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+THE POISSON-DIRICHLET LAW FOR PRIME PREDECESSORS
+=================================================
+
+A Reader's Guide for Curious Young Mathematicians
+
+Adapted from the paper by OpenAI, September 24, 2026
+
+----------------------------------------------------------------------
+
+NOTE TO THE READER
+
+This document explains a theorem about prime numbers. The theorem
+itself needs nothing more than the ideas you have already met in
+school: primes, multiplication, and a little probability. The proof
+uses much more, so we split this document into two halves.
+
+In the first half (Parts I and II) we explain what the theorem says,
+why anyone cares, and where it sits in the history of number theory.
+
+In the second half (Parts III, IV, V) we give the proof in plain
+language, using as few formulas as possible, and we are honest about
+what is technical and why.
+
+You do not need to read the second half to enjoy the first half. If
+you want to know what mathematicians are doing at the frontier, the
+second half tells you. If you want the punchline, the first half is
+enough.
+
+----------------------------------------------------------------------
+
+PART I. THE QUESTION AND THE ANSWER
+
+----------------------------------------------------------------------
+
+Chapter 1. A SMALL STORY
+
+Pick a prime number, say p = 101. Now look at p - 1 = 100.
+
+100 factors into primes: 100 = 2 * 2 * 5 * 5.
+
+Those factors tell us something about the prime. Every time you pick
+a prime, subtracting 1 gives a number whose prime factors play a
+hidden game.
+
+ p = 17 p - 1 = 16 = 2 * 2 * 2 * 2 (a power of two)
+ p = 97 p - 1 = 96 = 2 * 2 * 2 * 2 * 2 * 3 (mostly twos)
+ p = 101 p - 1 = 100 = 2 * 2 * 5 * 5 (mixed)
+ p = 257 p - 1 = 256 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 (a big power)
+ p = 2521 p - 1 = 2520 = 2^3 * 3^2 * 5 * 7 (several pieces)
+
+The question this document answers is:
+
+ If you pick a random prime p between 3 and x, and let x grow
+ large, what does the typical p - 1 look like as a product of
+ primes?
+
+A more careful way to ask: what is the "typical" list of prime
+factors of p - 1? Do they tend to be small? Large? Do they have a
+"shape" that is stable as x grows?
+
+We will see that the answer is YES: there is a very specific
+stable shape, called the Poisson-Dirichlet distribution with
+parameter one. That is the main theorem.
+
+Chapter 2. HOW DO WE MEASURE THE SHAPE OF A FACTORIZATION?
+
+For a number like 2520, the factorization is
+
+ 2520 = 2 * 2 * 2 * 3 * 3 * 5 * 7,
+
+which contains seven prime factors: three 2s, two 3s, one 5, one 7.
+Different primes p give different lists. To say what is "typical"
+we need to compare these lists.
+
+The key idea is to look at logarithms. Write the prime factors of
+p - 1, with multiplicity, in decreasing order:
+
+ q_1 >= q_2 >= q_3 >= ... >= 1.
+
+For p = 2521, we have p - 1 = 2520, so the list is
+
+ q_1 = 7, q_2 = 5, q_3 = 3, q_4 = 3, q_5 = 2, q_6 = 2, q_7 = 2.
+
+For p = 97, the list is
+
+ q_1 = 3, q_2 = 2, q_3 = 2, q_4 = 2, q_5 = 2, q_6 = 2.
+
+For p = 257, the list is
+
+ q_1 = q_2 = ... = q_8 = 2.
+
+Now define, for each index j >= 1,
+
+ V_j(p) = log q_j(p) / log(p - 1).
+
+These V_j are real numbers between 0 and 1. They measure how much
+of the "logarithmic size" of p - 1 each prime factor accounts for.
+Because a product of numbers has logarithm equal to a sum of the
+logarithms, the V_j add up:
+
+ V_1 + V_2 + V_3 + ... = 1.
+
+So the V's partition the interval [0, 1] of total mass 1 into
+pieces of different sizes. The list
+
+ V_1 >= V_2 >= V_3 >= ...
+
+is the shape of the factorization of p - 1.
+
+For example, for p = 2521 and p - 1 = 2520:
+
+ log q_1 = log 7 ~ 1.946
+ log q_2 = log 5 ~ 1.609
+ log q_3 = log 3 ~ 1.099
+ log q_4 = log 3 ~ 1.099
+ log q_5 = log 2 ~ 0.693
+ log q_6 = log 2 ~ 0.693
+ log q_7 = log 2 ~ 0.693
+ log q_1 + ... + log q_7 = log 2520 ~ 7.832
+
+So V_1 ~ 1.946 / 7.832 ~ 0.248, meaning the largest prime factor
+(7) accounts for roughly a quarter of the total logarithmic size.
+The largest piece is quite small, and the factorization is spread
+out.
+
+For p = 257, on the other hand, V_1 = ... = V_8 = 1/8 = 0.125, and
+the factorization is very concentrated in equally-sized small pieces.
+
+We want to know: as x grows, what is the distribution of the shape
+(V_1, V_2, V_3, ...) if p is chosen uniformly among primes 3 <= p <= x?
+
+Chapter 3 tells us what that distribution is.
+
+
+Chapter 3. THE POISSON-DIRICHLET DISTRIBUTION
+
+We now describe the mystery shape. It was invented by probabilists
+in the twentieth century and has a very simple construction that
+is usually called "stick-breaking."
+
+Imagine a stick of total length 1. We are going to break it into
+pieces, one after another. Here is the algorithm.
+
+ Step 1. Pick a random number U_1 uniformly from (0, 1). That is,
+ U_1 is any real in (0, 1) with equal chance. Snap off a
+ piece of length B_1 = 1 - U_1. We have used up
+ length 1 - U_1. The remaining stick has length U_1.
+
+ Step 2. Pick another independent random number U_2 uniformly
+ from (0, 1). Snap off a piece of length
+ B_2 = U_1 (1 - U_2). We have used up U_1 (1 - U_2) out
+ of the remaining U_1. The remaining stick has length
+ U_1 U_2.
+
+ Step 3. Pick U_3, snap off B_3 = U_1 U_2 (1 - U_3). The
+ remaining stick has length U_1 U_2 U_3.
+
+ And so on.
+
+ General formula: B_j = (U_1 U_2 ... U_{j-1}) (1 - U_j).
+
+The infinite sequence B_1, B_2, B_3, ... has the property that the
+pieces sum to one:
+
+ B_1 + B_2 + B_3 + ... = 1 (always, with probability 1).
+
+Exercise. Verify the telescoping:
+
+ (1 - U_1) + U_1 (1 - U_2) + U_1 U_2 (1 - U_3) + ...
+ = 1 - U_1 U_2 U_3 U_4 ...
+
+As the product U_1 U_2 U_3 ... shrinks to zero (almost surely), the
+sum tends to 1. Good.
+
+The pieces B_j are NOT, in general, in decreasing order. The
+largest piece might be B_1, but sometimes B_2 is larger, or B_3,
+and so on. The Poisson-Dirichlet distribution with parameter one,
+written PD(1), is defined as the distribution of the DECREASING
+REARRANGEMENT
+
+ L_1 >= L_2 >= L_3 >= ...
+
+of the pieces B_1, B_2, B_3, ... That is: L_1 is the largest B_j,
+L_2 is the second largest B_j, and so on. Since the B_j add to 1,
+so do the L_j:
+
+ L_1 + L_2 + L_3 + ... = 1.
+
+Some facts about PD(1).
+
+ - The first piece B_1 is a uniform random variable, so on
+ average it is 1/2. Actual largest piece L_1 is a bit larger
+ than that, because it might not be B_1.
+
+ - The largest piece L_1 has median roughly 0.606. That is, about
+ half the time L_1 is bigger than 0.606, and about half the time
+ it is smaller. So the shape of a typical prime's factorization
+ is dominated by one big piece that accounts for more than
+ half of the total logarithmic size.
+
+ - The second piece L_2 is also substantial, typically of order a
+ few tenths.
+
+ - There are infinitely many small pieces. For any t > 0 there are
+ only finitely many pieces larger than t, but the number of
+ pieces larger than t scales roughly like 1/t as t shrinks. So
+ small pieces are numerous; big pieces are rare.
+
+ - The largest piece L_1 has a very specific distribution called
+ the "Dickman distribution," and we will meet it again in
+ Chapter 5 when we discuss ordinary integers.
+
+For us, the important thing is: PD(1) is a very specific random
+"shape" for splitting a total of 1 into pieces. It is not uniform:
+it has a few big pieces and many small pieces. And it is going to
+be the shape of the prime factors of p - 1.
+
+Chapter 4. THE MAIN THEOREM
+
+Now we can state the answer.
+
+ THEOREM (Main theorem of the paper).
+ Fix an integer k >= 1 and a bounded continuous function F on
+ the cube [0, 1]^k. Then
+
+ (1 / (pi(x) - 1)) * sum_{3 <= p <= x, p prime}
+ F(V_1(p), V_2(p), ..., V_k(p))
+
+ converges, as x tends to infinity through all real values, to
+
+ the expectation of F(L_1, L_2, ..., L_k),
+
+ where L_1, L_2, L_3, ... is the PD(1) rearrangement of the
+ stick-breaking pieces B_1, B_2, B_3, ....
+
+In plain English: for any finite window of the prime-factor "shape"
+of p - 1, the distribution of that window matches the PD(1) shape
+as x grows.
+
+Since F was arbitrary, this also tells us that, for every fixed k,
+
+ (V_1, V_2, ..., V_k) converges IN DISTRIBUTION to (L_1, ..., L_k).
+
+The normalization uses log(p - 1), not log x, but this makes no
+difference in the limit, since p is close to x in the prime average.
+The prime average is taken over ordinary counting, not weighted in
+any special way.
+
+This theorem strengthens several earlier results and resolves a
+conjecture. Chapter 5 tells that story.
+
+PART II. WHAT WAS KNOWN, AND WHY IT IS HARD
+
+----------------------------------------------------------------------
+
+Chapter 5. A SHORT HISTORY
+
+For ordinary integers (not primes), the story starts in the 1930s.
