User blog:B1mb0w/Beta Function Code Version 1

Beta Function - Sequence Generating Code
The Beta Function has been defined using program code shown below.

A separate blog will be written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress.

Sequence Generating Code Version 1
The Beta Function is a significant re-design of the code used by the Alpha Function and will create long finite integer strings to define every large Veblen ordinal and FGH function for every finite integer (up to the size of \(f_{SV0}(n)\)). Refer to my other blogs on Unique Ordinal Representation and Program Code Version 4 for more background information.

Beta Function Description
 * \(\beta(r,v) = h^m(v)\) where
 * \(h^0(v) = f_{\gamma_0}^{u_0}(v)\) where \(u_0 < v\)
 * \(h^{n + 1}(v) = f_{\gamma_{n + 1}}^{u_{n + 1}}(h^n(v))\) where
 * \(u_{n + 1} < h^n(v)\)
 * \(\gamma_{n + 1} < \gamma_n\)
 * \(\gamma_m = 0\)

and

\(\beta(v^{v+1},v) = f_{\varphi(1,0_{[v]})}(v) = f_{SVO}(v)\) by definition

The Beta Function can then be uniquely represented by a sequence of finite integers as follows:

\(\beta(r,v) == \)

The Sequence Generating RuleSet is as follows:
 * \(h = (d,d(0:g_g,1:(h_g,h_U),(d,g_g)((0,0,0):,h_1)))\)
 * \(g = (q,p,R,g_{[q]},n_0,t_0(a))\)
 * \(n = (Q,t_0(A),P,g_{[Q]},S,n_1)\)
 * \(t(x) = (T,g_E,g_C,g_x)\)

The convention used for the parameters is:
 * Sequence function (e.g. \(h\)) defines the substitution sequences to replace any instance of the sequence function.
 * Instance of a Sequence function (e.g. \(g_0\)) is denoted by a lowercase letter with a subscript.
 * Lower case letter (e.g. \(q\)) with no subscript, denotes a finite integer greater or equal to 0
 * Upper case letter (e.g. \(R\)) with no subscript, denotes a finite integer greater or equal to 1

These rulesets are intended to be generated in a recursive way beginning with an initial h sequence. For example, a typical start for generating a sequence is:
 * \((v,h_0)\)
 * \((v,d,d(0:g_g,1:(h_g,h_U),(d,g_g)((0,0,0):,h_1))\)
 * Let \(d = 1\)
 * \((v,1,h_g,h_U,(d,g_g)((0,0,0):,h_1))\)
 * \((v,1,(d,d(0:g_g,1:(h_g,h_U),(d,g_g)((0,0,0):,h_1))),h_U,(d,g_g)((0,0,0):,h_1))\)
 * Let \(d = 0\)
 * \((v,1,(0,g_g,(d,g_g)((0,0,0):,h_1))),h_U,(d,g_g)((0,0,0):,h_1))\)
 * \((v,1,(0,(q,p,R,g_{[q]},n_0,t_0(a)),(d,g_g)((0,0,0):,h_1))),h_U,(d,g_g)((0,0,0):,h_1))\)
 * \((v,1,(0,(q,p,R,g_{[q]},(Q,t_0(A),P,g_{[Q]},S,n_1),t_0(a)),(d,g_g)((0,0,0):,h_1))),h_U,(d,g_g)((0,0,0):,h_1))\)
 * \((v,1,(0,(q,p,R,g_{[q]},(Q,(T,g_E,g_C,g_A),P,g_{[Q]},S,n_1),t_0(a))\) ... etc

The sequences explode quickly as seen in this example:
 * \((v,1,(0,(q,p,R,g_{[q]},(Q,(T,g_E,g_C,g_A),P,g_{[Q]},S,n_1),t_0(a))\) ... etc
 * Let \(q = 2\)
 * \((v,1,(0,(2,p,R,g_{[2]},(Q,(T,g_E,g_C,g_A),P,g_{[Q]},S,n_1),t_0(a))\) ... etc
 * \((v,1,(0,(2,p,R,g_1,g_2,(Q,(T,g_E,g_C,g_A),P,g_{[Q]},S,n_1),t_0(a))\) ... etc
 * \((v,1,(0,(2,p,R,(q,p,R,g_{[q]},n_0,t_0(a)),g_2,(Q,(T,g_E,g_C,g_A),P,g_{[Q]},S,n_1),t_0(a)\) ... etc

The sequences are not structured other than the implied sequence order of the elements. Therefore all internal brackets can be dropped:
 * \((v,1,0,2,p,R,q,p,R,g_{[q]},n_0,t_0(a),g_2,Q,T,g_E,g_C,g_A,P,g_{[Q]},S,n_1,t_0(a)\) ... etc

The sequences are guaranteed to be finite because of sequence limiting rules:
 * \(h = (d,d(0:g_g,1:(h_g,h_U),(d,g_g)((0,0,0):,h_1)))\) which limits the sequence as follows:
 * \(d = 0\) then \(h = (0,g_g)\)
 * \(d = 1\) then \(h = (1,h_g,h_U,),(d,g_g)((0,0,0):,h_1))\)
 * \(d = 0\) and \(g_g = (0,0)\) then \(h = (1,h_g,h_U)\)
 * else \(h = (1,h_g,h_U,h_1)\)
 * \(g = (q,q(0:a,q(1:,p,R,g_{[q]},n_0),t_0(a)))\) which limits the sequence as follows:
 * \(q = 0\) then \(g = (0,a)\)
 * \(q = 1\) then \(g = (1,t_0(a))\)
 * else \(g = (q,p,R,g_{[q]},n_0,t_0(a))\)
 * \(n = (Q,Q(1:,t_0(A),P,g_{[Q]},S,n_1))\)
 * \(Q = 0\) then \(n = (0)\)
 * else \(n = (Q,t_0(A),P,g_{[Q]},S,n_1)\)

