From: Thomas Walker Lynch <eknp9n@reasoningtechnology.com>
Date: Wed, 8 Jul 2026 10:45:02 +0000 (+0000)
Subject: .
X-Git-Url: https://git.reasoningtechnology.com/machine%20fig.png?a=commitdiff_plain;h=refs%2Fheads%2Fcore-developer_branch;p=TM-2026

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         This proof applies regardless of whether the finite difference table is infinitely deep or truncated. If a person evaluates a tenth degree polynomial but only programs the machine to perform two calls, the initial tape only needs an extent of 2, utilizing components 0, 1, and 2. Each extension consumes one cell of state. If a programmer intends to execute the extension function to reach call <RT·math>\omega</RT·math>, the initial tape must have an extent of <RT·math>\omega</RT·math>, meaning it holds <RT·math>\omega + 1</RT·math> values.
       </p>
 
-      <h3>Lemma, a polynomial function degree N will always be defined with a N + 1 column first row </h3>
+      <h3>Lemma: A polynomial function of degree <RT·math>\omega</RT·math> generates a difference vector in row 0 with an extent of <RT·math>\omega</RT·math></h3>
+
+      <p>
+        A polynomial of degree <RT·math>\omega</RT·math> is defined by a highest order term <RT·math>a_\omega t^\omega</RT·math>. The first forward difference operator, <RT·math>\Delta f(t) = f(t+1) - f(t)</RT·math>, inherently cancels the <RT·math>t^\omega</RT·math> term. The binomial expansion of <RT·math>(t+1)^\omega</RT·math> yields <RT·math>t^\omega</RT·math> as its leading term, which subtracts out, leaving a new polynomial of exactly degree <RT·math>\omega - 1</RT·math>.
+      </p>
+
+      <p>
+        Taking successive finite differences reduces the degree of the polynomial by exactly one at each step. Consequently, the <RT·math>\omega</RT·math>th difference evaluates to a constant, and the next difference evaluates to exactly zero. 
+      </p>
+
+      <p>
+        Because component 0 of the initial difference vector, <RT·math>D_{0, 0}</RT·math>, holds the function value itself, and the differences occupy components 1 through <RT·math>\omega</RT·math>, the initial vector <RT·math>D_0</RT·math> requires a component count of <RT·math>\text{sampi}</RT·math>, where <RT·math>\text{sampi} = \omega + 1</RT·math>. This defines a vector with a maximum index, or extent, of <RT·math>\omega</RT·math>.
+      </p>
+
+      <h3>Lemma: A difference vector in row 0 with an extent of <RT·math>\omega</RT·math> uniquely constructs a polynomial of degree <RT·math>\omega</RT·math></h3>
+
+      <p>
+        If an initial difference vector <RT·math>D_0</RT·math> has an extent of <RT·math>\omega</RT·math> (meaning the component <RT·math>D_{0, \omega}</RT·math> is constant and all higher differences are zero), the sequence of function values can be generated by a polynomial of degree <RT·math>\omega</RT·math>.
+      </p>
+
+      <p>
+        This is proven by constructing the function directly from the initial tape utilizing Newton's forward difference formula. The function value at any discrete step <RT·math>t</RT·math> is given by a linear combination of the initial difference components multiplied by binomial coefficients:
+      </p>
+
+      <p>
+        <RT·math>f(t) = \sum_{j=0}^{\omega} D_{0, j} \binom{t}{j}</RT·math>
+      </p>
+
+      <p>
+        To determine the degree of this constructed function, a person must evaluate the binomial coefficient, <RT·math>\binom{t}{j}</RT·math>. It expands into a polynomial in <RT·math>t</RT·math>:
+      </p>
+
+      <p>
+        <RT·math>\binom{t}{j} = \frac{t(t - 1)(t - 2) \cdots (t - j + 1)}{j!}</RT·math>
+      </p>
+
+      <p>
+        Because the numerator contains <RT·math>j</RT·math> factors of <RT·math>t</RT·math>, the term <RT·math>\binom{t}{j}</RT·math> is a polynomial of exactly degree <RT·math>j</RT·math>. 
+      </p>
+
+      <p>
+        The initial tape components, <RT·math>D_{0, j}</RT·math>, are static scalar constants. Therefore, the summation <RT·math>f(t)</RT·math> consists of adding together polynomials of increasing degrees from <RT·math>0</RT·math> up to <RT·math>\omega</RT·math>. Because the extent is <RT·math>\omega</RT·math>, the scalar <RT·math>D_{0, \omega}</RT·math> is nonzero, ensuring that the highest degree term in the summation does not vanish. Thus, the constructed function <RT·math>f(t)</RT·math> is strictly a polynomial of degree <RT·math>\omega</RT·math>.
+      </p>
+
+
+
 
-      <h3>A N+1 column first row means the table came from an N degree polynomial</h3>
-      
 
-      <h2>Finite difference table for an exponential function</h2>
+      <h2>Difference table for an exponential function</h2>
 
       <p>
         Here is the finite difference table for the function <RT·math>2^t</RT·math>.