# Sundog vs. Ghost Working hook: > Draw the circle. Find the ghost. Short version: > Sundog Ghost asks whether a finite inside can ever certify the order it > belongs to without importing an outside. Aperiodic tilings are the first hard > substrate: a tile patch is local, legal, and concrete, yet its full > explanation lives in a larger hierarchy the patch does not contain. Status: Brainstorm scaffold opened 2026-06-22 from the Inside/Outside Problem thread. Starter citation spine filed at [`GHOST_LITPASS_MEMO.md`](GHOST_LITPASS_MEMO.md). This is a reader-lane scaffold, not a theorem claim, public claim, or implementation commitment. Phase 1 disposition: Ghost is a reader/exhibit lane by default, and "outside debt" should be treated as distinct regimes with a Domino-Problem computability cliff, not as a smooth scalar gradient. Phase 2 start: internal toy closure workbench scaffolded at [`../ghost/workbench.html`](../ghost/workbench.html), with spec [`ghost/PHASE2_TOY_WORKBENCH_SPEC.md`](ghost/PHASE2_TOY_WORKBENCH_SPEC.md). The lane name is deliberately broader than `SUNDOG_V_APERIODICITY.md`. Aperiodicity is the mathematical pressure test. "Ghost" is the Sundog hook: the outside condition that must be present for an inside to make sense, but which the inside cannot fully derive from itself. ## Reader-Lane Disposition Default posture: > Ghost is an exposition/exhibit reframing of existing aperiodic-order theory > and engineering boundary discipline. It has no new invariant by default. Theorem-shaped expectations exit here. The lane may become research only if a later phase identifies a precise object that is not already recognizability, local derivability, patch complexity, extension complexity, hull structure, or another named formalism, and that object survives specialist review. Until then, the deliverable is a reader/workbench: - explain local rules versus global order; - make periodic closure and aperiodic non-closure visible; - translate "outside debt" into existing technical vocabulary; - show engineering analogues where modules depend on runtimes, trust bases, or human theories outside the code boundary. Public copy must remain true if every conjecture-shaped sentence is deleted. ## Thesis Every time we draw a circle around a system, we create an inside and an outside. The inside may be rich, lawful, and predictive. But when the system tries to explain itself completely, some condition moves to the boundary: - an axiom; - a runtime; - a measurement convention; - a coordinate frame; - a hidden hierarchy; - a boundary condition; - an observer position; - a larger geometry. That boundary condition is the **ghost**. The ghost is not supernatural. It is the dependency that becomes visible when local closure fails. For periodic tilings, the ghost can often be domesticated. Find the repeat cell, name the lattice, and the outside collapses into a finite generator. For aperiodic tilings, the ghost does not disappear. Every finite patch can be legal, but no finite patch is the whole explanation. The pattern is ordered, yet not loop-compressible. The outside keeps reappearing as scale. Candidate one-line thesis: > Aperiodicity is geometry with an unerasable outside. **Regime caveat (2026-06-27 lit pass).** This line is honest only for the computability-cliff regime (general Wang/SFT, where extension is undecidable). For substitution/hierarchical aperiodicity - including the Hat/Spectre anchors - the outside is *bounded and recoverable*: by Mosse-Solomyak, nonperiodicity is exactly what makes the supertile ancestry locally recoverable at a finite recognizability radius. Reserve "unerasable outside" for the undecidable regime; for Hat/Spectre prefer "the outside is finite but keeps moving outward as you zoom." See `GHOST_LITPASS_MEMO.md` Q2 Resolution. ## Claim Boundary This document does **not** claim: - a new aperiodic monotile; - a proof of a new tiling theorem; - a proof or paraphrase of Goedel's incompleteness theorem; - that all self-reference phenomena reduce to tiling; - that "ghost" is a formal invariant yet; - that a browser visualization is evidence for a mathematical claim; - that Sundog has discovered a new bridge between logic, cognition, and tiling theory. This document may stage: - a bounded reader/workbench lane around local rules, global order, and aperiodicity; - a vocabulary for "outside debt" in systems that cannot close themselves locally; - a careful public essay or interactive exhibit contrasting periodic and aperiodic closure; - a careful test of whether an existing finite-patch observable can carry the "ghost boundary" explanation without inventing new theory; - a dev-facing analogy between tilings, runtimes, tests, model priors, and hidden infrastructure. The hard stop: do not launder philosophical force into mathematical novelty. The philosophy is allowed to be generative. The math must be cited, bounded, and treated