WORK IN PROGRESS

Valid Sequence Counts
WORK IN PROGRESS

Granularity Examples \(\beta(0,3)\) to \(\beta(1.85,3)\)
The Beta Function is designed to generate any and every finite integer up to \(f_{SVO}(n)\) and will use nested FGH functions with descending ordinal values to access individual integers. Here is the first example using base v = 3:

\(\beta(0,3) = 0 = 0\)

\(\beta(0.5,3) = 1 = 1\)

\(\beta(0.75,3) = 2 = 2\)

\(\beta(1,3) = 3 = 3\)

\(\beta(1.25,3) = 4 = 4\)

\(\beta(1.4375,3) = 5 = 5\)

\(\beta(1.45,3) = 6 = 6\)

\(\beta(1.5,3) = 7 = 7\)

\(\beta(1.54,3) = 8 = 8\)

\(\beta(1.62,3) = 9 = 9\)

\(\beta(1.66,3) = 10 = 10\)

\(\beta(1.7,3) = 11 = 11\)

\(\beta(1.74,3) = f_{1}^{2}(3) = 12\)

\(\beta(1.79,3) = f_{1}^{2}(3) + 1 = 13\)

\(\beta(1.85,3) = f_{1}^{2}(3) + 2 = 14\)

Granularity Examples \(\beta(2.498,3)\) to \(\beta(2.6,3)\)
In these examples we can access individual integers around a larger ordinal:

\(\beta(2.49804953,3) = f_{1}^{f_{1}^{2}(3) + 11}(f_{2}(3)) + f_{1}^{f_{1}^{2}(3) + 10}(f_{2}(3)) + f_{1}^{f_{1}^{2}(3) + 4}(f_{2}(3)) + 4\)

\(= f_{1}^{f_{2}(3) - 1}(f_{2}(3)) + f_{1}^{f_{2}(3) - 2}(f_{2}(3)) + f_{1}^{f_{2}(3) - 8}(f_{2}(3)) + 4\)

\(< f_{2}^2(3) - 1\) A real number with far greater precision is required to access this finite integer.

\(\beta(2.5,3) = f_{2}^{2}(3)\)

\(\beta(2.55,3) = f_{2}^{2}(3) + 1\)

\(\beta(2.6,3) = f_{2}^{2}(3) + 2\)

Granularity Examples \(\beta(3,3)\) to \(\beta(3.01,3)\)
In these examples we can access individual integers around the ordinal \(\omega\):

\(\beta(2.99999999999999,3) = f_{1}^{f_{1}^{f_{1}^{2}(3) + 11}(f_{2}(3)) + f_{1}^{f_{1}^{2}(3) + 10}(f_{2}(3)) + f_{1}^{f_{1}^{2}(3) + 9}(f_{2}(3)) + f_{1}^{f_{1}^{2}(3) + 8}(f_{2}(3)) + f_{2}(3) + 2}(f_{2}^{2}(3)) + 2\)

\(< f_{\omega}(3) - 1\) A real number with far greater precision is required to access this finite integer.

\(\beta(3,3) = f_{\omega}(3)\)

\(\beta(3.005,3) = f_{\omega}(3) + 1\)

\(\beta(3.01,3) = f_{\omega}(3) + 2\)

Granularity Examples \(\beta(9,3)\) to \(\beta(9.000004,3)\)
This behaviour extends into the Veblen ordinals:

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(9.000002,3) = f_{\varphi(1,0)}(3) + 1\)

\(\beta(9.000004,3) = f_{\varphi(1,0)}(3) + 2\)

Granularity Examples \(\beta(?,3)\) to \(\beta(?,3)\)
Version 1 makes it possible to access Veblen functions of the form:
 * \(\varphi(\alpha,(\varphi(\beta,0)\uparrow\uparrow t)^{\gamma_e}.{\gamma_c} + \gamma_a)\) where
 * \(\beta > \alpha\)
 * \(0 < \gamma_e < \varphi(\beta,0)\uparrow\uparrow 2\)
 * \(0 < \gamma_c < \varphi(\beta,0)\uparrow\uparrow t\)
 * \(0 < \gamma_a < (\varphi(\beta,0)\uparrow\uparrow t)^{\gamma_e}.{\gamma_c}\)

Using the following Real Number inputs into the Beta Function generates these ordinals:

WORK IN PROGRESS

Test Bed for Version 1
Below is the test bed and various results using version 1.