+
+ 5.1 Dickman (1930) and the largest prime factor of n
+
+Dickman studied: for a uniformly chosen integer n <= x, what is the
+distribution of the largest prime factor P+(n)?
+
+Write P+(n) <= x^{1/u} for a parameter u >= 1. Dickman showed:
+
+ #{1 <= n <= x : P+(n) <= x^{1/u}} ~ x * rho(u),
+
+where rho is the "Dickman function:"
+
+ rho(u) = 1 for 0 <= u <= 1
+ u rho'(u) + rho(u-1) = 0 for u > 1.
+
+For example, rho(2) = 1 - log 2 ~ 0.307. So roughly 31 percent of
+integers up to x have largest prime factor at most sqrt(x).
+
+Dickman's work started a large field called "smooth number theory."
+N. G. de Bruijn (1951) gave the modern asymptotic form of rho(u)
+that we still use today.
+
+ 5.2 Billingsley (1972): the joint law
+
+Billingsley asked about the joint distribution of the ordered prime
+factors of a random integer. He proved: as x -> infinity, the
+decreasing sequence of logarithms of large prime divisors of a
+random integer n <= x, rearranged and normalized by log x, converges
+in distribution to a certain limit. That limit is precisely the
+Poisson-Dirichlet distribution.
+
+ 5.3 Donnelly and Grimmett (1993): the clean version
+
+Donnelly and Grimmett gave a "size-biased" treatment of the
+problem, which handles multiplicities cleanly and leads directly to
+the stick-breaking construction of PD(1). This is the version we use.
+
+ 5.4 Arratia, Kochman and Miller (2014): factorial intensities
+
+Arratia, Kochman and Miller recast the result in terms of "joint
+intensities" of factor tuples, and gave a factorial-moment form
+that is the cleanest analytic version.
+
+Combined, these results for ordinary integers form a complete proof
+that for a uniformly chosen n <= x, the shape of the prime
+factorization of n converges to PD(1).
+
+ 5.5 Ford, Konyagin and Luca (2010): the conjecture for primes
+
+The problem for primes is different. Ford, Konyagin and Luca, in
+their 2010 paper "Prime chains and Pratt trees," formulated the
+conjecture:
+
+ For a uniformly chosen prime p <= x, the shape of the prime
+ factorization of p - 1 also converges to PD(1).
+
+Exactly the same answer as for ordinary integers, but now on
+primes. FKL knew this was much harder than the ordinary-integer
+case: for ordinary integers, subtracting 1 is not a thing (you
+just factor n), but for primes, subtracting 1 destroys the
+structure of the number in a way that is hard to control.
+
+ 5.6 Granville (2008): the smoothness conjecture for primes
+
+Granville, in his survey "Smooth numbers: computational number
+theory and beyond," stated the conjectured asymptotic:
+
+ #{p <= x : P+(p-1) <= x^{1/u}} ~ pi(x) * rho(u) for u >= 1.
+
+This is the "shifted-prime Dickman conjecture." It is a
+consequence of the full PD(1) theorem.
+
+ 5.7 Partial results before 2026
+
+Some partial results were known.
+
+ - Baker and Harman (1998) proved lower bounds of the form
+ x / (log x)^C for primes p <= x with p - 1 having no large
+ prime factors. Threshold x^{0.2961}.
+
+ - Lichtman (2022) improved this to threshold x^{0.2844} for
+ primes in (x, 2x].
+
+ - Ford (2025) and Gorodetsky (2026) proved strong results about
+ small prime divisors of shifted primes, using sieve methods.
+
+ - Bharadwaj and Rodgers (2026) proved PD(1) for shifted primes
+ under the Elliott-Halberstam conjecture (a very strong but
+ unproven distributional hypothesis about primes in arithmetic
+ progressions). Unconditionally, they got "level 1/2": the
+ moment formulas hold on the part of the simplex where the
+ coordinates sum to less than 1/2.
+
+ 5.8 2026: the full unconditional proof
+
+The OpenAI paper "The Poisson-Dirichlet law for prime predecessors"
+(September 24, 2026) proves the full PD(1) theorem unconditionally.
+This is the paper explained in this guide. The proof uses a new
+"dilation-graph" method and a "memory" identity that we will try to
+explain in Parts III and IV.
+
+
+Chapter 6. WHY IS THIS HARD FOR PRIMES?
+
+For ordinary integers, Dickman, Billingsley, Donnelly-Grimmett and
+Arratia-Kochman-Miller all worked. For primes, why should it be any
+harder?
+
+The reason is subtle but important. Let's see what happens if we try
+to naively adapt the ordinary-integer proof.
+
+ Counting trick for integers.
+
+Suppose we want to count integers n <= x whose prime factors include
+a specific set of "small" primes q_1, ..., q_r. That is easy: the
+answer is just floor(x / (q_1 q_2 ... q_r)), which is x / Q where
+Q = q_1 q_2 ... q_r.
+
+ Counting trick for primes.
+
+Now suppose we want to count primes p <= x such that p - 1 is divisible
+by q_1 q_2 ... q_r = Q. That means p is congruent to 1 modulo Q. To
+count such primes, we need the prime number theorem for arithmetic
+progressions:
+
+ #{p <= x : p ≡ 1 mod Q} = Li(x) / phi(Q) + error.
+
+For FIXED Q, or Q growing slowly like a power of log x, the error
+term is smaller than Li(x)/phi(Q) by any fixed power of log x.
+That is Siegel-Walfisz.
+
+But for the full PD(1) theorem, we need Q up to x^{1-eps}, and no
+unconditional prime-counting tool is sharp enough at that scale.
+The best we can do unconditionally is Bombieri-Vinogradov: on
+average over Q up to x^{1/2 - eps}, the error is small. Beyond that,
+it is conjectural (Elliott-Halberstam).
+
+ The gap.
+
+Suppose we try to bound the number of primes p <= x with p - 1
+having all prime factors <= x^{1/u}. For u > 1, this is exactly
+Granville's conjecture. The naive count from PNT-APs is
+
+ sum over Q | m for some m <= x, m has only small prime factors
+ of (Li(x) / phi(Q)) + (error terms).
+
+If we ignore the errors, the main-term sum equals Li(x) times
+rho(u), which is exactly the Dickman/Granville answer. So the naive
+calculation WOULD give the right answer if the error terms were
+uniformly tiny.
+
+But the error term is x / (log x)^A for each individual modulus Q.
+We have to sum over many (potentially exp(some power of log x) many)
+moduli Q. The errors then ADD UP and swamp the main term. That is
+the whole problem.
+
+So we need a way to control these errors SIMULTANEOUSLY over all
+relevant moduli, using cancellation rather than PNT-APs. That is
+exactly what the FKL conjecture and Granville's conjecture ask for,
+and it is why the problem remained open for so long.
+
+The rest of the document explains the OpenAI proof of this: how,
+despite the failure of uniform PNT-APs, there is enough structure
+(via "signed sums" and "dilation graphs") to make the error terms
+cancel. That cancellation is the big new idea of the paper.
+
+
+----------------------------------------------------------------------
+
+PART III. THE PROOF
+
+----------------------------------------------------------------------
+
+Chapter 7. THE PROOF IN ONE PAGE
+
+Here is how the proof goes, in one page.
+
+The key realization: trying to directly control the small-prime
+divisors of p - 1 via PNT-APs is doomed, as we saw in Chapter 6.
+We have to find some structure that lets the errors cancel.
+
+The paper's strategy has four stages. Stages 1 and 2 are the two
+analytic inputs. Stage 3 uses them to extract the statistic. Stage
+4 is a purely probabilistic argument.
+
+ Step 1. One-sided marked correlation.
+
+We prove a "signed-moment" identity for a very special sum: the
+sum of a certain "mark" times the indicator of a prime factor of
+p - 1 being in a certain range.
+
+The technical heart of Step 1 is called the "determinant estimate."
+We construct a weighted bipartite graph between pairs of integers
+(m, n) with mn - 1 divisible by a product of small primes, and we
+show that a signed sum over the edges of this graph is very small
+(like x divided by (log x)^D for any fixed D).
+
+The method is Cauchy-Schwarz plus analytic number theory:
+
+ (a) Cauchy-Schwarz turns the one-variable-at-a-time problem into
+ a two-variable-at-a-time problem.
+
+ (b) That becomes a problem about two vectors of small-prime
+ products (one on each side).
+
+ (c) A "determinant" relation t = bmn - ars appears. The
+ "major" and "minor" arcs around rationals h/k pick out the
+ major contribution.
+
+ (d) The "minor" contribution is bounded by an "exact memory
+ identity": each prime's "lifespan" (its run of activity) is
+ tracked, and a signed sum over all lifespans is bounded by
+ the "probability" of picking its residue line.
+
+The signed-moment approach has precedents in the literature
+(Helfgott-Radziwill, Pilatte), but the state in the OpenAI paper
+is richer: it keeps track of an "active prime list" AND a primitive
+lattice vector, at each prime p. Root averaging gives a uniform
+projective line, and the memory preserves that line across gaps
+between active runs.
+
+The output of Step 1 is Theorem 6.2 of the paper: the "one-sided
+marked Type II estimate."
+
+ Step 2. Two-sided marked correlation.
+
+We prove a related estimate where both sides carry marks. The
+technical heart here is the "dilation-graph" method from the
+companion paper (S), with some new endpoint estimates.
+
+The idea is to represent the correlation statistic as a matrix
+element of an operator on L^2 of sequences. We split the operator
+into a "raw" piece (no shifts) and two "comparison" pieces (with a
+shift). Each comparison is "morally multiplicative" with respect to
+the mark structure, and the total sum is bounded via a finite
+character expansion and a sparse-frequency analysis.
+
+The output of Step 2 is Theorem 6.1 of the paper: the "two-sided
+marked correlation."
+
+ Step 3. Extraction.