as already-existing terrain until a real theorem says otherwise. ## Known Anchor The current aperiodic-monotile landscape gives the lane a real substrate rather than a loose metaphor. See [`GHOST_LITPASS_MEMO.md`](GHOST_LITPASS_MEMO.md) for the broader historical and current citation spine across tiling theory, aperiodic order, engineering trust boundaries, and philosophy of self-reference. - Smith, Myers, Kaplan, and Goodman-Strauss, ["An aperiodic monotile"](https://arxiv.org/abs/2303.10798), introduced the Hat family: topological disk tiles that tile the plane but force non-periodicity. Their proof route includes metatiles, substitution structure, and hierarchy. - The same authors, ["A chiral aperiodic monotile"](https://arxiv.org/abs/2305.17743), introduced the Spectre family, addressing the reflection/chirality issue and giving a strictly chiral aperiodic monotile family based on hierarchical substitution. - Bruneau and Whittaker, ["Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles"](https://arxiv.org/abs/2310.06759), provide a compact historical bridge from Wang tiles and the Domino Problem through Hat/Spectre. The Sundog point is not "the Hat proves the Inside/Outside Problem." The point is narrower and more useful: > Aperiodic monotiles make the local/global tension visible. A single local > shape can force lawful global order, but the explanation of that order lives > in a hierarchy no single bounded patch exhausts. That is a clean place to ask what the ghost is. ## Vocabulary Lock **Circle** The bounded region we choose to inspect: a finite patch, module, proof fragment, model, dataset, organism, observer, or theory. **Inside** The information available within the circle: local adjacency, visible geometry, observed behavior, source code, empirical measurements, or derived facts. **Outside** The information required to explain why the inside is legal, meaningful, or predictive: boundary conditions, axioms, runtime, hidden variables, evaluation criteria, or larger-scale context. **Ghost** The outside dependency made visible by failed closure. A ghost is not merely an unknown. It is an unknown with a job: the thing the inside needs in order to be explained as part of a larger lawful structure. **Closure** A system is locally closed when the bounded inside contains enough information to determine the relevant explanation. In a periodic tiling, a fundamental domain plus lattice vectors gives a strong closure certificate. **Outside Debt** The amount or kind of context a finite inside needs before its role in the larger order can be certified. **Ghost Boundary** The boundary layer where local legality becomes global dependency. In a finite tiling patch, this is the edge where many possible continuations remain live and where hidden hierarchy may be underdetermined. ## The Aperiodic Move The intuitive contrast: > Periodicity says: this inside explains the outside by repetition. > Aperiodicity says: this inside is lawful, but every attempted closure opens > into a larger law. That contrast is the center of the lane. A periodic pattern can often be compressed into: 1. a finite motif; 2. translation vectors; 3. a repeat rule. The patch becomes a map of the plane. An aperiodic pattern can still be generated by rules, but not by translational periodicity. The patch may imply legal continuations, but no finite circle around it yields a repeat cell for the whole plane. The explanation passes through substitution, matching constraints, inflation, metatiles, or another global/hierarchical mechanism. The ghost is that mechanism as seen from the patch boundary. Sundog phrasing: > Some shapes do not tile the plane by repeating themselves. They tile it by > continually revealing that they were part of a larger shape all along. ## Heuristic Questions, Not Conjectures These are reader/workbench questions. They are not public conjectures and should not be marketed as theorem-shaped claims. ### Ghost Boundary Heuristic > Any local rule system that forces non-periodic global order must leave, on > every sufficiently large finite circle, a detectable dependency on context > outside that circle. What this wants to become: - a statement about finite patches in finite-local-complexity tiling spaces; - a distinction between bounded outside debt and unbounded outside debt; - a way to compare periodic, quasiperiodic, substitution, and aperiodic systems; - a visualization of the "live" continuation classes at the boundary. Immediate danger: This may be false as written, too vague, or already covered by existing tiling dynamics language. Treat it as a reader prompt, not a claim. Research upgrade gate: - identify the closest existing formalism first; - write the question in that formalism's vocabulary; - name at least one example and one counterexample class; - get specialist review before any public theorem-shaped phrasing. ### Outside Debt Regimes > Systems