\(\beta(0,3) = 0\)

\(\beta(2,3) = 2\)

\(\beta(4,3) = 2\)

\(\beta(6,3) = 2\)

\(\beta(8,3) = 4\)

\(\beta(10,3) = 5\)

\(\beta(12,3) = 6\)

\(\beta(14,3) = f_{0}^{3}(6)\)

\(\beta(16,3) = f_{0}(f_{1}^{2}(3))\)

\(\beta(18,3) = f_{0}^{3}(f_{1}^{2}(3))\)

\(\beta(20,3) = f_{0}^{f_{0}^{4}(6)}(f_{1}^{2}(3))\)

\(\beta(22,3) = f_{0}^{2}(f_{2}(3))\)

\(\beta(24,3) = f_{0}^{5}(f_{2}(3))\)

Next attempt

\(\beta(0,3) = 0\)

\(\beta(0.25,3) = 0\)

\(\beta(0.5,3) = 1\)

\(\beta(0.75,3) = 2\)

\(\beta(1,3) = 3\)

\(\beta(1.25,3) = 4\)

\(\beta(1.3,3) = 5\)

\(\beta(1.45,3) = 6\)

\(\beta(1.5,3) = f_{0}(6)\)

\(\beta(1.75,3) = f_{1}^{2}(3)\)

\(\beta(2,3) = f_{0}^{6}(f_{1}^{2}(3))\)

\(\beta(4,3) = f_{0}(f_{\omega^{2}}^{3}(3))\)

\(\beta(6,3) = f_{0}^{f_{0}^{3}(f_{1}^{2}(3))}(f_{(\omega\uparrow\uparrow 2).(\omega) + 1}^{3}(3))\)

Next attempt - Base v = 2

\(\beta(0,2) = 0\)

\(\beta(0.5,2) = 1\)

\(\beta(1,2) = 2\)

\(\beta(1.2,2) = 3\)

\(\beta(1.5,2) = 4\)

\(\beta(2,2) = f_{0}^{3}(4)\)

\(\beta(2.0001,2) = f_{\omega}(2)\)

\(\beta(2.2,2) = f_{0}(f_{\omega}(2))\)

\(\beta(2.4,2) = f_{0}^{2}(f_{\omega}(2))\)

\(\beta(2.5,2) = f_{0}^{3}(f_{\omega}(2))\)

\(\beta(2.6,2) = f_{0}^{4}(f_{\omega}(2))\)

\(\beta(2.7,2) = f_{0}^{f_{0}(4)}(f_{\omega}(2))\)

\(\beta(2.75,2) = f_{0}^{f_{0}^{2}(4)}(f_{\omega}(2))\)

\(\beta(2.8,2) = f_{0}^{f_{0}^{3}(4)}(f_{\omega}(2))\)

\(\beta(2.85,2) = f_{\omega + 1}(2)\)

\(\beta(3.2,2) = f_{0}(f_{\omega + 1}(2))\)

\(\beta(3.4,2) = f_{0}^{2}(f_{\omega + 1}(2))\)

\(\beta(3.9,2) = f_{0}^{f_{0}^{3}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(3.95,2) = f_{0}^{f_{0}^{f_{0}(4)}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(4,2) = f_{\varphi(1,0)}(2)\)

\(\beta(4.2,2) = f_{0}^{f_{0}^{3}(f_{\omega + 1}(2))}(f_{\varphi(1,\omega + 1)^{\varphi(1,1).(\omega) + \omega}.(\varphi(1,1)^{\varphi(1,0) + \omega}.(\varphi(1,0)^{\omega} + \omega))}(2))\)

Next attempt - Base v = 3

\(\beta(0,3) = 0\)

\(\beta(0.25,3) = 0\)

\(\beta(0.5,3) = 1\)

\(\beta(0.75,3) = 2\)

\(\beta(1,3) = 3\)

\(\beta(1.25,3) = 4\)

\(\beta(1.3,3) = 5\)

\(\beta(1.45,3) = 6\)

\(\beta(1.5,3) = f_{0}(6)\)

\(\beta(1.75,3) = f_{1}^{2}(3)\)

\(\beta(2.1,3) = f_{2}(3)\)

\(\beta(2.5,3) = f_{2}^{2}(3)\)

\(\beta(3,3) = f_{\omega}(3)\)

\(\beta(3.08,3) = f_{\omega}^{2}(3)\)

\(\beta(3.15,3) = f_{\omega + 1}(3)\)

\(\beta(3.22,3) = f_{\omega + 1}^{2}(3)\)

\(\beta(3.3,3) = f_{\omega + 2}(3)\)

\(\beta(3.37,3) = f_{\omega + 2}^{2}(3)\)

\(\beta(3.45,3) = f_{\omega.2}(3)\)

\(\beta(3.53,3) = f_{\omega.2}^{2}(3)\)

\(\beta(3.61,3) = f_{\omega.2 + 1}(3)\)

\(\beta(3.785,3) = f_{\omega.2 + 2}(3)\)

\(\beta(3.949,3) = f_{\omega^{2}}(3)\)

\(\beta(4.045,3) = f_{\omega^{2} + 1}(3)\)

\(\beta(4.14,3) = f_{\omega^{2} + 2}(3)\)

\(\beta(4.38,3) = f_{\omega^{2} + \omega.2}(3)\)

\(\beta(4.53,3) = f_{\omega^{2}.2}(3)\)

\(\beta(5.2,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)

\(\beta(6,3) = f_{0}^{4}(f_{(\omega\uparrow\uparrow 2).(\omega) + 2}(3))\)

\(\beta(7,3) = f_{(\omega\uparrow\uparrow 2)^{2} + 2}(3)\)

\(\beta(8,3) = f_{0}^{f_{0}^{f_{0}^{3}(6)}(f_{1}^{2}(3))}(f_{(\omega\uparrow\uparrow 2)^{2}.(\omega + 1)}^{2}(3))\)

