Research / Questions and evidence
A pattern begins
a question.
The next step is to say exactly what is being studied, which construction is involved, and what would count as evidence.
Make the
question precise.
Row periods and permutations
The canonical TLF case has an elementary gcd answer, explained in the multiplication guide. Other constructions and interpretations need their own stated scope.
Phase relationships
As k changes, which values change and which relationships remain? Keep the seed, size and indexing convention fixed while comparing.
Orientation and symmetry
Which transformations relate two grids? Write down the transformation and its assumptions before treating visual similarity as equivalence.
Comparisons across sizes
Which proposed relationships can be stated consistently across N? A useful comparison needs a declared mapping between grids of different sizes.
These topics need explicit scope. Some have defined answers; others remain open. See the results and evidence status below.
Read the label.
Check the scope.
Use these labels to distinguish a definition from an observation or an established result. A claim needs its own evidence and declared scope.
- Defined
- A construction is specified, including its inputs and conventions.
- Observed
- A behavior has been seen in an example or experiment.
- Computationally verified
- A statement has been checked over an explicitly declared finite domain. That domain belongs with the result.
- Formally checked
- A proof assistant checked a stated theorem under its encoded definitions and assumptions. Source correspondence, software behavior and human peer review are separate questions.
- Proven
- A mathematical argument establishes a statement under stated assumptions.
- Hypothesis
- A specific proposition remains to be tested or proved.
- Exploratory
- An idea or analogy is being investigated without an established claim.
- Archived
- Material is retained as part of the project’s history.
Useful evidence includes the exact rule, input range, method, result and limitations. A polished visualization does not supply those details by itself.
Different layers.
Different questions.
NSquares.org is the mathematical home: definitions, examples and research questions about the structures themselves.
NS8-LAB supports research and benchmarking. A prototype or a successful check is not automatically a general mathematical result.
ArckN is the separate applied-technology layer, exploring structural signal analysis and computational tooling derived from NS8 research. An application needs evidence for its declared use.
Music mappings, cellular automata and quantum-inspired or physical analogies can motivate experiments. Their interpretation needs its own evidence; it does not follow from an arithmetic construction.
Results, evidence and open questions
The canonical TLF row-period and permutation question has a scoped elementary answer: with one-based coordinates, row r has period N/gcd(N, r−1), and is a permutation precisely when gcd(N, r−1) = 1. Replace r−1 with c−1 for a column. See the multiplication companion for phase offsets and the worked example. This arithmetic result does not establish every visible motif.
Construction and exact phase results
For an integer N ≥ 2 and coordinates and phase 1 ≤ r,c,k ≤ N, use the canonical TLF square:
U(r,c;k) = 1 + (((r−1)(c−1)+k−1) mod N).
dr(v) = 1 + ((v−1) mod 9), for positive v.The remainder is nonnegative. Apply the N wrap and final +1 before taking the digital root. The objects below retain different information.
The table compares complete trajectories: the same cyclic shift must preserve the selected object at every phase. Scroll the table horizontally on narrow screens.
| Object compared after the same cyclic shift at every phase | Least positive phase period | Method / status |
|---|---|---|
| Complete raw ordered square | N | Direct additive-phase derivation |
| Complete ordered positive digital-root square | 9 if 9 divides N, otherwise N | Recorded formally checked statement |
| Complete nine-frequency vector, fixed roots 1…9 with zero channels retained | 9 if 9 divides N, otherwise N | Recorded formally checked statement |
| Complete three-count vector, fixed 147/258/369 order | 3 if 3 divides N, otherwise N | Recorded formally checked statement |
| All maximizing-family labels, keeping every tie | General classification unresolved | Separate open question |
Equality between two individual phases and spatial row/column repetition are different questions. Histograms discard cell positions; equal periods do not mean equal information. The three groups are {1,4,7}, {2,5,8} and {3,6,9}; retain all tied groups when reporting a maximum.
An exact prime-square comparison
For an ordinary prime p ≥ 5, take order N = p². Let D(k) count all canonical cells into the fixed positive digital-root channels 1 through 9, including zero-frequency channels. For phases 1 ≤ k,l ≤ p², write:
k−1 = a+p*t, l−1 = c+p*u, 0≤a,t,c,u<p.
D(k)=D(l) exactly when a≡c (mod 9) and t≡u (mod 9).At p = 11, phases 1 and 10 have offset digits (0,0) and (9,0), so their nine-frequency states agree. Phase 12 has digits (0,1), so its state differs from phase 1. This compares frequency states; it does not assert equality of the cells in their positions. It is an individual-state criterion for prime-square orders, separate from the whole-trajectory periods above and from arbitrary composite orders or maximizing-family labels.
Records dated October 21, 2026 describe eight selected proof packages in a prepared mathematics candidate, with recorded prior Lean checks. The candidate is unpublished, and public source access and publication approval are pending. This site adds no proof check or peer review and provides no public reproduction link. Those records do not make every project statement proven; other candidates and open questions retain their individual status.
The prepared unpublished mathematics candidate contains eight selected proof packages. The current private reference separately includes a ninth checked prime-square state-equality result, outside that candidate’s selected source and license scope. Public proof/source access remains unavailable here. Recorded formal and AI checks do not establish human peer review, novelty, software certification or an application benefit.
Definitions, elementary derivations, recorded formally checked statements, finite computational observations and open questions have different evidence. Results must specify their domain, assumptions and object: raw ordered cells, ordered digital roots, nine-frequency histograms and three-family counts retain different information even when phase periods match.
NSquares presents mathematics, education and historical context. NS8-LAB maintains the research source and evidence; private research material remains outside this reader site. The Simple QCA Lab and Beat Explorer are parked legacy experiments, not validated quantum or external-signal capabilities. Their preservation does not reopen those research directions or establish an applied product. See the dated visual-motif archive and Guides & archive for context.