differ by what kind of outside a finite inside requires. That is not > necessarily a smooth gradient. There is an important discontinuity here. Periodic, quasiperiodic, substitution, and familiar hierarchical aperiodic examples can often be handled with finite, named recognition data inside a chosen formalism. The Domino Problem / SFT edge is different: extension can cross into computation and undecidability. Do not render that cliff as one more step on the same ladder. Bounded recognition / reader regime: | class | inside | ghost / outside debt | | --- | --- | --- | | finite object | bounded description | low once boundary is named | | periodic tiling | motif plus lattice | bounded by repeat cell | | quasiperiodic tiling | local patch plus projection/cut data | external phase/window data | | substitution tiling | patch plus hierarchy | recognizability / supertile ancestry | | hierarchical aperiodic monotile examples | local geometry plus forced hierarchy | finite recognition data inside a non-periodic hull | Computability cliff: | class | inside | cliff | | --- | --- | --- | | general Wang / SFT edge cases | local rules plus finite patch | extension and global-existence questions can encode computation and become undecidable | The reader lane may compare these regimes, but it should not imply that "outside debt" is a single scalar that smoothly increases from finite objects to undecidable tiling systems. If a specialist asks what the metric is, the honest answer is: there may not be one. The first public exhibit should show the contrast, not pretend to measure it. ### Local Closure Failure > Non-periodicity is not randomness. It is the failure of local closure under a > lawful global constraint. This is probably the safest public-facing line. It avoids pretending that aperiodicity means "unstructured" and keeps the focus on lawful order without repeat. ### Ghost as Hidden Ancestry > In hierarchical aperiodic tilings, the ghost of a finite patch is the > underdetermined ancestry of the larger supertile structure that contains it. This is the most visual formulation. It can become a workbench: click a patch, see all compatible supertile ancestries, watch the uncertainty move to the boundary as the circle expands. ## Why This Is Sundog Sundogs are not fake suns in the sense of mere illusion. They are real optical consequences of light, ice crystals, observer position, and atmosphere. The thing seen in the sky is a local appearance produced by a larger geometry. That maps well: - the visible sundog is the finite patch; - the atmosphere and observer geometry are the outside; - the halo is the ghost boundary made visible. The same pattern appears in the current Sundog corpus: - Geometry Workbench: the visible arc depends on the hidden parametric apparatus of sun altitude, crystal orientation, and projection. - Perception: sub-visible continuations depend on instrumented integration along predicted geometry. - Kakeya: direction-complete shadows hide body-resistance in a tiny set. - Mesa: behavior depends on a hidden field-like subspace, not just reward labels. - P vs. NP / certificates: local verification depends on a global witness structure. Ghost is the abstract lane that names the recurring boundary condition. ## Dev Hook The developer version is immediate: > Every program runs inside a circle it did not draw. Software is full of ghosts: - source code depends on compiler/runtime behavior; - tests depend on oracles of expected output; - sandboxes depend on a host; - APIs depend on contracts outside the function body; - models depend on training data, reward signals, and evaluation frames; - proof assistants depend on kernels, axioms, libraries, and trusted extraction boundaries; - distributed systems depend on timing, failure models, clocks, consensus assumptions, and operators. The dev-facing Sundog move: > Draw the module boundary. Find the assumption it cannot prove from inside. This gives Ghost a lane that can support both math-art and engineering inspection. A tiling patch and a code module become two versions of the same question: what outside condition makes this inside meaningful? ## Possible Artifact Build a small `ghost.html` internal workbench before any public page. Goal: > Let a viewer draw a circle around a pattern and see what the circle fails to > explain. Minimum viable substrate: - periodic square-grid pattern; - deterministic but non-periodic toy sequence or substitution stripe; - Penrose-style or Hat/Spectre-inspired patch viewer only if licensing and source discipline are clean; - no claim of novel tiling generation. Core interactions: 1. **Draw circle**: user selects a finite patch. 2. **Local facts**: show what can be known from the patch alone. 3. **Possible continuations**: show compatible outside contexts. 4. **Closure test**: identify whether a finite repeat cell is visible/certified. 