#ChatGPT 011 – 55

You said: They discovered three residual conditions, with two of them representing a known constraint and the third coming as a surprise.

What was the surprise?

HEre is the next !JohnNash #note from December 5, 2005:

The general idea here is that since an “input” for the equation
(when the vacuum term, as LHS is to be equated to an RHS of “input”)
is that this input MUST be a symmetric tensor of rank 2 which has
vanishing divergence.
So I/we have been concerned with how a natural input tensor
term, of the sort that would seem to fit well IF THE SPACE (or
space-time) WERE SIMPLY FLAT, could be modified slightly so that
it would remain divergence-free in the presence of general curvature.
We have observed that this is possible rather easily IF the
space-time has only non-Ricci curvature (if the Ricci tensor of it
vanishes).
But we need to find and appropriate adjustment (if possible)
for the general case.

next #Note from !JohnNash

Here the polynomials corresponding to the components of a tensor
or of a vector all have the factor of (1 – x1^2 -x2^2) or of the
square of this.
This is simply to adapt them to the theme or concept of the
localization, which, in this trial work, we chose to be simply
the domain of x1^2 + x2^2 <= +1 or simply a disc-like domain
bounded by the unit circle.
So integration was performed (nicely done by Mathematica) over
the domain of that disc of unit radius.

Here is some math from a note dec 5, 2005:

con11 = -48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40

con12 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +
10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50

con21 = -80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 +
10*k12 + 3*k14 + 30*k30 + 3*k32 + 15*k50

con22 = -48*k00 – 8*k02 – 3*k04 – 8*k20 – k22 – 3*k40

I will post the context after collecting the #math on these #notes from !JohnNash

let’s just look at the math first that is there then look at what he said to connect them

Here is more !JohnNash equations from Dec 5, 2005:

elimansr = {-48*j00 – 8*j02 – 3*j04 – 8*j20 – j22 – 3*j40,
-80*j01 – 30*j03 – 15*j05 – 10*j21 – 3*j23 – 3*j41 + 80*k10 + 10*k12 +
3*k14 + 30*k30 + 3*k32 + 15*k50, -48*k00 – 8*k02 – 3*k04 – 8*k20 –
k22 – 3*k40}

memo$588737699290$ = {first,component,confirmed,equal,to,a,multiple,
of,the,integral,of,K1,over,the,circle,of,unit,radius,in,x1$,x2$,
and,third,dually,would,be,the,same,for,K2}

memo$92876928$ = {mysteriously,the,middle,of,the three,conditions,from,
the,Eliminate,analysis,on,the,equations,on,K1$,and,K2$,seem,to,turn,
out,to,be,just,that,(x2*K1-x1*K2),has,vanishing,integral,
over,the,circle}

here is more #Math from !JohnNash about this research:

memo$958273673118$ =
{checks,to,proper,count,of,90,and,7,and,8,degs,came,from,older}

