12-fold geometry
Station k sits at 30°k. Each element forms a triangle at 120° intervals; each modality forms a square at 90° intervals.
Ring nodes illuminate when admitted by S = SAW ∩ SFE. Toggling a frame hides its overlay; the simulator below edits frame masks.
State and propagation simulator
A 12-bit mask uses bit k for sector k. Controls change antipodal pairs together, so every displayed mask remains closed.
Common constraint π
Partition masks
Pointer invariant: k = (n + r) mod 12. Anchor: kΩ = 0.
SFE,t+1 = SFE,t ∩ π
St+1 = SAW,t+1 ∩ SFE,t+1
Conflict and terminal evaluator
Conflicts are identified from the shortest separation on the ring. A 150° midpoint requires an explicit directed arc and is represented on the 144-position fine lattice.
Quincunx: choose a direction to locate its fine-grid midpoint.
Terminal signal for current S
No terminal condition.
Current tuple
Xₜ = ⟨k=0, r=0, n=0, P=∅, S=0xFFF, E=0⟩
Deterministic operator contract
| Operator | Implemented behavior | Boundary |
|---|---|---|
| K · Classify | Checks sector bounds and antipodal closure. | Daimonion interrupt is an explicit flag; no recognition predicate was specified. |
| U · Update | Intersects the same admitted mask into both frames. | Frames begin as the full ring; elemental labels are classifications, not initial filters. |
| L · Log | Appends JSONL event and extends a SHA-256 chain; then advances n. | Filesystem path is runtime configuration. |
| T · Tighten | Restricts transition relation to S × S; never reopens transitions. | Transition relation is finite and in-memory. |
| G · Govern | Square classification; directed fine-grid quincunx midpoint; explicit Palinode authority. | Gamma, maximum revision delta, and shadow semantics require authority inputs. |
Why the quincunx midpoint is not a sector
For endpoints five sectors apart, a sector midpoint would require 2m ≡ 2k + 5 (mod 12). The right side is odd, so no solution exists. A 144-step coordinate at 2.5° resolution supports the 75° midpoint; the two directed arcs yield different points. Choosing one without an explicit direction would be an arbitrary policy.
Mathematical reference
kₜ = (nₜ + rₜ) mod 12
nₜ₊₁ = (nₜ + 1) mod 12
k ∈ π ⇔ (k + 6) mod 12 ∈ π
Exactly 64 masks satisfy antipodal closure because each of the six antipodal pairs is either admitted or excluded. Each partition has a 12-bit admissible mask. Common constraints are applied to both masks, and the canonical set is their intersection.
Terminal rules: one antipodal pair containing station 0 gives Integration; one pair excluding station 0 gives Refusal. Empty S needs an explicit fast/slow mode to choose Exit or Collapse.
Proposed attention interface
Ãij = softmax((QiKjT + γlMij(𝒳ₜ))/√dk) V
This is a proposed empirical interface, not implemented attention behavior. The state encoding, matrix B, layer coefficient, tensor shapes, model boundary, training procedure, and evaluation data remain unspecified.
Falsification checklist
Software-level failures include admitting an unclosed mask; diverging AW and FE masks after a common constraint; violating k=(n+r) mod 12; reopening a removed transition; or failing to reproduce the append-only ledger digest. Empirical claims require separate tests against observations. The Lean theorem in the source bundle establishes the stated modular identity only; it is not a proof of the full machine.