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(12,3) = f_{0}(f_{(\varphi^{2}(\omega.2,2_*)\uparrow\uparrow 2)^{\varphi^{2}(\omega.2,1_*)^{2}.2}.(\varphi^{2}(\omega.2,2_*)^{(\omega\uparrow\uparrow 2)^{2}.(\omega.2 + 1) + 1}.(\omega^{2} + 1) + 1) + (\omega\uparrow\uparrow 2)^{2}.(\omega^{2}.2 + \omega.2 + 1) + \omega^{2} + \omega + 2}^{2}(3))\)

\(\beta(15,3) = f_{0}^{f_{\varphi(\varphi((\omega\uparrow\uparrow 2)^{2}.(\omega^{2}.2 + 1) + \omega,\varphi^{2}(1_*,2) + 1),\varphi^{2}((\omega\uparrow\uparrow 2).2 + 1_*,0) + 1)}(3)}(f_{(\varphi^{2}((\omega\uparrow\uparrow 2).2 + 1_*,\omega^{2} + 2)\uparrow\uparrow 2)^{\omega.2 + 1}.((\omega\uparrow\uparrow 2) + 1) + (\omega\uparrow\uparrow 2)}(3))\)

\(\beta(18,3) = f_{0}^{3}(f_{(\varphi^{3}((\omega\uparrow\uparrow 2).2_*,(\omega\uparrow\uparrow 2).(\omega.2) + 2)\uparrow\uparrow 2).(\omega.2) + (\varphi^{2}(2,\varphi^{3}(3_*,(\omega\uparrow\uparrow 2)^{2}.2) + \omega + 1_*)\uparrow\uparrow 2)^{\omega}.(\omega^{2} + 1) + 2}(3))\) '''This is the first error in this attempt. \(\varphi(3,x)\) is not valid for base v = 3.'''

\(\beta(21,3) = f_{(\varphi^{2}(\omega^{2},1)\uparrow\uparrow 2)^{(\varphi^{2}(1,\varphi(1,\omega^{2}.2 + \omega.2 + 2) + 1_*)\uparrow\uparrow 2)^{(\varphi(1,\omega + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)}.((\omega\uparrow\uparrow 2).(\omega^{2} + \omega.2) + 1)}.(\varphi((\omega\uparrow\uparrow 2).2 + \omega,\varphi(1,2) + 1))}}(3)\)

\(\beta(24,3) = f_{\varphi((\varphi(\varphi^{2}(\omega_*,2)^{\varphi^{2}(2_*,1)^{2}}.(\omega^{2} + \omega.2),\varphi(2,1) + \omega + 1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{2}.2 + \omega + 1}.((\varphi(2,1)\uparrow\uparrow 2)),\varphi^{3}(3,0) + 1)}(3)\)

\(\beta(27,3) = f_{\varphi(1,0,0)}(3)\)

Next attempt - Base v = 2

\(\beta(0,2) = 0\)

\(\beta(0.5,2) = 1\)

\(\beta(1,2) = 2\)

\(\beta(1.2,2) = 3\)

\(\beta(1.5,2) = 4\)

\(\beta(2,2) = f_{0}^{3}(4)\)

\(\beta(2.0001,2) = f_{\omega}(2)\)

\(\beta(2.2,2) = f_{0}(f_{\omega}(2))\)

\(\beta(2.4,2) = f_{0}^{2}(f_{\omega}(2))\)

\(\beta(2.5,2) = f_{0}^{3}(f_{\omega}(2))\)

\(\beta(2.6,2) = f_{0}^{4}(f_{\omega}(2))\)

\(\beta(2.7,2) = f_{0}^{f_{0}(4)}(f_{\omega}(2))\)

\(\beta(2.75,2) = f_{0}^{f_{0}^{2}(4)}(f_{\omega}(2))\)

\(\beta(2.8,2) = f_{0}^{f_{0}^{3}(4)}(f_{\omega}(2))\)

\(\beta(2.85,2) = f_{\omega + 1}(2)\)

\(\beta(3.2,2) = f_{0}(f_{\omega + 1}(2))\)

\(\beta(3.4,2) = f_{0}^{2}(f_{\omega + 1}(2))\)

\(\beta(3.9,2) = f_{0}^{f_{0}^{3}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(3.95,2) = f_{0}^{f_{0}^{f_{0}(4)}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(4,2) = f_{\varphi(1,0)}(2)\)

\(\beta(4.2,2) = f_{0}^{3}(f_{\varphi(\omega,\varphi(1,0))^{\varphi(1,0)^{\omega}}.(\omega) + \omega + 1}(2))\)

\(\beta(4.4,2) = f_{0}^{f_{0}^{2}(4)}(f_{\varphi(1,\varphi(1,\varphi(1,\omega)))^{\omega} + \omega + 1}(2))\)

\(\beta(4.6,2) = f_{\varphi(\omega,0)^{\varphi(1,\omega) + \omega}.(\varphi(1,\omega)^{\varphi(1,\varphi(1,0))^{\varphi(1,0)^{\omega}.(\omega)}}.(\varphi(1,0)))}(2)\)

\(\beta(4.8,2) = f_{0}^{f_{0}(f_{\omega}(2))}(f_{\varphi(\omega,\omega).(\varphi^{2}(1,\varphi(1,\varphi(\omega,0))_*)) + 1}(2))\) '''This is the first error in this attempt. \(\varphi^2\) is not valid for base v = 2.'''

