This document operationalizes the open questions of `insights.md` into a
concrete, prioritized research program. Where `insights.md` asks "what
are the open questions?", this document asks "what would it take to
answer them, and in what order?". Each item is presented as a
**research ticket** with explicit deliverables, dependencies, methods,
and success criteria.
---
## 0. Methodological Principles
Before listing tickets, three principles guide every entry below:
1. **Exact arithmetic is non-negotiable.** Every claim about
reconnection, holonomy, or distance multiplicity that relies on
floating-point comparisons is suspect. Use `Q(sqrt(d))` exact
arithmetic throughout (the `qcanon` / `qeq` pattern in
`experiment.mac`).
2. **Algebraic invariants come first; geometric realization second.**
The lesson of `insights.md` §1.1 is that the field
`F = Q(zeta_n + zeta_n^{-1})` and the fiber group `G` are the
true invariants. Each ticket should explicitly identify which
`(F, G)` it touches.
3. **Falsifiable predictions, not aesthetic claims.** Every ticket
must produce a numerical or combinatorial prediction that can be
checked against the existing `experiment.mac` / `sweep_ngon.mac`
pipeline (or a stated extension thereof).
---
## 1. Priority Tier A: Foundational Clarifications
These tickets unblock everything else by clarifying what the framework
*is*. They should be addressed first.
### Ticket A1: Equivalence of multi-sheeted and cut-and-project
**Question (from insights.md Tension 2.4, Question 20):**
Are the multi-sheeted covering space construction (`idea.md` §2.2)
and the cut-and-project construction (`idea.md` §2.4) functorially
equivalent for the pentagon?
**Deliverable:**
A worked proof or counterexample establishing, for the pentagon, a
functor `Phi: MultiSheet -> CutProject` and a natural isomorphism
`Phi(G_multi) ~ G_cap` of adjacency graphs.
**Method:**
1. Fix the pentagonal `Z^5` ambient lattice and the standard window
`W` (regular pentagon in `E_perp`, inradius `1`).
2. For each vertex `v` of the cut-and-project lattice, compute its
phantom-space coordinate `pi_perp(v)`.
3. Define the multi-sheet sheet index `s(v) = round( phi^{-1} *
<pi_perp(v), e_1_perp> )` and show that this assignment satisfies
the `signed3` vortex rule on edges.
4. Verify that the induced adjacency matches.
**Success criterion:**
Two clusters built at depth 4 from the same origin under each
construction differ in adjacency by zero edges, with explicit
isomorphism mapping written down.
**Estimated cost:** 1 week of focused work; uses existing
`experiment.mac` cap-window machinery (Section 17).
---
### Ticket A2: The unifying classification theorem
**Question (from insights.md Question 22):**
Is there a single theorem of the form **"reconnection type <->
algebraic invariant"** that subsumes the cube/pentagon/dodecahedron
/Sierpinski/Conway-Radin examples?
**Proposed theorem (to be proved or refined):**
> Let `P` be a regular polytope (or marked polygon `(P, E_A)`) with
> coordinate field `F = Q(F_P)`. Then `P` admits a locally finite
> reconnective expansion family iff
> (a) the orientation subgroup of the holonomy generated by
> reflections along active edges is finite as a subgroup of
> `O(F)`, **and**
> (b) the angular deficit `delta(P)` lies in the lattice `(2*pi)*Q`
> (giving Level 0) or in `R \ (2*pi)*Q` with finite winding
> (Level 1), and
> (c) the active-edge graph `(P, E_A)` is connected.
**Deliverable:**
Formal statement, complete proof or detailed counterexample, with
explicit tables for the cases enumerated in `polyhedra.md` and
`pinwheels.md`.
**Method:**
1. Make criteria (a)-(c) precise in terms of the field-theoretic
data.
2. For each test case (5 regular polytopes, ~6 pinwheel polygons,
Penrose tilings), verify the theorem.
3. Identify the minimal counterexample (if any).
**Success criterion:**
Either the theorem is proved, or a Level-0 reconnective polygon is
exhibited that violates one criterion (forcing a refinement).
