From: Thomas Walker Lynch Date: Thu, 9 Jul 2026 15:18:15 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/singularity_plot.png?a=commitdiff_plain;h=d25ff295066224acb89e4858affa86b773e73e42;p=TM-2026 . --- diff --git a/document/TM-2026.html b/document/TM-2026.html index ba465df..5328121 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -1453,7 +1453,7 @@

- A definitive pattern emerges. Because each successive forward difference operator annihilates the lowest power of t, the resulting algebraic system is strictly upper triangular. For any extent \omega, the \omegath difference equation reduces to D_{0,\omega} = \omega! a_\omega. This structural guarantee permits a person to reliably recover a_\omega = D_{0,\omega} / \omega!. Because an initial tape of extent \omega strictly dictates that D_{0,\omega} is nonzero, a_\omega is guaranteed to be nonzero. All subsequent lower order constants are then systematically resolved through cascading back substitution. Thus, extending this procedure \omega times definitively recovers the constants for exactly a polynomial of degree \omega. + A pattern emerges. Because each successive forward difference operator annihilates the lowest power of t, the resulting algebraic system is upper triangular. For any extent \omega, the \omegath difference equation reduces to D_{0,\omega} = \omega! a_\omega. This structural guarantee permits a person to reliably recover a_\omega = D_{0,\omega} / \omega!. Because an initial tape of extent \omega dictates that D_{0,\omega} is nonzero, a_\omega is guaranteed to be nonzero. All subsequent lower order constants are then systematically resolved through cascading back substitution. Thus, extending this procedure \omega times definitively recovers the constants for exactly a polynomial of degree \omega.

@@ -1658,109 +1658,118 @@ So then, perhaps we set the value at the singularity to x build out the difference table, then solve for x?

+ Figure with x variable. + +

+ As Newton pointed out, we can know the differences on the D_0 row of the table from the quotient function: +

-

- So let us consider the D_0 vector for the quotient h(t). -

+ + D_{0,n} = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} \frac{f(k)}{g(k)} + - - D_{0,n} = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} \frac{f(k)}{g(k)} - +

+ For k=5 the difference evaluates to be (divide 0 0), as it did for the quotient, and it also fails for all values above k=5 due to the summation. So we can not use this to solve for x. +

-

- The formula successfully computes the first several entries. If this were executing strictly in the first order, the arithmetic logic unit would blindly attempt the operation and at k=5 the result would be (divide 0 0), as it was for the quotient itself. But a second order machine can instead return (divide f(k)|k=5 g(k)|k=5), here the functional notation is used to refer to the f and g machines, so the entire machine definitions are being passed as arguments. This trapped expression is then given to L'Hôpital evaluator to resolve the indeterminate form. Note this is during the creation of the D_0 vector, so it is part of the simplification step, rather than being part of the evaluation of the quotient. The D_0 vector can then be used in extension calls to generate values. The h(t) will then be fully defined. -

+

+ The formula successfully computes the differences before k=5. As the second order evaluation returned (divide f(k)|k=5 g(k)|k=5), we can use L'Hôpital evaluator to resolve this indeterminate form D_{0,5}. Note this is during the creation of the D_0 vector, so it is part of the simplification step, rather than being part of the evaluation of the quotient. The result will thus be used in producing an extension machine that extends through the singularity, so the user will not see any exceptional higher order values while extending h(t). +

-

-We will make use use of this transcendental constant: -

+

+ The L'Hôpital evaluator will discover that a transcendental constant is required. It will be returned as a higher order object, a machine definition, because the value can not be written to the tape. +

- -T = \frac{32 \ln(2)}{3} - + + T = \frac{32 \ln(2)}{3} + -

The resulting D_0 vector for the quotient is:

+

The resulting D_0 vector for the quotient is:

- - D_{0} = \left[ \frac{31}{15}, \frac{13}{30}, \frac{8}{45}, \frac{10}{100}, \frac{1}{15}, \frac{32\ln(2)}{3}, 32/3, 64/3, 128/3, 256/3 \right] - + + D_{0} = \left[ \frac{31}{15}, \frac{13}{30}, \frac{8}{45}, \frac{10}{100}, \frac{1}{15}, \frac{32\ln(2)}{3}, 32/3, 64/3, 128/3, 256/3 \right] + -Figure D table for h(t) + Figure D table for h(t) -

- Note how the transcendental constants propagates down the table, only to cancel after it is used at the point of the singularity. The differential machinery is only capable of creating rational numbers, and at the one place that non rational is required in the sequence, that machine produces (divide 0 0). We then escalate to L'Hôpital evaluator to generate the number T, which then, can only be represented with a machine. -

+

+ I find it fascinating to watch the transcendental difference values march down the table, then all cancel out after h(5) is generated. +

-

Starting with an initial tape of the extended quotient vector and extending results in these quotients:

+

+ So the value of h(5) is transcendental, but all the machinery we were using, up until using L'Hôpital evaluator was closed over the rational field. That is why no matter how we tried to organize the difference tables, there was no way to find h(5), or to find a D table for extending through it. This situation occurred due to introducing the 2^t function. Had we remained with finite length D tables, i.e. finite degree polynomials, we could have used deconvolution of the D_0 vectors, or synthetic division on the polynomials, to create the quotient. +

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Count (t)Direct Evaluation (f(t)/g(t))Extended Quotient Vector (H(t))
031/1531/15
130/1230/12
228/928/9
324/624/6
416/316/3
5(divide 0 0)\frac{32 \ln(2)}{3}
632/332/3
796/696/6
8224/9224/9
9480/12480/12
+ +

And for the victory lap, evaluation versus higher order computation and function extension:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Count (t)Direct Evaluation (f(t)/g(t))Extended Quotient Vector (H(t))
031/1531/15
130/1230/12
228/928/9
324/624/6
416/316/3
5(divide 0 0)\frac{32 \ln(2)}{3}
632/332/3
796/696/6
8224/9224/9
9480/12480/12
Address

Unary Representation address

-

We can define a Turing Machine that is identical to the recursive definition of Natural Numbers as given by Peano. Giuseppe Peano, Arithmetices principia, nova methodo exposita (Turin: Fratres Bocca, 1889).

If we were to run the Natural Numbers Machine and watch as it writes to the tape, we would watch as the Natural Numbers are printed one after another, '·s·ss·sss·ssss· ...'. As the Natural Numbers Machine never halts, we cannot use the Natural Number Machine to initialize a tape, but we can analyze the machine. When the leftmost cell holds a terminator, we say it has the value 'zero'. We call '·s·' the number 'one'. Each set of 's' adjacent symbols surrounded by the terminators, and zero, is said to be a Natural Number. diff --git a/document/with_x_variable.png b/document/with_x_variable.png new file mode 100644 index 0000000..6294bd2 Binary files /dev/null and b/document/with_x_variable.png differ