+
+Now we use the two correlation estimates to prove the interior
+prime statistic theorem:
+
+ sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
+ = o(x / log x),
+
+where f_x counts the number of ways a chosen set of large primes
+divides n and C_x is the average.
+
+The proof is a sieve argument with four sub-steps:
+
+ (a) Presieve: restrict x to p where p - 1 has no very small
+ prime factors, using the block sieve.
+
+ (b) Apply the one-sided estimate to replace composite values
+ in the sieve by "rough" values with the right density.
+
+ (c) Additional marks let the two-sided estimate bound the
+ resulting centered sum.
+
+ (d) A mean-square average over many disjoint "marking arrays"
+ (Chapter 14) removes the marks.
+
+The output: the prime average of any suitable factor statistic of
+p - 1 matches the expected value.
+
+ Step 4. The probabilistic finish.
+
+At this point we know:
+
+ For every fixed k and every bounded continuous F on [0, 1]^k,
+ as x -> infinity,
+
+ (1/pi(x)) * sum over primes p <= x of F(V_1(p), ..., V_k(p))
+ converges to the "factorial intensity limit" of the
+ expected joint count of the k primal balls.
+
+The last stage is to identify this limit as PD(1). The argument
+goes through "size-biased sampling":
+
+ (a) From the average of the factor statistics we can compute
+ the expected "joint intensity" of factor tuples. That is,
+ the expected counting measure of ordered k-tuples of factor
+ balls.
+
+ (b) Size-biased sampling turns those joint intensities into the
+ distribution of the largest, second largest, etc., via the
+ stick-breaking map.
+
+ (c) A total-mass-one argument shows no mass leaks to the boundary.
+ That is, the pieces really add up to 1.
+
+The output is exactly the PD(1) shape. That is the theorem.
+
+Parts IV, V go into the details of Steps 1, 2, 3, 4, and the tools
+they use.
+
+
+----------------------------------------------------------------------
+
+PART IV. THE DETAILS OF THE PROOF
+
+----------------------------------------------------------------------
+
+Chapter 8. MARKS: A WAY TO PROBE THE FACTORIZATION
+
+Recall from Chapter 6: we cannot directly count primes using the
+prime number theorem for arithmetic progressions at the scale we
+need. The paper gets around this by using a different strategy
+called "marks."
+
+A mark is a small extra piece of information about a factorization
+that we attach, which lets us split the problem into tractable
+parts. Think of a mark as a "colored sticker" that we put on
+specific small primes.
+
+ How marks work.
+
+Fix a small set of "bands": intervals of primes that all have size
+exp(L^{a_i}) for some fixed 0.1 < a_i < 0.2. For example, if
+x = 10^1000, then a band might be all primes between exp(100) and
+exp(200), or between exp(200) and exp(400), etc.
+
+For a positive integer n, define
+
+ omega_i(n) = the number of prime factors of n (with multiplicity)
+ that lie in band i.
+
+Define also the "inverse harmonic mass" of band i:
+
+ V_i = sum over primes p in band i of 1/p,
+
+which is approximately log 2, by a prime-counting estimate.
+
+The "mark weight" is
+
+ W(n) = product over i of [ q^{omega_i(n) - 1} * omega_i(n) / V_i ],
+
+where 0 < q < 1 is a fixed small damping factor.
+
+The factor omega_i(n) / V_i roughly counts the primes in band i.
+The factor q^{omega_i - 1} damps cases with very many primes. And
+the whole product is ZERO if some band has zero primes, i.e., if n
+has no prime factors in some band.
+
+So the weight W(n) "locks onto" the presence of one or more primes
+in each band. It is bounded (by a constant depending only on K and
+q), and it is what we will use to probe the structure of n.
+
+ Why marks help.
+
+The weight W(n) is a bounded function of n that behaves like a
+"local indicator" of band structure. It is "almost multiplicative":
+W(d*e) approximately factors as W(d)*W(e) up to the damping.
+
+With W in hand we can phrase our target statistic in terms of
+marked averages, and marked averages can be attacked by
+"correlation estimates" that we introduce next.
+
+ Two types of marks in the paper.
+
+The paper uses two closely related kinds of marks.
+
+ - "Fixed marking bands," as above, with K of them. These appear
+ in the one-sided marked Type II estimate (Chapter 9).
+
+ - "Active groups," which are up to O(log L) small groups of
+ primes scattered in ranges like [exp(L^.27), exp(L^.36)] and
+ [exp(L^.39), exp(L^.46)]. These appear in the two-sided
+ marked correlation (Chapter 13), where they are used because
+ they can be "flipped on and off" to give a family of tests
+ that we can average over.
+
+Both kinds of marks play the same role: they give us a "structured
+probe" of the values we're averaging, which cancels what we can't
+control directly.
+
+
+Chapter 9. THE ONE-SIDED MARKED ESTIMATE, IN WORDS
+
+The first analytic input is a "correlation estimate" for marked
+sums with a prime condition on one side.
+
+ The setup.
+
+Fix an accuracy parameter D_* (say, we want to save a factor of
+L^{-D_*} for some fixed D_*). Fix K "marking bands" as in Chapter 8,
+with 0.1 < a_1 < ... < a_K < 0.2, and 0 < q < 1.
+
+ - alpha_m = m^{iv} * ( 1_{m prime} - 1_{m W-rough} / (V(W) log m) )
+ with |v| <= L^C for a fixed C. Here "W-rough" means m has no
+ prime factor <= W = exp(L^{0.24}), and V(W) is the product over
+ primes p <= W of (1 - 1/p).
+
+ In words: alpha_m is a "centered prime indicator." It is the
+ same as 1_{m prime}, minus a smooth predicted density
+ (1/(V(W) log m)) that we subtract off.
+
+ - beta_n is supported on W-rough n, with |beta_n| <= L^C.
+
+ - F : Z -> C is a bounded function invariant under band prime
+ multiplication: F(ph) = F(h) for every prime p in a marking
+ band.
+
+ - W(h) is the mark weight from Chapter 8.
+
+ - X = H_m * H_n ~ x, the total size of the two-variable box.
+
+ The theorem (paper Theorem 6.2, one-sided marked Type II).
+
+For every fixed D_* > 0, if K is large enough (depending only on
+D_* and the fixed parameters, not on the band exponents a_i), then
+
+ | sum over m, n of alpha_m beta_n F(mn - 1) W(mn - 1) |
+ <= X L^{-D_*}.
+
+Here the sum is over the supports of alpha and beta. The bound is
+uniform in the chosen marking bands, in |v| <= L^C, and in the
+coefficient bounds on beta. The threshold for x may depend on the
+a_i and all fixed parameters, but the required lower bound on K
+does not.
+
+ What this says, in plain words.
+
+The variables m and n are constrained so that mn - 1 is divisible
+by the products that the mark weight W rewards. That is a strong
+arithmetic constraint on mn - 1.
+
+The theorem says: after subtracting off the "expected" contribution
+(the smooth proxy 1/(V(W) log m)), the m-side has almost no
+correlation with the rest of the sum. In other words, the smooth
+proxy is a very good approximation to the prime behavior, even
+after multiplying by the marked weight on mn - 1.
+
+ Why is this hard?
+
+Because one side of the sum carries a genuine prime indicator.
+The proposition is saying: the sum over primes m of a specific
+function, weighted by the marked statistics of mn - 1, essentially
+cancels against the smooth proxy.
+
+The reason this could fail (and would fail with few bands) is that
+F and W might correlate with the arithmetic of mn - 1. For example,
+if F were very sensitive to whether mn - 1 is prime, we'd get a bias.
+
+The "marks" W are the trick: they're built only from SMALL prime
+divisors of mn - 1, so they're "arithmetically cheap" to correlate
+with. After enough bands, the correlation with the genuine prime
+variable m is negligible.
+
+Chapter 10. THE DETERMINANT METHOD, PART 1: THE SHAPE OF THE ARGUMENT
+
+Here is how the one-sided estimate is proved, at a high level.
+
+ Step A. Reduce band structure to "choose one prime per band."
+
+The mark weight W(h) is built from the number of band primes
+dividing h. We first reduce W to a sum over choices of one prime
+per band, by expanding omega_i(h) as a sum over primes. What
+remains is a sum over m, n, and a selected product b_0 of primes.
+This is a standard expansion.
+
+ Step B. Cauchy-Schwarz.
+
+The m-variable in alpha is a genuine prime indicator. We use
+Cauchy-Schwarz to exchange it for a "two-sided" sum over pairs
+(m, r) of test variables. The resulting expression is:
+
+ | sum_m ... |^2 <= (something) * Q_Y
+
+where Q_Y is a nonnegative expanded sum over (a, b) products of
+marking-band primes, m, r, n, s, and a multiplicity h.
+
+ Step C. The determinant identity.
+
+The constraint that both (m, n) pairs satisfy the mark condition
+turns into a condition on a DETERMINANT. Two products (a, b) from
+marking bands and two "first coordinates" (m, r) with their "second
+coordinates" (n, s) get coupled through:
+
+ t = bmn - ars.
+
+The exact "diagonal" condition for the marked structure to survive is
+
+ t = b - a.
+
+This t = b - a relation is the "determinant identity" that gives
+the method its name. It says: a specific 2x2 determinant of
+variables has a specific value.
+
+ Step D. Split major and minor.
+
+The exact determinant identity t = b - a is too sharp to work
+with directly. We replace it by a "smeared" version: instead of
+imposing "t equals b - a exactly," we impose "t is close to b - a
+modulo 1/k for many small denominators k."
+
+This is the classical major/minor arcs decomposition from analytic
+number theory. The major arcs (small denominators k) give the MAIN
+term, and the minor arcs (large denominators) give the correction.
+
+ Step E. Control both terms.
+
+The minor term is small by Fourier analysis alone: the smeared
+version has a very small "off-diagonal" contribution by standard
+exponential-sum bounds.
+
+The major term is where the interesting arithmetic lives: it is
+here that the marking bands interact with the prime condition on
+m. The "memory identity" (Chapter 11) gives an exact accounting
+of this interaction. The "major-arc evaluation" (Chapter 12) then
+evaluates it.
