guide_list
List available modules and their methods. No args = all 22 modules. With name = methods in that module.
factorize
Exact prime factorization of any integer. Handles arbitrarily large numbers.
compute
Exact integer arithmetic: add, subtract, multiply, divide, gcd, lcm, power. No floating point.
exact_sqrt
Exact symbolic square root via the computation layer. No decimal approximation.
rational_sqrt
Exact symbolic square root of any rational number (integers, fractions, decimals).
exact_trig
Exact trig for special angles, IEEE 754 for others (clearly flagged).
compare
Compare two integers: exact GCD, LCM, shared factors, and similarity.
verify
Cross-verify an integer's properties: factorization roundtrip, divisor count, consistency checks.
project
Classify a value's mathematical domain and structural properties.
route
Show domain path between two values.
landmarks
List reference values in a domain: integer, rational, radical, trig, complex, or all.
recommend
Describe a problem in natural language, get the optimal tool call. Use when unsure which of the 202 methods to use.
field_create
Create or reset the computation workspace. Only needed when building a persistent model with multiple related values.
field_add
Store a single raw value (urchin) in the workspace. Use for constants, rates, and single inputs.
field_read
Read workspace state or compare two stored urchins (GCD, LCM, ratio, divisibility).
field_gcd
Exact GCD of 3+ values in one call. Takes raw numbers — no workspace needed.
field_lcm
Exact LCM of 3+ values in one call. Takes raw numbers — no workspace needed.
field_lattice
Full divisibility lattice from a set of values. All pairwise GCDs and LCMs.
field_ratios
All pairwise exact ratios for a set of values. For comparing proportions across a dataset.
field_derive
Compute from workspace urchins with tracked dependencies and auto-cascade.
field_update
Update an urchin's value — all dependents auto-recompute through the full dependency graph.
field_sensitivity_all
Compute sensitivity gradients for ALL tunable inputs against all outputs.
field_optimize
Iterative gradient-based optimization on the urchin field.
field_links
View dependency graph between workspace urchins. Shows what derives from what.
field_query
Search workspace by structural properties: dimension, similarity, domain, tag, or all.
field_cluster
Auto-group workspace urchins by similarity. Only useful with 5+ stored urchins.
field_snapshot
Save, restore, list, or delete named workspace states. For branching scenarios and what-if analysis.
field_export
Export workspace + snapshots to JSON. For persisting a model across sessions.
field_import
Import workspace state from a previous JSON export. For resuming work on a saved model.
field_diff
Compare two workspace snapshots. Shows added, removed, and modified urchins.
field_note
Annotate workspace urchins, pairs, or the field itself.
field_notes
Read annotations from the workspace log. Optional filter by target.
field_clear_notes
Clear annotations. No args = all. With target = matching only.
scratch_math
★ SCRATCH PAD — Multi-step exact calculations in one call.
scratch_math_units
scratch_math with dimensional analysis — tracks units through every step.
slide
Exact addition and subtraction for raw values. Use for leaf inputs, not for combining urchins.
fraction_math
Exact fraction arithmetic — add, subtract, multiply, divide.
graph_define
Define a complete computational graph in one call.
graph_observe
Read full computational graph state — all inputs, intermediates, and outputs with values and lineage.
graph_forward
Forward pass: update inputs, cascade through graph, return all outputs.
graph_sensitivity
Compute exact gradients: how much each output changes per unit change in an input.
graph_solve
Inverse solve: find the input value that produces a target output via bisection search.
graph_what_if
Run multiple scenarios against the graph and compare outcomes.
graph_learn
Iterative gradient-based adjustment toward target outputs using exact-fraction gradients.
graph_derivative
Compute exact d(output)/d(input) using reverse-mode automatic differentiation.
graph_gradient
Compute all partial derivatives of an output w.r.t. all tunable inputs in one pass.
graph_jacobian
Full Jacobian matrix: d(output_i)/d(input_j) for all output-input pairs. All entries exact fractions.
graph_hessian
Second derivative d²(output)/d(input)² plus curvature classification (concave up/down/flat).
graph_taylor
Compute exact Taylor series coefficients around current evaluation point.
graph_integral
Exact definite integral using Simpson's rule with exact fraction arithmetic.
graph_critical_points
Find critical points where f'(x) = 0. Classifies each as local min, max, or inflection.
graph_tangent_line
Compute exact tangent line at current evaluation point. Returns slope, intercept, and equation.
graph_curve_analysis
Full calculus analysis at current point: f(a), f'(a), f''(a), tangent line, curvature, and point type.
unit_check
Check dimensional compatibility before computing. Catches mismatches like adding dollars to square feet.
unit_factor
Return the exact conversion factor between two compatible units as a Fraction.
list_units
List available units, optionally filtered by dimension. 35 dimensions supported.