$$c(a)+c(b)\ \ge\ \alpha\log a+\alpha\log b\ =\ \alpha\log(ab)\ =\ \alpha\log n .$$
Both families are bounded below by $\alpha\log n$, hence so is their minimum. $\square$
〔Grade T1. The entire content of the induction is two elementary facts: $\alpha\log\frac54<1$, and additivity in the sense of (2.2).〕
〔Grade C check: no violation for $n\le2^{17}$; within that range equality holds exactly at $4^1,\ldots,4^8$. This is a computational sanity check; for the complete characterization of the equality set see Theorem 74.A′.〕
2.1.3 Exact values and the liminf
Corollary 74.A.1. For every integer $k\ge1$,
$$c(4^k)=5k,\qquad \rho_{\mathcal D}(4^k)=5k-2,$$
and
$$\liminf_{n\to\infty}\frac{\rho_{\mathcal D}(n)}{\log n}=\frac{5}{\log4}. \tag{2.4}$$
Proof. Upper bound: repeated use of (2.2) with $a=4$ gives $c(4^k)\le k\,c(4)=5k$. Lower bound: Theorem 74.A gives $c(4^k)\ge\alpha\log4^k=5k$. The two agree.
For (2.4): by Theorem 74.A, for every $n\ge2$,
$$\frac{\rho_{\mathcal D}(n)}{\log n}=\frac{c(n)-2}{\log n}\ \ge\ \alpha-\frac{2}{\log n},$$
so the liminf is at least $\alpha$; and along $n=4^k$,
$$\frac{\rho_{\mathcal D}(4^k)}{\log4^k}=\frac{5k-2}{k\log4}\longrightarrow\alpha ,$$
so it is at most $\alpha$. $\square$
〔The restriction $k\ge1$ is not removable: $c(4^0)=c(1)=2\ne0$.〕
2.1.3a Equality rigidity
Theorem 74.A′. For $n\ge1$: $\ c(n)=\alpha\log n\iff n=4^k$ for some $k\ge1$.
Proof. ($\Leftarrow$) Corollary 74.A.1.
($\Rightarrow$) Strong induction. For $n=1,2,3$ the inequality is strict ($2>0$, $3>2.500$, $4>3.962$); at $n=4$ equality holds. Let $n\ge5$ with equality. The successor candidate is strictly above the bound: $c(n-1)+1\ge\alpha\log(n-1)+1>\alpha\log n$, strictness coming from $1>\alpha\log\frac{n}{n-1}$, whose maximum over $n\ge5$ is $0.8048$. So equality cannot come from the successor candidate. Hence $n=ab$ with $a,b\ge2$ and $c(a)+c(b)=\alpha\log a+\alpha\log b$; by Theorem 74.A both differences $c(a)-\alpha\log a$ and $c(b)-\alpha\log b$ are nonnegative and sum to zero, hence each vanishes. By hypothesis $a=4^r$ and $b=4^s$ with $r,s\ge1$, so $n=4^{r+s}$. $\square$
〔Grade T1. Grade C check: for $n\le2^{20}$ the equality set is exactly $4^1,\ldots,4^{10}$.〕
2.1.4 The capacity constant
Corollary 74.A.2. For integers $K\ge2$ set
$$A(K):=\max\{\,n\ge1:\ c(n)\le K\,\}.$$
The maximum exists: the set is nonempty (it contains $n=1$, since $c(1)=2\le K$) and bounded by Theorem 74.A ($c(n)\le K$ forces $\log n\le K/\alpha$). Then for $K,L\ge2$
$$A(K+L)\ \ge\ A(K)\,A(L), \tag{2.5a}$$
and
$$\gamma:=\lim_{K\to\infty}\frac{\log A(K)}{K}=\sup_{K\ge2}\frac{\log A(K)}{K}=\frac{\log4}{5}=\frac1\alpha . \tag{2.6}$$
Proof. For (2.5a) write $a=A(K)$, $b=A(L)$. If $a,b\ge2$ then $c(ab)\le c(a)+c(b)\le K+L$ by (2.2), so $ab\le A(K+L)$. If $a=1$ then $A(K)A(L)=A(L)\le A(K+L)$ since $A$ is nondecreasing; similarly for $b=1$.
