The generic ALM linear content (LC) algebras associated with 2:282, (𝕊, ∘) and its mirror dual (𝕋, *), were obtained with respect to the invertible total internal LC, ILC = 107 A + 66 L + 31 M = (107, 66, 31) of the verse. The ILC, however, only tracks the counts of ALM letters. To include additional information the algebras will be enhanced accordingly to also account for the positions of letters, textual structure, and word boundaries. To simplify, the algebras will be confined to the first sentence as before.
As previously described, the internal ALM content of the first sentence of
2:282 are of additive and multiplicative types: AdLC and MuLC:
ILC = AdLC + MuLC1 X + MuLC2 X2
with AdLC = 11 A + 3 L + 4 M = (11, 3, 4).
The AdLC counts the
letters while the MuLCs keep track of the textual structures and word
boundaries. They are bracketed expressions that are subsequently
evaluated in the generic left and right LC algebras:
MuLC1 =
( ( ( (AA)(AL) ) (AM.A) ) ( (AA)(AM) ) ) ( ( (AL) ( (AL)(MM) ) )A )
= (5569, 716, 5863)
(Left)
= (2910, 4749, 9441)
(Right)
MuLC2 =
( ( (AA) ( (AL)(AM.A) ) ) ( (AA)(AM) ) ) ( ( (AL) ( (AL)(MM) ) )A )
= (10347, 414, 11924)
(Left)
= (7646, 7190, 2315)
(Right)
The left and right total ALM contents of sentence one are:
ILC = (11, 3, 4) + (5569, 716, 5863) X + (10347, 414, 11924) X2
(Left)
ILC = (11, 3, 4) + (2910, 4749, 9441) X + (7646, 7190, 2315) X2
(Right)
which simplify to:
ILC = (10347 X2 + 5569 X + 11,
414 X2 + 716 X + 3, 11924 X2 + 5863 X + 4)
(Left)
ILC = (7646 X2 + 2910 X + 11,
7190 X2 + 4749 X + 3, 2315 X2 + 9441 X + 4)
(Right)
The ILCs are vectors with components that are quadratic polynomials
in X. Once the internal content is known, the isotopes are obtained like
before with U = ILC and its left and right inverse multiplication maps.
The linear maps LU(•), RU(•) are
matrices with polynomial entries. However, their inverses
LU−1(•),
RU−1(•) consist of entries that are
rational functions (ratio of polynomials) of X. This can be avoided
if the vector components are from F12697[X]/p(X) with
a suitably chosen modulus polynomial p(X). The choice of p(X)
determines the structural properties of the polynomial quotients.
To constrain the vector components to quadratic polynomials,
p(X) will be an appropriately chosen cubic polynomial:
p(X) = (X − a)(X − b)(X − c). To provide a link
between p(X) and 2:282, the three constants will be taken to be the total
ALM counts of the verse: a = 107, b = 66, c = 31.
Let U be the ILC as before. The left and right multiplication
maps of U, LU(•) and RU(•), and their
inverses LU−1(•) and
RU−1(•)
are determined via their actions on the basis letter {A, L, M}
in the generic left and right LC algebras:
(Left Generic LC Algebra)
U =
(10347 X2 + 5569 X + 11,
414 X2 + 716 X + 3, 11924 X2 + 5863 X + 4)
LU(A) = UA =
(8692 X2 + 348 X + 10033,
718 X2 + 3701 X + 4627,
11929 X2 + 3432 X + 10487)
LU(L) = UL =
(10418 X2 + 9301 X + 7431,
7942 X2 + 5112 X + 9026,
1878 X2 + 10945 X + 12511)
LU(M) = UM =
(9182 X2 + 11848 X + 5162,
11067 X2 + 5444 X + 7100,
1788 X2 + 4978 X + 859)
RU(A) = AU =
(699 X2 + 11180 X + 5732,
11906 X2 + 707 X + 2948,
9029 X2 + 1885 X + 5869)
RU(L) = LU =
(5537 X2 + 8789 X + 3797,
1912 X2 + 2311 X + 6840,
5206 X2 + 7865 X + 9628)
RU(M) = MU =
(10089 X2 + 10949 X + 3844,
9222 X2 + 8827 X + 3834,
5143 X2 + 1115 X + 6198)
And their inverses:
LU−1(A) =
(5070 X2 + 7141 X + 6088,
10691 X2 + 10627 X + 1205,
1140 X2 + 7075 X + 8686)
LU−1(L) =
(9623 X2 + 12453 X + 9662,
7073 X2 + 1845 X + 533,
2317 X2 + 12500 X + 5225)
LU−1(M) =
(7929 X2 + 11240 X + 6354,
2878 X2 + 1894 X + 8001,
5900 X2 + 4360 X + 8820)
RU−1(A) =
(6714 X2 + 9871 X + 7692,
5161 X2 + 11496 X + 9687,
6908 X2 + 4447 X + 8691)
RU−1(L) =
(3548 X2 + 43 X + 7859,
6973 X2 + 2134 X + 4735,
9773 X2 + 7831 X + 7565)
RU−1(M) =
(5138 X2 + 3273 X + 4638,
