Sparse interpolation preserving almost all prime positions

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_proof: Proof of L228. Sparse positive corrections to halved gaps are matched with later gaps that absorb equal negative corrections. A prime-pair sieve bound makes the affected position intervals have density zero; a greedy expansion controls the value of the binary series.

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This is Plasma AI's author-recorded interpolation theorem. The proof is complete but has not been independently reviewed. Its external inputs are the prime number theorem and the classical prime-pair upper bound stated at the beginning of McNew's proof of Theorem 4.2, p. 10 of the preprint identified on the source card. That displayed bound was checked against the page image; its underlying Brun-sieve proof is an external premise, not reconstructed here. The proof derives the small-gap density estimate it uses from this precise bound.

For Problem 251, the result strengthens the comparison-sequence obstruction in the critical-length notes: the comparison can retain the actual value of the nth prime at density one, not only its growth and short-block statistics. Taking $q_n=2^{K_n}$, with $K_n$ nondecreasing and $2^{K_n}=o(\log n)$, also retains every residue pattern of halved gaps of length at most $K_n$ at each start $n$. The proof sketch explains which prime-specific estimate remains missing.

The result does not settle Problem 251. The interpolating sequences are increasing integer sequences, and a nontrivial one cannot consist entirely of primes: any increasing prime sequence starting at $2$ that skips a prime has its nth term strictly above $p_n$ at every later index. Agreement with $p_n$ at density one excludes that possibility.

The sublogarithmic condition has a precise role in this interpolation statement. If instead $q_n\ge c\log n$ eventually for some $c>0$, the divisibility $q_n\mid(h'_n-h_n)$ and $h'_n-h_n=o(\log n)$ force all sufficiently late gap changes to vanish. Agreement with $p_n$ at density one then forces all sufficiently late position changes to vanish too. The attainable differences of series values are consequently dyadic rationals and cannot fill an interval. This does not exclude any one specified rational value. Even without the small-gap-change requirement, L229 gives a stronger interval obstruction: at these logarithmic moduli all values from density-one agreement with the prime positions form a set of Hausdorff dimension zero. Independently of congruences, L230 limits how sparse the common exceptional set can be: an interval construction cannot have $|D\cap[1,N]|=o(N/\log\log N)$. The quantitative clause of L228 attains $O(N/\log\log N)$ whenever $q_n=O((\log n)^\alpha)$ for some fixed $\alpha<1$, including a fixed modulus. Thus the achievable big-oh bound cannot be replaced by little-oh for interval interpolation.

To check the claim, follow the proof's small-gap estimate, the choice of paired changes, the density-zero argument and the greedy expansion. The argument needs no numerical search or finite certificate. Its mathematical obligations are closed at author-recorded standing; independent verification remains absent.