list0to8 = {c10 + d01 – j00, d10 + e01 – k00, -4*c00 + 2*c20 + d11 – j10,
-4*d00 + 2*d20 + e11 – k10, c11 – 4*d00 + 2*d02 – j01,
d11 – 4*e00 + 2*e02 – k01, -5*c10 + 3*c30 – d01 + d21 – j20,
-5*d10 + 3*d30 – e01 + e21 – k20, -4*c01 + 2*c21 – 4*d10 + 2*d12 – j11,
-4*d01 + 2*d21 – 4*e10 + 2*e12 – k11, -c10 + c12 – 5*d01 + 3*d03 – j02,
-d10 + d12 – 5*e01 + 3*e03 – k02, -6*c20 + 4*c40 – d11 + d31 – j30,
-6*d20 + 4*d40 – e11 + e31 – k30, -5*c11 + 3*c31 – 2*d02 – 4*d20 +
2*d22 – j21, -5*d11 + 3*d31 – 2*e02 – 4*e20 + 2*e22 – k21,
-4*c02 – 2*c20 + 2*c22 – 5*d11 + 3*d13 – j12, -4*d02 – 2*d20 + 2*d22 –
5*e11 + 3*e13 – k12, -c11 + c13 – 6*d02 + 4*d04 – j03,
-d11 + d13 – 6*e02 + 4*e04 – k03, -7*c30 + 5*c50 – d21 + d41 – j40,
-7*d30 + 5*d50 – e21 + e41 – k40, -6*c21 + 4*c41 – 2*d12 – 4*d30 +
2*d32 – j31, -6*d21 + 4*d41 – 2*e12 – 4*e30 + 2*e32 – k31,
-5*c12 – 3*c30 + 3*c32 – 3*d03 – 5*d21 + 3*d23 – j22,
-5*d12 – 3*d30 + 3*d32 – 3*e03 – 5*e21 + 3*e23 – k22,
-4*c03 – 2*c21 + 2*c23 – 6*d12 + 4*d14 – j13, -4*d03 – 2*d21 + 2*d23 –
6*e12 + 4*e14 – k13, -c12 + c14 – 7*d03 + 5*d05 – j04,
-d12 + d14 – 7*e03 + 5*e05 – k04, -8*c40 + 6*c60 – d31 + d51 – j50,
-8*d40 + 6*d60 – e31 + e51 – k50, -7*c31 + 5*c51 – 2*d22 – 4*d40 +
2*d42 – j41, -7*d31 + 5*d51 – 2*e22 – 4*e40 + 2*e42 – k41,
-6*c22 – 4*c40 + 4*c42 – 3*d13 – 5*d31 + 3*d33 – j32,
-6*d22 – 4*d40 + 4*d42 – 3*e13 – 5*e31 + 3*e33 – k32,
-5*c13 – 3*c31 + 3*c33 – 4*d04 – 6*d22 + 4*d24 – j23,
-5*d13 – 3*d31 + 3*d33 – 4*e04 – 6*e22 + 4*e24 – k23,
-4*c04 – 2*c22 + 2*c24 – 7*d13 + 5*d15 – j14, -4*d04 – 2*d22 + 2*d24 –
7*e13 + 5*e15 – k14, -c13 + c15 – 8*d04 + 6*d06 – j05,
-d13 + d15 – 8*e04 + 6*e06 – k05, -9*c50 + 7*c70 – d41 + d61,
-9*d50 + 7*d70 – e41 + e61, -8*c41 + 6*c61 – 2*d32 – 4*d50 + 2*d52,
-8*d41 + 6*d61 – 2*e32 – 4*e50 + 2*e52, -7*c32 – 5*c50 + 5*c52 – 3*d23 –
5*d41 + 3*d43, -7*d32 – 5*d50 + 5*d52 – 3*e23 – 5*e41 + 3*e43,
-6*c23 – 4*c41 + 4*c43 – 4*d14 – 6*d32 + 4*d34,
-6*d23 – 4*d41 + 4*d43 – 4*e14 – 6*e32 + 4*e34,
-5*c14 – 3*c32 + 3*c34 – 5*d05 – 7*d23 + 5*d25,
-5*d14 – 3*d32 + 3*d34 – 5*e05 – 7*e23 + 5*e25,
-4*c05 – 2*c23 + 2*c25 – 8*d14 + 6*d16, -4*d05 – 2*d23 + 2*d25 – 8*e14 +
6*e16, -c14 + c16 – 9*d05 + 7*d07, -d14 + d16 – 9*e05 + 7*e07,
-10*c60 – d51, -10*d60 – e51, -9*c51 – 2*d42 – 4*d60,
-9*d51 – 2*e42 – 4*e60, -8*c42 – 6*c60 – 3*d33 – 5*d51,
-8*d42 – 6*d60 – 3*e33 – 5*e51, -7*c33 – 5*c51 – 4*d24 – 6*d42,
-7*d33 – 5*d51 – 4*e24 – 6*e42, -6*c24 – 4*c42 – 5*d15 – 7*d33,
-6*d24 – 4*d42 – 5*e15 – 7*e33, -5*c15 – 3*c33 – 6*d06 – 8*d24,
-5*d15 – 3*d33 – 6*e06 – 8*e24, -4*c06 – 2*c24 – 9*d15,
-4*d06 – 2*d24 – 9*e15, -c15 – 10*d06, -d15 – 10*e06, -11*c70 – d61,
-11*d70 – e61, -10*c61 – 2*d52 – 4*d70, -10*d61 – 2*e52 – 4*e70,
-9*c52 – 7*c70 – 3*d43 – 5*d61, -9*d52 – 7*d70 – 3*e43 – 5*e61,
-8*c43 – 6*c61 – 4*d34 – 6*d52, -8*d43 – 6*d61 – 4*e34 – 6*e52,
-7*c34 – 5*c52 – 5*d25 – 7*d43, -7*d34 – 5*d52 – 5*e25 – 7*e43,
-6*c25 – 4*c43 – 6*d16 – 8*d34, -6*d25 – 4*d43 – 6*e16 – 8*e34,
-5*c16 – 3*c34 – 7*d07 – 9*d25, -5*d16 – 3*d34 – 7*e07 – 9*e25,
-4*c07 – 2*c25 – 10*d16, -4*d07 – 2*d25 – 10*e16, -c16 – 11*d07,
-d16 – 11*e07}