For (2.6): by (2.5a), $K\mapsto\log A(K)$ is superadditive for $K\ge2$, and Fekete's lemma (in its eventually superadditive form, since (2.5a) is stated only for $K,L\ge2$) gives existence of the limit and its equality with the supremum. Upper bound: Theorem 74.A gives $\log A(K)\le K/\alpha$, so the supremum is at most $1/\alpha$. Lower bound: by Corollary 74.A.1, $c(4^k)=5k$, so $A(5k)\ge4^k$ and
$$\frac{\log A(5k)}{5k}\ \ge\ \frac{k\log4}{5k}=\frac1\alpha .\qquad\square$$
Corollary 74.A.2′ (exact capacity points). For $k\ge1$: $\ A(5k)=4^k$.
Proof. $c(n)\le5k$ with Theorem 74.A gives $\alpha\log n\le5k$, i.e. $\log n\le5k/\alpha=k\log4$, so $n\le4^k$; and $c(4^k)=5k$ shows the value is attained. $\square$ 〔Grade C check: exact for $k=1,\ldots,8$.〕
Register. $\gamma$ is the capacity constant of this min-plus weighted grammar: it measures the exponential growth rate of the largest integer reachable within a given budget $K$. It is the reciprocal of $\alpha$.
〔Its independence must be located precisely. The existence of $\gamma$ follows from the superadditivity (2.5a) together with Fekete, a source independent of the induction of Theorem 74.A; but its exact value uses Theorem 74.A for the upper bound and $c(4^k)=5k$ — itself a consequence of 74.A — for the lower. It is therefore not an independent proof of the value of $\alpha$, but the capacity-dual formulation of one and the same sharp lower bound.〕
2.1.5 Closed form on powers of two
Corollary 74.A.3. For every integer $m\ge1$,
$$c(2^m)=\Bigl\lceil\frac{5m}{2}\Bigr\rceil,\qquad \rho_{\mathcal D}(2^m)=\Bigl\lceil\frac{5m}{2}\Bigr\rceil-2. \tag{2.7}$$
Proof. The lower bound is uniform: Theorem 74.A gives $c(2^m)\ge\alpha\log2^m=\frac{5m}{2}$, and $c$ is integer-valued, so $c(2^m)\ge\lceil5m/2\rceil$.
The upper bound splits into three cases.
$m=1$: directly, $c(2)=3=\lceil5/2\rceil$.
$m=2k$, $k\ge1$: by Corollary 74.A.1, $c(2^{2k})=c(4^k)=5k=\frac{5m}{2}$.
$m=2k+1$, $k\ge1$: by (2.2) with $a=4^k\ge4$ and $b=2$,
$$c(2^{2k+1})\le c(4^k)+c(2)=5k+3,$$
while $\lceil5(2k+1)/2\rceil=\lceil5k+\tfrac52\rceil=5k+3$.
In each case the bounds agree. $\square$
〔The case $m=1$ must be treated separately. It cannot be written as $c(2^{2\cdot0+1})\le c(4^0)+c(2)$: first, $c(4^0)=c(1)=2\ne0$; second, $2=1\cdot2$ is not a legal factor candidate in (2.1), which requires $a,b\ge2$.〕
〔Grade T1. Grade C check: exact for $m=1,\ldots,21$. Corollary 74.A.1 in the even case is a special case of this.〕
Corollary 74.A.4 (increments). From (2.7),
$$c(2^m)-c(2^{m-1})=\begin{cases}1,& m=1\\[2pt] 2,& m\ge2\ \text{even}\\[2pt] 3,& m\ge3\ \text{odd}\end{cases} \tag{2.8}$$
Write $c_\Delta(m)$ for this value when $m\ge2$.
Reading. Apart from the initial step, entering an even $m$ costs $2$ and entering an odd $m$ costs $3$. The mechanism: $2^{\text{even}}$ divides into whole copies of $4$, and $c(4)/2=2.5$ is the least cost per doubling; $2^{\text{odd}}$ necessarily leaves one isolated factor $2$, at cost $3$.
〔The exception at $m=1$ must be stated. It does not affect §4 (the relevant increments begin at $m\ge2$), but the theorem must be accurate.〕
§2.2 Two slopes: $h_0$ and $\lambda$
2.2.1 Two constants in the literature
Paper XXXII uses a constant $h_0$; from $h_0\log2=2.48$ there, $h_0=3.578$. Paper LIX §2.1 uses $\lambda=3.856763$, describing it as the value "under the P32 convention". Paper L §1.2, citing the same convention, writes $\lambda\approx3.86$.
The observation of this section is that $h_0$ and $\lambda$ are two different constants serving two different objects.
Grade C measurement (self-built table, (F3)):
$$\frac{\rho_{\mathcal D}(10^7)}{\log10^7}=3.5984,\qquad \frac{\rho_{\mathcal D}(10^8)}{\log10^8}=3.5829 .$$
These agree with $h_0=3.578$ and not with $\lambda=3.856763$. Thus $h_0$ is a pointwise ratio at a given scale, while $\lambda$ is the centering coefficient used at the prime layer (see the definition of $\eta(p)$ in §0.2).
