likelihoods produce misleading certainty.","Selection bias is not removed by Bayes rule."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["calibration-and-reliability","proper-scoring-rules","causal-inference"],"doesNotEstablish":"Bayesian updating can measure how evidence changes belief under a model; it cannot turn repeated post-hoc astrological correlations into prospective evidence."},{"id":"mathematics-calibration-and-reliability","slug":"calibration-and-reliability","name":"Calibration and reliability","category":"Probability and statistics","proofStatus":"method","description":"Test whether stated probabilities agree with observed frequencies and remain stable across relevant groups and time.","definition":"A probabilistic forecaster is calibrated when events assigned probability p occur at approximately frequency p over an appropriate reference class. Reliability diagrams and calibration error summarize agreement, but calibration must be evaluated with discrimination, sample size, dependence, and subgroup stability.","notation":["P(Y=1 | p̂=p) ≈ p","calibration error = observed frequency − forecast probability"],"assumptions":["Forecasts are locked before outcomes.","Outcome definitions and horizons are stable.","Reference classes are large enough to estimate frequencies."],"invariants":["A constant base-rate forecast can be calibrated but uninformative.","Calibration depends on the evaluated population.","Retrospective relabeling invalidates the test."],"procedure":["Bin or smooth locked forecasts without viewing outcomes during design.","Compare predicted and observed frequencies with uncertainty.","Assess discrimination, sharpness, and subgroup drift."],"errorBounds":["Small bins create noisy estimates.","Adaptive binning can bias summaries.","Non-stationarity can make historical calibration stale."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["proper-scoring-rules","bayesian-updating","change-point-detection"],"doesNotEstablish":"Calibration alone does not show useful skill over a baseline, causation, or transportability to a new decision context."},{"id":"mathematics-proper-scoring-rules","slug":"proper-scoring-rules","name":"Proper scoring rules and multiplicity","category":"Probability and statistics","proofStatus":"method","description":"Score probabilistic forecasts honestly while controlling the many-comparisons problem created by large rule libraries.","definition":"A proper scoring rule gives an expected optimum when the forecaster reports its true probability distribution. Forecast evaluation must also account for multiplicity: searching many planets, windows, outcomes, and subgroups inflates false discoveries unless the analysis plan, correction, or held-out evaluation is fixed in advance.","notation":["Brier = (p − y)²","log score = −log p(y)"],"assumptions":["Forecast probabilities and outcomes are valid.","The score and baseline are selected before outcomes.","The family of tested hypotheses is declared."],"invariants":["Proper scores reward honest probabilities in expectation.","Lower Brier and log loss are better under their standard definitions.","Adding undisclosed tests changes the error budget."],"procedure":["Lock forecast, outcome, horizon, score, and baseline.","Compute paired score differences.","Report uncertainty, multiplicity controls, and all registered analyses."],"errorBounds":["Rare outcomes require large samples.","Log loss is sensitive to overconfident errors.","Repeated observations may violate independence."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["calibration-and-reliability","bayesian-updating","cryptographic-commitments"],"doesNotEstablish":"A score measures performance on a specified task; it does not prove a universal mechanism or justify claims beyond the registered population and horizon."},{"id":"mathematics-graph-theory","slug":"graph-theory","name":"Graph theory","category":"Systems and networks","proofStatus":"framework","description":"Represent entities and typed relationships as nodes and edges while keeping topology distinct from causality.","definition":"A graph consists of vertices connected by edges, optionally directed, weighted, temporal, or typed. Graph algorithms expose reachability, dependency, centrality, paths, and communities, but those structural properties inherit the meaning and quality of the encoded edges.","notation":["G = (V, E)","Aᵢⱼ = edge weight from i to j"],"assumptions":["Node and edge semantics are explicit.","Direction and time are represented where material.","Missing edges are not automatically negative evidence."],"invariants":["Isomorphic graphs preserve topology.","A directed path encodes reachability under edge semantics.","Graph metrics depend on graph construction."],"procedure":["Define a typed node and edge schema.","Build the graph from provenance-bearing records.","Run algorithms appropriate to edge meaning and validate sensitivity."],"errorBounds":["Incomplete graphs bias centrality.","Projection can erase edge types.","Correlation edges must not be rendered as causal links."],"sourceIds":["nist-dads"],"relatedSlugs":["causal-inference","constraint-satisfaction","information-theory"],"doesNotEstablish":"A knowledge graph connects claims and methods; visual proximity or centrality does not establish truth, causation, or predictive power."},{"id":"mathematics-dynamical-systems","slug":"dynamical-systems","name":"Dynamical systems","category":"Systems and networks","proofStatus":"framework","description":"Model how state evolves under explicit equations, parameters, inputs, and boundary conditions.","definition":"A dynamical system specifies a state space and a rule for evolution through continuous or discrete time. Stability, attractors, sensitivity, and bifurcations are properties of that model; applying them to a real system requires measured state variables and validated equations.","notation":["dx/dt = F(x,t;θ)","xₜ₊₁ = F(xₜ,uₜ)"],"assumptions":["State variables are sufficient for the intended scale.","Evolution law and boundary conditions are declared.","Parameter stability is tested."],"invariants":["State plus inputs determines modeled evolution.","Conserved quantities follow only from the equations.","Sensitivity depends on metric and initial uncertainty."],"procedure":["Define state, inputs, parameters, and observation model.","Estimate or derive the evolution rule.","Test trajectories and stability out of sample."],"errorBounds":["Chaotic sensitivity limits long-range point forecasts.","Unobserved state creates apparent noise.","Structural model error can dominate numerical error."],"sourceIds":["nist-dlmf-numerical","nist-statistical-handbook"],"relatedSlugs":["numerical-integration","time-series-models","change-point-detection"],"doesNotEstablish":"Calling a business, sky, or symbolic system dynamic does not supply equations or evidence that celestial variables drive its evolution."