5. **Ghost layer**: highlight the boundary where unresolved context lives. 6. **Scale up**: enlarge the circle and show whether the ghost disappears, stabilizes, or recedes. For periodic examples, the ghost should shrink once the repeat cell is captured. For aperiodic or substitution examples, the ghost should migrate upward: the inside grows, but the explanation keeps moving to a larger hierarchy. The exhibit is successful if a non-specialist comes away with: > Ordered does not mean repeating. Local does not mean closed. ## Workbench Metric Sketches These are possible measurements to prototype. They are not settled math. **Continuation Count** Given a finite patch and a local rule set, estimate the number of distinct legal continuations to a larger bounding box. Compare growth across periodic and non-periodic systems. Risk: boundary growth may dominate and make the metric visually noisy. **Ancestry Ambiguity** For substitution tilings, count or visualize the possible supertile ancestry assignments compatible with the selected patch. Risk: requires careful substrate choice and known substitution rules. **Repeat-Cell Capture Radius** For periodic patterns, measure the smallest circle that captures enough information to infer the period lattice. Use this as the "ghost collapse" baseline. Risk: inference can be brittle unless the periodic family is deliberately simple. **Boundary Dependency Ratio** Track how much of the explanation depends on edge choices versus interior choices as the selected circle grows. Risk: needs a formal definition before it deserves a serious name. **Compression Delta** Compare direct description length of a patch to description length under an allowed generator class: periodic generator, substitution generator, local-rule solver, etc. Risk: Kolmogorov-complexity language gets slippery fast. Keep it operational: actual encoders, actual generator families, actual finite examples. ## Lit-Pass Questions Before this becomes public: 1. What existing tiling-dynamics terms already name this? 2. Is "outside debt" just recognizability, repetitivity, local derivability, mutual local derivability, cohomology, hull structure, or extension complexity in disguise? 3. Which examples best separate: - periodic closure; - quasiperiodic order; - substitution hierarchy; - matching-rule aperiodicity; - computationally hard extension? 4. Is there an existing finite-patch metric for extension complexity that can be visualized honestly? 5. Can the Hat/Spectre proof machinery be explained without recreating or misrepresenting copyrighted figures? 6. What is the safest toy substrate for a browser artifact? Candidate reading spine: - Wang's Domino Problem and Wang tiles; - Berger/Robinson aperiodic tile sets; - Penrose tilings and local matching rules; - substitution tilings, recognizability, and local derivability; - Hat and Spectre monotile papers; - finite local complexity tiling spaces and tiling dynamical systems; - symbolic dynamics / shifts of finite type for local-rule analogues. ## Public Framing Candidates Title candidates: - Draw the Circle, Find the Ghost - The Ghost Boundary - Aperiodicity and the Hidden Outside of Order - Geometry With an Unerasable Outside (regime-restricted: honest for the undecidable Wang/SFT substrate, misleading for Hat/Spectre where the outside is bounded - see Q2 Resolution in `GHOST_LITPASS_MEMO.md`) - The Map Inside the Territory Hero lines: > A repeating pattern closes its circle. An aperiodic pattern keeps the circle > open. > Every finite patch is lawful. None is the whole law. > The ghost is the outside a system needs in order to make sense. > Some order cannot be explained by copying a smaller order. > Draw a boundary around the pattern. The unsolved part appears at the edge. Longer public paragraph: > A periodic tiling explains the plane by repeating a finite unit. Aperiodic > tilings are stranger: they are ordered, rule-governed, and locally concrete, > yet no finite unit repeats to give the whole plane. Sundog Ghost treats this > as a visible form of the inside/outside problem. The patch is the inside. The > hidden hierarchy is the outside. The boundary between them is where the ghost > appears. ## Failure Modes **Mathematical overclaim** Saying "we found a principle behind aperiodicity" without a theorem is failure. The acceptable claim is "we are building a reader/workbench for existing aperiodic-order principles and testing whether any existing observable is a good public explanation surface." **Research-lane drag** Letting the reader lane imply a pending theorem is failure. The first public version must be valuable even if "outside debt" reduces entirely to recognizability, local derivability, patch complexity, extension complexity, or hull structure. **Goedel laundering** Loose Goedel references sound profound and become inaccurate quickly. If Goedel appears in public copy, it must be framed as philosophical ancestry, not as