Avoiding 5/4-powers on the alphabet of nonnegative integers

computation notebook

Eric Rowland and Manon Stipulanti
https://ericrowland.github.io/papers.html#Avoiding_5_over_4-powers_on_the_alphabet_of_nonnegative_integers
https://sites.google.com/view/manonstipulanti/research

This notebook contains the computations performed in the paper “Avoiding 5/4-powers on the alphabet of nonnegative integers” by Eric Rowland and Manon Stipulanti.

Initialization code

morphism and coding

The morphism φ:

phi[(Subscript[n_, 0])]:=Thread[Subscript[{0,1,0,0,1,n+3},Range[0,5]]]
phi[(Subscript[n_, 1])]:=Thread[Subscript[{1,1,0,0,0,n+2},Mod[Range[0,5]+6,8]]]
phi[(Subscript[n_, 2])]:=Thread[Subscript[{1,1,1,0,0,n+3},Mod[Range[0,5]+12,8]]]
phi[(Subscript[n_, 3])]:=Thread[Subscript[{0,1,1,0,1,n+2},Mod[Range[0,5]+18,8]]]
phi[(Subscript[n_, 4])]:=Thread[Subscript[{0,1,0,0,1,n+1},Mod[Range[0,5]+24,8]]]
phi[(Subscript[n_, 5])]:=Thread[Subscript[{1,1,0,0,0,n+2},Mod[Range[0,5]+30,8]]]
phi[(Subscript[n_, 6])]:=Thread[Subscript[{1,1,1,0,0,n+1},Mod[Range[0,5]+36,8]]]
phi[(Subscript[n_, 7])]:=Thread[Subscript[{0,1,1,0,1,n+2},Mod[Range[0,5]+42,8]]]

For example, iterating φ twice starting from (Subscript[0, 2]) gives the following.

Column[NestList[Join@@(phi/@#)&,{(Subscript[0, 2])},2]]

The coding τ gets rid of subscripts:

tau[(Subscript[n_, _])]:=n

For example:

tau/@{(Subscript[1, 4]),(Subscript[1, 5]),(Subscript[1, 6]),(Subscript[0, 7]),(Subscript[0, 0]),(Subscript[3, 1])}

phiinverses[{a,b,c,d,e,f}, j] gives the list of letters that map under φ to {(Subscript[a, _]),(Subscript[b, _]),(Subscript[c, _]),(Subscript[d, _]),(Subscript[e, _]),(Subscript[f, j])}.

phiinverses[{0,1,0,0,1,n_},5]:={Subscript[n-3,0],Subscript[n-1,4]}
phiinverses[{1,1,0,0,0,n_},3]:={Subscript[n-2,1],Subscript[n-2,5]}
phiinverses[{1,1,1,0,0,n_},1]:={Subscript[n-3,2],Subscript[n-1,6]}
phiinverses[{0,1,1,0,1,n_},7]:={Subscript[n-2,3],Subscript[n-2,7]}

For example:

phiinverses[{0,1,0,0,1,13},5]

phi[(Subscript[10, 0])]
phi[(Subscript[12, 4])]