+
+Chapters 11 and 12 spell out these two pieces.
+
+
+Chapter 11. THE DETERMINANT METHOD, PART 2: THE SIGNED MOMENT AND THE MEMORY
+
+We now explain how the "minor" part (complement of major arcs) of
+the determinant estimate is bounded. This is the technical heart
+of Step 1.
+
+The main tool is a "signed moment bound" on a certain operator.
+We construct a Hilbert space of "physical states" and prove that a
+signed moment of the operator is exponentially small. Then a
+standard Hilbert-space argument bounds the pairing we care about.
+
+ The physical state space.
+
+A "physical state" is a pair P = (P_1, P_2) of primitive integers,
+together with an ordered list of M primes from each marking band,
+all dividing P_1.
+
+The measure on this space is counting measure in P, times
+prod_i V_i^{-M} (one factor V_i^{-M} per band). This makes the
+state space a discrete Hilbert space.
+
+Concretely: P_1 is a "first coordinate," P_2 is a "second
+coordinate," and the "prime lists" record which primes of each
+marking band divide P_1.
+
+ The row operation.
+
+We define an operator T that moves from one physical state to
+another, in a specific way:
+
+ - Copy the first J slots of each band's list from source to
+ target (so the "shared" primes are the same on both sides).
+ - The target's last slot is a fresh prime from the same band.
+ - The source's last slot product is "b" (product of one
+ fresh prime per band from source).
+ - The target's last slot product is "a" (product of one
+ fresh prime per band from target).
+ - The shift equation is: b(mn - 1) = a(rs - 1) for source pair
+ (m, n) and target pair (r, s).
+
+The multiplier is a specific combination of:
+
+ - The "det/damping": q^{(omega(P_1) - KM)/2 + (omega(Q_1) - KM)/2}.
+ - A "sign" from the determinant.
+ - The "arc integral" or "delta" from the smeared major-arc
+ version of Chapter 10.
+
+ "Good positions" and the G projection.
+
+Two technical conditions ("good" and "separated") are imposed on
+positions P. These ensure that the lattice spanned by the "prime
+lists" is not too thin, and that the "phase" of the omitted label
+is discriminating.
+
+The projection G onto good states is essentially guaranteed by
+the geometry of the lattice.
+
+ The moment.
+
+The key ingredient in the proof is the moment bound:
+
+ (1/UV) * sum over primitive P of < u_P, (A A*)^R u_P >
+ <= L^{-E_0 * N}
+
+where A = G S T S G is the restricted, symmetrized operator, R is
+about L^{0.5}, N = 2R, and u_P is a "delta state" at P. Here U and
+V are the first and second coordinate box lengths, respectively.
+
+This says: after R successive applications of A (and its adjoint),
+the "return probability" from any state is exponentially small.
+
+ Why this is true.
+
+Because each step of A is "highly unlikely to return" in a specific
+sense. The determinant condition t = b - a, together with the
+good-state conditions, force the trajectory to progress through a
+specific geometric structure that quickly escapes from any given
+starting state.
+
+The "memory identity" (below) is the technical machinery that lets
+us track this quickly-escaping structure exactly. Without it, the
+analysis becomes a tangle of estimates for "shared primes."
+
+ From the moment to the pairing bound.
+
+Given the moment bound, a standard Hilbert-space argument
+(essentially Cauchy-Schwarz with an intermediate spectral bound)
+gives the pairing bound:
+
+ |<f, A g>| <= (some constant) * UV * L^{-E_0 + O_C(1)}
+
+for suitable test vectors f, g. That is the desired answer, up to a
+log-power loss factor.
+
+ The memory identity.
+
+Here is the key identity that makes the moment bound work.
+
+A prime p can be active at several "runs" of the trajectory. If
+the same p appears in two CONSECUTIVE active runs, the physical
+constraint "shared slot" forces the same projective line. But
+between runs, the prime is "pending": it is not active, but we
+remember its line from before.
+
+The memory object "stores" each pending prime, along with its
+projective line, so that when it becomes active again we can
+retrieve the same line. The accounting is:
+
+ - When a prime becomes "pending" (leaves the active run), we store
+ it in the memory, paying a factor V_i^{-1}.
+ - When a prime becomes "active from memory," we retrieve it, also
+ paying V_i^{-1}.
+ - For each "pending hit" (the line at a pending prime coincides
+ with the current projective line), we pay a factor q_j (or
+ sqrt(q) at the endpoints j = 0, N).
+ - A "ghost" (memory birth or death that creates or destroys a
+ pending prime not tied to an active run) has a similar cost.
+
+The upshot: each lifespan of each prime is charged exactly once,
+and the total cost is tracked by an adjoint operator on the memory
+space.
+
+This is where the q_j and eta_j from the setup come in. The paper
+proves the operations are adjoints (in the memory Hilbert space),
+and the moment bound reduces to standard operator-norm bounds on
+the memory subspace.
+
+For our purposes here, the important takeaway is: the memory
+identity is an EXACT accounting of the shared-prime structure, and
+it is what makes the moment bound possible.
+
+Figure "lifespan" in the paper gives a picture of a single
+prime's trajectory through active and pending states. The picture
+is worth a thousand words:
+
+ A "lifespan" of a prime p is the segment of the trajectory
+ between its birth (when it first becomes active) and its death
+ (when it disappears from the marking lists). During its life,
+ p can be active, pending, or in "ghost" states at the ends.
+ Each intermediate visit costs a factor q_j (or hits its
+ line).
+
+Chapter 12. THE DETERMINANT METHOD, PART 3: THE MAJOR ARCS
+
+We now explain how the "major arcs" are evaluated. This is
+essentially paper section 5.
+
+Recall: we need to bound
+
+ Q_Y^{maj} = sum over a, b, m, n, r, s of
+ alpha_m alpha_r-bar beta_n beta_s-bar H_M(bmn - ars; a, b)
+
+where H_M is the "smeared diagonal" (arc integral) from Chapter
+10, X = H_m * H_n ~ x, and Y is a dyadic scale with
+log Y in [L^{0.1}, L^{0.2}].
+
+We want: |Q_Y^{maj}| <= X Y L^{-D} for any fixed D.
+
+ Why naive bounds fail.
+
+The arc integral only gives a bounded multiplier, so the naive
+bound on the sum is way too big. We need CANCELLATION in the
+alpha factors to make the sum small.
+
+The challenge: alpha has cancellation (prime vs smooth proxy),
+but the sum over ars pairs gives the "m, r" variables a lot of
+freedom. A naive bound on |alpha| loses all cancellation.
+
+ The strategy.
+
+Split the sum into three factors:
+
+ - M_alpha(t): the "long coefficient" alpha, twisted by t.
+ This should be small (like L^{-A} for every A) uniformly in t.
+ - N_beta(t): the "long coefficient" beta, twisted by t.
+ This can be as large as L^C, but has controlled mean square.
+ - B_lambda(t): the "small prime" factor, coming from the
+ product of marking band primes. This is bounded by 1 in
+ absolute value but can give strong information about
+ frequencies.
+
+Then Mellin/Fourier separation writes the arc integral as a
+product of these three factors, and chooses an optimal splitting
+between "small prime averaged," "M_alpha is small," and
+"N_beta has controlled mean-square."
+
+ The long coefficient is small.
+
+Paper Lemma 5.1 ("The long coefficient") says:
+
+ |M_alpha(t)| <= L^{-A} uniformly for |t| <= 2 X L^B,
+
+for any fixed A, B. This is a "long coefficient" cancellation
+bound.
+
+The proof uses:
+
+ - The Vinogradov mean-value theorem (Chapter 18).
+ - The truncated-rough approximation to alpha (replacing the
+ prime indicator by a Bonferroni-style divisor sum).
+ - The prime number theorem for arithmetic progressions mod q
+ for q up to L^C (Siegel-Walfisz).
+ - The long-prime logarithmic-phase estimate (Chapter 18) for
+ the qualitative cancellation at high frequencies.
+
+ The sparse-frequency analysis.
+
+If B_lambda(t) is LARGE for many t (which it can be), we use a
+sparse-frequency argument: divide the frequency axis into unit
+intervals; the intervals where |B_lambda(t)| > L^{-A_s} form a
+sparse set (of size exp(O(L^{0.91}))); on the sparse set we use
+the mean-square bound for N_beta.
+
+ Paper Lemma 5.2 ("A small prime factor and sparse frequencies").
+
+For P_s(t) = P^{-1} sum of band-prime contributions, and
+N_beta(t) = H_n^{-1} sum of beta coefficients, the "sparse set"
+of times where |P_s(t)| > L^{-A_s} has measure exp(O(L^{0.91})),
+and on it the supremum of |N_beta(t)|^2 sums to at most L^{2C+1}.
+
+ Combining.
+
+Mellin/Fourier separation writes the major-arc sum as an integral
+of F_lambda(t/X, s') * Q(t) * Q'(s'-t), where Q = B M_alpha N_beta.
+Then:
+
+ - Split into |t| <= X L^{B_1} (main) and tail.
+ - For |t| <= L^{B_2} (low frequencies), use the "constant-term"
+ cancellation that comes from the alpha-prime-rough identity.
+ - For L^{B_2} <= |t| <= X L^{B_1} (high frequencies), use the
+ fact that M_alpha is small by Lemma 5.1.
+
+ - On the "sparse" set (where B_lambda is large), use N_beta
+ mean-square to control it.
+ - Off the "sparse" set, use M_alpha * B_lambda * N_beta
+ uniform smallness.
+
+The result: |Q_Y^{maj}| <= X Y L^{-D} for any fixed D. This
+finishes the one-sided estimate.
+
+
+Chapter 13. THE TWO-SIDED MARKED CORRELATION
+
+The second analytic input is a "two-sided correlation" where BOTH
+sides of the equation carry marks. This is paper Theorem 6.1.
+
+ What it says.
+
+We use a family of "active groups" of primes. These are:
+
+ - S, a family of "small" groups in [exp(L^.27), exp(L^.36)].