Next attempt - Base v = 2

\(\beta(0,2) = 0\)

\(\beta(0.5,2) = 1\)

\(\beta(1,2) = 2\)

\(\beta(1.2,2) = 3\)

\(\beta(1.5,2) = 4\)

\(\beta(2,2) = 4 + 3\)

\(\beta(2.0001,2) = f_{\omega}(2)\)

\(\beta(2.2,2) = f_{\omega}(2) + 1\)

\(\beta(2.4,2) = f_{\omega}(2) + 2\)

\(\beta(2.5,2) = f_{\omega}(2) + 3\)

\(\beta(2.6,2) = f_{\omega}(2) + 4\)

\(\beta(2.7,2) = f_{0}^{f_{0}(4)}(f_{\omega}(2))\)

\(\beta(2.75,2) = f_{0}^{f_{0}^{2}(4)}(f_{\omega}(2))\)

\(\beta(2.8,2) = f_{0}^{f_{0}^{3}(4)}(f_{\omega}(2))\)

\(\beta(2.85,2) = f_{\omega + 1}(2)\)

\(\beta(3.2,2) = f_{\omega + 1}(2) + 1\)

\(\beta(3.4,2) = f_{\omega + 1}(2) + 2\)

\(\beta(3.9,2) = f_{0}^{f_{0}^{3}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(3.95,2) = f_{0}^{f_{0}^{f_{0}(4)}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(4,2) = f_{\varphi(1,0)}(2)\)

\(\beta(4.2,2) = f_{\varphi(\omega,\varphi(1,0))^{\varphi(1,0)^{\omega}}.(\omega) + \omega + 1}(2) + 3\)

\(\beta(4.4,2) = f_{0}^{f_{0}^{2}(4)}(f_{\varphi(1,\varphi(1,\varphi(1,\omega)))^{\omega} + \omega + 1}(2))\)

\(\beta(4.6,2) = f_{\varphi(\omega,0)^{\varphi(1,\omega) + \omega}.(\varphi(1,\omega)^{\varphi(1,\varphi(1,0))^{\varphi(1,0)^{\omega}.(\omega)}}.(\varphi(1,0)))}(2)\)

\(\beta(4.8,2) = f_{0}^{f_{0}(f_{\omega}(2))}(f_{\varphi(\omega,\omega).(\varphi^{2}(1,\varphi(1,\varphi(\omega,0))_*)) + 1}(2))\) '''This is the first error in this attempt. \(\varphi^2\) is not valid for base v = 2.'''

\(\beta(5,2) = f_{\varphi(\varphi(1,0)^{\omega} + \varphi(1,0).(\omega),\varphi(\omega,\omega))^{\varphi(1,0)} + \varphi(\varphi^{2}(1,\varphi(1,\varphi(\omega,\varphi(\omega,\omega)))_*),\varphi(1,0))^{\varphi(1,\varphi(\omega,\varphi(1,0)))^{\omega}.(\varphi(\omega,\varphi(1,0))^{\varphi^{2}(1,\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))_*)}.(\omega) + \omega)}.(\varphi^{2}(1,\varphi(1,\varphi(\varphi(\omega,0) + \omega,\varphi(1,0)))_*))}(2)\)

\(\beta(5.2,2) = f_{\varphi(1,\varphi^{2}(1_*,0))^{\omega}.(\omega) + \omega + 1}(2) + 1\)

\(\beta(5.4,2) = f_{\varphi^{2}(1_*,\omega)^{\omega} + \omega + 1}(2)\)

Next attempt - Base v = 2

\(\beta(0,2) = 0\)

\(\beta(0.5,2) = 1\)

\(\beta(1,2) = 2\)

\(\beta(1.2,2) = 3\)

\(\beta(1.5,2) = 4\)

\(\beta(2,2) = 4 + 3\)

\(\beta(2.0001,2) = f_{\omega}(2)\)

\(\beta(2.2,2) = f_{\omega}(2) + 1\)

\(\beta(2.4,2) = f_{\omega}(2) + 2\)

\(\beta(2.5,2) = f_{\omega}(2) + 3\)

\(\beta(2.6,2) = f_{\omega}(2) + 4\)

\(\beta(2.7,2) = f_{0}^{f_{0}(4)}(f_{\omega}(2))\)

\(\beta(2.75,2) = f_{0}^{f_{0}^{2}(4)}(f_{\omega}(2))\)

\(\beta(2.8,2) = f_{0}^{f_{0}^{3}(4)}(f_{\omega}(2))\)

\(\beta(2.85,2) = f_{\omega + 1}(2)\)

\(\beta(3.2,2) = f_{\omega + 1}(2) + 1\)

\(\beta(3.4,2) = f_{\omega + 1}(2) + 2\)

\(\beta(3.9,2) = f_{0}^{f_{0}^{3}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(3.95,2) = f_{0}^{f_{0}^{f_{0}(4)}(f_{\omega}(2))}(f_{\omega + 1}(2))\)

\(\beta(4,2) = f_{\varphi(1,0)}(2)\)

\(\beta(4.2,2) = f_{\varphi(1,(\varphi(1,0)\uparrow\uparrow 2)^{\omega} + \omega)^{\omega}.(\omega)}(2) + 1\) '''This is the first error in this attempt. \(\varphi\uparrow\uparrow 2\) is not valid for base v = 2.'''