length-400000 prefix of the lexicographically least 5/4-power-free word

word54prefix=FromDigits/@Characters["0000111102021010010112120000131101021013020011110002121001011012020013110003111002011013010012110002111003011012010013110003121001011013020011110002121003011012020013110003111003021011010012120001111002021013010013110002111003011012010012110003111002011013010013120001111003021011010012120003111002021013010013110003121001011012020011110002121003011013010012110003111002011012010013110002111003011013020011110003121001011012020013110002121003011013010013120001111002021011010012120003111003011012010013110002111002011013010012110003111003021011010013120001111002021013010012120003111003011013020011110002121001011012020013110003111002011013010012110002111003011012010013110003121001011013020011110002121003011012020013110003141001011013030011110002131001031011020011120001121001021011030011130001131001031012020011110002121001011012030011130002131001031011020011140001121001021011030011120001131001031012020011130001121001041011020013120001111002021011010012120003111003011012010013110002111002011013010012110003111003021011010013120001111002021013030011110002121001011012020011130001121003021011010012120001111002021014010013110002111003011012010012110003111002011013010013110004111003011014010012110002111002011012010013110003111003011013010012110004111002011014010013110002111003011012010012130001111002021011010013120001121001021013020011110002121003011012010012110003111002011013010013110002111003011012020011110003121001011013020013110002121003011012010013130001111002021011010012120001131001021013020011110002121001011012020013110003111002011013010012110002111003011012010013110003121001011013020011110002121003011012020013110003111003021011010012120001111002021013010013110002111003011012010012110003111002011013010013120001111003021011010012120003111002021013010013110003121001011012020011110002121003011013010012110003111002011012010013110002111003011212000011110202101001011213000013110102101302001111000212100101101202001311000311100201101301001211000211100301101201001311000312100102101102001112000112100302101301001211000213100101101202001111000312100102101302001311000211100201101201001211000311100301101301001311000212100101101202001111000313100102101102001312000111100202101301001211000211100301101201001311000311100201101301001212000111100302101101001312000311100202101301001211000313100101101202001111000212100103101102001312000111100202101101001212000311100301101201001311000211100201101301001211000311100302101101001312000111100202101301001212000311100301101302001111000212100101101202001311000311100201101301001211000211100301101201001311000312100101101302001111000212100301101202001311000311100302101101001212000111100202101301001311000211100301101201001211000311100201101301001312000111100302101302001111000212100101101202001311000311100201101401001211000211100301101201001311000312100101101302001111000412100301101202001311000211100201101301001211000311100301101201001311000212100101101302001111000312100301101202001311000411100201101301001211000211100301101201001311000311100201101401001211000411100301101301001311000211100201101201001211000311100301101401001311000411100201101301001211000213100101101202001111000312100102101102001312000111100202101301001211000211100301101201001311000311100201101301001212000111100302101101001312000311100202101301001211000313100101101202001111000212100103101102001312000111100202101101001212000311100301101201001311000211100203101101001213000111100302101102001112000112100102101302001311000211100201101201001211000311100301101301001311000212100101101202001111000312100301101302001311000211100203101101001212000111100302101102001112000312100101101202001311000211100301101301001211000311100201101201001311000212100101101302001111000312100301101201001211000211100201101301001311000311100301101202001111000212100101101302001311000312100301101201001313000111100202101101001212000113100102101302001111000212100101101202001311000311100201101301001211000211100301101201001311000312100101101302001111000212100301101202001311000311100302101101001212000111100202101301001311000211100301101201001211000311100201101301001312000111100302101101001212000311100202101301001311000312100101101202001111000212100301101301001211000311100201101201001311000211100301101302001111000312100101101202001311000212100301101301001312000111100202101101001212000311100301101201001311000211100201101301001211000311100302101101001312000111100202101301001212000311100301101302001111000212100101101202001311000311100201101301001211000211100301101201001311000312100101101302001111000212100301101202001311000314100101101303001111000213100103101102001112000112100102101103001113000113100103101202001111000212100101101203001113000213100103101102001114000112100102101103001112000113100103101202001113000112100104101102001312000111100202101101001212000311100301101201001313000111100203101101001213000211100301101301001311000212100101101202001111000312100301101302001311000211100201101201001211000311100301101301001311000411100201101401001211000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1101301001311000211100301101401001211"];

the prefix p and the preimage z

prefixp=Take[word54prefix,6764];

prefixp//Short

preimagez=Module[{blocks,inversepairs,z},
(* Partition the appropriate factor into blocks of length 6. *)
blocks=Partition[Drop[word54prefix,Length[prefixp]],6];
(* Add an odd subscript to each block, and compute the two possible preimages of each block. *)
inversepairs=phiinverses@@@Thread[{blocks,Table[Mod[1+6j,8],{j,0,Length[blocks]-1}]}];
(* Construct the subscript-increasing preimage starting with (Subscript[0, 2]). *)
Table[If[0≤Mod[j,8]-3≤3,inversepairs[[j,2]],inversepairs[[j,1]]],{j,1,3371 6}]
];

preimagez

Length[preimagez]

the alphabet Γ

alphabetΓ={(Subscript[-3, 0]),(Subscript[-3, 2]),(Subscript[-2, 0]),(Subscript[-2, 1]),(Subscript[-2, 2]),(Subscript[-2, 3]),(Subscript[-2, 5]),(Subscript[-2, 7]),(Subscript[-1, 1]),(Subscript[-1, 3]),(Subscript[-1, 4]),(Subscript[-1, 5]),(Subscript[-1, 6]),(Subscript[-1, 7]),(Subscript[0, 4]),(Subscript[0, 6])};