},{"id":"mathematics-time-series-models","slug":"time-series-models","name":"Time-series models","category":"Systems and networks","proofStatus":"method","description":"Model ordered observations while preserving trend, seasonality, dependence, interventions, and forecast origin.","definition":"A time series is an ordered sequence whose observations may depend on prior values, time-varying inputs, seasonality, and structural changes. Valid forecasting separates training from future evaluation and compares against simple persistence, seasonal, and base-rate baselines.","notation":["yₜ = f(yₜ₋₁,…,xₜ)+εₜ","forecast made at origin t₀"],"assumptions":["Timestamps and observation intervals are trustworthy.","Missingness and revisions are modeled.","Evaluation uses information available at forecast time."],"invariants":["Temporal order cannot be shuffled without changing the problem.","Lagged features must precede the target.","Backtests must reproduce historical information sets."],"procedure":["Define target, cadence, horizon, and forecast origin.","Split data chronologically and fit candidate models.","Evaluate against naive baselines across rolling origins."],"errorBounds":["Autocorrelation reduces effective sample size.","Revisions can leak future data.","Regime changes can invalidate fitted parameters."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["periodic-functions-and-phase","change-point-detection","proper-scoring-rules"],"doesNotEstablish":"A historical fit or attractive overlay is not a reliable forecast; prospective, leakage-free performance is required."},{"id":"mathematics-change-point-detection","slug":"change-point-detection","name":"Change-point detection","category":"Systems and networks","proofStatus":"method","description":"Detect candidate shifts in level, variance, trend, or model parameters with controlled false alarms.","definition":"Change-point methods search an ordered series for times where a statistical property changes. Offline methods segment a completed record; online methods monitor sequentially. Penalties, priors, minimum segment lengths, and detection delays determine what counts as a change.","notation":["τ = argmin segmented loss + penalty","H₀: θ₁ = θ₂; H₁: θ₁ ≠ θ₂"],"assumptions":["The changing property is defined.","Noise and dependence are modeled.","Detection thresholds are selected without outcome cherry-picking."],"invariants":["A detected point is conditional on model and threshold.","Online detection occurs after evidence accumulates.","Multiple candidate points consume an error budget."],"procedure":["Specify change type and minimum duration.","Fit null and segmented alternatives.","Validate with sensitivity, false-alarm simulation, and external evidence."],"errorBounds":["Gradual drift can resemble multiple abrupt changes.","Outliers can trigger false points.","Small samples make location uncertainty wide."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["time-series-models","calibration-and-reliability","periodic-functions-and-phase"],"doesNotEstablish":"A change near a transit or ingress does not establish correspondence beyond chance; event windows and controls must be pre-registered."},{"id":"mathematics-optimization","slug":"optimization","name":"Optimization","category":"Decisions and computation","proofStatus":"framework","description":"Choose feasible inputs that maximize or minimize a declared objective, with sensitivity to assumptions and constraints.","definition":"Optimization selects a decision variable to minimize or maximize an objective subject to constraints. A solution is only as meaningful as the objective, feasible set, data, and uncertainty model; mathematical optimality is not the same as safety, fairness, causal effectiveness, or business value.","notation":["minimize f(x) subject to g(x) ≤ 0","x* ∈ argminₓ f(x)"],"assumptions":["Objective and constraints represent the decision.","Feasibility can be evaluated.","Uncertainty and trade-offs are declared."],"invariants":["An optimum is relative to a feasible set and objective.","Adding constraints cannot improve a minimization optimum value.","Equivalent scaling should preserve the decision when modeled consistently."],"procedure":["Define variables, objective, constraints, and uncertainty.","Solve with an algorithm matched to structure.","Test feasibility, sensitivity, and alternative objectives."],"errorBounds":["Local methods may miss global optima.","Data uncertainty can reorder candidates.","Proxy objectives can induce harmful solutions."],"sourceIds":["nist-dads","nist-statistical-handbook"],"relatedSlugs":["constraint-satisfaction","proper-scoring-rules","causal-inference"],"doesNotEstablish":"An optimized electional time is optimal only under the encoded tradition rules and weights; it is not thereby proven to improve real outcomes."},{"id":"mathematics-constraint-satisfaction","slug":"constraint-satisfaction","name":"Constraint satisfaction","category":"Decisions and computation","proofStatus":"framework","description":"Find assignments that satisfy explicit hard rules while distinguishing them from preferences and evidence weights.","definition":"A constraint-satisfaction problem defines variables, domains, and constraints, then seeks assignments satisfying all hard constraints. Soft constraints and weighted preferences require a separate optimization or ranking layer so that exceptions and disagreements remain visible.","notation":["find x such that Cᵢ(x)=true for all hard constraints","xᵢ ∈ Dᵢ"],"assumptions":["Variables and domains are finite or searchable.","Hard and soft rules are distinguished.","Conflicts have a declared resolution policy."],"invariants":["Every returned assignment satisfies all active hard constraints.","An unsatisfiable core identifies jointly conflicting constraints.","Removing constraints cannot reduce the feasible set."],"procedure":["Compile rules into typed predicates.","Check consistency and extract conflicts.","Enumerate or optimize feasible assignments with a complete audit trail."],"errorBounds":["Natural-language rules may be mistranscribed.","Incomplete constraints create false feasibility.","Search cutoffs can hide valid assignments."],"sourceIds":["nist-dads"],"relatedSlugs":["optimization","graph-theory","formal-logic-and-rule-compilation"],"doesNotEstablish":"Faithfully satisfying traditional rules establishes internal consistency, not that those rules predict or cause the desired outcome."