+ - B, a family of "big" groups in [exp(L^.39), exp(L^.46)].
+ - Sizes bounded: c_0 log L <= |S|, |B| <= C_0 log L.
+
+The point of having O(log L) groups is that we can "flip them on
+and off" - choose any subset - and still control the total number
+of subsets (which is exp(O(log L)) = L^{O(1)}).
+
+For each choice of active groups, we can define a weight
+W_t^b(n) that records which primes of the groups divide n, with
+damping q_0 and harmonic normalization V_g^{-t_g}.
+
+The "centered core" F_x is:
+
+ F_x(n) = W(n) * ( sum_{l in long-prime products} 1_{l | n} - C_x )
+
+where l runs over products of "long primes" from intervals
+[x^{a_1}, x^{b_1}], ..., [x^{a_d}, x^{b_d}] with sum b_i <= 1 - eps,
+and C_x is the expected density (of order O_eps(1)).
+
+In words: F_x(n) is centered around zero for a "random" n. It picks
+out how many "long prime factors" n has, relative to the average.
+
+ The theorem.
+
+Let G(n) = G(n_*) where n_* is n with active primes removed, with
+|G(n)| <= L^C * tau(n_*)^C. Then for any fixed smooth
+compactly-supported Psi and any fixed A > 0:
+
+ sum over w >= 1 of Psi(w/x) / w * F_x(w) G(w+1) W_1(w) W_1(w+1)
+ <= L^{-A}.
+
+The estimate is uniform over the active groups and slot intervals
+satisfying the stated bounds.
+
+ Why is this hard?
+
+The statistic involves a centered shift indicator (an indicator of
+the long-prime-product structure), plus marks on both w and w+1.
+The shift from w to w+1 can be correlated with the marks, and we
+need that correlation to cancel.
+
+ The strategy.
+
+The proof uses the "dilation-graph" method from the companion
+paper S, plus some new endpoint estimates specific to this paper.
+
+The idea: represent the correlation statistic as a matrix element
+of an operator on L^2 of sequences. Split the operator into:
+
+ - The "raw" piece: no shift.
+ - Two "comparison" pieces: shift by a product of small primes.
+
+Each comparison piece is "morally multiplicative" with respect to
+the mark structure. This lets us bound it via a finite character
+expansion: expand in characters modulo the small primes, using
+the fact that each character is multiplicative.
+
+Then the raw piece is what we want, and the comparison pieces are
+bounded by the L^2 operator bound.
+
+ What we need from S.
+
+From the companion paper S, we use:
+
+ - The "general graph transference" theorem: an operator bound
+ in an "ideal Hilbert space" transfers to a bound in the
+ "physical Hilbert space" in which the marks live.
+ - The "residual ideal family" theorem: the specific signed
+ combination of patterns we use satisfies the operator bound
+ in the ideal Hilbert space.
+ - The character expansion and cell decomposition inputs.
+
+ What is new in this paper.
+
+The endpoint estimates: verifying that our specific F_x and G
+satisfy the endpoint hypotheses of the S-graphs, including their
+growth and centeredness. These are proved in paper section 6 and
+are the main new work for Step 2.
+
+Chapter 18 has the specific lemma statements for the analytic
+estimates needed to make the endpoint bounds work.
+
+
+Chapter 14. EXTRACTION, PART 1: SETUP AND PRESIEVE
+
+Now we combine the two correlation estimates to prove the "interior
+prime statistic" theorem (paper Theorem 7.1).
+
+ The statement.
+
+Fix d >= 1 and eps > 0. Fix intervals I_1, ..., I_d of primes
+with lower endpoints at least x^eps and upper endpoints multiplying
+to at most x^{1 - eps}.
+
+For a positive integer u, define
+
+ f_x(u) = sum over l_i in I_i (i=1..d) with l_1,...,l_d distinct
+ of 1_{l_1 l_2 ... l_d | u},
+
+and
+
+ C_x = sum over the same l-tuples of 1 / (l_1 l_2 ... l_d).
+
+Then C_x is O_eps(1) by the prime number theorem.
+
+The theorem: for every nonnegative smooth Phi : (0, infinity) -> R
+with compact support, as x goes to infinity,
+
+ sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
+ = o(x / log x).
+
+ What f_x means.
+
+For a specific tuple of intervals, f_x(u) counts how many "ways" u
+is divisible by "one prime from each slot." The C_x is the expected
+count.
+
+The theorem says: on average over primes p, the count f_x(p-1)
+matches the expected count C_x, up to o(1) in the average weighted
+by x/log x.
+
+ Why this implies PD(1).
+
+The full PD(1) convergence will follow from a probabilistic
+argument that converts "interior average matches PD(1) expected
+value" into "distribution matches PD(1)." That argument is Step
+4 (Chapter 17).
+
+ The proof structure.
+
+We prove the interior theorem via:
+
+ 1. Presieve: bound the contribution from primes p where p-1 has
+ a "very small" prime factor. This reduces to a sum over p
+ where p-1 has NO small prime factors.
+
+ 2. Replace composite values in the sieve by rough values: this
+ uses the One-Sided Estimate (Chapter 9).
+
+ 3. Bound the resulting centered sum: this uses the Two-Sided
+ Estimate (Chapter 13).
+
+ 4. Remove the marks by averaging over many arrays: mean-square
+ argument (Chapter 16).
+
+ Presieve details.
+
+We fix a large even integer h (the sieve depth). Let
+
+ gamma = c_0 / (4h + 3), z = x^gamma,
+
+where c_0 is a small constant (chosen below eps/2).
+
+Apply the Block Sieve (Chapter 18) to the "bad conditions"
+r' | u+1 for primes r' <= z. The sieve density is g(r') = 1/r',
+and the required density bounds hold by a prime-counting estimate.
+
+The block sieve bounds the "total weight" of u for which u+1 has
+any prime factor <= z. Because the block sieve is a standard
+small-sieve argument, this "sieved out" weight is very small -
+like a factor of L^{-A} for any fixed A.
+
+ The presift formula.
+
+More precisely, applying the Block Sieve to Phi(u/x) W(u) f_x(u)
+and to Phi(u/x) W(u) separately and subtracting, one gets:
+
+ | sum over u with P^-(u+1) > z of
+ Phi(u/x) W(u) (f_x(u) - C_x) |
+ <= (e^{-h}/gamma) * x * m_W / L * (1 + o(1)),
+
+where m_W is the model mean of W (which is bounded above and
+below by positive constants). This is the "presift formula." The
+factor e^{-h} is the sieve error; it goes to zero as h -> infinity.
+
+ What's left.
+
+After presieving, u+1 has no prime factors <= z.
+
+If u+1 is prime, we're directly in the "prime contributions" we
+want. If u+1 is COMPOSITE, then u+1 = p_1 p_2 ... p_j with all
+p_i > z and j >= 2.
+
+We need to show: composite u+1 contribute only o(x / log x).
+
+ Strategy for composite u+1.
+
+For each j >= 2, we want to show:
+
+ sum over p_1, ..., p_j > z of
+ Phi((p_1...p_j - 1)/x) * W(p_1...p_j - 1) *
+ (f_x(p_1...p_j - 1) - C_x) = o(x / L).
+
+We do this in three stages:
+
+ Stage A. Replace each prime variable p_i by a "rough proxy."
+
+Fix the "proxy coefficient":
+
+ a(m) = 1_{m > z and m W-rough} / (V(W) log m).
+
+This is a smooth approximation to the "prime indicator" 1_{p_i}.
+The One-Sided Estimate (Chapter 9) with the choice F(h) =
+f_x(h) - C_x (which is band-prime-invariant because f_x only
+depends on divisor structure) and beta_n = the appropriate
+coefficient lets us replace p_i by a(m) up to o(x/L). Chaining
+through all j variables gives:
+
+ Stage B. Apply the Two-Sided Estimate.
+
+After the replacement, we have a sum of the form:
+
+ sum over m_1, ..., m_j (all W-rough) of
+ Phi((m_1...m_j - 1)/x) *
+ W(m_1...m_j - 1) *
+ (f_x(m_1...m_j - 1) - C_x) *
+ prod_i a(m_i).
+
+Now we can choose the active groups of Theorem 6.1 (Chapter 13)
+to match the marking weight W that appears in this sum. The
+"active one-mark weight" (from the allocation of the primes of v
+= m_1...m_j among the j factors) exactly reproduces the "mark
+weight at u+1" in the two-sided estimate.
+
+Summing over all j of the two-sided estimate, we get o(x/L).
+Summing over j from 2 to 2/gamma gives o(x/L).
+
+ Stage C. The allocation identity.
+
+The key combinatorial identity is:
+
+ sum over ways to assign the active primes of v to the j slots
+ = j^{omega_P(v)} * (q_0/j)^{omega_P(v) - s_P}
+ = j^{s_P} * q_0^{omega_P(v) - s_P},
+
+where omega_P(v) counts active prime hits (all active primes are
+> W by definition) and s_P = |P| is the number of active groups.
+
+This identity is what turns the "allocation over factorizations
+of v" (which appears naturally in the m_1...m_j decomposition
+of u+1) into the "one-mark weight W_1(u+1)" of the two-sided
+theorem. That is the central combinatorial step of the extraction
+argument.
+
+Chapter 15 fills in some of the technical details of stages A-C,
+and Chapter 16 covers the final "averaging" step (removing the
+marks W from the interior statistic).
+
+
+Chapter 15. EXTRACTION, PART 2: THE COIN-PARTITION AND SUCCESSES
+
+Getting from the "proxy-replacement sum" of Chapter 14 to a genuine
+application of the Two-Sided Estimate (Theorem 6.1) is more delicate
+than it looks. There is a "coin toss" that must succeed often enough.
+
+ The candidate groups.
+
+The "active groups" in Theorem 6.1 have to be chosen. They come from:
+
+ - Small groups P_g in [exp(L^.27), exp(L^.36)].
+ - Big groups P_g in [exp(L^.39), exp(L^.46)].
