Coefficient heights in the finite MZV construction

The finite construction in Finite MZV constructions for leading constants already applies to every fixed integer index \(q\geq3\). This chapter bounds its symbolic rational coefficients, including stored intermediate rational results. Numerical MZV prefixes are a separate cost. The algorithm and its existing operational work limit are unchanged.

Theorem 43 (A common denominator and polynomial coefficient-bit bound)

Put \(Q=q+1\), \(L=q!\), and let \(b=\operatorname{bit\_length}(q)\). Throughout the coefficient arithmetic of leading_constant_mzv._construct(q, budget), every stored rational denominator divides \(L^{5Q}\). Every stored rational numerator and denominator has at most

\[B_q=5qQb+8Q(b+1)+2\]

bits, where the sign is excluded and zero has bit length zero. The unreduced integer products and sums for a single rational addition, subtraction, multiplication or nonzero division have at most \(2B_q+1\) bits in the elementary arithmetic model.

This does not bound the number of words by a polynomial: the previous \(F_q\) term bound and \(128Q^8 2^q\) unit-cost arithmetic bound still apply. It does not bound Python instruction counts, internal GCD/factorial/binomial buffers, peak RSS, or the rational arithmetic of numerical MZV prefixes.

Proof. Initial factors. All shifts are nonnegative integers below \(q\), and pole orders are one or two. Differences of distinct shifts are nonzero integers of absolute value at most \(q\). In _partial_fractions the highest-pole coefficient is a product of at most \(2q\) such reciprocal factors. Its denominator divides \(L^{2q}\) and its absolute value is at most one. The logarithmic derivative is a sum of at most \(q\) terms of size at most two, with denominator dividing \(L\). Thus every initial partial fraction, including the simple coefficient of a double pole, has denominator dividing \(L^{2q+1}\) and absolute value at most \(2q\). These statements also hold during the partial products and partial sums.

Each intermediate elementary symmetric polynomial in _phi_polynomial(a) has denominator dividing \((a-1)!\). A term is a reciprocal product of a subset of the distinct integers \(1,\ldots,a-1\), and its denominator divides that factorial. At the end, multiplication by \(a!(a-1)!\) gives an integer. The sum of the absolute coefficients is

\[a!(a-1)!\prod_{j=1}^{a-1}(1+1/j)=(a!)^2\leq L^2.\]

The finite product-of-harmonic-words coefficient is a binomial integer at most \(2^q\). So is \(\binom{q-1}{\ell}\). For each fixed word and pole, _initial adds at most \(q^3\) contributions. All its accumulated coefficients and partial sums therefore have denominator dividing \(L^{2q+1}\) and absolute value at most

\[I_q=2q^5 4^q L^4.\]

The elementary-polynomial intermediates and integer prefactors also satisfy this absolute bound.

Recursion. Suppose the current pole coefficients have common denominator dividing \(L^E\) and absolute value at most \(U\). _telescope changes no denominator. Its residual and each antiderivative coefficient sum at most \(q\) coefficients, giving absolute bound \(qU\).

_shift_divide multiplies these by partial fractions with exactly two poles, at zero and \(h\in\{1,\ldots,q\}\), both of order at most two. Directly from the partial-fraction formulas, their denominators divide \(h^3\), and their absolute values are at most two. Consequently the new denominator divides \(L^{E+3}\). There are at most \(2q\) antiderivative entries contributing to each new coefficient, so its absolute value, including partial accumulations, is at most \(4q^2U\).

Each recursive call removes the first harmonic index. The initial word length is at most \(q-3\). At the empty word, the final boundary divides antiderivative coefficients by \((1+h)^p\), where \(1+h\leq q\) and \(p\leq2\), adding at most two to the denominator exponent and giving absolute bound \(2q^2U\). Hence throughout a branch the denominator exponent is at most \(2q+1+3(q-3)+2=5q-6<5Q\).

With \(T_q=8q^3\), the bound \(I_qT_q^{q+1}\) covers all branch coefficients, residuals, boundaries and intermediate products. Residual keys within a branch have strictly decreasing weights, so copying the recursively returned dictionary does not add different levels to the same key. Finally, there are fewer than \(2^q\) initial words. Even a partial final accumulation has absolute value at most

\[H_q=2^q I_q(8q^3)^{q+1}.\]

Since \(\log_2L\leq qb\) and \(\log_2H_q\leq8Q(b+1)\), a reduced denominator is at most \(2^{5qQb}\), and a numerator is at most this denominator times \(H_q\). The displayed \(B_q\) includes spare bits for zero and equality at a power of two. Rational reduction cannot increase either length. A single elementary rational operation on operands with at most \(B_q\) bits uses integer products of at most \(2B_q\) bits and their sums of at most \(2B_q+1\) bits.

The old generic bound was \((4Q^2+1)2^{128Q^8 2^q}-1\). The new bound is strictly smaller for every \(q\geq3\): using \(b\leq q\), \(B_q\leq16Q^3\), whereas the old exponent alone exceeds \(128Q^8\), hence exceeds \(16Q^3\). This bounds bits of each coefficient, not the exponential expression size or the bit complexity of the whole construction.

Records and reproducibility

tools.risk_research.leading_constant_height computes the integer bounds without constructing the enormous old power. Its saved record also constructs the original finite expression with the original selected-work budget and reports the largest final coefficient bits. Those measurements are not measurements of intermediate coefficients; the theorem above covers intermediates. The final denominator-divisibility check is an implementation check, not a proof of the theorem.

Records bind the source and proof bytes, restore the index and work limit, and compare the complete typed JSON record. The mathematical index domain has no upper bound; operational budgets can reject an input without returning a partial record. Numerical intervals and finite-degree errors are outside this record’s scope.