Note the structure of the letters:

alphabetΓ//InputForm

Basic properties of the words p and z

Lemma 9

Part 1

Take[prefixp,-844]==Take[tau/@preimagez,-844]

844 is maximal:

Take[prefixp,-845]==Take[tau/@preimagez,-845]

Part 2

z is subscript-increasing:

FindRepeat[Last/@preimagez]

Alphabet:

Union[preimagez]
Length[%]

z is a word on (Subscript[Σ, 8])\Γ:

Intersection[preimagez,alphabetΓ]

The last letter of the form -(Subscript[1, j]) in z appears at position 80 (when indexing letters starting with 0):

First/@Position[preimagez,Subscript[-1,_]]-1

Part 3

The subsequence of letters in z with even subscripts starting at position 86 is periodic:

FindRepeat[Take[preimagez,{86+1,-1,2}]]

Part 4

Letters with even subscripts in images under φ:

Union[Cases[Flatten[Table[phi[(Subscript[n_, j])],{j,0,7}]],(Subscript[_, _?EvenQ])]]

Part 5

Set the last letter of τ(φ((Subscript[n, j]))) equal to 0 and 1 and solve for n:

Union[Table[Subscript[n/.First[Solve[tau[Last[phi[(Subscript[n, j])]]]==0,n]],j],{j,0,7}],Table[Subscript[n/.First[Solve[tau[Last[phi[(Subscript[n, j])]]]==1,n]],j],{j,0,7}]]
%==alphabetΓ

Pre-5/4-power-freeness

code for checking pre-5/4-power-freeness

Pre54PowerFreeQ[word:{___Subscript}]:=
AllTrue[
Range[1,Floor[((Length[word])/5)]],
Function[xlength,
AllTrue[
Range[0,Length[word]-5xlength],
Function[initialposition,
AnyTrue[
(* positions of the letters in x *)
Range[1,xlength],
Function[position,
If[
(* the subscripts are equal *)
Last[word[[position]]]==Last[word[[position+4xlength]]],
tau[word[[position]]]≠tau[word[[position+4xlength]]],
!MemberQ[
If[EvenQ[Last[word[[position]]]],{-2,2},{0}],
tau[word[[position]]]-tau[word[[position+4xlength]]]
]
]
]
]
]
]
]
]

For example:

Pre54PowerFreeQ[{(Subscript[0, 0]),(Subscript[n, 1]),(Subscript[(OverBar[n]), 2]),(Subscript[(OverBar[(OverBar[n])]), 3]),(Subscript[2, 4])}]

Pre54PowerFreeQ[{(Subscript[0, 0]),(Subscript[n, 1]),(Subscript[(OverBar[n]), 2]),(Subscript[(OverBar[(OverBar[n])]), 3]),(Subscript[0, 4])}]

Pre54PowerFreeQ[{(Subscript[0, 1]),(Subscript[n, 2]),(Subscript[(OverBar[n]), 3]),(Subscript[(OverBar[(OverBar[n])]), 4]),(Subscript[2, 5])}]

Pre54PowerFreeQ[{(Subscript[0, 1]),(Subscript[n, 2]),(Subscript[(OverBar[n]), 3]),(Subscript[(OverBar[(OverBar[n])]), 4]),(Subscript[0, 5])}]

For words that are not subscript-increasing, Pre54PowerQ can output True for words that are not pre-5/4-powers:

Pre54PowerFreeQ[{(Subscript[0, 1]),(Subscript[n, 2]),(Subscript[(OverBar[n]), 3]),(Subscript[(OverBar[(OverBar[n])]), 4]),(Subscript[0, 2])}]

phi/@{(Subscript[0, 1]),(Subscript[n, 2]),(Subscript[(OverBar[n]), 3]),(Subscript[(OverBar[(OverBar[n])]), 4]),(Subscript[0, 2])}

Check that z is pre-5/4-power-free:

AbsoluteTiming[Pre54PowerFreeQ[preimagez]]

Lemma 18

For each (d,i), compute the values of j for which (Subscript[(n+d), i]) is the last letter of φ((Subscript[n, j])):

phiinverselookup=Sort[GroupBy[Table[Replace[Last[phi[(Subscript[n, i])]],(Subscript[n+d_, subscript_]):>({d,subscript}->i)],{i,0,7}],First->Last]]