},{"id":"mathematics-information-theory","slug":"information-theory","name":"Information theory","category":"Decisions and computation","proofStatus":"framework","description":"Quantify uncertainty, coding cost, and predictive information without confusing compression with understanding.","definition":"Information theory measures uncertainty and dependence through quantities such as entropy, cross-entropy, and mutual information. Estimates depend on distributions, sample size, discretization, and conditioning; apparent information can arise from leakage or shared trends.","notation":["H(X) = −Σ p(x) log p(x)","I(X;Y) = H(Y) − H(Y|X)"],"assumptions":["Probability distributions are defined.","Sampling supports the estimator.","Conditioning variables prevent obvious confounding where possible."],"invariants":["Entropy is nonnegative for discrete variables.","Mutual information is symmetric and nonnegative.","Deterministic invertible recoding preserves information."],"procedure":["Define variables and estimation method.","Estimate against shuffled and simple baselines.","Use held-out data and report estimator bias."],"errorBounds":["High-dimensional estimates are sample hungry.","Binning changes estimates.","Mutual information does not identify causal direction."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["proper-scoring-rules","graph-theory","causal-inference"],"doesNotEstablish":"Statistical dependence between planetary features and outcomes does not establish a causal celestial mechanism or robust future utility."},{"id":"mathematics-cryptographic-commitments","slug":"cryptographic-commitments","name":"Cryptographic commitments","category":"Decisions and computation","proofStatus":"method","description":"Commit to canonical inputs, rules, and forecasts before outcomes while making later changes detectable.","definition":"A cryptographic commitment workflow canonicalizes a record, hashes the exact bytes, and binds the digest to an external timestamp or signed registry entry. Later disclosure can prove that the opened record matches the commitment, provided the canonicalization and identity protocol are preserved.","notation":["digest = SHA-256(canonical(record))","verify(opening) = committed digest"],"assumptions":["Canonical serialization is deterministic.","Hash and signature algorithms are appropriate.","Timestamp and identity controls are independently auditable."],"invariants":["Identical canonical bytes produce identical digests.","Any byte change should change the digest with overwhelming probability.","Verification does not require trusting the opening party."],"procedure":["Validate and canonicalize the full record.","Hash, sign, and timestamp the digest.","Publish the commitment before outcomes and preserve the opening bundle."],"errorBounds":["A commitment cannot reveal omitted fields.","Weak identity controls permit attribution disputes.","Hashing false data preserves false data."],"sourceIds":["fips-180-4","rfc-8785"],"relatedSlugs":["proper-scoring-rules","calendar-and-timescale-mappings","formal-logic-and-rule-compilation"],"doesNotEstablish":"Immutability prevents hindsight editing; it does not establish that the committed prediction is accurate or the underlying theory is valid."},{"id":"mathematics-formal-logic-and-rule-compilation","slug":"formal-logic-and-rule-compilation","name":"Formal logic and rule compilation","category":"Decisions and computation","proofStatus":"method","description":"Translate bounded source statements into typed conditions, conclusions, exceptions, and conflicts that can be audited.","definition":"Rule compilation maps a source-bounded statement into explicit predicates and outputs while retaining scope, exceptions, provenance, and disagreements. A compiler can test applicability and contradictions, but fidelity requires human source review and does not imply truth of the proposition.","notation":["conditions ∧ scope ∧ ¬exception → interpretation","rule = (predicate, output, provenance)"],"assumptions":["Source scope and translation are known.","Predicates preserve the source meaning.","Conflict policy is published."],"invariants":["A rule cannot fire outside its declared scope.","Every output retains its source and version.","Withheld rules remain visible to audit."],"procedure":["Extract bounded source claim and context.","Encode typed conditions, exceptions, and output.","Review fidelity, test fixtures, and conflict behavior."],"errorBounds":["Ambiguous language may resist deterministic encoding.","OCR and translation errors propagate.","Rule coverage can create false completeness."],"sourceIds":["nist-dads"],"relatedSlugs":["constraint-satisfaction","graph-theory","cryptographic-commitments"],"doesNotEstablish":"Faithful formalization establishes provenance and reproducibility only; the empirical status of an astrological proposition remains unvalidated until tested."},{"id":"mathematics-causal-inference","slug":"causal-inference","name":"Causal inference and counterfactuals","category":"Decisions and computation","proofStatus":"framework","description":"Distinguish prediction and association from claims about what would happen under an intervention.","definition":"Causal inference asks how an outcome would differ under alternative interventions, using a declared causal graph, identification assumptions, and study design. Randomization can identify effects under compliance and measurement conditions; observational analyses require stronger, testable and untestable assumptions.","notation":["ATE = E[Y(1) − Y(0)]","Y ⟂ T | X under conditional exchangeability"],"assumptions":["Treatment, outcome, and intervention are well defined.","Confounders required for identification are addressed.","Interference and selection are considered."],"invariants":["Association alone does not identify intervention effect.","Adjustment follows the causal graph rather than predictive importance.","A counterfactual contrast requires a target population."],"procedure":["Draw the assumed causal structure.","Choose a design and identification strategy.","Estimate effects with falsification and sensitivity analyses."],"errorBounds":["Unmeasured confounding may dominate.","Positivity failures prevent comparison.","Measurement and selection bias can reverse estimates."],"sourceIds":["nist-statistical-handbook"],"relatedSlugs":["graph-theory","time-series-models","optimization"],"doesNotEstablish":"Unless celestial timing is manipulated or otherwise identified under a defensible design, predictive association must not be described as celestial causation."