2.2.2 Consistency with Theorem 74.A
The measured values lie below $\alpha=3.6067$. This does not contradict Theorem 74.A: the finite-scale bound the theorem gives for $\rho_{\mathcal D}/\log n$ is
$$\frac{\rho_{\mathcal D}(n)}{\log n}\ \ge\ \alpha-\frac{2}{\log n},$$
which at $n=10^8$ is $3.6067-0.1086=3.498$, and $3.5829\ge3.498$.
〔Limit on phrasing. Only "does not contradict the theorem" may be written; not "lies below $\alpha$ merely because of the $c=\rho+2$ offset". Consistency has been earned by the display above; "merely" would assert a complete attribution of the gap, which is not argued here.〕
2.2.3 Relation to the prime penalty
Paper XXXII §3 records the residual difference between general integers and primes ($E[R(p)]=4.57$ against $E[R(n)]=2.76$, a gap of about $1.81$) and names it the prime penalty.
The exact relation. $h_0$ is the pointwise ratio $\rho_{\mathcal D}(N)/\log N$; $\lambda$ is the slope of an OLS regression with intercept (Appendix A.1.1). The two differ by an intercept term exactly:
$$\frac{\rho_{\mathcal D}(N)}{\log N}=\lambda+\frac{\mu_N}{\log N},\qquad \mu_N:=\rho_{\mathcal D}(N)-\lambda\log N . \tag{2.5b}$$
Numerically at $N=10^8$: $\rho_{\mathcal D}(10^8)=66$, $\mu_N=66-3.856763\times18.4207=-5.0405$, $\mu_N/\log N=-0.27363$; and $\lambda-h_0=3.856763-3.58290=0.27386$.
〔Note that $\mu_N$ is the residual at that point, not an OLS intercept, and that (2.5b) is an algebraic decomposition of the definition $h_0:=\rho_{\mathcal D}(N)/\log N$: the identity $\lambda-h_0=-\mu_N/\log N$ holds by definition, so the numerical agreement above is rounding, not an independent check.〕
〔The prime penalty is a side remark only. It supplies qualitative background for the fact that primes cost systematically more, but (2.5b) already gives the exact source of the difference between $h_0$ and $\lambda$. The two are also not to be equated: $\lambda-h_0\approx0.279$ is a difference between a slope and a pointwise ratio, of different dimension from a difference of residual means.〕
2.2.4 The one possible value of the full-$n$ limit
Corollary 74.B. If $\lim_{n\to\infty}\rho_{\mathcal D}(n)/\log n$ exists, it equals $5/\log4=3.6067\ldots$
Proof. An existing limit equals the liminf, which is $5/\log4$ by Corollary 74.A.1. $\square$
〔Grade T1, under the hypothesis that the limit exists (a frozen premise in the sense of (F2)). This corollary does not assert that the limit exists.〕
Register. The corollary pins down the only possible value of the full-$n$ limit, consistent with §2.2.1: $\lambda=3.856763$ was never a candidate for that limit, and the upstream literature does not use it as one.
The open item that must be stated alongside it. The existence of the limit along the primes,
$$\lim_{p\to\infty,\ p\ \text{prime}}\frac{\rho_{\mathcal D}(p)}{\log p},$$
is unproved — and it is precisely the limit on which the definitions of $\eta$ and $\mu_{12}$ in §0.2 depend. The method of Theorem 74.A is unavailable for it: the factor step of the induction degenerates to a vacuous truth at primes ($p$ has only the divisors $1$ and $p$, while (2.1) requires $a,b\ge2$), so the entire load falls on the successor candidate, which gives a pointwise recursion and no asymptotics.
〔Compare: for integer complexity $\|n\|$ the liminf and limsup remain separated to this day — the classical form of the same gap.〕
§2.3 An exact identity and centering rigidity
2.3.1 The identity
In the notation of §0.2, set for each block
$$H_{1,j}=\sum_{p\in I_j}\frac1p,\qquad H_{2,j}=\sum_{p\in I_j}\frac1{p^2},$$
$$\bar f^{(1)}_j=\frac{1}{H_{1,j}}\sum_{p\in I_j}\frac{h(p)}{p},\qquad Q^{(2)}_j=\frac{1}{H_{2,j}}\sum_{p\in I_j}\frac{h(p)^2}{p^2},$$
so that $\bar f^{(1)}_j$ is the $1/p$-weighted mean of $h$ and $Q^{(2)}_j$ the $1/p^2$-weighted mean of $h^2$.
