I wanted a family of rational approximations to that I could actually write down.

Start with . Then

That’s just expanding and using . Also .

Fix a positive rational and expand around it. Truncating at gives an explicit rational

Write in lowest terms, . For even the error is positive:

The usual plan would be: look at the integer linear forms and hope they go to zero. They don’t. They go to infinity.

For any fixed positive rational ,

So you never get small integer forms this way, even after reducing completely. That’s a real obstruction for this family, not “the unreduced denominator looked big.”

Denominators first. Write coprime, and ignore the finite set of primes dividing . Any leftover prime can only divide a when . Mod , the only surviving summand is , and you get . So divides the reduced . Prime number theorem then gives .

Error next.

Let

Then along even . That rate is always at least , and is optimal. Combine the two steps and you get .

What this does not cover: mixing truncations, letting depend on , other families. It also doesn’t say anything about whether is actually irrational.

I did write an exact enclosure at so I could check signs without ever evaluating as a float. The integrals can be pulled out of a generating product (the coefficient) using only exact rationals. Checks at and match the predicted valuations and give . The theorem doesn’t need those; they’re just to make sure I didn’t make something up.

Different idea. Rational functions with a shared pole set, even :

After reflection and decay you only keep and a rational. The hope was that canceling the and coefficients across different would slow the denominators down.

I took cross products of the nuisance coefficient triples and stripped content, and also looked for integer relations after making each four-coefficient row primitive. Two grids, both at : one with for , and one with next to multiples of so I wasn’t stuck with a parity restriction.

Neither looks like it’s decaying after full normalization. At there are a few forms below 1. Every tested form at is bigger than 1 in absolute value, that’s exact rationals, not floats. At the smallest sizes I saw were about and on the two grids. That’s just scale; the comparison with 1 is exact.

Finite failures don’t kill the whole family. Other parameters, other linear combinations: still open.

The obstruction above does not apply once you mix centers. Let and . Take

The coefficient in the error is still 1. Fermat then gives for large primes in , so the reduced valuation of in is at most 6 instead of exactly 7. That’s a real improvement. Upper bound of 6, not a claim it’s always 6.

Simplest example: . Exact tests at : valuation 6 for most of those primes, 5 for some. All four primitive errors still bigger than 1.

Then Lagrange at nodes clustered near the fastest single-center , canceling more moments. Extra moment conditions don’t buy extra prime savings, Fermat only eats the first layer. Actual valuations in the tests stay in , mostly 6:

All twenty of those extra normalized errors still exceed 1, certified with a 4000-term exact Euler enclosure (no floating ). Largest reduced denominator in the tests was 2543 bits, so the enclosure is still big.

Still no family of combinations that I’ve proved gives nonzero integer forms going to zero. Finite failures aren’t a universal obstruction, and I can’t apply the single-center theorem to these.

Irrationality is still unproved. This time I can prove positivity and quadratic decay for an infinite family of rational-coefficient polynomials in . Clearing denominators and stripping all coefficient content still overpowers that decay in the tests.

First, the derivative idea from the accelerated approximations. Put , with mass . For , the error for is

For even , normalized derivatives give

The empty denominator is 1. These entries form a positive Gram matrix, but changing the polynomial basis removes the extra freedom. Write and use . Both changes, from these polynomials to powers of and then to powers of , have determinant 1. So

For rational , this is times the ordinary Gram determinant with weight . That rational factor disappears in the primitive integer polynomial, even if and vary with . At , the determinant is exactly independent of . Extra derivatives don’t help.

I tested 65 weighted Gram cases at : constant weight, repeated roots at , and shifted Chebyshev polynomials, using polynomial degrees and . Their determinants are affine in because its coefficient matrix has rank one. Only one size-one case is below 1 after primitive normalization; every tested case at exceeds 1. Another 18 exact checks confirmed the basis identity at , , .

Next idea: take two copies of the measure. Its moments are

At , these equal . For the squared difference , define

For , the quadratic coefficient cancels because . Each of those moments is affine in , but their coefficient matrix can have higher rank. Meanwhile . For example, . Form

Degree at most for , and at most for , where only the entry can be quadratic. No other zeta values enter.

Nonvanishing holds for the whole family. Let be the pushforward of under . The matrix is the moment Gram matrix for . At a nonzero coefficient vector its quadratic form is

Every open subinterval of has positive mass: its preimage contains an open rectangle where both source densities are positive. A nonzero polynomial cannot vanish on an interval. That proves positive definiteness and for every , independently of numerics or any assumption about irrationality.

There is a uniform decay bound too. Use the monic polynomials

Here , so on . The basis change has determinant 1, and because . Hadamard’s inequality gives

Now if we write each nonzero rational polynomial as , with integer coefficients and . Divide by the gcd of all coefficients:

This preserves the sign and removes the entire polynomial’s content. A missed common coefficient factor cannot explain the tested failures. I scanned product moments , sum moments , and squared differences at both offsets, with positive source weights at . The weighted source moments stay affine in . Product and sum determinants have degree at most because their quadratic coefficient matrices have rank one.

48 cases, twelve each at . Six size-one cases are below 1; every tested case at exceeds 1 after normalization. For the unweighted source at , the natural logarithms divided by are:

The first two columns add to the third. These are finite results, not asymptotic bounds. The first primitive value is about , despite the small positive raw determinant. I also factored the primitive convolution polynomials, evaluated every irreducible factor with an exact interval, and multiplied the factors back to check the saved polynomials. A finite factor below 1 makes no asymptotic claim.

The precise missing estimate: for one fixed , it would suffice to prove

Combined with the real bound, this would give eventually for some . If with , then would be positive integers tending to zero, a contradiction. That arithmetic estimate is unproved, and the finite normalizers don’t support it. Another family could change the balance, but i haven't yet looked into it.