}],"bridges":[{"id":"bridge-semiconductor-overlay-registration","conceptId":"mathematics-reference-frame-transformations","domain":"semiconductor","title":"Lithography overlay registration","application":"Represent translation, rotation, magnification, and distortion corrections between wafer and reticle coordinate systems.","inputs":["alignment marks","measured residual vectors","tool coordinate basis"],"transformation":"Fit a declared linear or higher-order registration model and preserve residuals.","outputs":["correction matrix","overlay residual map","condition diagnostics"],"evidenceRole":"measurement","targetPath":"/knowledge/processes/photolithography","limitations":"Higher-order wafer distortion and tool drift can violate a simple linear model."},{"id":"bridge-semiconductor-yield-spc","conceptId":"mathematics-calibration-and-reliability","domain":"semiconductor","title":"Yield learning and process control","application":"Compare measured defect and yield behavior with stable process limits and calibrated metrology.","inputs":["lot measurements","control limits","tool and recipe identifiers"],"transformation":"Estimate reliability and monitor departures from the qualified process distribution.","outputs":["control signals","calibration status","subgroup diagnostics"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/yield-learning-and-statistical-process-control","limitations":"Control limits detect distributional change; they do not identify the physical root cause."},{"id":"bridge-semiconductor-process-dependency-graph","conceptId":"mathematics-graph-theory","domain":"semiconductor","title":"Manufacturing dependency graph","application":"Connect process steps, equipment, materials, defects, metrology, suppliers, and downstream failure modes as typed edges.","inputs":["process nodes","typed dependencies","source records"],"transformation":"Build a directed provenance graph and compute paths only under declared edge semantics.","outputs":["dependency paths","critical interfaces","evidence gaps"],"evidenceRole":"formalization-only","targetPath":"/knowledge/maps/semiconductor-manufacturing-process-map","limitations":"Graph centrality is not proof of physical causation or commercial importance."},{"id":"bridge-semiconductor-thermal-integration","conceptId":"mathematics-numerical-integration","domain":"semiconductor","title":"Accumulated thermal exposure","application":"Integrate a time-varying temperature or power profile to compare qualified process exposure.","inputs":["temperature time series","time intervals","response model"],"transformation":"Apply declared quadrature over valid segments and propagate sensor uncertainty.","outputs":["integrated exposure","numerical error estimate","coverage gaps"],"evidenceRole":"physical-model","targetPath":"/knowledge/processes/thermal-oxidation-diffusion-and-furnace-processing","limitations":"Equal integrated exposure need not imply equal material response when kinetics are nonlinear."},{"id":"bridge-semiconductor-fab-change-points","conceptId":"mathematics-change-point-detection","domain":"semiconductor","title":"Tool and lot regime shifts","application":"Detect candidate shifts in defect rate, critical dimension, or equipment telemetry.","inputs":["ordered fab measurements","lot boundaries","maintenance events"],"transformation":"Compare stable and segmented process models under a preselected false-alarm policy.","outputs":["candidate shift times","location uncertainty","external-evidence checklist"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/semiconductor-metrology-and-defect-inspection","limitations":"A detected shift localizes a change but does not identify the responsible chamber, material, or recipe."},{"id":"bridge-semiconductor-package-optimization","conceptId":"mathematics-optimization","domain":"semiconductor","title":"Packaging trade-space optimization","application":"Explore thermal, electrical, mechanical, yield, cost, and supply constraints without collapsing them into one hidden score.","inputs":["design variables","constraint models","declared objectives"],"transformation":"Construct a constrained multi-objective optimization and report Pareto alternatives.","outputs":["feasible designs","trade-off frontier","sensitivity results"],"evidenceRole":"decision-method","targetPath":"/knowledge/processes/advanced-packaging-and-heterogeneous-integration","limitations":"A computed optimum depends on proxy models and cannot replace qualification evidence."},{"id":"bridge-semiconductor-metrology-uncertainty","conceptId":"mathematics-uncertainty-propagation","domain":"semiconductor","title":"Metrology uncertainty budget","application":"Propagate instrument, calibration, sampling, and model components into a reported process measurement.","inputs":["instrument readings","calibration covariance","sampling model"],"transformation":"Combine correlated uncertainty components through the measurement equation.","outputs":["measurand estimate","combined uncertainty","dominant sensitivities"],"evidenceRole":"measurement","targetPath":"/knowledge/concepts/semiconductor-metrology-and-defect-inspection","limitations":"Unmodeled systematic effects remain outside the reported budget."