Slide a window through a symbolic image under φ…

Check for 5/4-powers of length 5l where the prefix of length l and the suffix of length l have no letters that depend on n:

Table[
(* For each factor, compare the prefix and suffix and check that they are unequal. *)
And@@Function[factor,tau/@Take[factor,length]≠tau/@Take[factor,-length]]/@
Select[
(* Slide a window of length 5*length. *)
Partition[Join@@Table[phi[(Subscript[n, j])],{j,0,7}],5length,1,1],
(* Pick out the factors with prefixes and suffixes that don't depend on n. *)
FreeQ[{Take[#,length],Take[#,-length]},n]&
],
{length,{2,4}}
]

Identify letters that create 5/4-powers of length 5l where the prefix of length l or the suffix of length l has a letter that depends on n:

Module[{prefix,suffix,nsolution,n},
Union@@Table[
(* For each factor, compare the prefix and suffix and solve for (the at most one occurrence of) n. *)
Union@@Function[factor,
prefix=Take[factor,length];
suffix=Take[factor,-length];
Replace[
tau/@prefix==tau/@suffix,
{
(* If there is a pair of unequal integer letters, then the factor is not a 5/4-power. *)
False->{},
(* Otherwise, look at each pair of letters, and solve for n in the pairs involving n. *)
_Equal:>Union@@Function[letterequation,
nsolution=n/.First[Solve[(* Remove subscripts. *)First/@letterequation,n]];
(* For the solution, add the possible subscripts. *)
FirstCase[
List@@letterequation,
Subscript[n+d_,subscript_]:>Thread[(Subscript[nsolution, phiinverselookup[{d,subscript}]])]
]
]/@Select[Thread[prefix==suffix],!FreeQ[#,n]&],
_:>Print[Style["Problem: The equation didn't match False or _Equal.",Red]]
}
]
]/@DeleteDuplicates[
Select[
(* Slide a window of length 5*length. *)
Partition[Join@@Table[phi[(Subscript[n, j])],{j,0,7}],5length,1,1],
(* Pick out the factors with a prefix or suffix that depends on n. *)
!FreeQ[{Take[#,length],Take[#,-length]},n]&
]
],
{length,{2,4}}
]
]
%==alphabetΓ

Theorem 20	s is pre-5/4-power-free.

base case

Check that z φ(z) is pre-5/4-power-free:

AbsoluteTiming[Pre54PowerFreeQ[Join[preimagez,Join@@phi/@preimagez]]]

one small part of the induction

Positions of the letters (Subscript[3, 4]) and (Subscript[1, 0]) in z:

First/@Position[preimagez,(Subscript[3, 4])]

First/@Position[preimagez,(Subscript[1, 0])]

Function[position,Check[Take[preimagez,{position-3,position}],"too early in the word",Take::take]]/@Join[First/@Position[preimagez,(Subscript[3, 4])],First/@Position[preimagez,(Subscript[1, 0])]]

5/4-power-freeness

Theorem 27

short factors

Check programmatically that there are no 5/4-powers of length <5 952 starting in p:

AbsoluteTiming[
Counts[Join@@Monitor[Table[
AnyTrue[
Range[1,xlength],
Function[position,word54prefix[[initialposition+position-1]]≠word54prefix[[initialposition+4xlength+position-1]]]
],
{xlength,1,951},
{initialposition,1,Length[prefixp]}
],xlength]]
]

alphabet of a prefix

Union[Take[word54prefix,Length[prefixp]+952-1]]

code for long factors

PossiblyEqualFactorClasses finds classes of initial positions among positionset where factors beginning at those positions are possibly equal when n takes values from nvalues.
If a third argument is provided, the