Proposition 74.0 (identity). Whenever $D_j\ne0$,
$$\boxed{\ R_{\rm wt}(j)=\frac{H_{1,j}^2}{j\,H_{2,j}}\cdot\frac{\bigl(\bar f^{(1)}_j\bigr)^2}{Q^{(2)}_j}\ } \tag{2.9}$$
Proof. By definition $B_j=H_{1,j}\bar f^{(1)}_j$ and $D_j=H_{2,j}Q^{(2)}_j$; substitute into $R_{\rm wt}=B_j^2/(jD_j)$. $\square$
〔Grade T1. This is a rearrangement of definitions, with no hypotheses. Its value lies precisely there: it decomposes $R_{\rm wt}$ into a purely arithmetic factor $H_{1,j}^2/(jH_{2,j})$ and a purely statistical factor $(\bar f^{(1)}_j)^2/Q^{(2)}_j$, and the separation depends on no block-internal independence or distributional assumption whatever.〕
Asymptotics. By Mertens' theorem and the prime number theorem, $H_{1,j}\sim1/j$ and $H_{2,j}\sim\bigl(2^{j+1}j\log2\bigr)^{-1}$, so
$$\frac{H_{1,j}^2}{j\,H_{2,j}}\ \sim\ 2\log2\cdot\frac{2^j}{j^2}. \tag{2.10}$$
〔Grade C check: for $j=18,\ldots,27$ the ratio of the left factor of (2.9) to the right-hand side of (2.10) runs $0.970\to0.980$, increasing monotonically toward $1$.〕
2.3.2 Scales under three residual structures
By (2.9) the behaviour of $R_{\rm wt}$ is determined entirely by the statistical factor.
| # |
Residual structure |
$(\bar f^{(1)})^2/Q^{(2)}$ |
Scale of $R_{\rm wt}$ |
For 59.1 |
| 1 | $h$ carries a nonzero uniform block mean $m_j\to m\ne0$ | $\to m^2/(m^2+\sigma^2)$, a constant | $\asymp2^j/j^2$ | false |
| 2 | $h$ carries a slowly growing uniform block mean $m_j\to\infty$ | $\to1$ | $\asymp2^j/j^2$ | false |
| 3 | Pseudorandom fluctuation with vanishing weighted mean | $\approx H_{2,j}/H_{1,j}^2$ | $O_{\mathbb P}(1/j)$ | true |
〔Grade T2 (a reframing of established results). The three rows are a model reading, not consequences of the identity; the identity itself is T1 (§2.3.1). The two must be graded separately.〕
Two technical points.
First, row 2 carries no additional factor of $\log^2 j$. If $h(p)=m_j+\xi_p$ with $m_j\asymp\log j$ and the weighted second moment of $\xi$ bounded, then $Q^{(2)}_j\asymp m_j^2+\operatorname{Var}(\xi)$ is of the same order as $(\bar f^{(1)}_j)^2\asymp m_j^2$, the ratio tends to $1$, and the leading scale is still $2^j/j^2$. A factor $\log^2j$ appears only if $Q^{(2)}$ is misread as a centered variance; but $D_j$ uses the uncentered second moment. The conclusion (59.1 false) is unchanged; only the rate is as tabulated.
Second, the constant in row 3 is not fixed. In an independent, weighted-mean-zero, homoscedastic model $E[B_j^2]\approx\sigma^2H_{2,j}$ while $D_j\approx\sigma^2H_{2,j}$, so the typical scale is $R_{\rm wt}=O_{\mathbb P}(1/j)$ with a first-order expected constant near $1$. A constant belonging to one particular random model must not be written as a determinate conclusion.
2.3.3 Centering rigidity
From rows 1 and 2:
> Exact centering is a necessary condition for 59.1, and an extremely rigid one.
Its limit must be stated at the same time:
> Even with centering exactly correct (that is, with the numerator of the statistical factor pressed down to the pseudorandom scale), 59.1 still additionally requires the block sums to exhibit sufficient cancellation. Correct centering does not entail 59.1.
2.3.4 The precision requirement, inverted from published data
Let $\varepsilon$ be the difference between the registered value of $\lambda$ and the true one. Then $h$ acquires an increment $-\varepsilon\log p$, so
$$\bar f^{(1)}_j\ \longmapsto\ \bar f^{(1)}_j-\varepsilon\cdot\frac{1}{H_{1,j}}\sum_{p\in I_j}\frac{\log p}{p}\ \approx\ \bar f^{(1)}_j-\varepsilon\,(j+\tfrac12)\log2 . \tag{2.11}$$
That increment grows linearly in $j$, while the arithmetic factor of (2.10) grows like $2^j/j^2$.