},{"id":"bridge-celestial-facts-longitude-normalization","conceptId":"mathematics-angle-normalization","domain":"celestial-facts","title":"Canonical celestial longitude","application":"Normalize continuous ecliptic longitude while retaining full precision, frame, and unwrapped motion.","inputs":["raw longitude","angular unit","reference frame"],"transformation":"Apply floor-modulo to the declared interval and retain provenance.","outputs":["canonical longitude","boundary distance","unwrapped companion value"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/longitude-normalization","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-coordinate-frame","conceptId":"mathematics-reference-frame-transformations","domain":"celestial-facts","title":"Ecliptic and equatorial frame conversion","application":"Transform one physical direction between frame, epoch, origin, and correction conventions.","inputs":["source coordinates","time and epoch","frame conventions"],"transformation":"Apply versioned SOFA-compatible transformation sequence.","outputs":["target coordinates","round-trip residual","convention record"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/ecliptic-vs-equatorial","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-timescale-map","conceptId":"mathematics-calendar-and-timescale-mappings","domain":"celestial-facts","title":"Civil time to ephemeris time","application":"Resolve a local civil label and map the instant to the scale required by an ephemeris.","inputs":["local date and time","IANA timezone","fold or gap policy"],"transformation":"Resolve the instant, then apply versioned leap-second and scale offsets.","outputs":["UTC instant","ephemeris-scale argument","resolution audit"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/civil-time-to-utc","limitations":"Historical zone uncertainty can be wider than numerical precision."},{"id":"bridge-celestial-facts-ephemeris-interpolation","conceptId":"mathematics-interpolation","domain":"celestial-facts","title":"Ephemeris state interpolation","application":"Estimate a body state between tabulated or integrated ephemeris samples.","inputs":["bracketing state vectors","target epoch","interpolation method"],"transformation":"Interpolate within the supported interval and compare against denser reference output.","outputs":["state estimate","method identifier","residual bound"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/ephemeris-versioning","limitations":"Interpolation cannot repair an incorrect ephemeris, timescale, or observer origin."},{"id":"bridge-celestial-facts-event-root","conceptId":"mathematics-root-finding","domain":"celestial-facts","title":"Ingress and station event location","application":"Locate when a continuous angular residual reaches a boundary or apparent speed reaches zero.","inputs":["ephemeris function","search interval","target residual"],"transformation":"Bracket and refine each crossing with direction and duplicate controls.","outputs":["event instant","crossing direction","time tolerance"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/jupiter-station-reference","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-celestial-facts-calculation-uncertainty","conceptId":"mathematics-convergence-precision-and-error","domain":"celestial-facts","title":"Calculation conformance and precision","application":"Separate solver tolerance, floating-point precision, source uncertainty, and convention disagreement.","inputs":["implementation output","independent reference","declared tolerances"],"transformation":"Compare continuous values before classifications and attribute discrepancies.","outputs":["residuals","pass or explain verdict","disagreement category"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/precision-rounding-and-uncertainty","limitations":"Conformance within tolerance validates implementation behavior, not interpretive claims."},{"id":"bridge-celestial-facts-fact-digest","conceptId":"mathematics-cryptographic-commitments","domain":"celestial-facts","title":"Reproducibility digest","application":"Bind exact inputs, versions, conventions, and outputs to a deterministic record digest.","inputs":["canonical fact bundle","software version","calculation conventions"],"transformation":"Canonicalize the record and compute a standard cryptographic digest.","outputs":["digest","canonical payload","verification metadata"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/calculations/reproducibility-digests","limitations":"The digest proves byte-level integrity, not truth of the submitted time or location."},{"id":"bridge-astronomy-sky-separation","conceptId":"mathematics-spherical-coordinates","domain":"astronomy","title":"Angular separation on the celestial sphere","application":"Compute observer-relative separation between measured or modeled directions.","inputs":["two direction vectors","reference frame","epoch"],"transformation":"Convert to a common Cartesian frame and evaluate the stable great-circle separation.","outputs":["angular separation","frame record","uncertainty estimate"],"evidenceRole":"measurement","targetPath":"/knowledge/astronomy/coordinates-reference-frames-and-sky-position","limitations":"Angular proximity does not imply physical proximity."},{"id":"bridge-astronomy-astrometric-covariance","conceptId":"mathematics-uncertainty-propagation","domain":"astronomy","title":"Astrometric covariance propagation","application":"Carry correlated position and motion uncertainty to another epoch or frame.","inputs":["state estimate","covariance matrix","transformation Jacobian"],"transformation":"Propagate covariance through the declared astrometric model.","outputs":["transformed state","output covariance","sensitivity diagnostics"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/astrometry-parallax-and-proper-motion","limitations":"Linear covariance propagation may fail across nonlinear or multimodal uncertainty."},{"id":"bridge-astronomy-orbit-dynamics","conceptId":"mathematics-dynamical-systems","domain":"astronomy","title":"Orbital state evolution","application":"Evolve a physical state under a declared gravitational model and initial conditions.","inputs":["initial state","force model","integration interval"],"transformation":"Numerically integrate equations of motion with convergence checks.","outputs":["trajectory","state uncertainty","model version"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/orbits-gravity-and-ephemerides","limitations":"Long-horizon uncertainty includes both initial-state and force-model error."},{"id":"bridge-astronomy-light-curve-series","conceptId":"mathematics-time-series-models","domain":"astronomy","title":"Time-domain 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data.","outputs":["candidate period","phase model","false-alarm assessment"],"evidenceRole":"measurement","targetPath":"/knowledge/astronomy/time-domain-and-multimessenger-astronomy","limitations":"A periodogram peak is not a physical mechanism and may be an alias."