PossiblyEqualFactorClasses[positionset_List,nvalues_List,factorlengthbound_Integer:Length[prefixp]]:=
Module[{longprefix,periodcharacters,prefixlength,periodlength,factorlength,classes,newletter,m},
longprefix=word54prefix;
periodcharacters=Join@@Table[tau/@phi[(Subscript[n, Mod[i,8]])],{i,2,9}];
prefixlength=Length[prefixp];
periodlength=Length[periodcharacters];
factorlength=1;
(* Maintain a list of classes where each class consists of initial positions of factors which are possibly equal. *)
classes={positionset};
Monitor[
While[
(* The same class can arise in multiple ways when we partition previous classes, so we delete duplicates. *)
classes=DeleteDuplicates[
Join@@Function[class,
(* The factors of length length-1 in the period starting at the positions in  class  are identical (for some values of n). Determine which factors are still identical when we extend by one letter. *)
DeleteCases[
(* If the last letter in multiple factors is n+d for a given d, then multiple copies of a class get created (one for each value of n). So we delete duplicates. *)
DeleteDuplicates[
(* Group {position, newletter} pairs by their new letter, and then forget about the letters. *)
Values[GroupBy[
(* For each position in the current class, we extend the factor to the right and record the possible new letters. *)
Join@@(Function[position,
newletter=If[position≤prefixlength,
longprefix[[position+factorlength-1]],
(* The position repesents a general position modulo 48 in the repeating period. *)
periodcharacters[[Mod[position+factorlength-1-prefixlength,periodlength,1]]]
];
(* Check whether the new letter is symbolic in n. *)
If[FreeQ[newletter,n],
{{position,newletter}},
Table[{position,newletter/.n->m},{m,nvalues}]
]
]/@class),
Last->First
]]
],
(* A one-element class corresponds to a uniquely-occurring factor, so we delete them. *)
{_}
]
]/@classes
];
classes≠{}&&factorlength≤factorlengthbound
,
factorlength++
],
Row[{"Minimum length to distinguish factors overlapping the prefix is at least ",factorlength,"; ",Tab,Total[Length/@classes]," positions (in ",Length[classes]," classes) still have possibly equal factors"}]
];
If[factorlengthbound==Length[prefixp],
Print["Minimum length to distinguish factors overlapping the prefix: ",factorlength]
];
classes
]

long factors — first run (with n taking values from {0,1,2,3,4})

first set of positions (Subscript[S, 1])={0,1,…,(Abs[p])-1}⋃{(Abs[p]),(Abs[p])+1,…,(Abs[p])+23}

AbsoluteTiming[PossiblyEqualFactorClasses[Range[1,Length[prefixp]+24],{0,1,2,3,4}]]

second set of positions (Subscript[S, 2])={0,1,…,(Abs[p])-1}⋃{(Abs[p])+24,(Abs[p])+25,…,(Abs[p])+47}

AbsoluteTiming[PossiblyEqualFactorClasses[Join[Range[1,Length[prefixp]],Range[Length[prefixp]+25,Length[prefixp]+48]],{0,1,2,3,4}]]

long factors — second run (with n taking values from {-1,0,1,2,3,4})

For these we artificially stop the computation once we reach factors of length 952.

first set of positions (Subscript[S, 1])

AbsoluteTiming[PossiblyEqualFactorClasses[Range[1,Length[prefixp]+24],{-1,0,1,2,3,4},952]]

In this case, there are 4 nonempty classes of positions.
We examine those pairs of positions.

Grid[{
Take[word54prefix,6761+{0,37-1}],
With[{periodcharacters=Join@@Table[tau/@phi[(Subscript[n, Mod[i,8]])],{i,2,9}]},
Table[
periodcharacters[[Mod[i,Length[periodcharacters],1]]],
{i,6785-Length[prefixp],(6785-Length[prefixp])+37-1}
]
]
}]

In each pair of factors, we see a pair of letters (2,3+n). For the two factors to be equal, we in particular need those letters to be equal. When solving 2==n+3, we find n==-1.

Solve[2==3+n,n]

The letters -1 only appear in s in its prefix z and only at the nine positions 6,14,16,32,40,48,56,70,80 as shown below.

First/@Position[preimagez,Subscript[-1,_]]
Length[%]

The following finite check shows that the factors in each pair are different.

({3,3,2,4,2,2}≠tau/@preimagez[[#-5;;#]]+{2,1,2,3,2,3}&)/@{7,15,17,33,41,49,57,71,81}
And@@%

second set of positions (Subscript[S, 2])

AbsoluteTiming[PossiblyEqualFactorClasses[Join[Range[1,Length[prefixp]],Range[Length[prefixp]+25,Length[prefixp]+48]],{-1,0,1,2,3,4},952]]

As before, in each pair, we see a pair of letters (2,3+n).

Grid[{
Take[word54prefix,6761+{0,49-1}],
With[{periodcharacters=Join@@Table[tau/@phi[(Subscript[n, Mod[i,8]])],{i,2,9}]},
Table[
periodcharacters[[Mod[i,Length[periodcharacters],1]]],
{i,6809-Length[prefixp],(6809-Length[prefixp])+49-1}
]
]
}]

The following finite check shows that the factors in each pair are different.

