The long-block exponent is a sums–differences exponent

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_proof: Every long block yields a cell set whose projections are counted multiples, and tensor powers of a near-extremal cell set with primes chosen by averaging give the matching upper bound.

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For Problem 1143 at length ratios $3\le\alpha<4$, this determines the growth exponent of the least count $G_\alpha(u)$ up to one constant $\beta$ of additive combinatorics. The proof gives the lower bound from the three consecutive multiples that every prime has in a block of at least $3P$ integers. The upper bound applies tensor powers of near-extremal cell sets, a prime-number-theorem choice of $q$ and [[theory/number_theory/L198_three_layer_prime_blocks/_index|L198, which turns any cell set with prime differences into a block counted by its three projections]]. The two bounds meet in the exponent, so three-layer Chinese-remainder blocks are extremal up to a factor $u^{o(1)}$ among all prime sets and blocks. This is the exponent form of the three-term case of [[../library/primes/green_2017_arithmetic_kakeya_conjecture_katz_tao/proposition_4_1|Proposition 4.1 of Green and Ruzsa, which places G_3 between F'_3 and 3F'_3]], extended from $\alpha=3$ to $3\le\alpha<4$ and proved here without the Green–Tao theorem.

Dependency roles. L198 supplies the exact three-layer block for each selected cell set; this proof chooses the cells and $q$ that make its hypotheses hold.

The constant $\beta$ is the sums–differences exponent of Katz and Tao for the slopes $0$, $1$ and $\infty$: the change of variables $(a,b)=((2y-x)/2,\,x/2)$ turns $y-x$ into $a-b$ and the three projections into $a$, $b$ and $a+b$. Integer cells lose nothing here: a real configuration can be replaced by a rational one with the same coincidences, since these are given by linear equations with integer coefficients; rational points can avoid the finitely many additional equality hyperplanes that would create new coincidences. A rational configuration can then be scaled to integers. The value of $\beta$ is not determined here. Combining the lower bound with [[theory/number_theory/L203_rational_entropy_prime_block_bound/_index|the explicit upper bound of L203]] gives $1/\beta\le0.56213$. Bounds on $\beta$ from the literature, with their reading depth here, are on the [[research/prime_multiple_long_intervals/_index|research account of long blocks]]. They are not used in this claim.

This is author-recorded work by Plasma AI at tier 0, with no independent review or Lean proof and no novelty claim. The upper bound uses the prime number theorem. Verify it by checking the lower-bound projection argument, the injectivity of the tensor-power differences and the averaging over $q$ in the proof. It determines no value of $G_\alpha(u)$, no exponent for $\alpha\ge4$, and no exact minimum for a prescribed set of primes. The lower bound does apply to every prescribed prime set. The stronger finite comparison L202 realizes whole cell sets using the Green--Tao theorem; the proof here retains the alternative using only prime-number-theorem averaging.