},{"id":"bridge-astronomy-model-graph","conceptId":"mathematics-graph-theory","domain":"astronomy","title":"Observation-to-model provenance graph","application":"Connect detector products, calibrations, measured quantities, model assumptions, and inferred claims.","inputs":["data products","calibration records","claims and citations"],"transformation":"Build typed provenance paths that keep observation and inference edges distinct.","outputs":["claim lineage","missing dependencies","model boundaries"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astronomy/evidence-uncertainty-and-model-comparison","limitations":"Complete lineage improves auditability but does not guarantee a model is correct."},{"id":"bridge-astronomy-model-comparison","conceptId":"mathematics-bayesian-updating","domain":"astronomy","title":"Model-conditioned parameter inference","application":"Update parameter distributions from calibrated observations under explicit physical likelihoods and priors.","inputs":["calibrated data","physical likelihood","parameter priors"],"transformation":"Compute posterior distributions and predictive diagnostics.","outputs":["posterior estimates","model checks","prior sensitivity"],"evidenceRole":"physical-model","targetPath":"/knowledge/astronomy/evidence-uncertainty-and-model-comparison","limitations":"Posterior precision is conditional on the selected model family."},{"id":"bridge-astrology-traditions-rule-predicates","conceptId":"mathematics-formal-logic-and-rule-compilation","domain":"astrology-traditions","title":"Source-bound interpretation predicates","application":"Compile a bounded passage into explicit chart conditions, scope, exceptions, and attributed output.","inputs":["source passage","tradition identifier","chart facts"],"transformation":"Encode reviewed predicates without adding unstated conditions.","outputs":["applicable rules","withheld rules","source lineage"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-astrology-traditions-rule-conflicts","conceptId":"mathematics-constraint-satisfaction","domain":"astrology-traditions","title":"Rule conflict and satisfiability audit","application":"Expose incompatible conditions, exceptions, and precedence policies across named traditions.","inputs":["typed rules","active chart facts","conflict policy"],"transformation":"Evaluate hard predicates and extract unsatisfied or conflicting cores.","outputs":["applicable set","conflict set","resolution trace"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/registry","limitations":"Logical consistency is not empirical confirmation."},{"id":"bridge-astrology-traditions-tradition-graph","conceptId":"mathematics-graph-theory","domain":"astrology-traditions","title":"Tradition and source lineage graph","application":"Connect rules to passages, editions, reviewers, variants, techniques, and report modules.","inputs":["source catalog","rule records","review records"],"transformation":"Create typed provenance edges without merging disagreements.","outputs":["lineage paths","variant clusters","coverage gaps"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology","limitations":"Historical influence and graph proximity do not prove predictive efficacy."},{"id":"bridge-astrology-traditions-zodiac-frame","conceptId":"mathematics-reference-frame-transformations","domain":"astrology-traditions","title":"Tropical and sidereal label comparison","application":"Show how one celestial direction receives different longitude labels under declared zero points.","inputs":["continuous direction","tropical origin","sidereal ayanāṁśa"],"transformation":"Apply each named frame conversion before any interpretation rules.","outputs":["parallel labels","offset record","boundary flags"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/tropical-vs-sidereal","limitations":"Coordinate conversion explains disagreement but cannot adjudicate symbolic correctness."},{"id":"bridge-astrology-traditions-house-boundaries","conceptId":"mathematics-angle-normalization","domain":"astrology-traditions","title":"House and aspect boundary classification","application":"Classify continuous angular geometry under a named house or aspect convention.","inputs":["continuous longitudes","house system","orb policy"],"transformation":"Normalize and classify only after preserving boundary distance.","outputs":["derived labels","boundary uncertainty","convention ID"],"evidenceRole":"formalization-only","targetPath":"/knowledge/astrology/calculations/house-cusp-boundaries","limitations":"Classification is convention-dependent and is not a measured physical property."},{"id":"bridge-astrology-traditions-rule-information","conceptId":"mathematics-information-theory","domain":"astrology-traditions","title":"Incremental rule information audit","application":"Measure whether a rule pack adds out-of-sample information beyond base rates and ordinary covariates.","inputs":["locked rule features","outcomes","baseline predictions"],"transformation":"Estimate held-out score or information gain with leakage controls.","outputs":["incremental information","uncertainty","null comparison"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Dependence does not identify cause and must replicate prospectively."},{"id":"bridge-astrology-traditions-corporate-rules","conceptId":"mathematics-optimization","domain":"astrology-traditions","title":"Corporate-report decision boundary","application":"Separate reflective interpretation from operational actions, losses, and prohibited high-stakes uses.","inputs":["report claims","decision context","risk policy"],"transformation":"Apply explicit utility and safety constraints outside the interpretation compiler.","outputs":["reflective statements","withheld directives","escalation conditions"],"evidenceRole":"decision-method","targetPath":"/knowledge/astrology/corporate-mundane","limitations":"Astrological interpretation must not be the sole basis for investment, legal, employment, or safety decisions."},{"id":"bridge-panchanga-timing-limb-modulo","conceptId":"mathematics-modular-arithmetic","domain":"panchanga-timing","title":"Pañcāṅga limb indexing","application":"Convert continuous Sun–Moon geometry and weekday cycles into tithi, yoga, karaṇa, nakṣatra, and vāra indices.","inputs":["sidereal longitudes","local day boundary","index convention"],"transformation":"Apply declared modular divisions while preserving continuous parent values.","outputs":["limb indices","fractional progress","boundary distance"],"evidenceRole":"calculation","targetPath":"/knowledge/panchanga","limitations":"The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity."},{"id":"bridge-panchanga-timing-limb-boundary","conceptId":"mathematics-root-finding","domain":"panchanga-timing","title":"Limb boundary time search","application":"Locate when a tithi, nakṣatra, yoga, or karaṇa boundary occurs.","inputs":["continuous limb phase","search window","ephemeris convention"],"transformation":"Bracket each modular crossing and refine its instant.","outputs":["boundary instant","direction","time tolerance"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/tithi","limitations":"A computed boundary does not establish auspiciousness."},{"id":"bridge-panchanga-timing-dasha-intervals","conceptId":"mathematics-uncertainty-propagation","domain":"panchanga-timing","title":"Daśā timing under uncertain birth time","application":"Propagate a birth-time interval through lunar mansion position and period-boundary calculations.","inputs":["birth-time interval","Moon longitude function","daśā convention"],"transformation":"Evaluate boundary ranges across the admissible input interval.","outputs":["period date ranges","boundary sensitivity","stable and unstable labels"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/vimshottari-birth-balance-reference","limitations":"Timing ranges quantify input sensitivity, not predictive validity."},{"id":"bridge-panchanga-timing-local-day","conceptId":"mathematics-calendar-and-timescale-mappings","domain":"panchanga-timing","title":"Local sunrise and civil-day mapping","application":"Map an instant to a locality-specific calendrical day under explicit sunrise and timezone rules.","inputs":["observer location","timezone history","sunrise convention"],"transformation":"Resolve civil time and calculate the relevant local boundary.","outputs":["local day label","boundary instant","uncertainty flag"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/calculations/sunrise-day-boundary","limitations":"Polar and near-boundary cases require an explicit fallback convention."},{"id":"bridge-panchanga-timing-election-constraints","conceptId":"mathematics-constraint-satisfaction","domain":"panchanga-timing","title":"Muhūrta eligibility compiler","application":"Find candidate windows satisfying a named tradition’s hard conditions while exposing withheld and conflicting rules.","inputs":["candidate intervals","tradition rule pack","activity type"],"transformation":"Evaluate typed predicates and compute feasible interval intersections.","outputs":["eligible windows","failed conditions","conflicts"],"evidenceRole":"formalization-only","targetPath":"/knowledge/muhurta","limitations":"Eligibility means conformity to encoded tradition, not demonstrated outcome improvement."},{"id":"bridge-panchanga-timing-election-optimization","conceptId":"mathematics-optimization","domain":"panchanga-timing","title":"Transparent candidate-window ranking","application":"Rank feasible windows under published soft preferences without hiding rule trade-offs in one mystical score.","inputs":["feasible windows","declared preferences","uncertainty penalties"],"transformation":"Compute a multi-criteria ranking and retain component contributions.","outputs":["ranked windows","component scores","sensitivity"],"evidenceRole":"decision-method","targetPath":"/knowledge/muhurta","limitations":"Ranking optimizes encoded preferences only and must not be marketed as proven auspiciousness."},{"id":"bridge-panchanga-timing-transit-phase","conceptId":"mathematics-periodic-functions-and-phase","domain":"panchanga-timing","title":"Transit cycle and repeated crossing model","application":"Represent direct, retrograde, and return crossings without treating mean orbital periods as exact event schedules.","inputs":["unwrapped longitude series","target boundary","motion direction"],"transformation":"Track phase continuously and segment each distinct crossing.","outputs":["crossing sequence","cycle phase","retrograde loop markers"],"evidenceRole":"calculation","targetPath":"/knowledge/astrology/timing/jupiter-ingress-reference","limitations":"Cycle phase is descriptive geometry and does not imply repeated life outcomes."},{"id":"bridge-empirical-validation-forecast-lock","conceptId":"mathematics-cryptographic-commitments","domain":"empirical-validation","title":"Pre-outcome forecast commitment","application":"Lock hypotheses, features, rules, horizons, outcomes, and analysis plans before results are observed.","inputs":["forecast record","protocol version","identity and timestamp evidence"],"transformation":"Canonicalize, hash, sign, and publish the commitment.","outputs":["public digest","sealed payload","later verification bundle"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Pre-registration prevents some hindsight bias but cannot guarantee adherence or data quality."},{"id":"bridge-empirical-validation-forecast-score","conceptId":"mathematics-proper-scoring-rules","domain":"empirical-validation","title":"Prospective forecast scoring","application":"Compare locked probabilistic forecasts with outcomes using a preselected proper score and paired baselines.","inputs":["locked probabilities","resolved outcomes","baseline forecasts"],"transformation":"Compute paired score differences with multiplicity and dependence controls.","outputs":["Brier or log score","baseline delta","confidence or credible interval"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Results apply only to the registered task, population, and horizon."},{"id":"bridge-empirical-validation-forecast-calibration","conceptId":"mathematics-calibration-and-reliability","domain":"empirical-validation","title":"Probability calibration audit","application":"Test whether events forecast at a stated probability occur at that frequency prospectively.","inputs":["locked forecasts","binary outcomes","forecast strata"],"transformation":"Estimate reliability curves and uncertainty alongside discrimination.","outputs":["calibration curve","calibration error","subgroup stability"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"A base-rate forecaster may be calibrated without adding useful discrimination."},{"id":"bridge-empirical-validation-rule-pruning","conceptId":"mathematics-bayesian-updating","domain":"empirical-validation","title":"Bayesian rule-pruning protocol","application":"Update bounded rule-effect estimates while shrinking noisy factors and preserving null results.","inputs":["pre-registered rule features","objective outcomes","hierarchical prior"],"transformation":"Fit a versioned hierarchical model and run prior and posterior predictive checks.","outputs":["posterior effect distributions","shrinkage diagnostics","retained and pruned candidates"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Pruning on one dataset can overfit; retained rules require prospective replication."