({3,3,2,4,2,2,3,2}≠tau/@preimagez[[#-7;;#]]+{2,3,2,1,2,1,2,3}&)/@Rest[{7,15,17,33,41,49,57,71,81}]
And@@%

The sequence of letters in (Subscript[w, 5/4])

Corollary 5

We first show that the equation in the corollary holds for small values of i, namely 0≤ i ≤ 843.

FindRepeat[Table[word54prefix[[6 i+123061+1]]-word54prefix[[i+5920+1]],{i,0,843}]]

Proposition 29

For the proof of the proposition, we need the following check:

word54prefix[[6 *1127+0+1]]==0

function for the nth letter of (Subscript[w, 5/4])

word54letter[n_]:=
If[n≤6761,
word54prefix[[n+1]],
With[{quotientremainder=QuotientRemainder[n-123056,6]},
If[quotientremainder[[2]]==5,
If[quotientremainder[[1]]≤-1,
word54prefix[[n+1]],
(* Here we store letters computed recursively, but this can lead to memory problems when computing large numbers of terms. *)
word54letter[n]=word54letter[quotientremainder[[1]]+5920]+{3,2,3,2,1,2,1,2}[[Mod[quotientremainder[[1]],8]+1]]
],
(* We use Null for letters that we don't sample. *)
{0,1,0,0,1,Null,1,1,0,0,0,Null,1,1,1,0,0,Null,0,1,1,0,1,Null}[[Mod[6 quotientremainder[[1]]+quotientremainder[[2]],24]+1]]
]
]
]

Check:

Table[word54letter[n],{n,0,Length[word54prefix]-1}]==word54prefix

Corollary 32

For our particular case, we have  k=6,  r=123056, and  s=5920, so we find  E=7 and  (Subscript[q, E])=23428:

Module[{k=6,r=123056,s=5920},
Table[{e,Floor[(((k^e)-1)/((k^e)(k-1)))(r-s+k-1)+(s/(k^e))]},{e,0,15}]
]

So the dimension of the (Rationals)-vector space is at most this:

Module[{k=6,l=8,E=7,qE=23428},
Sum[(k^e),{e,0,E-1}]+k*l+(qE+1)+(E+1)
]

Proposition 34

f[e_,t_,j_,i_]:=(k^(e-t))i+((j-s)/(k^t))-(((k^t)-1)/((k^t)(k-1)))(r-s+k-1)

The sequence (Subscript[((Subscript[q, e])), 0≤e≤E-1]):

Module[{k=6,r=123056,s=5920,bigE=7},
Table[Floor[(((k^e)-1)/((k^e)(k-1)))(r-s+k-1)+(s/(k^e))],{e,0,bigE-1}]
]

Compute a limit for large e:

With[{k=6,r=123056,s=5920},Ceiling[Limit[(((k^(e-1))-1)/((k^e)(k-1)))(r-s+k-1)+(s/(k^e))+((Length[prefixp]-2-s)/k),e->Infinity]]]

Check for small e:

Block[{k=6,r=123056,s=5920,bigE=7},
Table[
i/.First[Solve[f[e,e-1,0,i]+s==Length[prefixp]-2,i]],
{e,1,bigE-1}
]
]
Ceiling[Max[%]]

Upper bound on the rank:

With[{k=6,l=8,bigE=7,H4046size=4046},
(1/2)k*l+H4046size+(bigE+1)
]

Theorem 33

This is code for computing a linearly independent subset of a set of vectors:

FlatPick[list_,selection_,pattern_:True,levelspec_:{1}]:=(list[[##]]&)@@@Position[selection,pattern,levelspec,Heads->False]

LinearlyIndependentSubset[{}]:={}
LinearlyIndependentSubset[vectors_List]:=
With[{uniquevectors=DeleteDuplicates[vectors]},
FlatPick[
uniquevectors,
Transpose[RowReduce[Transpose[uniquevectors]]],
{(0|0.)...,1|1.,(0|0.)...}
]
]

We compute a linearly independent subset of a set of kernel sequences. This gives the dimension of a vector space V.
This code uses word54letter, which is defined in the subsection “function for the nth letter of (Subscript[w, 5/4])” above.
We break the computation up into chunks to avoid row-reducing a huge matrix at the end.
We run this computation in parallel.