a technical warrant for tiling claims. **Aperiodic equals random** Aperiodic tilings are ordered. The exhibit must make this unmistakable. **Ghost equals mysticism** The ghost is a boundary dependency, not a spooky entity. Keep the language poetic in the hook and operational in the method. **Visualization-as-proof** A beautiful patch viewer is not evidence. It is an intuition surface. Any formal claim needs cited math or a separate proof artifact. **Boundary-only triviality** Every finite patch of every infinite tiling has an outside. The lane only earns its keep if it distinguishes kinds of outside debt: bounded, collapsing, recursive, unbounded, undecidable, etc. ## Phase Ladder ### Phase 0 - Vocabulary and Fence Lock - Keep this document as the governing scaffold. - Decide whether "ghost", "outside debt", and "ghost boundary" survive contact with the literature. - Write a one-page claim boundary before any visual work. - Replace conjecture-shaped phrasing with reader/workbench phrasing unless a formal upgrade gate is explicitly passed. Exit gate: - The lane can be explained in three sentences without claiming or implying a theorem. ### Phase 1 - Literature Pass - Read the Hat and Spectre papers closely enough to summarize the proof shape without flattening it. - Identify the existing formal terms closest to "outside debt." - Expand and maintain [`GHOST_LITPASS_MEMO.md`](GHOST_LITPASS_MEMO.md) as the citation spine, with vocabulary replacements and current-publication updates. Exit gate: - We know which existing technical terms should replace "outside debt" in specialist-facing copy, and the public reader still works with no new invariant. ### Phase 2 - Toy Model Workbench Status: started 2026-06-22; **COMPLETE 2026-06-27** (acceptance suite `pass=13 fail=0`; browser QA passed desktop + mobile). - Spec: [`ghost/PHASE2_TOY_WORKBENCH_SPEC.md`](ghost/PHASE2_TOY_WORKBENCH_SPEC.md). - Internal artifact: [`../ghost/workbench.html`](../ghost/workbench.html) served locally as `/ghost/workbench.html` from a static dev server. - Pure core: `ghost/ghost-core.js`. - Acceptance test: `npm run ghost:test`. - Local QA server: `npm run ghost:serve` (static; the full-site Vite dev server is not needed and its dependency optimizer hangs on this one page). The Phase 2 start artifact uses one-dimensional symbolic stripes rather than planar tiles: - periodic control: repeated `A B C D`, where local closure can collapse to a repeat cell; - substitution control: Fibonacci `A -> AB`, `B -> A`, where bounded ancestry can be shown without a translational repeat cell; - computability cliff: named as a separate non-measured regime, not rendered as another rung on a gradient. Hat/Spectre implementation remains deferred until source/licensing and proof vocabulary are clean. Exit gate: - The ghost boundary is visible as an interaction, not merely explained in prose. - The periodic/substitution contrast is visible without implying a new invariant. - The Domino/SFT cliff remains a separate panel, not a smooth metric row. Exit gate MET 2026-06-27: window + boundary edges are interactive; periodic closes to a repeat cell while Fibonacci shows ancestry without a global period; the cliff is a separate panel. Next is Phase 3 (one real aperiodic substrate), which stays gated on clean public assets and cited substitution rules. ### Phase 3 - Aperiodic Reader Status: **COMPLETE 2026-06-27** (acceptance suite + browser QA passed; exit gate met; owner sign-off). - Substrate: **rhombic Penrose (P3)**, generated from the published Robinson- triangle deflation rule (no figure reuse). Owner-selected over Ammann-Beenker / chair / Hat-Spectre. - Spec: [`ghost/PHASE3_APERIODIC_READER_SPEC.md`](ghost/PHASE3_APERIODIC_READER_SPEC.md). - Internal artifact: [`../ghost/aperiodic.html`](../ghost/aperiodic.html) (`ghost/aperiodic-core.js` + `ghost/aperiodic-ui.js`), served at `/ghost/aperiodic.html` via `npm run ghost:serve`. - Acceptance test: `npm run ghost:aperiodic:test` (`pass=13 fail=0`). What it does: - Adds a reader mode for one real aperiodic substrate (Penrose). - Substrate has clean public substitution rules and a citable explanation; the geometry is self-generated, so no asset/licensing reuse (Penrose name caveat for any future public page recorded in the spec, section 9). - Makes the inflation hierarchy visible as hierarchy: raising the ancestry level merges tiles into larger supertiles; a circular window shows supertiles that are contained (ancestry recoverable, ghost collapsed) vs crossing (ghost at the boundary). Growing the window collapses more ghost; raising the level recedes it upward - the recognizability-radius idea (Q2) made visual. Exit gate: - A specialist would not object to the phrasing "this visualizes one aspect of the known proof structure." Exit gate MET 2026-06-27: combinatorics verified against the substitution rule (phi^2 count growth, thick:thin -> phi); hierarchy