**Estimated cost:** 1-2 months; mostly analytical.
---
### Ticket A3: The "true" basic object
**Question (from insights.md Question 21):**
Should the basic object of the framework be:
(a) the polygon `P`,
(b) the pair `(P, E_A)`,
(c) the triple `(F, G, tau)`, or
(d) an equivalence class thereof?
**Deliverable:**
A category-theoretic definition with explicit objects and morphisms,
showing that (c) is the universal choice and that (a), (b) project
onto subcategories.
**Method:**
1. Define the category `Recon` with objects `(F, G, tau)` triples
satisfying the consistency axioms of `idea.md` §5.
2. Define functors `Geom: Polygons -> Recon` (geometric realization)
and `Pin: PinwheelPolygons -> Recon`.
3. Show that `Geom` and `Pin` are faithful but not full, exhibiting
multiple polygons mapping to the same `(F, G, tau)`.
**Success criterion:**
Worked example: at least two distinct polygons over `Q(sqrt(5))`
with the same fiber group `Z_10` and the same vortex rule, shown to
produce isomorphic adjacency graphs.
**Estimated cost:** 3 weeks.
---
## 2. Priority Tier B: Computational Predictions
These tickets test specific quantitative predictions of the framework
using the existing `experiment.mac` / `sweep_ngon.mac` pipeline,
extended as noted.
### Ticket B1: Erdős O(log n) confirmation
**Question (from insights.md Question 14):**
Is `D(P_n) = Theta(log n)` for `n` points on the pentagonal lattice?
**Deliverable:**
Plot of `D(P_n)` vs `log(n)` over `n in [100, 10000]` showing
linear scaling with explicit slope close to the predicted
`2/(d_eff * log(phi^2)) ~ 0.32` for `d_eff = 2.3`.
**Method:**
1. Extend `experiment.mac` Section 23 to support cluster sizes up to
`N = 10000` (huge preset, BFS depth 6).
2. Run the `signed3` rule on the pentagon at depths 4, 5, 6, 7.
3. Tabulate `D(N)` (number of distinct squared distances) at each
depth.
4. Fit `log D` vs `log log N` (testing power-of-log corrections).
**Success criterion:**
Slope `D / log(n)` in `[0.2, 0.5]` and tightly clustered across
depths, confirming O(log n). A slope that grows polynomially with
`log n` would falsify the framework.
**Estimated cost:** 1 week of compute + analysis; existing tooling
suffices.
---
### Ticket B2: Field-universality of d_spec
**Question (from insights.md Tension 2.2, Question 8):**
Is `d_spec ~ 1.1` specific to `Q(sqrt(5))` or universal across all
real quadratic fields under the `signed3` rule?
**Deliverable:**
Table of `d_spec` for n-gons with `n in {5, 8, 12}` (corresponding
to `Q(sqrt(5)), Q(sqrt(2)), Q(sqrt(3))`), measured at three preset
sizes.
**Method:**
1. Use `sweep_ngon.mac` with `SWEEP_NS = [5, 8, 12]`,
`SWEEP_PRESET = "huge"`, `SWEEP_TAU = "signed3"`.
2. Extract `d_spec_KPM` and `d_spec_KPM_CDF` from each run.
3. Compute the standard deviation across runs; if `< 0.05`, declare
universality.
**Success criterion:**
Either `d_spec(5) ~ d_spec(8) ~ d_spec(12)` to within `0.05`
(universality), or a monotone trend in `n` (suggesting a flow with
the field discriminant).
**Estimated cost:** 2-3 days of compute time.
---
### Ticket B3: Dimensional flow with diffusion time
**Question (from insights.md Tension 2.1):**
Does `d_spec(t)` show genuine dimensional flow between UV and IR
regimes, or is it constant?
**Deliverable:**
Plot of `d_spec(t)` extracted from `P_0(t)` decay in three time
windows: `t in [1, 10]`, `[10, 100]`, `[100, 1000]`.
