Universal Generative Deformation Calculus

Universal Generative Deformation Calculus

Author: Komorebi P.
Version: 1.1.0

This release implements the following construction:

object
  -> every admissible direction at that object
  -> every admissible direction at every child
  -> infinite raw path forest
  -> quotient by exact path and object relations
  -> completion of convergent infinite routes
  -> certified mathematical families

The central point is state dependence. A child is not assigned a copy of the root's direction list. The system calls directions(child, depth) again, so each generated object exposes its own abstract direction fiber.

Mathematical layers

The package keeps three layers separate.

1. Raw paths store every finite construction route, including different routes with the same endpoint.
2. Quotient nodes use an exact canonical key to identify equivalent endpoints while preserving edge and path multiplicity.
3. Completed branches add limits only under a declared metric, topology, inverse limit, direct limit, or other completion rule.

Unary directions produce a rooted path forest. Multi-input operations such as multiplication, tensoring, gluing, or nonlinear interaction produce an operadic hyperforest.

Main deliverables

• paper/Universal_Generative_Deformation_Calculus.pdf
• paper/Universal_Generative_Deformation_Calculus.tex
• generative_deformation/ reference Python package
• tests/ independent test suite
• results/results.json
• results/checks.csv
• examples/generated/ machine-readable explored forests
• paper/figures/ Python-generated PDF and PNG figures
• paper/tables/ reproducible tables
• dist/komorebi_generative_deformation-1.1.0-py3-none-any.whl

Visual atlas edition

Version 1.1.0 replaces the original sparse figure set with a coherent visual atlas. It includes:

• the seed-to-forest-to-quotient-to-completion architecture
• state-dependent direction fibers in four domains
• raw-tree versus quotient-graph comparison
• certified continuous direction covers
• infinite-branch completion and stability regimes
• multi-input operadic hyperforests
• growth and path-partition phase diagrams
• family-certificate and exploration-frontier workflows
• polynomial, analytic, sequence, Gaussian, qubit, matrix, shape, graph, and congruence-lift atlases
• a cross-domain comparison of raw-route and quotient-object growth

All figures are generated by Python in vector PDF and matching PNG formats.

Installation

From PowerShell in the extracted release directory:

python -m pip install -e .

Or install the included wheel:

python -m pip install .\dist\komorebi_generative_deformation-1.1.0-py3-none-any.whl

Complete verification

.\Run_All.ps1

This installs the package in editable mode, runs all experiments, executes the test suite, and recompiles the paper.

The recorded release has:

• 50 of 50 independent tests passed
• 18 of 18 experiment checks passed
• 59-page compiled paper
• 20 Python-generated figure pairs in PDF and PNG
• 11 worked finite realizations spanning algebra, analysis, probability, geometry, graphs, quantum states, sequences, and congruence lifting
• no LaTeX overfull or underfull boxes
• no undefined references
• clean PDF preflight

Explore a built-in deformation system

Integer walk:

python -m generative_deformation explore --example walk --depth 12 --max-paths 9000 --out .\walk_output

Analytic grammar from cosh(x):

python -m generative_deformation explore --example analytic --depth 3 --max-paths 1800 --out .\analytic_output

Polynomial grammar from x**2 + 1:

python -m generative_deformation explore --example polynomial --depth 4 --max-paths 4000 --out .\polynomial_output

Additional exact reference systems:

python -m generative_deformation explore --example sequence --depth 3 --max-paths 800 --out .\sequence_output
python -m generative_deformation explore --example gaussian --depth 4 --max-paths 5000 --out .\gaussian_output
python -m generative_deformation explore --example qubit --depth 5 --max-paths 1800 --out .\qubit_output
python -m generative_deformation explore --example matrix --depth 3 --max-paths 800 --out .\matrix_output
python -m generative_deformation explore --example shape --depth 3 --max-paths 600 --out .\shape_output
python -m generative_deformation explore --example graph --depth 3 --max-paths 700 --out .\graph_output
python -m generative_deformation explore --example adic-simple --depth 4 --max-paths 100 --out .\adic_simple_output
python -m generative_deformation explore --example adic-singular --depth 5 --max-paths 300 --out .\adic_singular_output

Each output directory contains forest.json, nodes.csv, and edges.csv.

Define a new domain

Subclass DeformationSystem and implement:

from generative_deformation import DeformationSystem, Direction

class MySystem(DeformationSystem):
    name = "my-system"

    def directions(self, state, depth):
        return [Direction("first_move"), Direction("second_move")]

    def apply(self, state, direction):
        ...

    def canonical_key(self, state):
        ...

Optional methods provide validation, invariants, serialization, and a metric.

from generative_deformation import ExplorationConfig, PerturbationForest

forest = PerturbationForest(MySystem(), [seed]).explore(
    ExplorationConfig(max_depth=6, max_paths=100000)
)
forest.export("my_output")

Continuous directions

An uncountable direction fiber is not enumerated point by point. It is represented as a parameter family and explored through certified finite covers. IntervalDirectionFamily supplies exact dyadic nets. propagated_cover_error computes the pathwise endpoint bound

e[k+1] <= L[k] e[k] + M[k] epsilon[k].

The paper proves the closed product-sum bound.

Multi-input interactions

InteractionRule and generate_interactions implement finite prefixes of the operadic extension. The built-in scalar realization includes addition, multiplication, and averaging.

Family certificates

FamilyCertificate records:

• member nodes
• parameters
• closure directions
• invariant fields
• exact relations
• verification failures

The certificate proves internal closure relative to the declared grammar. It does not prove historical novelty.

Scope boundary

“Every direction” means every direction in the declared admissible fiber. The framework does not pretend to enumerate every conceivable target-dependent map. A grammar that hides the target inside a custom operation is rejected by the anti-encoding condition.

Finite exploration is a prefix of an infinite object. Completeness requires a theorem about finite branching, fair enumeration, continuous coverage, convergence, or compactness. A large finite run is not a substitute for such a theorem.