\(\beta(4.4,2) = f_{\varphi(1,\varphi(1,\omega)^{\omega}.(\varphi(1,(\varphi(1,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,(\varphi(1,0)\uparrow\uparrow 3) + 1).2 + 1)^{\omega}.(\omega) + \varphi(1,(\varphi(1,0)\uparrow\uparrow 2)^{\varphi(1,\varphi(1,\varphi(1,0)^{\omega} + \omega) + \varphi(1,0).(\omega))^{\omega}.(\omega)} + \omega)^{\omega} + \omega}.(\omega) + \omega)^{\varphi(1,(\varphi(1,0)\uparrow\uparrow 2)^{\varphi(1,0)^{\omega} + \omega}.(\varphi(1,\omega).(\varphi(1,(\varphi(1,0)\uparrow\uparrow 3) + 1)) + \omega) + 1)^{\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0) + \omega) + \varphi(1,\varphi(1,0) + 1))}}) + 1)}(2)\)

\(\beta(4.6,2) = f_{\varphi(\omega,0)^{\varphi(1,\omega) + \omega}.(\varphi(1,\omega)^{\varphi(1,\varphi(1,0)^{\omega}.(\omega) + 1)^{\varphi(1,0)^{\omega}.(\omega) + \omega}})}(2) + 1\)

\(\beta(4.8,2) = f_{\varphi(\omega,\omega).(\varphi(1,\varphi(\omega,0)^{\varphi(1,\omega)^{\varphi(1,\varphi(1,0).(\omega) + \varphi(1,\omega)^{\varphi(1,0).(\omega)})^{\omega} + \varphi(1,0)}.(\omega) + \varphi(1,\omega).(\varphi(1,0)) + \varphi(1,0) + \omega}.(\varphi(1,(\varphi(1,0)\uparrow\uparrow 2).(\varphi(1,(\varphi(1,0)\uparrow\uparrow 3) + \varphi(1,0)^{\omega}.(\omega) + \omega)) + \omega)^{\omega}.(\omega) + \varphi(1,0)^{\omega} + \varphi(1,0) + \omega) + 1))}(2)\)

\(\beta(5,2) = f_{\varphi(1,\varphi(1,(\varphi(\omega,\omega)\uparrow\uparrow 3) + 1) + \omega)^{\omega}.(\varphi(1,\varphi(\omega,0).(\omega) + \omega)^{\omega}.(\omega) + \varphi(1,(\varphi(1,\omega)\uparrow\uparrow 2).(\varphi(1,(\varphi(\omega,0)\uparrow\uparrow 2)^{\omega}.(\varphi(1,0)^{\omega} + \varphi(1,0).(\omega)) + \varphi(1,(\varphi(\omega,\omega)\uparrow\uparrow 2).(\varphi(\omega,\omega).(\varphi(1,(\varphi(1,0)\uparrow\uparrow 3) + \omega)^{\varphi(1,0)^{\omega}}.(\omega) + \varphi(1,\varphi(1,0)^{\varphi(1,\varphi(\omega,0) + 1)^{\varphi(\omega,0)^{\varphi(1,0).(\omega)}.(\varphi(1,\varphi(1,0) + 1))}} + 1))) + 1))) + 1))}(2)\)

\(\beta(5.2,2) = f_{\varphi(1,\varphi^{2}(1_*,0).(\varphi(\omega,0)^{\omega} + \varphi(1,0) + \omega) + \varphi^{2}(1_*,0)^{\omega} + \varphi^{2}(1_*,0) + \varphi(1,\omega).(\omega))^{\varphi(\omega,\omega)^{\omega}.(\varphi(\omega,0)^{\omega} + \varphi(1,\omega).(\omega)) + \varphi(1,(\varphi(1,\omega)\uparrow\uparrow 2)^{\omega} + \omega)^{\omega}.(\omega) + \varphi(1,(\varphi(1,0)\uparrow\uparrow 3)^{\omega}.(\varphi(1,(\varphi(1,\varphi^{2}(1_*,0).(\omega) + 1)\uparrow\uparrow 2) + 1)^{\omega}.(\varphi(1,\omega)^{\varphi(1,0) + \omega})) + 1)}}(2)\)

\(\beta(5.4,2) = f_{\varphi^{2}(1_*,\omega)^{\omega} + \omega + 1}(2)\)

\(\beta(5.6,2) = f_{\varphi(1,(\varphi^{2}(1_*,\omega)\uparrow\uparrow 2)^{\varphi^{2}(1_*,\omega)^{\varphi(1,\varphi(\omega,\omega)^{\varphi(1,\omega)^{\varphi(1,(\varphi(1,(\varphi(1,0)\uparrow\uparrow 3).(\omega) + 1)\uparrow\uparrow 2).2 + 1)}} + 1)^{\varphi(\omega,0) + \omega}}.(\omega) + \varphi^{2}(1_*,\omega)^{\omega}.(\omega) + \omega}.(\varphi(1,(\varphi(\omega,\omega)\uparrow\uparrow 2)^{\varphi(1,0)} + \omega).(\omega) + \varphi(1,(\varphi(\omega,0)\uparrow\uparrow 2)^{\omega} + \omega) + \omega) + 1).(\varphi(1,(\varphi^{2}(1_*,\omega)\uparrow\uparrow 2).(\varphi(\omega,0)) + 1))}(2)\)