**Method:**
1. Extend `experiment.mac` Section 6 to fit `P_0(t)` slopes in
multiple windows.
2. Run at depth 6 with `M_WALKS = 50000`, `T_STEPS = 2000`.
3. Report slopes per window.
**Success criterion:**
Either monotone interpolation from `d_spec_UV ~ d_eff` to
`d_spec_IR ~ 1.1` (confirming flow), or constancy (confirming
`d_spec` is scale-independent).
**Estimated cost:** 1 week of compute.
---
### Ticket B4: CA universality vs polygon
**Question (from insights.md Tension 2.5):**
Under a **fixed** (non-`n`-scaling) CA rule, does CA behavior track
spectral universality across `n`?
**Deliverable:**
Phase diagram of CA fate (extinct / still / periodic / growing) for
rules `B2/S12` and `B3/S23` across `n in {3, 4, 5, 6, 7, 8, 10, 12}`.
**Method:**
1. Modify `experiment.mac` Section 10 to disable the
`ca_b_thresh = round(n/3)` scaling and use fixed `B/S` sets.
2. Sweep over `n` and seed shapes, classifying each by fate.
3. Look for a phase boundary as a function of `n`.
**Success criterion:**
If the phase boundary lies at the same `n` regardless of polygon
geometry beyond degree, the spectral universality of `signed3`
extends to CA dynamics.
**Estimated cost:** 1 week.
---
## 3. Priority Tier C: Geometric Extensions
### Ticket C1: Dodecahedral multi-sheeted cover
**Question (from insights.md Question 5):**
What is `d_eff` for the dodecahedral multi-sheeted cover, and does
it lie in `(3, 4)` as predicted?
**Deliverable:**
Implementation of the 3D adjacency oracle in exact `Q(sqrt(5))`
arithmetic, with a measurement of `d_eff` from BFS volume growth.
**Method:**
1. Define `dodec_vertices` as 20 exact vectors in `Q(sqrt(5))^3`.
2. Implement `reflect_dodec_face(c, k)` analogous to
`reflected_neighbor_center` for n-gons.
3. Extend `experiment.mac` BFS to 3D, with a 3D fingerprint hash.
4. Run at depth 3 (already ~thousands of cells) and fit `d_eff`.
**Success criterion:**
`d_eff in (3, 4)` to two decimal places.
**Estimated cost:** 3-4 weeks (mostly implementation).
---
### Ticket C2: Pinwheel polygon catalog
**Question (from insights.md Insight 1.4, Question 7):**
What is the catalog of pinwheel polygons for the square, triangular,
and Penrose lattices?
**Deliverable:**
For each base lattice, a list of pinwheel polygons `(P, E_A)`
realizing it, with measured `d_eff` and `d_spec` for each.
**Method:**
1. Implement the pinwheel oracle of `pinwheels.md` §6.1.
2. Enumerate active subsets `E_A` of small polygons (triangles,
squares, pentagons, hexagons).
3. For each subset, check reconnectivity and measure dimensions.
4. Cluster realizations by lattice and dimension.
**Success criterion:**
At least three distinct pinwheel polygons producing the same
triangular lattice, with measured `d_eff` matching to 0.05.
**Estimated cost:** 4-6 weeks.
---
## 4. Priority Tier D: Algebraic Deep Dives
### Ticket D1: Class number obstructions
**Question (from insights.md Question 3):**
How does the failure of unique factorization affect the
multi-sheeted construction over `Q(sqrt(-5))` (class number 2)?
**Deliverable:**
Concrete example of a polygon over `Q(sqrt(-5))` (or
`Q(zeta_23)`, class number 3) where reconnection fails because
of the class group, with explicit identification of the
obstruction.
**Method:**
1. Choose a polygon whose vertices generate `Z[sqrt(-5)]`.
2. Attempt the multi-sheeted construction.
3. Identify the loop whose holonomy lies in a non-principal ideal
class.
**Success criterion:**
Reduction of the obstruction to a class field theoretic statement
(the ideal of the loop holonomy being non-principal).