+
+We choose them as ALL primes in consecutive "dyadic bands." That is,
+for small groups: [exp(y), exp(2y)] for y = L^.27, 2 L^.27, ...,
+2^8 L^.27, giving about 8 or 9 groups. Similarly for big groups.
+
+ The coin toss.
+
+Fix a tuple (m_1, ..., m_j) with product v and u = v - 1. For each
+candidate group g, define a "success probability":
+
+ s_g(u, v) = c_1 * q_0^{omega_g(u) - 1} * omega_g(u) / V_g
+ * (q_0/j)^{omega_g(v) - 1} * omega_g(v) / V_g,
+
+where c_1 is a small enough constant (depending on j, q_0, and the
+bounds on V_g).
+
+The probability s_g(u, v) is nonzero ONLY if both u and v have at
+least one prime in group g. So a "success" in group g requires:
+
+ - u = v - 1 has a prime in band g ("u-hit"),
+ - v has a prime in band g ("v-hit").
+
+ A second-moment argument on successes.
+
+Consider each group g as an independent "coin." Toss it with
+probability s_g. Let P be the set of groups where the coin came up
+heads (succeeded).
+
+We show: with positive probability (in fact, with 1 - o(1)
+probability, uniformly in the tuple), at least c_2 log |S|
+successes occur in the S-family and at least c_2 log |B| successes
+in the B-family, for some small c_2 > 0.
+
+The key facts needed:
+
+ - Each s_g has expectation bounded below by a constant (in the
+ model where u and v are "random rough tuples").
+ - Different groups have almost no correlation in their
+ u-hit / v-hit structure.
+
+This gives the "many successes with high probability" result via
+Chebyshev's inequality. The technical input is a uniform
+residue-class equidistribution of the u-hit / v-hit structure
+across groups.
+
+ Why successes are important.
+
+The successes are what make the "allocation identity" of Chapter
+14 Stage C work. Because:
+
+ - A success in group g means both u and v have a prime in g,
+ with the "damping" q_0.
+
+ - In the ALLOCATION over factorizations of v into m_1 ... m_j,
+ that prime of v appears in exactly one factor m_i.
+
+ So each v-hit is "allocated" to one of the j slots. Grouping
+ by "which prime goes to which slot" produces the factor
+ j^{omega_P(v)} that appears in the allocation identity.
+
+ - After conditioning on the successes, the mark weight at v
+ is that j^{omega_P(v)} times the base mark weight W.
+
+ The coin-partition expansion.
+
+We expand:
+
+ 1 = sum over subsets P of C of prod_{g in P} s_g * prod_{g not in P} (1 - s_g).
+
+Restrict to |P cap S| >= c_2 log |S| and |P cap B| >= c_2 log |B|.
+This has mass >= 1 - o(1) by the Chebyshev argument.
+
+Then for a fixed P, the sum over (m_1, ..., m_j) with prod s_g
+has the form:
+
+ (constant) * sum over u of (Phi(u/x) W(u) (f_x(u) - C_x))
+ * W_1(u) * W_1(u+1) * G_P(u+1),
+
+where:
+
+ - W_1 is the "one-mark active weight" (as in Chapter 13).
+ - G_P(u+1) is a bounded "core" that sums over factorizations of
+ u+1 into j rough factors with the right mark structure.
+
+This is EXACTLY the form that the Two-Sided Estimate (Theorem 6.1)
+handles!
+
+So Chapter 14's "proxy-replacement sum" is bounded as:
+
+ <= (sum over P of retained mass) * (L^{-A} from Theorem 6.1)
+ + (mass of discarded P) * (bounded coefficient)
+
+ = 0 * L^{-A} + o(1) * (bounded)
+ = o(x/L), as desired.
+
+ Exceptions and their bounds.
+
+Two things have to be checked carefully:
+
+ - The discarded (few successes) cases have mass o(1) uniformly
+ over the tuple. The total tuple mass (weighted by prod a(m_i))
+ is bounded, so the resulting contribution is o(x/L).
+
+ - If some candidate band prime p appears SQUARED in v = m_1...m_j,
+ the "allocation" over factorizations gives an extra factor j
+ of overcounting. We remove these cases by a bound: they involve
+ v being divisible by p^2 for p >= exp(L^.27). The number of such
+ v in the range is at most x * sum(p^{-2}) << x * exp(-c L^.27),
+ so negligible.
+
+ Note on the CORE.
+
+The G_P(u+1) core is bounded by a fixed log-power times a fixed
+divisor power. It behaves like a "smooth distribution" of the
+marked structure of u+1. That's exactly the "G(n_*) bounded by
+L^C tau(n_*)^C" from Theorem 6.1.
+
+Chapter 16 covers the final averaging step: removing the marks W
+from the interior statistic.
+
+
+Chapter 16. EXTRACTION, PART 3: AVERAGING TO REMOVE THE MARKS
+
+We are now almost done with extraction. The previous two chapters
+gave us a "marked" version of the interior statistic. We want the
+UNMARKED version:
+
+ (L/x) * | sum over primes p of Phi((p-1)/x) * (f_x(p-1) - C_x) |
+ <= small.
+
+To remove the mark W we use a "variance reduction" argument: if we
+average over MANY independent marking arrays, the average weight
+(W_a/m_a) should be close to 1.
+
+ Setup.
+
+Fix b independent marking arrays. Each array a in 1..b gets its
+own weight W_a and its own model mean m_a. Define the squared
+deviation:
+
+ Z_b(u) = ( (1/b) sum_{a=1}^b W_a(u) / m_a - 1 )^2.
+
+Z_b(u) is nonnegative and equals zero when the averaged weight
+equals 1.
+
+ Model variance.
+
+In the "independent divisor model" (Chapter 7 of the paper), the
+mark weights W_a / m_a are independent across a, with mean 1 and
+bounded second moment. Hence:
+
+ E_model[Z_b] = (1/b^2) * sum_a Var(W_a / m_a) <= C_K / b.
+
+So the model variance of Z_b is O(1/b).
+
+ Applying the sieve to Z_b.
+
+Apply the Block Sieve at fixed depth (h' = 2, z' = x^{c_0/20}) to
+the weight Phi(u/x) Z_b(u). The sieve keeps only primes u+1 > z'.
+
+The result (a variant of the presieve formula):
+
+ limsup_{x -> infinity} (L/x) *
+ sum over primes p of Phi((p-1)/x) Z_b(p-1) <= C_K / b.
+
+So for large b, the "deviation from the mean" is small on primes.
+
+ Cauchy-Schwarz.
+
+We compare the AVERAGED marked sum to the UNMARKED sum:
+
+ | avg_a sum_p Phi((p-1)/x) * (W_a(p-1)/m_a) * (f_x(p-1) - C_x)
+ - sum_p Phi((p-1)/x) * (f_x(p-1) - C_x) |
+ = | sum_p Phi((p-1)/x) * (avg_a(W_a(p-1)/m_a) - 1)
+ * (f_x(p-1) - C_x) |
+ <= sqrt( sum_p Phi((p-1)/x) * Z_b(p-1) )
+ * sqrt( sum_p Phi((p-1)/x) * |f_x(p-1) - C_x|^2 ),
+
+where the last inequality is Cauchy-Schwarz in p.
+
+The first factor is <= (C_K/b)^{1/2} * (x/L)^{1/2} by the Z_b
+bound. The second factor is <= C * (x/L)^{1/2} by the boundedness
+of f_x - C_x and the prime number theorem.
+
+So the product is:
+
+ <= C_K^{1/2} * C * (x/L) * b^{-1/2} * (1 + o(1)).
+
+Dividing by x/L, we get:
+
+ | avg_a (marked sums) - (unmarked sum) | / (x/L)
+ <= C * C_K^{1/2} * b^{-1/2}.
+
+ Finishing the interior theorem.
+
+Combining all the errors:
+
+ | sum_p Phi((p-1)/x) * (f_x(p-1) - C_x) | / (x/L)
+ <= (e^{-h}/gamma) [presieve error]
+ + C_1 * (C_K/b)^{1/2} [averaging error]
+ + o(1) [all other errors].
+
+Take h -> infinity first (with b, K, marking arrays all fixed),
+then b -> infinity. Both errors go to zero, and the o(1) term
+also does. This proves Theorem 7.1 (interior prime statistic
+theorem).
+
+The order of the limits is important: all fixed parameters (h,
+K, b, marking arrays) are chosen BEFORE taking x -> infinity, but
+h and b can be taken to infinity after the real x-limit.
+
+Chapter 17. THE PROBABILISTIC FINISH: FROM INTERIOR TO FULL LAW
+
+We now have:
+
+ sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
+ = o(x / log x)
+
+for any smooth Phi with compact support and any set of intervals
+I_1, ..., I_d with lower endpoints >= x^eps and upper endpoints
+multiplying to <= x^{1 - eps}.
+
+The remaining task: show that this implies the convergence of the
+prime-factor shape to PD(1).
+
+ The plan.
+
+We proceed in six steps:
+
+ 1. Define the joint intensity mu_{x,d} of ordered d-tuples of
+ factor balls.
+
+ 2. Convert the interior statistic into convergence of mu_{x,d}
+ to a reference measure on the open simplex.
+
+ 3. Size-biased sample: sample factor balls with probability
+ proportional to their mass. This turns the joint intensity
+ into the stick-breaking distribution.
+
+ 4. Show that the resulting probability density has total mass 1
+ on the open simplex (no mass escapes to the boundary).
+
+ 5. Parameterize by the "A-coordinates" via a triangular map to
+ identify the density with the PD(1) stick-breaking law.
+
+ 6. Sort and pad to get convergence of the DECREASING
+ rearrangement.
+
+ Step 1: Joint intensity.
+
+For a prime p, the prime factors of p - 1, WITH multiplicity, form
+a multiset of "labelled balls." A ball corresponding to a prime q
+has mass t_q = log q / log(p - 1). The balls add up to 1.
