topographo

Executable tools for Topographical Graph Theory (TGT): reusable Cayley-Dickson and sedenion settlement math.

Topographo packages the reusable mathematical core behind the Occurrence Theory research notes. The package is intentionally narrower than the paper: it exposes the Cayley-Dickson and sedenion settlement computations needed for TGT without importing the interpretive narrative layer or printing audit output at import time.

In this package, TGT means the computational study of the sedenion zero-divisor crack as a topological, graph-like, and operator-theoretic object. The basic workflow is:

1. Build the 16-dimensional Cayley-Dickson algebra, the sedenions.
2. Enumerate or sample unit zero divisors, the distinguished singular locus called the crack.
3. Turn those events into multiplication operators and diagnostics.
4. Build finite graphs or channels whose edges/transitions are defined by multiplication, annihilation, metric transport, or settlement strain.
5. Check every result against validation gates before treating numerical output as evidence.

Sedenion Settlement Dynamics (SSD)

The central reusable object is Sedenion Settlement Dynamics. SSD studies the 16-dimensional Cayley-Dickson algebra, its unit zero-divisor locus, and operators built from left multiplication:

• L_x: left multiplication by an algebra element.
• M_x = L_x.T @ L_x: the metric operator measuring norm transport.
• T_x = L_{x^2} - L_x^2: the alternator used to express settlement strain.

These operators are enough to express the main graph-theoretic and channel computations. For example, a zero-divisor graph can use crack samples as vertices and connect two vertices when a multiplication-derived test such as rank(L_z @ L_w) or z * w == 0 detects annihilation. A settlement channel averages the operator action L_z.T @ X @ L_z over crack events.

Package layout

The reusable TGT/SSD layer is separated from the interpretive Occurrence Theory layer, which lives outside this package.

• topographo.core — one exact signed-basis specification for Cayley-Dickson multiplication, the derived NumPy structure tensor, multiplication operators, and mandatory validation gates. It does not know about Occurrence Theory, event/state language, or report formatting.
• topographo.ssd — the sedenion-specific wrapper (SedenionAlgebra), small channel diagnostics, and layered exact-rational execution. Exact algebra, typed programs, raw execution, projective state, observers, and JSON codecs have separate modules; exact_machine remains as a compatibility facade.
• topographo.exceptional — the exceptional-algebra layer: the 27-dimensional Albert algebra J3(O) and its F4/G2 structure (Peirce/Hessian analysis, determinant invariants, anisotropy). Generic exceptional-algebra math, not Occurrence-Theory-specific.

Installation

pip install topographo

Requires Python 3.11+ and NumPy.

Minimal use

import numpy as np

from topographo.core import verify_gates
from topographo.ssd import SedenionAlgebra, average_metric_operator

assert all(result.passed for result in verify_gates())

algebra = SedenionAlgebra()
events = algebra.basis_zero_divisors()
mean_metric = average_metric_operator(algebra, events)

equilibrium_error = np.linalg.norm(mean_metric - np.eye(algebra.dim))

Sketch of a TGT zero-divisor graph

import numpy as np

from topographo.ssd import SedenionAlgebra

algebra = SedenionAlgebra()
events = algebra.basis_zero_divisors()
operators = [algebra.left_operator(z) for z in events]

graph = {i: [] for i in range(len(events))}
for i, left_i in enumerate(operators):
    for j, left_j in enumerate(operators):
        if i == j:
            continue
        if np.linalg.svd(left_i @ left_j, compute_uv=False)[-1] < 1e-9:
            graph[i].append(j)

For exact finite crack certificates, use SedenionAlgebra.basis_zero_divisors() to enumerate the full 84-point design. sample_crack(n) samples from that design with replacement and is intended for stochastic diagnostics, not machine-zero theorem gates. These dimension-16 helpers belong to SedenionAlgebra; generic CayleyDicksonAlgebra supplies only algebra and operator operations.

Exact ordered-event machine

The exact implementation has explicit responsibility boundaries:

• topographo.core.exact derives exact rational arithmetic from the canonical signed-basis table.
• topographo.ssd.exact adapts that backend to the pinned 061.11 coordinates.
• topographo.ssd.program defines typed, fully parenthesized expressions and immutable ordered event programs.
• topographo.ssd.machine performs repeated left action through one policy-driven transition engine; raw execution continues through zero.
• topographo.ssd.projective identifies every nonzero rational scaling, including negative scaling, and returns Annihilated for a zero product.
• topographo.ssd.observers derives measurements such as squared norm, while topographo.ssd.codec owns JSON encoding and exact replay validation.

topographo.ssd.exact_machine remains as a compatibility facade for the 0.4 API. Both exact and NumPy backends use the same algebraic specification. The map Phi(a, b) = (conj(a), b) converts 061.11 values to core coordinates; to_core_coordinates() and from_core_coordinates() provide the checked boundary.

from topographo.ssd import exact, machine, program, projective

initial = exact.basis(4)
events = program.Program([exact.basis(2), exact.basis(1)])
result = machine.run(initial, events)

assert result.state == exact.mul(
    exact.basis(1), exact.mul(exact.basis(2), initial)
)
assert projective.run_projective(initial, events).state == result.state

Validation gates

The validation gates are deliberately conservative. They catch sign-convention or tensor-indexing errors early: they certify that the implementation uses the intended Cayley-Dickson convention rather than acting as broad theorem tests. Any independent implementation should pass composition, antisymmetry, quadratic, and Moufang checks before its numerical certificates are trusted.

Links

• Source: https://github.com/TheSwanFactory/occurrence
• API documentation: https://theswanfactory.github.io/occurrence/

License

MIT.