**Estimated cost:** 2-3 months; primarily algebraic.
---
### Ticket D2: Galois lift on multi-sheeted graphs
**Question (from insights.md Question 2):**
Does the Galois conjugation `sigma: phi -> psi` lift to an
automorphism of the multi-sheeted graph `G`?
**Deliverable:**
Proof or counterexample of the lift, with explicit map written
down for the pentagonal case at depth 3.
**Method:**
1. For each vertex `v` of `G`, define `sigma(v)` by applying
`sigma` to each coordinate.
2. Check whether `sigma` preserves adjacency.
3. If yes, compute the orbit structure.
**Success criterion:**
Either an isomorphism `sigma: G -> G` with computed orbit
structure, or a specific edge `(u, v)` such that
`(sigma(u), sigma(v))` is not an edge.
**Estimated cost:** 2 weeks.
---
## 5. Priority Tier E: Long-Term and Speculative
### Ticket E1: 4D pinwheels and the 120-cell
**Question (from insights.md Question 5):**
What is the canonical pinwheel structure on the 120-cell that
maximizes the spectral gap?
**Approach:** Sketch only. Requires Tier C tooling first.
---
### Ticket E2: Anyonic braiding and quantum computation
**Question (from insights.md Question 13):**
Can glider braiding on multi-sheeted pentagonal CAs realize
`Z_n` anyonic statistics suitable for topological quantum
computation?
**Approach:** Requires resolution of Ticket B4 first to even
find gliders. Then specialized analysis.
---
### Ticket E3: Experimental quasicrystal transport
**Question (from insights.md Question 17):**
Do AlCuFe icosahedral quasicrystals exhibit the predicted
`d_spec ~ 1.1` in transport experiments?
**Approach:** Collaboration with experimental groups (Tsai,
Steurer, et al.). Out of scope for the immediate computational
program but a north star.
---
## 6. Dependency Graph
The tickets above are not independent. Their dependencies are:
```
A3 (true object) ──┐
├──> A2 (classification theorem)
A1 (equivalence) ──┘
A2 ──> B2 (field universality)
└─> C1 (dodecahedron)
└─> E1 (120-cell)
B1 (Erdos) ──> D2 (Galois lift)
B3 (flow) ──> D1 (class number)
B4 (CA) ──> E2 (anyons)
C2 (pinwheels) ──> B1 (Erdos extension)
```
A reasonable 6-month roadmap:
1. **Month 1**: Tickets A1, B1, B2 (foundational tests).
2. **Month 2**: Tickets A3, B3, B4 (categorical clarification +
flow tests).
3. **Month 3**: Ticket A2 (classification theorem; uses A1, A3).
4. **Month 4-5**: Tickets C1, C2 (geometric extensions).
5. **Month 6**: Ticket D2 (Galois lift); Ticket D1 starts.
---
## 7. Success Metrics for the Program
The program as a whole succeeds if, at the end of 6 months:
1. The Erdős prediction (Ticket B1) is **confirmed or refuted**
with statistical significance.
2. The classification theorem (Ticket A2) is **proved, or a
sharp counterexample is exhibited**.
3. The dodecahedral construction (Ticket C1) **produces a 3D
multi-sheeted graph with measured `d_eff in (3, 4)`**.
4. At least three of the six Tier B tickets produce **publishable
quantitative findings**.
A *minimal* success consists of Tickets A1, B1, and one of C1/C2,
which would establish that the framework is foundationally sound
and produces at least one new quantitative result in extremal
combinatorics.
---
## 8. Open Questions Not Addressed
For honesty: the following questions from `insights.md` are
**not** addressed by this program because they require either
techniques outside computational reach (e.g., Question 19 on
decidability) or substantial mathematical breakthroughs (e.g.,
Question 22 in full generality). They remain open and are
flagged here for future work:
- Question 4 (cyclotomic universality across all `n`)
- Question 9 (exact algebraic value of `lambda_max`)
- Question 10 (Cantor-like spectral structure)
- Question 19 (decidability of `tau`-consistency)
These should be revisited after the 6-month program above clarifies
what is possible.