is interactive; window shows bounded recoverable ancestry vs boundary ghost; cliff stays a separate regime. Verified via acceptance suite + accessibility snapshot + computed styles + DOM reads on desktop and mobile (environment screenshot pipeline unavailable). ### Phase 4 - Optional Metric Probe Status: **COMPLETE 2026-06-27** (optional, non-gating). Rigorous falsification battery run; acceptance suite `npm run ghost:metric:test` = `pass=23 fail=0`. - Spec (pre-registration): [`ghost/PHASE4_METRIC_PROBE_SPEC.md`](ghost/PHASE4_METRIC_PROBE_SPEC.md). - Results memo: [`ghost/GHOST_PHASE4_METRIC_MEMO.md`](ghost/GHOST_PHASE4_METRIC_MEMO.md). - Core: `ghost/metric-probe-core.js`. This phase was optional and did not gate the reader (Phase 3 was already complete). - Prototyped the **recognizability radius** (Mosse's constant of recognizability) as the finite-patch observable, after mapping it to existing vocabulary in Q2; repeat-cell capture radius as the periodic control. - Falsified the **unbounded** reading of the Ghost Boundary Heuristic on every substrate: outside debt is a finite, bounded recognizability radius. Measured: Fibonacci 1, period-doubling 1, Thue-Morse 2 (letters, exactly depth-stable); periodic `ABCD` capture radius 5; Penrose P3 ~0.978 finest-edge units (converges from below by depth 6). The unbounded regime is undecidable Wang/SFT extension, which is not simulable and is recorded as the boundary, not measured. - The probe collapsed into known theory, which is the recorded success outcome. Exit gate: - Either the metric is identified as known vocabulary, dies cleanly, or earns a narrow technical memo with an explicit "not a new invariant" default. Exit gate MET 2026-06-27: the metric is **identified as known vocabulary** (the constant of recognizability; Mosse, Durand-Leroy arXiv:1610.05577), bounded, **not a new invariant**. The conjecture arc closes: "outside debt" had a real, finite, citable referent all along. ### Phase 5 - Public Page Candidate Status: **BUILT + WIRED + QA passed 2026-06-27; launch-ready, NOT deployed.** Owner deploy + asset/license check still gate the public launch. - Page: `ghost.html` (indexable public page), reader module `js/ghost.mjs` importing the shared core `ghost/aperiodic-core.js` (single source; no new math, no figure reuse). - Wired: registered in `site-pages.json` (kind `workbench`); sitemap regenerated and `npm run seo:sitemap:check` = OK (25 URLs, `/ghost` indexable); full OG/Twitter/JSON-LD + canonical `https://sundog.cc/ghost`. - Leads with the interactive Penrose reader, not conjectural grandeur. - Carries the one-sentence payload: > Ordered does not mean repeating. Local does not mean closed. - Copy reflects the Phase 4 result: the missing context is a FINITE recognizability radius, so the "unerasable outside" framings are deliberately NOT used as the headline. Penrose attribution (Penrose 1974 / Gardner 1977) + recognizability citation (Mosse; Durand-Leroy) are on-page. A fences card states it is not a theorem / not a new invariant / aperiodic != random / no Goedel claim. Exit gate: - Public copy is true even if every heuristic/probe sentence is deleted. Exit gate MET 2026-06-27 (copy is exhibit-led and true without any heuristic/probe sentence). QA: reader renders (890 tiles at depth 5, thick:thin = phi), the window shows recoverable vs crossing supertiles, widening the circle collapses more ghost and raising the level recedes it upward, theme applied, responsive desktop + mobile, no console errors. Pre-deploy checklist (SEO/webdev items DONE 2026-06-27): - `og/ghost.png` 1200x630 generated via the public/og pipeline (added `viz_ghost` + `ghost` card to `public/og/_gen2.py`; Penrose rosette + drawn circle, on brand); rasterized through the sharp fallback. DONE. - Clean-URL redirect `/ghost.html -> /ghost 301` added to `public/_redirects`. DONE. - Inbound link added: `geometry.html` hero actions ("Open Ghost reader"). DONE. - Sitemap regenerated; `npm run seo:sitemap:check` = OK (25 URLs, `/ghost` indexable). DONE. - Penrose asset/license diligence: US patent (1979) expired; usage is self-generated geometry from published rules, no figure reuse, on-page attribution = the clean path. Researched, NOT legal advice; owner makes the final sign-off. PENDING owner call. Still owner-gated (NOT done): owner asset/license sign-off; `npm run deploy` (publishes https://sundog.cc/ghost); post-deploy social-validator pass. Assistant will not deploy without an explicit go. ## Current Best Shape The right near-term move is not "Sundog vs. Aperiodicity" as a theorem lane. It is: > Sundog Ghost: a reader/workbench for the hidden outside of local order, with > aperiodic tilings as the first hard test. That keeps the hook alive: > Draw the circle. Find the ghost. And it keeps the reader question honest: > Can we classify systems by how much outside a finite inside needs in order to > explain itself?