How few directions can avoid every collinear triple?
Two sharp minima connect a positive unit-step word problem to a fixed-menu walk in three dimensions. The target is the exact ordered pair \((d_*,s_*)\), not merely another construction.
The joint minimum problem
Positive-basis model. Let \(\mathcal B(d)\) mean that there is an infinite sequence \(P_0,P_1,\ldots\in\mathbb Z^d\), starting at zero, whose steps lie in \(\{e_1,\ldots,e_d\}\) and whose vertices contain no collinear triple. Define \[d_*=\min\{d\ge1:\mathcal B(d)\}.\]
Fixed-menu 3D model. Let \(\mathcal E(3,s)\) mean that one can choose a single fixed set \(S\subset\mathbb Z^3\setminus\{0\}\), with \(|S|\le s\), and then take an infinite self-avoiding walk from zero using only steps in \(S\), with no three vertices collinear. Define \[s_*=\min\{s\ge1:\mathcal E(3,s)\}.\]
Replacing each spatial step type by its own basis vector gives Shallit’s encoding implication \(\mathcal E(3,s)\Rightarrow\mathcal B(s)\). Therefore
\[d_*\le s_*.\]Subject to independent review of the six-step draft, the current working range is
\[\boxed{4\le d_*\le s_*\le6}.\]Possible pairs: (4,4), (4,5), (4,6), (5,5), (5,6), (6,6).
Quantifiers matter. The menu defining \(s_*\) is fixed once and must work for the whole infinite walk. A basis construction in dimension \(d\) does not automatically project to three dimensions with \(d\) fixed integer steps. Thus equality \(d_*=s_*\) is itself part of the problem, not an assumption.
The alternating six-step construction
The rule g85 permits eight state transitions that merge, through Cambie’s offsets, into six spatial vectors:
{(1,0,5), (0,3,5), (−1,−2,5), (0,−1,1), (1,1,6), (−1,−1,2)}.
The sign-reversed companion g170 has a different six-vector menu. Shallit’s encoding maps the six spatial types to six positive basis directions, so the same draft would establish both \(s_*\le6\) and \(d_*\le6\).
Read the current 6D proof draft (PDF)
Shared with Cambie and Shallit; independent review remains pending.
A family minimum, not the global answer
The exact period-eight audit finds 226 rules with 14 vectors, 28 with 10, and only g85/g170 with six. Period 16 again has only the two alternating minimizers.
A separate analytic argument rules out fewer than six throughout this particular four-state tagging scheme, even if its integer offset representatives change. That is construction-specific optimality. It does not rule out an unrelated four- or five-step walk.
This construction proposes reducing the 14-vector upper bound in Adenwalla’s forum question to six; it does not alter Adenwalla’s particular subsequence.
What would settle the pair?
- A four-step 3D construction would prove \((d_*,s_*)=(4,4)\).
- Impossibility of every 5D positive-basis walk would prove \((d_*,s_*)=(6,6)\), once the six-dimensional draft is reviewed.
- A 4D or 5D basis construction would improve \(d_*\), but need not improve the three-dimensional step bound \(s_*\).
- Impossibility of every five-step 3D walk would prove \(s_*=6\), while \(d_*\) could still be 4, 5, or 6.
Long finite prefixes and failed fixed recodings are evidence, not infinite constructions or global lower bounds.
How the thread developed
The original finite-step \(\mathbb Z^3\) theorem is by Stijn Cambie and Erik Kalviainen. Jeffrey Shallit introduced the positive-basis formulation and encoding in this context. Cambie supplied offsets reducing Shallit’s 16D construction to 14D. Kalviainen’s alternating signed-Gaussian draft combines those ingredients to reach six.
- Six-step/6D draft PDF — unverified follow-up argument
- Full joint-minimum formulation
- Six-vector derivation and scoped optimality
- Research timeline entry