LaunchKernels[8];

AbsoluteTiming[Module[{k=6,r=123056,s=5920,bigE=7,termcount=4100,je,e,part,j,i},
Length[
basisV=LinearlyIndependentSubset[
Monitor[
Join[
(* exponents 0 through 3 *)
LinearlyIndependentSubset[
Join@@Table[
je=Mod[(((k^e)-1)/(k-1))(r-s+k-1)+s,(k^e)];
ParallelTable[
Table[word54letter[(k^e)i+j],{i,0,termcount-1}],
{j,DeleteCases[Range[0,(k^e)-1],je]}
],
{e,0,3}
]
],
(* exponents 4 through 7 *)
Join@@Table[
je=Mod[(((k^e)-1)/(k-1))(r-s+k-1)+s,(k^e)];
Join@@Table[
LinearlyIndependentSubset[
ParallelTable[
Table[word54letter[(k^e)i+j],{i,0,termcount-1}],
{j,DeleteCases[Range[part (k^4),(part+1) (k^4)-1],je]}
]
],
{part,0,(k^(e-4))-1}
]
,
{e,4,bigE}
]
],
Column[{
"Computing kernel sequences for exponent "<>ToString[e]<>" of "<>ToString[bigE],
If[TrueQ[part≥5],ProgressIndicator[part,{0,(k^(e-4))-1}],Nothing]
}]
]
]
]
]]

Let (Subscript[G, 4]) be the standard basis for periodic sequences with period length dividing 4.
Check that (Subscript[G, 4])⊂V:

AbsoluteTiming[With[{termcount=4100},
Table[
MatrixRank[Append[
basisV,
PadRight[{},termcount,UnitVector[4,m]]
]],
{m,1,4}
]
]]

Relations:

With[{w=word54letter},And@@Table[-w[6 i]+w[36 i]+w[2+6 i]-2 w[4+6 i]+2 {1,0,0,0}[[Mod[i,4]+1]]==0&&-w[6 i]-w[36 i]+w[2+6 i]+2{0,1,0,0}[[Mod[i,4]+1]]==0&&-w[6 i]+w[36 i]-w[2+6 i]+2{0,0,1,0}[[Mod[i,4]+1]]==0&&w[6 i]-w[36 i]-w[2+6 i]+2{0,0,0,1}[[Mod[i,4]+1]]==0,{i,0,4100}]]

Let (Subscript[H, 1]) be the set containing the eventually constant sequence 1,0,0,0,….
Check that (Subscript[H, 1])⊂V:

AbsoluteTiming[With[{termcount=4100},
MatrixRank[Append[
basisV,
UnitVector[termcount,1]
]]
]]

Relation:

With[{w=word54letter},And@@Table[-w[23+216 i]-w[29+216 i]+w[35+216 i]+w[41+216 i]+If[i==0,1,0]==0,{i,0,4100}]]

Compute the dimension of the vector space generated by the previous set and some of the missing kernel sequences:

AbsoluteTiming[Module[{k=6,r=123056,s=5920,bigE=7,termcount=4100,je,jE,e,i},
MatrixRank[fullbasis=
Join[
(* the basis from before *)
basisV,
(* new kernel sequences *)
Table[
je=Mod[(((k^e)-1)/(k-1))(r-s+k-1)+s,(k^e)];
Table[word54letter[(k^e)i+je],{i,0,termcount-1}]
,
{e,0,bigE}
],
(* really the last new kernel sequence (we swear!) *)
{
jE=Mod[(((k^bigE)-1)/(k-1))(r-s+k-1)+s,(k^bigE)];
Table[word54letter[(k^(bigE+1))i+jE],{i,0,termcount-1}]
}
]
]
]]

For the proof, we use that sufficiently large powers of k=6 are congruent to 0 modulo 24 (and in particular modulo l=8):

With[{k=6},Table[Mod[(k^m),24],{m,0,5}]]

We consider 0≤j≤(k^e)-1 such that j≠(Subscript[j, e]).
If j≡1 mod k, then j-r-k+1≡0 mod k:

With[{k=6,r=123056,s=5920},Mod[-r-k+1,k]]

Cutoff point between the two subcases:

With[{k=6,r=123056,s=5920},r-k s+k-1]

For 8≤e≤12 and j<r-k s+k-1 with j≡1 mod k, check that the kernel sequence w((k^e)i+j) is a linear combination of previous kernel sequences:

AbsoluteTiming[Module[{k=6,r=123056,s=5920,termcount=4100,bigE=7,jE,j,e,i},
jE=Mod[(((k^bigE)-1)/(k-1))(r-s+k-1)+s,(k^bigE)];
Monitor[Table[
And@@ParallelTable[
MatrixRank[Append[fullbasis,Table[word54letter[(k^e)i+j],{i,0,termcount-1}]]]==Length[fullbasis],
{j,DeleteCases[Range[1,r-k s+k-1-1,6],jE/;e==8]}
],
{e,8,12}
],e]
]]

For e≥13, we compute a bound on i:

With[{k=6,r=123056,s=5920,e=13},
(s/(k^e))+(((k^7)-1)/((k^e)(k-1)))(r-s+k-1)+((k^7)/(k^e))(Length[prefixp]-2-s)
]

Code for computing (Subscript[w, a/b])

PeriodicFunctionQ[f,{nmin,nmax},l] checks whether the values f[nmin],f[nmin+1],...,f[nmin+nmax-l] agree with f[nmin+l],f[nmin+1+l],...,f[nmin+nmax].

PeriodicFunctionQ[f_,{nmin_,nmax_},periodlength_]:=
Catch[
Do[
If[f[n]=!=f[n+periodlength],Throw[False]],
{n,nmin,nmax-periodlength}
];
True
]

PeriodicFunctionQ[{a,b,c,a,b,c,a}[[#1]]&,{1,7},3]

PeriodicFunctionQ[{a,b,c,a,b,c,a}[[#1]]&,{1,7},4]

PeriodicFunctionQ[{a,b,c,a,b,c,a}[[#1]]&,{1,3},4]

PowerSuffixedFunctionQ[f,{nmin,nmax},a/b] checks whether the values of f contain an a/b-power ending at f[nmax].

PowerSuffixedFunctionQ[f_,{nmin_,nmax_},power_]:=
With[{numerator=Numerator[power]},
Catch[
Do[
If[PeriodicFunctionQ[f,{nmax-length+1,nmax},(length/power)],Throw[True]],
{length,numerator,nmax-nmin+1,numerator}
];
False
]
]

PowerSuffixedFunctionQ[{a,b,c,a,b}[[#1]]&,{1,5},(5/3)]

PowerSuffixedFunctionQ[{a,b,c,a,b,c,a,b}[[#1]]&,{1,8},(5/2)]

PowerSuffixedFunctionQ[{0}[[#1]]&,{1,1},(3/2)]

PowerSuffixedFunctionQ[{0,0}[[#1]]&,{1,2},2]

PowerSuffixedFunctionQ[{0,1}[[#1]]&,{1,2},2]

PowerSuffixedFunctionQ[{0,1,0}[[#1]]&,{1,3},(3/2)]

PowerSuffixedFunctionQ[{0,1,1}[[#1]]&,{1,3},(3/2)]

PowerSuffixedFunctionQ[{0,1,0,0}[[#1]]&,{1,4},(4/3)]

LexLeastWord[a/b,n] computes the first n letters of (Subscript[w, a/b]).

LexLeastWord[power_,maxlength_]:=
Module[{word},
word[0]=0;
Do[
word[n]=0;
While[PowerSuffixedFunctionQ[word,{0,n},power],word[n]++]
,
{n,1,maxlength-1}
];
word/@Range[0,maxlength-1]
]

LexLeastWord[(5/4),32]

Check:

AbsoluteTiming[LexLeastWord[(5/4),1000]==Take[word54prefix,1000]]

GrowLexLeastWord[a/b,n,prefix] computes the first n letters of the lexicographically least a/b-power-free word with a given prefix.
Given a prefix of (Subscript[w, a/b]), it computes a longer prefix.

GrowLexLeastWord[power_,maxlength_,initialword_]:=
Module[{word},
MapIndexed[(word[First[#2]-1]=#1)&,initialword];
Do[
word[n]=0;
While[PowerSuffixedFunctionQ[word,{0,n},power],word[n]++]
,
{n,Length[initialword],maxlength-1}
];
word/@Range[0,maxlength-1]
]

LexLeastWord[(5/4),32]
GrowLexLeastWord[(5/4),64,%]

Check:

AbsoluteTiming[GrowLexLeastWord[(5/4),1000,LexLeastWord[(5/4),200]]==LexLeastWord[(5/4),1000]]