PRD-04 Root Dicycle Completion

A narrow construction research PRD for completing 12-dicycles from the b1-1 base configuration, beginning with the root's 1-nodes b1..b4 and then moving outward level by level.

This HTML is the human-review draft. After approval, the accepted content should be converted into an agent-facing Markdown PRD and, if needed, a separate execution package.
ID: prd-04-root-dicycle-completion
Status: proposed / open
Created: 2026-07-04
Source line: narrowed from PRD-03 terminology and constraints

Reading Order

  1. Read Goal and Scope to understand the narrower research target.
  2. Read Terms Borrowed From PRD-03 and Hard Constraints before interpreting diagrams.
  3. Use Source Figures for the starting geometry and examples of pink, green, X-node, and Y-node reuse.
  4. Use Task 04-01 as the first executable research task.

Goal

Starting from the source diagram b1-1. BaseConfig_1stType.jpg, construct 12-dicycles around the root's four 1-nodes b1, b2, b3, and b4. The first research line is to reuse existing nodes whenever possible while preserving C1-C4, and to prefer completions where each parallel pair of di-links shared by two dicycles has one X-type middle point and one Y-type middle point.

This PRD does not try to settle full aperiodicity or C5 immediately. C5 becomes meaningful only after the construction can produce completed shells beyond level 10, so PRD-04 treats it as a future global check.

Scope

Included Now Deferred
Root-local completion around b1..b4; construction of similar 12-dicycles for all four root 1-nodes; explicit accounting of new versus reused 3-nodes. Full all-node infinite construction; proof of aperiodicity; C5 proof or finite-radius C5 evidence past level 10.
The first line of research: for every existing pair of dicycles around b1..b4, test completions where middle points on the two parallel di-links are of different types: one X-node and one Y-node. Relaxations that change the loop budget, such as 6-loop plus 8-loop models, unless a later PRD-04 amendment explicitly adds them.
A level-by-level growth plan: after root 1-nodes, apply the same construction discipline to 2-nodes and onward. Treating a local root sketch as proof that all later shells can be completed. Every shell must carry its own C1-C4 evidence.

Terms Borrowed From PRD-03

PRD-04 reuses the PRD-03 graph vocabulary, with only the narrower construction target changed.

Term Meaning in PRD-04
node A graph vertex. Every completed non-frontier node must have degree 4.
n-node A node at graph distance exactly n from the chosen center. Around the root, b1..b4 are the 1-nodes.
bundle The group of three 2-nodes hanging off a given 1-node.
di-link A two-edge chain 2-node -> 3-node -> 2-node joining two 2-nodes of different bundles.
dicycle A cycle alternating 2-nodes and 3-nodes. A 12-dicycle has 12 di-links and 24 graph edges.
X-node For a pair of dicycles, a reused middle 3-node where the two dicycles use different incident edge-pairs of the node, so all four incident edges are used across the pair.
Y-node For a pair of dicycles, a reused middle 3-node where the two dicycles share one adjacent incident edge at that node.

Hard Constraints

ID Constraint PRD-04 interpretation
C1 4-regularity Every completed node has exactly four incident edges. Frontier incompleteness must be explicit and counted, not hidden.
C2 No small loops No cycle of length 5 or less. In particular, same-bundle di-links are forbidden because they would form 4-loops.
C3 12-hexagon rule For each node, each of the six edge-pairs must lie in exactly two 6-loops. PRD-04 must track this locally whenever a new dicycle is completed.
C4 Di-link / dicycle structure Every valid di-link belongs to a dicycle, and PRD-04's first target is a 12-dicycle around each root 1-node.
C5 Quadratic graph-distance growth Tracked as future evidence only. Do not accept or reject PRD-04 Task 04-01 on C5 before level-over-10 completions exist.