\(\beta(5.8,2) = f_{\varphi^{2}(\omega_*,0)^{\varphi^{2}(1_*,0)^{\varphi(1,\omega)^{\omega} + \omega}.(\varphi(1,0) + \omega)}.(\omega) + \varphi(1,\varphi(1,(\varphi(\omega,0)\uparrow\uparrow 3).(\omega) + \omega)^{\omega}.(\omega) + \omega)^{\varphi(1,(\varphi(\omega,0)\uparrow\uparrow 3).(\omega) + 1)^{\omega}.(\varphi(1,(\varphi(1,\omega)\uparrow\uparrow 3).(\varphi^{2}(\omega_*,0)^{\varphi(1,0) + \omega}.(\varphi(1,(\varphi(1,\varphi^{2}(1_*,0).(\omega) + 1)\uparrow\uparrow 2).(\omega) + 1)^{\varphi(\omega,0)})) + 1))}}(2)\)

\(\beta(6,2) = f_{\varphi^{2}(\omega_*,\omega) + \omega}(2)\)

Next attempt - Base v = 2

\(\beta(4.17,2) = f_{0}^{f_{\varphi(1,\varphi(1,0)^{\omega} + 1).(\varphi(1,\varphi(1,0).(\varphi(2,\varphi(1,\varphi(1,\varphi(1,0) + \varphi(1,0)) + 1) + 2) + \omega) + \varphi(1,\varphi(1,0)^{\omega}.(\omega) + \varphi(1,\varphi(1,\varphi(1,0).(\omega) + \omega) + 1))))}(2)}(f_{\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0).(\omega)) + \omega) + \varphi(1,\varphi(1,\varphi(1,0) + \varphi(1,0)^{\omega} + \varphi(1,0).(\omega) + \omega) + 1)^{\varphi(1,\varphi(1,0) + \omega)^{\omega}} + \omega}(2))\)

\(\beta(4.171,2) = f_{0}^{f_{0}(f_{\varphi(1,0).(\omega) + 1}(2))}(f_{\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0).(\omega) + \omega) + 1)^{\varphi(2,\varphi(1,\varphi(1,\varphi(1,0).(\omega) + \omega) + 1) + 2) + \omega} + \varphi(1,0)^{\omega}.(\omega) + \omega + 1}(2))\)

\(\beta(4.172,2) = f_{\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0).(\omega) + \omega) + \varphi(1,\varphi(1,0).(\omega) + \varphi(2,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)) + 1) + 2) + \omega).(\omega) + \varphi(1,\varphi(1,\varphi(1,0).(\omega) + \omega) + 2)^{\varphi(1,0)^{\omega} + \omega}.(\omega) + \varphi(2,\varphi(1,\varphi(1,0).(\omega) + 1) + 2) + \varphi(1,0)) + 1)^{\varphi(1,0)^{\omega}.(\omega) + \omega}.(\omega) + 1}(2) + 1\)

Next attempt - Base v = 2

\(\beta(4.17701,2) = f_{\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0).(\omega) + \omega) + \omega) + 2)^{\omega}.(\omega) + \omega}(2) + 1\)

\(\beta(4.177011,2) = f_{\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0).(\omega) + \omega) + \omega) + 2)^{\varphi(1,0).(\omega)}.(\omega) + \omega + 1}(2)\)

\(\beta(4.177012,2) = f_{\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0).(\omega) + \omega) + \omega) + 2)^{\varphi(1,\varphi(1,0)^{\omega}.(\omega) + 1)^{\varphi(1,0)^{\omega}.(\omega) + \omega}.(\omega) + \varphi(1,0)^{\omega}.(\omega) + \omega}.(\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0).(\omega) + \omega) + 1)^{\omega} + \varphi(1,0).(\omega) + \omega) + 1}(2) + 1\)

\(\beta(4.177013,2) = f_{\varphi(1,\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0).(\omega) + \omega) + \omega) + 2) + 1)^{\varphi(1,0).(\omega)} + 1}(2)\)

\(\beta(4.177014,2) = f_{\varphi(1,\varphi(1,\varphi(1,\varphi(1,\varphi(1,\varphi(1,0)^{\omega}.(\varphi(1,0)^{\omega}.(\omega) + \varphi(1,0).(\omega) + \omega) + \omega) + 2) + 1) + 2) + 1) + \omega}(2) + 1\)

Next attempt - Base v = 2

\(\beta(0,2) = 0\)

\(\beta(0.5,2) = 1\)

\(\beta(1,2) = 2\)

\(\beta(1.25,2) = 3\)

\(\beta(1.45,2) = f_{1}(2)\)

\(\beta(1.55,2) = f_{1}(2) + 1\)

\(\beta(1.7,2) = f_{1}(2) + 2\)

\(\beta(1.84,2) = f_{1}(2) + 3\)

\(\beta(2.001,2) = f_{\omega}(2)\)

\(\beta(2.09,2) = f_{\omega}(2) + 1\)

\(\beta(2.19,2) = f_{\omega}(2) + 2\)

\(\beta(2.23,2) = f_{\omega}(2) + 3\)

\(\beta(2.28,2) = f_{\omega}(2) + f_{1}(2)\)

\(\beta(2.31,2) = f_{\omega}(2) + f_{1}(2) + 1\)

\(\beta(2.33,2) = f_{\omega}(2) + f_{1}(2) + 2\)

\(\beta(2.36,2) = f_{\omega}(2) + f_{1}(2) + 3\)

