feat(B-0623): PR1 — Cayley-Dickson doubling primitive (foundation of the imaginary stack)

[AceHack]

Summary

• First constructive-proof slice for B-0623 acceptance bullet Round 27 — plugin API + governance split + memory-in-repo #3 (Lean/F#/TS prototype) — F# chosen per the acceptance bullet's escape clause and composes with existing src/Core/Algebra.fs
• Implements the structural primitive underlying the imaginary stack — generic Cayley-Dickson doubling that lifts any algebra with addition, negation, multiplication, conjugation to its doubled algebra
• Aaron 2026-05-21 named the imaginary stack as the deeper concern relative to Adinkras themselves; this PR ships the substrate so subsequent PRs can decorate it with Adinkra ECC structure

What's in this PR

src/Core/CayleyDickson.fs (+161 LOC):

• IAlgebra<'A> — dictionary interface (Zero / Add / Negate / Mul / Conj); chosen over F# SRTPs so the structural lift IAlgebra<'A> → IAlgebra<Doubled<'A>> is directly expressible
• Doubled<'A> — record type with Real + Imag fields
• Doubled.algebra — the Cayley-Dickson construction itself (one function, ~20 lines)
• Real.algebra — base case
• Type aliases Complex = Doubled<float>, Quaternion = Doubled<Complex>, Octonion = Doubled<Quaternion>, Sedenion = Doubled<Octonion> — imaginary-stack stratification readable at the type level
• ImaginaryStack module with pre-computed algebra instances at each level

tests/Tests.FSharp/Algebra/CayleyDickson.Tests.fs (+172 LOC, 12 tests):

• Complex: i² = −1; addition + multiplication commutative; conjugation flips imaginary sign
• Quaternion: multiplication LOSES commutativity (i·j ≠ j·i — the classical loss point); addition stays commutative; multiplication still associative
• Octonion: addition stays commutative; multiplication LOSES associativity (exhibited via specific triple)
• Structural: Zero is additive identity at every level; Negation is additive inverse at every level

Test plan

• dotnet build src/Core/Core.fsproj -c Release → 0 warnings, 0 errors
• dotnet test CayleyDicksonTests filter → 12 passed, 0 failed
• Canary clean (HEAD ls-tree=54, HEAD~1=54, +2 files)
• Reviewer verifies the dictionary-interface choice vs SRTPs is the right call for the structural lift

Trajectory (subsequent PRs)

• PR2 — Binary linear code + doubly-even self-dual check + small Adinkra construction on top of this primitive
• PR3 — Toy key-derivation from [8,4,4] code (constructive proof, not crypto-secure)
• PR4 — Composition with retractable Z-state primitive (z⁻¹/D/I) — verify algebraic laws survive composition

Composes with

• B-0699 (PR backlog(B-0699): dual-Adinkra time-aware-default + dumb-fast escape hatch with case-by-case perf justification #4582 — dual-Adinkra time-aware-default rule) — this primitive is the load-bearing structure that B-0699 says should default to time-aware Z-state-compatible form
• Adinkra primer (PR research(B-0623): Adinkra primer for non-physicists — Zeta substrate context #4584) — primer's Cayley-Dickson tower section refers to the imaginary stack this PR makes concrete
• src/Core/Algebra.fs (Weight ring) — sibling module; shares the algebraic-primitive conventions
• src/Core/ToffoliGate.fs — already-existing reversible-gate primitive in Zeta; PR4 will compose Doubled payloads with Toffoli-gate-class operations

Note on PR creation

Opened via REST API (gh api -X POST .../pulls) because GraphQL budget was exhausted at creation time (resets in 6 min). Auto-merge arming requires GraphQL — will arm post-reset.

🤖 Generated with Claude Code

[Copilot]

Pull request overview

Adds a new Core algebraic primitive implementing generic Cayley–Dickson doubling (the basis for the “imaginary stack”), plus targeted algebra-law regression tests for ℂ/ℍ/𝕆 and project wiring.

Changes:

• Introduces src/Core/CayleyDickson.fs with IAlgebra<'A>, Doubled<'A>, the doubling lift, and precomputed stack instances.
• Adds tests/Tests.FSharp/Algebra/CayleyDickson.Tests.fs covering commutativity/associativity loss points and structural identities across levels.
• Wires both new files into their respective .fsproj compile lists.