+
+Define the "joint intensity" of ordered d-tuples of distinct balls:
+
+ mu_{x,d}(A_1 x ... x A_d) = (1/(pi(x) - 1)) *
+ sum over primes p <= x of
+ sum over ordered d-tuples of DISTINCT balls b_1, ..., b_d
+ of 1_{(t(b_1), ..., t(b_d)) in A_1 x ... x A_d}.
+
+In words: mu_{x,d} is the expected counting measure of ordered
+d-tuples of distinct factor balls, mapped to their masses.
+
+ Step 2: Convergence of the intensity.
+
+Suppose we want the "probability" that (t_1, ..., t_d) lies in a
+rectangle R = (a_1, b_1] x ... x (a_d, b_d] with 0 < a_i < b_i and
+sum b_i < 1. Set I_i = (x^{a_i}, x^{b_i}]. Then:
+
+ f_x(p-1) = sum over l_i in I_i of 1_{l_1 ... l_d | p-1}
+ = number of ordered d-tuples of distinct factor-primes
+ of p-1 with log-masses in the range (a_i, b_i].
+
+So "mu_{x,d}(R) * (pi(x) - 1)" equals "sum over primes p <= x of
+f_x(p-1)" in a small band. That is exactly what the interior
+theorem controls.
+
+By the interior theorem:
+
+ sum over primes p <= x of (f_x(p-1) - C_x) = o(x/log x).
+
+Where C_x = sum over the same l-tuples of 1/(l_1 ... l_d). By the
+prime number theorem and partial summation:
+
+ C_x ~ prod_i log(b_i / a_i).
+
+So:
+
+ mu_{x,d}(R) * (pi(x) - 1)
+ = sum over primes p <= x of f_x(p-1)
+ = pi(x) * prod_i log(b_i/a_i) + o(pi(x)).
+
+Therefore:
+
+ mu_{x,d}(R) -> prod_i log(b_i / a_i)
+ = int_R prod_i (dt_i / t_i).
+
+So the joint intensity converges to the reference measure:
+
+ dt_1/t_1 * dt_2/t_2 * ... * dt_d/t_d
+
+on the open simplex D_d = {(t_1, ..., t_d): t_i > 0, sum t_i < 1}.
+
+ Step 3: Size-biased sampling.
+
+Given the balls with masses t_1, ..., t_N (adding to 1), sample
+them successively WITHOUT replacement: at each step, choose a
+remaining ball with probability equal to its mass divided by the
+remaining total mass.
+
+Let T_{x,i} be the mass selected at step i. Define T_{x,i} = 0
+for i > N.
+
+The probability that the first d samples are the specific ordered
+tuple (t_1, ..., t_d) is:
+
+ w_d(t_1, ..., t_d) = prod_{i=1}^d t_i / (1 - t_1 - ... - t_{i-1}).
+
+This is the "size-bias weight" of the tuple.
+
+ Step 4: The joint probability density.
+
+Combining the intensity limit with the size-bias weight gives:
+
+ nu_d(A) = int_A w_d(t) d(mu_{x,d}) -> int_A w_d(t) prod_i (dt_i/t_i).
+
+So the limit measure on D_d has density:
+
+ h_d(t_1, ..., t_d) = w_d(t) / (t_1 t_2 ... t_d)
+ = 1 / prod_i (1 - t_1 - ... - t_{i-1}).
+
+Explicitly: h_d(t_1, ..., t_d) = prod_{i=1}^d
+ 1 / (1 - t_1 - ... - t_{i-1}).
+
+ Step 5: Total mass and the A-coordinates.
+
+Direct computation gives:
+
+ int_{D_d} h_d(t) dt_1 ... dt_d = 1.
+
+So h_d is a PROBABILITY DENSITY on the open simplex.
+
+The critical computation: under the triangular change of variables
+
+ A_i = t_i / (1 - t_1 - ... - t_{i-1}),
+ t_i = A_i * prod_{r < i} (1 - A_r),
+
+the density h_d transforms to:
+
+ dA_1 * dA_2 * ... * dA_d,
+
+which is the UNIFORM measure on (0, 1)^d. And (0, 1)^d has volume 1.
+So the total mass is 1, and this confirms "no mass escapes to the
+boundary."
+
+ Step 6: Stick-breaking and sorting.
+
+Set U_i = 1 - A_i. Then (U_1, ..., U_d) are iid uniform on (0, 1),
+and:
+
+ t_i = (1 - U_i) * prod_{r < i} U_r for i = 1..d.
+
+Compare to Chapter 3: the stick-breaking construction B_j =
+(prod_{r<j} U_r)(1-U_j) is EXACTLY the same formula, applied with
+independent uniform U_j.
+
+So the limit measure on D_d (from the "size-biased sampling" step)
+equals the law of the first d stick-breaking pieces (B_1, ..., B_d).
+
+ From stick-breaking to PD(1).
+
+We want the DECREASING REARRANGEMENT, not the natural ordering
+B_1, B_2, B_3, ...
+
+Let R_d = 1 - sum_{i=1}^d B_i = prod_{i=1}^d U_i. Then
+E[R_d] = 2^{-d} -> 0. So the remaining mass beyond the first d
+sticks is small on average, as d grows.
+
+For a finite or summable sequence with total mass 1, sorting the
+first d coordinates gives order statistics S_{d,1} >= S_{d,2} >=
+... >= S_{d,d}. The difference |V_i - S_{d,i}| is bounded above
+by the unmatched mass R_d (a standard rearrangement inequality).
+
+So:
+
+ | Average over primes of F(V_1(p), ..., V_k(p))
+ - Average over primes of F(sorted first d terms of p-1) |
+ <= omega_F(eta) + 2 ||F||_inf * E[R_d] / eta,
+
+where omega_F is the modulus of continuity of F.
+
+Take x -> infinity (interior convergence), then d -> infinity (with
+the R_d bound above), then eta -> 0 (continuity of F). We get:
+
+ (1/pi(x)) * sum over primes p <= x of F(V_1(p), ..., V_k(p))
+ converges to E[F(L_1, ..., L_k)]
+
+for every fixed bounded continuous F. This is the main theorem.
+
+ All real x.
+
+The same argument applies to any real-valued x parameter. Attach
+the prime 2 with a small correction (measured by "Pi(x) - 1" vs
+"Pi(x)") to fix the exact count. Done.
+
+The theorem holds.
+
+
+----------------------------------------------------------------------
+
+PART V. THE ANALYTIC TOOLBOX
+
+----------------------------------------------------------------------
+
+Chapter 18. THE ANALYTIC TOOLBOX
+
+The proof uses several analytic estimates as black boxes. We
+collect them here for reference. The full statements are in
+paper sections 2 and 9.
+
+ 18.1 Prime estimates (paper Lemma 2.1).
+
+Assume the prime number theorem with error smaller than any fixed
+negative power of the logarithm. Also Mertens-type estimates:
+
+ sum_{p <= y} 1/p ~ log log y + c_M,
+ prod_{p <= y} (1 - 1/p) ~ exp(-gamma_E) / log y.
+
+And Siegel-Walfisz: for fixed A, C > 0, uniformly for r <= (log y)^C
+and (a, r) = 1,
+
+ #{p <= y : p = a mod r} = Li(y)/phi(r) + O(y / (log y)^A).
+
+All standard, from the PNT and its refinements.
+
+ 18.2 Divisor moments (paper Lemma 2.2).
+
+For every fixed nonnegative integer k, there is C_k > 0 such that:
+
+ sum_{n <= Y} tau(n)^k << Y (log(2Y))^{C_k},
+ sum_{n <= Y} tau(n)^k / n << (log(2Y))^{C_k}.
+
+Also main-value inequalities for Dirichlet polynomials. For
+P(t) = sum_{n <= Y} c_n n^{it}:
+
+ int_I |P(t)|^2 dt << (T + Y log(2Y)) sum |c_n|^2,
+ sum_{t in T} |P(t)|^2 << (T + 1 + Y log(2Y)) (log(2Y))^2
+ sum |c_n|^2,
+
+for I of length T and T subset of I with mutual spacing >= 1.
+
+These are used everywhere.
+
+ 18.3 Rough counts (paper Lemma 2.3).
+
+Fix a "smoothness cutoff" W = exp(L^{0.24}), where L = log x. For
+a W-rough integer H in [x^delta, x^C] with delta > 0:
+
+ #{n in I : P^-(n) > W} = V(W) |I| + O(H L^{-A}),
+
+for any A > 0 and any interval I subset of [H, 2H]. There are also
+progression and character versions:
+
+ #{n in I : n = b mod d, P^-(n) > W} = (V(W)/d) |I| + O(H L^{-A}/d)
+
+for d <= exp(L^a) with all prime divisors > W, and
+
+ sum_{n in I, P^-(n) > W} chi(n) = 1_{chi principal} V(W) |I|
+ + O(H L^{-A})
+
+for chi mod q <= L^C.
+
+These are proved by Bonferroni approximations to the rough indicator.
+
+ 18.4 The long prime polynomial (paper Lemma 2.4).
+
+Fix 0 < tau < eta < 1 and C, A > 0. There is B_0 = B_0(tau, eta,
+C, A) such that for N in [x^tau/2, x^eta], q <= L^C, and |t| in
+[L^{B_0}, x^2]:
+
+ |sum_{p in I} chi(p) p^{-1 + it}| <<_{tau,eta,C,A} L^{-A}
+
+for every Dirichlet character chi mod q and every interval
+I subset of [N, 2N].
+
+Proved in Appendix 9 (paper section 9). The proof uses:
+
+ - A "smooth cutoff" convolution to smooth the interval.
+ - Mellin inversion to a contour integral.
+ - A zero-free strip from the logarithmic-phase estimate.
+ - Standard PNT-AP for low frequencies.
+
+ 18.5 The logarithmic-phase estimate (paper Lemma 2.5).
+
+Fix C > 0. There is an absolute constant C_3 > 0 such that for
+sufficiently large x, for
+
+ exp(L/(log L)^2) <= N <= 2 x^5,
+ q <= L^C,
+ exp(L/(2(log L)^2)) <= |t| <= 4 x^3,
+
+and for every residue class a mod q and interval I subset of
+[N, 2N]:
+
+ |sum_{n in I, n = a mod q} n^{it}| << (N/q) * exp(-L/(log L)^{C_3}).
