SpiralCore

SpiralCore is an exact-geometry source, credited to Matthew (versions v63 and v69). From the multiplication table of the octonions (an eight-part number system) it produces a small set of exact moves on eight numbers: each move shuffles the eight coordinates and flips some signs. Every move can be undone exactly. The model stores earlier words in a tree with three branches per node (the "ternary FBS" tree, the project's name for it; see the glossary), and it carries a chosen eight-number block along a short path of these moves. No learned multiplication is involved, only swaps and sign flips. This chapter explains the moves, the tree, and how they connect to the other geometry. The integer form is the core of the current model. Terms are also defined in the glossary.

A ternary tree picks one stored block. The block then travels along exact edges between the 16 roots of an E8 orbit.

The exact moves

The table. spiralcore_operator.rs reproduces the SpiralCore v63 octonion and Cl(0,6) convention exactly. The octonions' seven oriented Fano lines are

\[(124)(235)(346)(457)(561)(672)(713).\]

The Clifford algebra Cl(0,6) has six generators that each square to \(-1\). Products of two distinct generators give the 15 bivectors \(B_{ij}\) (one per pair of the six, \(\binom62=15\)), with \(B_{ij}B_{jk}=-B_{ik}\). They mirror the way the semiprimes \(p_ip_j\cdot p_jp_k\to p_ip_k\) combine. The \(B_{ij}\) generate a group of order 64 that maps onto the 32 square-free even prime products, with kernel \(\pm1\). Together this gives a 64-state composition table. Each \(B\) is a signed coordinate permutation of \(\mathbb{R}^8\) with \(B^2=-I\).

The orbit. One anchor orbit holds 16 roots and 64 signed operator states, and an output block's carrier is moved by the composed edge history. A breadth-first search (BFS) over the labelled edges gives an ordered history between any two roots, composed by left-composition.

Exact inverse (proof). From \(B^2=-I\) it follows that \(B^4=I\) and so \(B^3=B^{-1}\). The inverse of a history is therefore the reversed sequence with each step cubed.

A worked step. The projection chooses one input block \(x_{\sigma(j)}\) for each output block \(j\) and sets \(y_j=A_jx_{\sigma(j)}\), where \(A_j\) is the composed signed action. The forward pass is a gather and a sign flip. There is no learned coefficient for any input-output pair, and the adjoint is the exact inverse action \(A_j^{-1}\delta_j\) (SpiralCore/FBS projection, proof by construction).

How the tree and the moves connect to the rest

The code is in spiralcore_operator.rs and cd_space.rs. The remaining links (icosians, Z[φ], Hopf) are in the other chapters. The semiprime-to-axis step needs a labelling choice, covered in E8, H4 and the 600-cell.

What carries route history

Octonion-signed binding, a design

A historical proposal: bind with a signed XOR whose sign depends on operand order, unlike plain XOR.

On the Fano plane, \(e_a\cdot e_b=\pm e_{a\oplus b}\): a signed XOR whose sign depends on operand order, unlike plain XOR (which is commutative and associative, so it loses order and grouping). Incidence alone does not define coherent octonion signs. The design is not on the current path, and it needs its own pre-registration and a TEST FITNESS: FIT review before compute. The canonical v63 Fano table in spiralcore_operator.rs is the control for it. It has not been tested against plain XOR (VSA mechanism page).

The 9 October substrate review

Mark (N3mesis)'s supplied Octonian Substrate Architecture and discussion PDFs motivate separating state, algebra, routing and identity. The review of that draft found the following.

• Its literal positive cycles \((123)(145)(167)(246)(257)(347)(356)\) fail alternativity and quadratic norm multiplicativity. For example, \((e_1+e_2)(e_4+e_7)=0\) although each factor has squared length 2. Fano incidence is not a complete sign convention.
• For the draft's XOR labelling, an exhaustive search of all \(2^7=128\) orientations of those seven lines finds 16 that give a valid octonion algebra (the enumeration is in the historical geometry tour; the review record states only the counterexample). One of them reverses three of the draft's lines: \((123)(145)(167)(246)(527)(437)(536)\).
• The sample store route discards a coordinate after right multiplication and projection, and its packing function returns empty bytes. A pre-route embed/project inverse does not establish post-route preservation.
• Modular \(\mathbb{Z}/2^{256}\) coefficients are neither the real normed division algebra nor a replacement for exact \(\mathbb{Z}[\varphi]\) pairs.

These findings concern the supplied draft. They are not a negative language experiment and not a retirement of octonion binding. The canonical Rust table is unchanged, and any alternative convention needs a signed basis correspondence, not just the same seven lines. The orientation in the draft is inconsistent with the canonical table and must not replace it (full review).

Where the model uses this

Where it stands

Go deeper

• Code: spiralcore_operator.rs, discrete.cu, build.rs, spiralcore_fbs.rs.
• Back to the overview, or on to the mechanism board.