},{"id":"bridge-empirical-validation-ordinary-periods","conceptId":"mathematics-time-series-models","domain":"empirical-validation","title":"Milestones plus ordinary non-event periods","application":"Evaluate celestial features across complete chronological exposure rather than selected memorable events.","inputs":["timestamped outcomes","ordinary comparison periods","features available at each origin"],"transformation":"Build rolling, leakage-free forecasts across the entire observation calendar.","outputs":["prospective predictions","baseline comparisons","coverage diagnostics"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"A single organization supplies limited, dependent observations and weak external validity."},{"id":"bridge-empirical-validation-regime-analysis","conceptId":"mathematics-change-point-detection","domain":"empirical-validation","title":"Outcome regime-change audit","application":"Detect business or measurement shifts independently before testing whether registered celestial windows add information.","inputs":["objective outcome series","operational covariates","detection protocol"],"transformation":"Identify candidate changes under a controlled false-alarm process.","outputs":["change windows","location uncertainty","non-celestial explanations"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/corporate-mundane/corporate-outcome-preregistration","limitations":"Temporal coincidence with a planetary event is not sufficient evidence of correspondence."},{"id":"bridge-empirical-validation-causal-boundary","conceptId":"mathematics-causal-inference","domain":"empirical-validation","title":"Prediction-versus-causation boundary","application":"Specify when a benchmark can support predictive skill and why it usually cannot identify celestial causation.","inputs":["study design","assignment mechanism","outcomes and covariates"],"transformation":"Map identification assumptions and test observable implications.","outputs":["supported claim type","unidentified paths","sensitivity analysis"],"evidenceRole":"empirical-test","targetPath":"/knowledge/astrology/tropical-vs-sidereal/comparisons/prospective-model-scoring","limitations":"Most observational celestial-timing studies can test incremental prediction, not physical causation."}],"sources":[{"id":"nist-dlmf-numerical","title":"DLMF Chapter 3: Numerical Methods","publisher":"National Institute of Standards and Technology","url":"https://dlmf.nist.gov/3","establishes":"Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.","boundary":"A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation."},{"id":"nist-statistical-handbook","title":"NIST/SEMATECH e-Handbook of Statistical Methods","publisher":"National Institute of Standards and Technology","url":"https://www.itl.nist.gov/div898/handbook/","establishes":"Methods for uncertainty analysis, calibration, time-series modeling, process monitoring, experimental design, reliability, and statistical comparison.","boundary":"Statistical procedures quantify evidence under a design and model; they do not repair biased sampling, outcome leakage, post-hoc hypotheses, or unmeasured confounding."},{"id":"nist-gum","title":"NIST Technical Note 1297: Guidelines for Evaluating and Expressing Measurement Uncertainty","publisher":"National Institute of Standards and Technology","url":"https://www.nist.gov/pml/nist-technical-note-1297","establishes":"A measurement framework for identifying uncertainty components, combining standard uncertainties, and reporting expanded uncertainty with declared coverage.","boundary":"Reported uncertainty describes the measurement model and included components. It is not a guarantee that all systematic errors or model inadequacies were found."},{"id":"iau-sofa","title":"Standards of Fundamental Astronomy","publisher":"International Astronomical Union","url":"https://www.iausofa.org/","establishes":"Authoritative algorithms and conventions for astronomical timescales, Earth orientation, reference systems, astrometry, and celestial-coordinate transformations.","boundary":"SOFA standardizes astronomical computation. It supplies no astrological symbols, meanings, auspiciousness judgments, or evidence of predictive validity."},{"id":"jpl-horizons","title":"JPL Horizons System","publisher":"NASA Jet Propulsion Laboratory","url":"https://ssd.jpl.nasa.gov/horizons/","establishes":"Ephemeris products and documented observer, target, timescale, coordinate, and output conventions for reproducible Solar System state and observable calculations.","boundary":"Ephemeris agreement validates positions under declared conventions; it does not validate downstream symbolic classifications or interpretations."},{"id":"nist-dads","title":"Dictionary of Algorithms and Data Structures","publisher":"National Institute of Standards and Technology","url":"https://xlinux.nist.gov/dads/","establishes":"Reference vocabulary for graphs, optimization, search, data structures, complexity, and computational methods used to make algorithms explicit.","boundary":"A formal data structure can represent domain relationships without establishing that the represented causal or interpretive relationships are true."},{"id":"fips-180-4","title":"FIPS PUB 180-4: Secure Hash Standard","publisher":"National Institute of Standards and Technology","url":"https://csrc.nist.gov/pubs/fips/180-4/upd1/final","establishes":"Standard secure hash algorithms that map messages to fixed-length digests and support integrity controls when used in an appropriate protocol.","boundary":"A digest can detect changed bytes relative to a committed value; it does not prove that the original input was truthful, complete, or independently observed."},{"id":"rfc-8785","title":"RFC 8785: JSON Canonicalization Scheme","publisher":"RFC Editor","url":"https://www.rfc-editor.org/rfc/rfc8785","establishes":"A deterministic JSON representation suitable for repeatable hashing and signing through constrained serialization and property ordering.","boundary":"Canonicalization makes equivalent data serialize consistently. It does not provide authentication, secrecy, timestamp authority, or semantic correctness by itself."}]}