Source Figures

Figures are embedded here as clickable images. Click any figure to open the original source file.

b1-1 base configuration, first type
Starting surface: b1-1. BaseConfig_1stType.jpg. PRD-04 begins here, using the root 1-nodes b1..b4 as the first completion targets.
pink 12-dicycle around b1
Reference dicycle: pink 12-dicycle around b1. This is the first pattern to imitate for the other root 1-nodes.
green 12-dicycle around b2
Similar dicycle: green 12-dicycle around b2. PRD-04 asks for analogous completions around b1, b2, b3, and b4.
blue and green 12-dicycles with x12 as X-node
X-node example: blue and green dicycles use different incident edge-pairs at x12, so all four incident edges are used across the pair.
blue and green 12-dicycles with x12 as Y-node
Y-node example: blue and green dicycles reuse one adjacent incident edge at x12.
pink and green 12-dicycles creating x12 and y12
Pair target: pink dicycle around b1 and green dicycle around b2 create an X-node at x12 and a Y-node at y12, the desired mixed-type pattern for parallel di-links.

X/Y Reuse Model

For a pair of dicycles, the target first search family is not merely "reuse nodes" or "create new nodes." It is specifically: when two parallel di-links occur between the same pair of bundles, use different middle-point types across the pair, one X and one Y.

Parallel di-link X/Y target One pair of root dicycles: mixed middle-point reuse c13 c24 x12 X-node: different edge-pairs used c14 c23 y12 Y-node: one adjacent edge reused parallel partner

Construction Rules

  1. Start from b1-1 and preserve all existing edges and node identities unless a candidate completion proves that reuse violates C1-C4.
  2. Construct or validate the known pink 12-dicycle around b1 first, then construct analogous 12-dicycles around b2, b3, and b4.
  3. For each pair among {b1,b2,b3,b4}, inspect the two parallel di-links between the same bundle pair. The preferred completion has one X-node and one Y-node across that parallel pair.
  4. When adding a dicycle, explicitly classify every intermediate 3-node as existing reused, existing unused, or newly created.
  5. Reject a candidate immediately if it creates a degree greater than 4, a same-bundle di-link, a loop of length 5 or less, or an unaccounted C3 edge-pair count.
  6. After all four root 1-node dicycles are complete, repeat the same process for eligible 2-nodes, then for later shells.

Task 04-01: Root 1-Node 12-Dicycle Completion

first task research + construction C1-C4 only

Objective

Produce a finite root-level completion candidate from b1-1 in which each root 1-node b1..b4 has a 12-dicycle, and every pair of dicycles among those four follows the mixed X/Y target on parallel di-links when possible.

Required Inputs

Acceptance Criteria

ID Criterion Evidence required
AC1 Candidate contains 12-dicycles around all four root 1-nodes. Machine-readable graph plus human-readable cycle listing for b1, b2, b3, and b4.
AC2 Each candidate node and edge has a provenance tag. Table with existing, reused, new, and frontier/unfilled counts.
AC3 Every pair among the four root dicycles is checked for mixed X/Y parallel middle points. Six pair reports: b1-b2, b1-b3, b1-b4, b2-b3, b2-b4, b3-b4.
AC4 C1-C4 hold for all completed non-frontier nodes in the root-local candidate. Verifier output with exact pass/fail counts. If C3 is only partially checkable, report the exact unchecked surface.
AC5 Failure is useful if no candidate exists. Minimal blocker report naming the first violated constraint, the involved nodes/di-links, and whether the blocker comes from X/Y reuse, degree saturation, small loops, or C3 counts.
C5 is explicitly out of scope for Task 04-01 acceptance. The task should preserve the data needed for future C5 measurement, but it must not claim C5 until level-over-10 completions exist.

Future Task Ladder

Task Target Entry condition
04-01 Complete root 1-node dicycles around b1..b4. Starts from this PRD.
04-02 Apply the same mixed X/Y reuse discipline to selected 2-nodes. Requires a Task 04-01 candidate or blocker report.
04-03 Grow shell-by-shell while keeping exact C1-C4 evidence at each shell. Requires at least one repeatable 2-node completion strategy.
04-04 Run C5 evidence checks once completed levels exceed 10. Requires a finite graph with completed shells beyond level 10.

Relationship To PRD-03

PRD-04 borrows basic terminology, C1-C5, and the source diagram vocabulary from PRD-03, but it is intentionally narrower. PRD-03 remains the broader aperiodic-web research line and record of prior Z4, coordinate-free, export, and dense-construction work. PRD-04 starts a focused line around one concrete completion discipline: root 12-dicycles with mixed X/Y reuse.

Source Index