\(\beta(2.4,2) = f_{1}(f_{\omega}(2))\)

\(\beta(2.45,2) = f_{1}(f_{\omega}(2)) + 1\)

\(\beta(2.49,2) = f_{1}(f_{\omega}(2)) + 2\)

\(\beta(2.5,2) = f_{1}(f_{\omega}(2)) + 3\)

\(\beta(2.511,2) = f_{1}(f_{\omega}(2)) + f_{1}(2)\)

\(\beta(2.515,2) = f_{1}(f_{\omega}(2)) + f_{1}(2)\)

\(\beta(2.518,2) = f_{1}(f_{\omega}(2)) + f_{1}(2) + 1\)

\(\beta(2.525,2) = f_{1}(f_{\omega}(2)) + f_{1}(2) + 2\)

\(\beta(2.532,2) = f_{1}(f_{\omega}(2)) + f_{1}(2) + 3\)

\(\beta(2.539,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2)\)

\(\beta(2.552,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 1\)

\(\beta(2.566,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 2\)

\(\beta(2.573,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 3\)

\(\beta(2.58,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + f_{1}(2)\)

\(\beta(2.584,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + f_{1}(2) + 1\)

\(\beta(2.587,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + f_{1}(2) + 2\)

\(\beta(2.591,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + f_{1}(2) + 3\)

\(\beta(2.594,2) = f_{1}^{3}(f_{\omega}(2))\)

\(\beta(2.608,2) = f_{1}^{3}(f_{\omega}(2)) + 1\)

\(\beta(2.622,2) = f_{1}^{3}(f_{\omega}(2)) + 2\)

\(\beta(2.626,2) = f_{1}^{3}(f_{\omega}(2)) + 3\)

\(\beta(2.63,2) = f_{1}^{3}(f_{\omega}(2)) + f_{1}(2)\)

\(\beta(2.631,2) = f_{1}^{3}(f_{\omega}(2)) + f_{1}(2) + 1\)

\(\beta(2.633,2) = f_{1}^{3}(f_{\omega}(2)) + f_{1}(2) + 2\)

\(\beta(2.635,2) = f_{1}^{3}(f_{\omega}(2)) + f_{1}(2) + 3\)

\(\beta(2.637,2) = f_{1}^{3}(f_{\omega}(2)) + f_{\omega}(2)\)

\(\beta(2.638,2) = f_{1}^{3}(f_{\omega}(2)) + f_{\omega}(2) + 1\)

Next attempt - Base v = 2

\(\beta(2.001,2) = f_{\omega}(2)\)

\(\beta(2.09,2) = f_{\omega}(2) + 1\)

\(\beta(2.19,2) = f_{\omega}(2) + 2\)

\(\beta(2.23,2) = f_{\omega}(2) + 3\)

\(\beta(2.28,2) = f_{\omega}(2) + 4\)

\(\beta(2.31,2) = f_{\omega}(2) + 5\)

\(\beta(2.33,2) = f_{\omega}(2) + 6\)

\(\beta(2.36,2) = f_{\omega}(2) + 7\)

\(\beta(2.4,2) = f_{1}(f_{\omega}(2))\)

\(\beta(2.45,2) = f_{1}(f_{\omega}(2)) + 1\)

\(\beta(2.49,2) = f_{1}(f_{\omega}(2)) + 2\)

\(\beta(2.5,2) = f_{1}(f_{\omega}(2)) + 3\)

\(\beta(2.511,2) = f_{1}(f_{\omega}(2)) + 4\)

\(\beta(2.515,2) = f_{1}(f_{\omega}(2)) + 4\)

\(\beta(2.518,2) = f_{1}(f_{\omega}(2)) + 5\)

\(\beta(2.525,2) = f_{1}(f_{\omega}(2)) + 6\)

\(\beta(2.532,2) = f_{1}(f_{\omega}(2)) + 7\)

\(\beta(2.539,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2)\)

\(\beta(2.552,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 1\)

\(\beta(2.566,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 2\)

\(\beta(2.573,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 3\)

\(\beta(2.58,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 4\)

\(\beta(2.584,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 5\)

\(\beta(2.587,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 6\)

\(\beta(2.591,2) = f_{1}(f_{\omega}(2)) + f_{\omega}(2) + 7\)

\(\beta(2.594,2) = f_{1}^{2}(f_{\omega}(2))\)

\(\beta(2.608,2) = f_{1}^{2}(f_{\omega}(2)) + 1\)

\(\beta(2.622,2) = f_{1}^{2}(f_{\omega}(2)) + 2\)

\(\beta(2.626,2) = f_{1}^{2}(f_{\omega}(2)) + 3\)

\(\beta(2.63,2) = f_{1}^{2}(f_{\omega}(2)) + 4\)

\(\beta(2.631,2) = f_{1}^{2}(f_{\omega}(2)) + 5\)

\(\beta(2.633,2) = f_{1}^{2}(f_{\omega}(2)) + 6\)

\(\beta(2.635,2) = f_{1}^{2}(f_{\omega}(2)) + 7\)

\(\beta(2.637,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2)\)

\(\beta(2.638,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) + 1\)

\(\beta(2.64,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) + 2\)

\(\beta(2.641,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) + 3\)

\(\beta(2.64175,2) = f_{1}^{2}(f_{\omega}(2)) + f_{\omega}(2) + 4\)

WORK IN PROGRESS