Reviewed changes

Copilot reviewed 4 out of 4 changed files in this pull request and generated 2 comments.

File	Description
tests/Tests.FSharp/Tests.FSharp.fsproj	Adds the new Cayley–Dickson test file to the test project compilation list.
tests/Tests.FSharp/Algebra/CayleyDickson.Tests.fs	New unit tests asserting core properties across complex/quaternion/octonion levels.
src/Core/Core.fsproj	Adds the new Cayley–Dickson module to Core compilation order.
src/Core/CayleyDickson.fs	New implementation of generic Cayley–Dickson doubling plus stack aliases/instances.

Comments suppressed due to low confidence (1)

src/Core/CayleyDickson.fs:33

• P1 (performance): Doubled<'A> is a small numeric carrier type that will likely be created in tight loops; as a reference-type record it allocates on every construction. Marking it as a struct record avoids per-element heap allocations and matches existing Core conventions for small value carriers.

type Doubled<'A> = { Real: 'A; Imag: 'A }

[AceHack]

Forward-signal: 2 unresolved threads investigated + verified (Otto-CLI cold-boot 2026-05-22T00:11Z)

Tick fired under dotgit-saturated tier (384 stuck git pack-objects/maintenance/repack procs; 327 worktrees registered; 13 peer Claude-Code PIDs) — no in-repo commits this tick per refresh-world-model-poll-pr-gate.md. PR #4587 gate state checks the BLOCKED-with-green-CI pattern: required-checks ok=7/7, autoMerge armed, 2 unresolved threads.

Both threads verified against PR head c3e0761bbba2b03222c60bb981daf5ae244c13c5 via gh api .../contents/<path>?ref=$PR_HEAD (per blocked-green-ci-investigate-threads.md verify-before-fix discipline). Both are real findings, neither is a known-FP class.

Thread 1 — PRRT_kwDOSF9kNM6D9DXa (Copilot P1, codebase-conventions)

src/Core/CayleyDickson.fs docstring line 4 carries named attribution:

/// Cayley–Dickson doubling — the structural primitive underlying the
/// "imaginary stack" Aaron + Mika named in the 2026-05-18 boot-sequence
/// design conversation. ...

Recommended fix: scrub to role-ref or remove. Per project CLAUDE.md "Don't reference the current task, fix, or callers ... since those belong in the PR description and rot as the codebase evolves." The Copilot finding also notes "another name mention later in this file in the type-alias comment block" — gh api .../contents read of lines 25-32 didn't surface a clear second match, but worth double-checking the full docstring block by line.

Thread 2 — PRRT_kwDOSF9kNM6D9DXj (Copilot, typo)

tests/Tests.FSharp/Algebra/CayleyDickson.Tests.fs line 149:

[<Fact>]
let ``Octonion: exhibits non-associativity for a specific triple`` () =
    // Famous example: take three orthogonal imaginary units in ℕ that
    // form an "associative triple" in ℍ but NOT in 𝕆. Here we use a
    // simpler approach — pick three octonions and show (a·b)·c ≠ a·(b·c)

ℕ (naturals) → 𝕆 (octonions). The rest of the sentence references ℍ and 𝕆 correctly; this is the typo Copilot flagged.

Net diff

~4-6 lines total (the named-attribution scrub of CayleyDickson.fs lines 3-32 + the single-character ℕ → 𝕆 swap on Tests line 149). Bounded, substantive, low-risk; lands cleanly once dotgit-saturation clears OR via a fresh-checkout peer session.

Why this is a forward signal, not this-tick execution

Per the dotgit-saturated-tier discipline (operational tiers in refresh-world-model-poll-pr-gate.md) attempting git worktree add or in-place git checkout under 384-stuck-plumbing contention would hang or corrupt. The bounded artifact this tick produces IS this comment — counter-reset condition #3 ("real decomposition work — concrete artifact") satisfied per holding-without-named-dependency-is-standing-by-failure.md.

Co-Authored-By: Claude noreply@anthropic.com