11266 X2 + 932 X + 6396,
5425 X2 + 9331 X + 12548)
(Right Generic LC Algebra)
U =
(7646 X2 + 2910 X + 11,
7190 X2 + 4749 X + 3, 2315 X2 + 9441 X + 4)
LU(A) = UA =
(7554 X2 + 3856 X + 10140,
10054 X2 + 655 X + 6074,
6697 X2 + 10473 X + 7672)
LU(L) = UL =
(10599 X2 + 2160 X + 5533,
12528 X2 + 3789 X + 12191,
5552 X2 + 8139 X + 11718)
LU(M) = UM =
(8068 X2 + 12288 X + 3237,
5511 X2 + 11652 X + 7190,
9066 X2 + 7921 X + 5631)
RU(A) = AU =
(6321 X2 + 3290 X + 11248,
1158 X2 + 8980 X + 12325,
6368 X2 + 3090 X + 4989)
RU(L) = LU =
(11167 X2 + 3060 X + 764,
8558 X2 + 11889 X + 778,
1290 X2 + 1129 X + 6757)
RU(M) = MU =
(1524 X2 + 2488 X + 6941,
3384 X2 + 4600 X + 4908,
4676 X2 + 3842 X + 4033)
And their inverses:
LU−1(A) =
(3513 X2 + 5970 X + 1093,
904 X2 + 7182 X + 8753,
7499 X2 + 7983 X + 3077)
LU−1(L) =
(7108 X2 + 450 X + 750,
503 X2 + 4280 X + 4466,
12591 X2 + 11683 X + 12543)
LU−1(M) =
(7353 X2 + 12531 X + 3362,
1170 X2 + 6023 X + 5760,
10184 X2 + 1376 X + 8581)
RU−1(A) =
(4412 X2 + 60 X + 3723,
9022 X2 + 5524 X + 9539,
1688 X2 + 3051 X + 11454)
RU−1(L) =
(7051 X2 + 3519 X + 4393,
6527 X2 + 9897 X + 2216,
8383 X2 + 8531 X + 10097)
RU−1(M) =
(10788 X2 + 4901 X + 1876,
5615 X2 + 1253 X + 2692,
3858 X2 + 5514 X + 12042)
The actions will then extend linearly to all vectors. The multiplication
(∘) in the principal isotope is defined as before:
V ∘ W = RU−1(V) ⋅
LU−1(W)
in terms of the original multiplication (⋅).
The pairwise products (∘) of the basis letters {A, L, M}
give the multiplication table (the structure constants)
for the left/right principal isotopes. The
multiplicative identities, e(X),
accordingly are vectors with polynomial components or
equivalently, polynomials with vector coefficients:
(2:282 Principal Isotope From Left LC Algebra)
AA =
(6988, 12609, 9269) X2 +
(8601, 2297, 764) X + (7965, 4045, 11015)
AL =
(4344, 10250, 6682) X2 +
(1806, 644, 3976) X + (4009, 9749, 4018)
AM =
(5775, 1735, 7810) X2 +
(8478, 7491, 7289) X + (10609, 9959, 7271)
LA =
(1075, 3393, 11141) X2 +
(10685, 4527, 12196) X + (5222, 1630, 6794)
LL =
(4014, 11325, 8403) X2 +
(2385, 12310, 12366) X + (5150, 8831, 3826)
LM =
(1126, 2947, 2463) X2 +
(5260, 1256, 117) X + (879, 5414, 6769)
MA =
(2383, 1383, 11607) X2 +
(5427, 52, 8592) X + (72, 12563, 2369)
ML =
(2366, 9152, 1141) X2 +
(1392, 4180, 11122) X + (387, 1916, 3202)
MM =
(4704, 6057, 3420) X2 +
(511, 10479, 9004) X + (5443, 7276, 5888)
Multiplicative Identity:
e = (4433, 4409, 11176) X2 +
(2729, 7631, 9819) X + (985, 2558, 10000)
(2:282 Principal Isotope From Right LC Algebra)
AA =
(10277, 7186, 12371) X2 +
(8308, 11063, 12379) X + (11806, 9630, 6915)
AL =
(404, 9002, 10360) X2 +
(3169, 11222, 4254) X + (1250, 5073, 7655)
AM =
(4119, 4772, 7084) X2 +
(8908, 12473, 2721) X + (8184, 2268, 3367)
LA =
(683, 3500, 10386) X2 +
(9395, 7901, 2277) X + (7126, 9798, 2399)
LL =
(5339, 6217, 6486) X2 +
(3226, 8576, 10369) X + (11556, 10208, 11181)
LM =
(7124, 10692, 5510) X2 +
(4840, 11905, 6144) X + (4254, 8887, 11346)
MA =
(12441, 241, 1752) X2 +
(6862, 7808, 10915) X + (823, 9689, 9340)
ML =
(4239, 2158, 12642) X2 +
(8477, 10508, 1463) X + (4698, 7666, 11064)
MM =
(3432, 1558, 4565) X2 +
(10064, 507, 12083) X + (6547, 10743, 6479)
Multiplicative Identity:
e = (1730, 792, 2730) X2 +
(2798, 4691, 3831) X + (976, 10802, 9309)
The infix multiplication sign (∘) between the basis
letters is as usual suppressed and the pairwise products are
implicitly displayed with juxtaposition.
The left and right principal isotopes are the unital algebras constructed with the ALM content of the sentence-one in verse 2:282. The literal composition of the ALM letters, their textual structures and word boundaries are manifested in the algebra in the form of polynomials in F12697[X]/p(X) with p(X), a cubic polynomial whose roots are the total counts of the ALM letters in the verse.