+
+The proof uses the Vinogradov mean-value theorem with growing
+degree. After Taylor-expanding logarithm to degree k ~ L/(log L)^2,
+the phase becomes a polynomial, and the Vinogradov theorem bounds
+the moments.
+
+ 18.6 Vinogradov mean-value theorem (paper Lemma 9.2).
+
+There are absolute constants C_1, C_2 > 0 such that for every
+integer k >= 2, some integer s with k <= s <= C_1 k^4 satisfies:
+
+ J_{s,k}(M) <= exp(C_1 k^{C_2}) * M^{2s - K + 1/100}
+
+where K = k(k+1)/2 and J_{s,k}(M) counts solutions of the
+Vinogradov system:
+
+ u_1^j + ... + u_s^j = v_1^j + ... + v_s^j (1 <= j <= k),
+ 1 <= u_i, v_i <= M.
+
+The proof uses the classical Vinogradov mean-value method in
+Linnik's p-adic form. Modern proofs using efficient congruencing
+(Wooley) give much stronger bounds, but the classical statement
+suffices for the logarithmic-phase estimate.
+
+ 18.7 The block sieve (paper Lemma 2.6, from S's Lemma 2.9).
+
+Given a family of "bad conditions" at primes p <= z with density
+g(p) = 1/p, and a weight W, define:
+
+ X g(d) + r(d) = the total weight on which all bad conditions
+ at primes dividing d hold.
+
+Then the "good" weight (avoiding all bad conditions) is:
+
+ X prod_{p<=z} (1 - g(p)) * (1 + O(e^{-h}))
+ + O( sum_{d <= z^{4h+2}, d squarefree} |r(d)| ).
+
+This is essentially the Erdos-Renyi sieve with two Bonferroni
+polynomials (upper and lower) that differ in the "tail" of
+degree > h. The h-error gives the O(e^{-h}) factor.
+
+ 18.8 What we don't cover here.
+
+Two important bookkeeping lemmas from the paper that we do not
+fully reproduce:
+
+ - The "summed progression remainder" (paper Lemma 7.2): a
+ progression-sum bound like x * C_x * M_x(H) with error
+ x L^{-A}.
+ - The "truncated divisor model" (paper Lemma 8.4): a weight
+ H_marks can be approximated by a "divisor model" H_0 with
+ H_0(n) = sum_{d' | n} A_{d'}, d' <= exp(O(L^{.47})),
+ sum |A_{d'}| / d' <= L^{C_8}.
+
+Both are "bookkeeping" lemmas that essentially say "the smooth
+part behaves as expected." Their proofs are in the paper's
+sections 7 and 8 and are largely routine, but they are nonempty
+work.
+
+
+Chapter 19. REFERENCES AND FURTHER READING
+
+ 19.1 About prime factor distributions.
+
+ - Karl Dickman, "On the frequency of numbers containing prime
+ factors of a certain relative magnitude," Arkiv for Matematik,
+ Astronomi och Fysik 22A (1930), 1-14. The original work on
+ smooth numbers.
+
+ - N. G. de Bruijn, "On the number of positive integers <= x and
+ free of prime factors > y," Proc. Kon. Ned. Akad. Wetensch. 54
+ (1951), 50-60. Modern asymptotics for the Dickman function.
+
+ - Patrick Billingsley, "On the distribution of large prime
+ divisors," Periodica Mathematica Hungarica 2 (1972), 283-289.
+ The joint law for ordinary integers.
+
+ - Peter Donnelly and Geoffrey Grimmett, "On the asymptotic
+ distribution of large prime factors," J. London Math. Soc. 47
+ (1993), 395-404. The size-biased / stick-breaking approach.
+
+ - Richard Arratia, Fred Kochman, Victor S. Miller, "Extensions
+ of Billingsley's theorem via multi-intensities,"
+ arxiv:1401.1555 (2014). The factorial-intensity form.
+
+ 19.2 About shifted primes and the FKL conjecture.
+
+ - Kevin Ford, Sergei V. Konyagin, Florian Luca, "Prime chains
+ and Pratt trees," GAFA 20 (2010), 1231-1258. Contains the
+ original conjecture about shifted primes.
+
+ - Andrew Granville, "Smooth numbers: computational number theory
+ and beyond," in Algorithmic Number Theory (MSRI Publ. 44),
+ 2008, 267-323. Contains the shifted-Dickman conjecture.
+
+ - R. C. Baker, Glyn Harman, "Shifted primes without large prime
+ factors," Acta Arith. 83 (1998), 331-361. Early partial
+ results.
+
+ - Jared Duker Lichtman, "Primes in arithmetic progressions to
+ large moduli, and shifted primes without large prime factors,"
+ arxiv:2211.09641 (2022). Improved partial results.
+
+ - Kevin Ford, "Poisson approximation of prime divisors of
+ shifted primes," IMRN 2025, rnaf079. Small-prime divisors
+ of shifted primes.
+
+ - Ofir Gorodetsky, "A Kubilius model for sieve-theoretic
+ sequences," Analysis Math. (2026). Small-prime models.
+
+ - Abhishek Bharadwaj, Brad Rodgers, "Large prime factors of
+ well-distributed sequences," Canadian Math. Bull. (2026).
+ PD(1) for shifted primes under Elliott-Halberstam; level 1/2
+ unconditionally.
+
+ 19.3 About the analytic tools.
+
+ - H. L. Montgomery and R.C. Vaughan, "Hilbert's inequality,"
+ J. London Math. Soc. 8 (1974), 73-82. Mean-value theorems.
+
+ - Trevor D. Wooley, "Vinogradov's mean value theorem via
+ efficient congruencing," Annals 175 (2012), 1575-1627.
+
+ - Kannan Soundararajan, "Moments of the Riemann zeta function,"
+ Annals 170 (2009), 981-993. Factorial prime polynomial
+ coefficient bounds.
+
+ - Kevin Ford, "Vinogradov's integral and bounds for the Riemann
+ zeta function," Proc. London Math. Soc. 85 (2002), 565-633.
+
+ - Harald Andres Helfgott and Maksym Radziwill, "Expansion,
+ divisibility and parity," arxiv:2103.06853 (2021). Precursor
+ of the signed-moment method.
+
+ - Cedric Pilatte, "Improved bounds for the two-point logarithmic
+ Chowla conjecture," arxiv:2310.19357 (2026). More signed-moment
+ development.
+
+ - Kaisa Matomaki and Maksym Radziwill, "Multiplicative functions
+ in short intervals," arxiv:1501.04585 (2017). Technique for
+ sparse frequencies.
+
+ 19.4 The companion paper.
+
+ - OpenAI, "Weighted dilation graphs, smooth shifted primes and
+ totient fibers," September 24, 2026. The companion paper S,
+ which supplies the graph-theoretic apparatus for Step 2.
+
+ 19.5 About the present paper.
+
+ - OpenAI, "The Poisson-Dirichlet law for prime predecessors,"
+ September 24, 2026. The primary source.
+
+ 19.6 Other related reading.
+
+ - W. R. Alford, Andrew Granville, Carl Pomerance, "There are
+ infinitely many Carmichael numbers," Annals of Math. 139
+ (1994), 703-722.
+
+ - Paul Erdos, "On the normal number of prime factors of p-1 and
+ some related problems concerning Euler's phi-function,"
+ Quart. J. Math. 6 (1935), 205-213.
+
+ - Carl Pomerance, "Popular values of Euler's function,"
+ Mathematika 27 (1980), 84-89.
+
+----------------------------------------------------------------------
+
+A CLOSING NOTE
+
+You have just read a plain-English guide to a research paper that
+is itself about 100 dense pages of mathematics. The guide is not
+the paper; it omits many epsilons, constants, and "routine"
+arguments. But the LOGICAL structure is complete: every step in
+the proof is at least named, and the main ideas are described.
+
+If you want to read the actual paper, the LaTeX source is
+available in the same directory as this file. The paper is
+structured in nine sections that roughly correspond to this
+guide's chapters, as follows.
+
+ Paper section This guide
+ -------------------------------- ----------------------------------
+ 1. Introduction Chapters 1-6
+ 2. Conventions / analytic inputs Chapter 18
+ 3. Determinant estimate Chapters 9-12 (setup / proof)
+ 4. Memory Chapter 11 (subsection)
+ 5. Major arcs Chapter 12 (subsection)
+ 6. Two-sided correlations Chapter 13
+ 7. Extraction Chapters 14-16
+ 8. Poisson-Dirichlet limit Chapter 17
+ 9. Long prime appendix Chapter 18
+
+Two figures in the paper are worth looking at side by side with
+this guide:
+
+ - figures/lifespan.tex shows a schematic "lifespan" of a prime
+ through active, pending, and ghost states. Chapter 11 explains
+ the physical meaning.
+
+ - figures/proof-map.tex shows the dependency graph of the whole
+ proof. Chapter 7 tries to convey the same picture in words.
+
+A final thought.
+
+The theorem says: the prime factors of a random prime's
+predecessor follow the same universal shape as the prime factors
+of a random integer, but with a very different proof. The same
+answer, reached by entirely different methods. In the end, the
+"shape" of a factorization doesn't care about whether the number
+came from an ordinary integer or from a predecessor of a prime.
+That is the beautiful fact at the heart of this paper.
+
+If the reader remembers nothing else from this document, they
+should remember: p - 1 factors most of the time into ONE big
+piece (log-size > half of the total) and many tiny pieces, and
+the shape of that split, when you sort it from big to small, is
+ALWAYS the same distribution, called Poisson-Dirichlet with
+parameter 1.
+
+End of document.
+
+----------------------------------------------------------------------
+
+A note about the author of the guide. This file was written by an
+AI assistant (halp) as a plain-English companion to the original
+LaTeX paper. Errors introduced in the guide are the assistant's
+own; the mathematical content is due to OpenAI, September 2026.
+
+----------------------------------------------------------------------
+
+(End of file.)