Table 1 of Paper LIX gives measured $R_{\rm wt}$ for $j=14,\ldots,32$. The tightest constraint comes from $j=32$ ($R_{\rm wt}=2.70$, not the tabulated maximum $25.80$); substituting into (2.9)–(2.11),
$$\boxed{\ \Bigl|\frac{\varepsilon}{\sqrt{Q^{(2)}_{32}}}\Bigr|\ \lesssim\ 3.0\times10^{-5}\ } \tag{2.12}$$
Conversion to an absolute bound, necessarily conditional. $\sqrt{Q^{(2)}_j}$ is the root of the $1/p^2$-weighted second moment of $h$. Grade C measurement (this paper's truncation, $j=18,\ldots,27$): $0.7356\to0.7551$, increasing slowly, with mean $0.7523$ over the last four blocks. If $\sqrt{Q^{(2)}_{32}}$ is of the same order (taking $0.75$), then
$$|\varepsilon|\ \lesssim\ 2.3\times10^{-5}. \tag{2.12b}$$
〔The conversion is conditional. $Q^{(2)}_{32}$ under the calibration of Paper LIX is not published; the value here is extrapolated from this paper's own truncation. Wherever absolute quantities are compared "by orders of magnitude" below (the $0.05$ of §2.5.2, the $0.0153$ of §2.6.4), the comparison passes through (2.12b) and its factor depends on this extrapolation.〕
〔Grade T2, conditional. Three limits: (a) the principal form must be written as a ratio, the absolute bound depending on $\sqrt{Q^{(2)}_{32}}$; (b) the inversion assumes that block-mean errors from other sources have already been correctly removed, so the whole of $\bar f^{(1)}$ must not be attributed unconditionally to $\lambda$; (c) it uses the asymptotics (2.10), whose arithmetic factor at $j=32$ is about $2\%$ below the asymptotic value, so (2.12) is slightly conservative.〕
2.3.5 A second reason for the failure of the Bartlett family
Paper LXXIII §7.5 records a bandwidth envelope $m_j^{0.33}\lesssim F_W\lesssim m_j^{0.63}$ against a requirement of $O(\log m_j)$, and reads that failure as a polynomial-against-logarithmic gap.
From §2.3.2–§2.3.4 a reason independent of that argument is available:
> The Bartlett family contests a polynomial gap, $m_j^{0.33}$ against $\log m_j$, whereas by (2.10) the sensitivity of the target quantity to the centering constant is exponential, of order $2^j$. The bandwidth was not chosen badly; the order class is wrong.
〔This paper does not claim that the argument of P73 §7.5 is in error — it holds within its own framework. What is claimed is that the conclusion has a second, coarser source, prior to any choice of window. The coexistence of two mutually independent routes to one conclusion is itself evidence, of greater robustness than either alone.〕
§2.4 Uniqueness of the calibration
2.4.1 An elementary upper bound
Lemma 74.C.0. For every $m\ge2$,
$$\rho_{\mathcal D}(2m)\le\rho_{\mathcal D}(m)+3,\qquad \rho_{\mathcal D}(2m+1)\le\rho_{\mathcal D}(m)+4 .$$
Hence $\rho_{\mathcal D}(n)=O(\log n)$; concretely $\rho_{\mathcal D}(n)\le4\log_2n$ for $n\ge2$.
Proof. Take $a=2$, $b=m$ (both $\ge2$) in the factor candidate of (2.1): $\rho_{\mathcal D}(2m)\le\rho_{\mathcal D}(2)+\rho_{\mathcal D}(m)+2=\rho_{\mathcal D}(m)+3$. Then $\rho_{\mathcal D}(2m+1)\le\rho_{\mathcal D}(2m)+1$ by the successor candidate. Recursing on the binary expansion of $n$ gives $O(\log n)$. $\square$
〔Grade T1, elementary. Grade C check: $\max_n\rho_{\mathcal D}(n)/\log n=4.126$ for $n\le2^{17}$.〕
〔Method note. Paper LIX §7.1 obtains $D_j\le C'j/2^j$ from a Guy-type bound together with Chebyshev. The lemma above gives a conclusion of the same type without invoking any prime counting (see below). This is not a new result; it is a cheaper route.〕
2.4.2 Uniqueness
Proposition 74.C. Fix the four real values of $\mu_{12}$. Then there is at most one $\lambda\in\mathbb R$ for which there exist $M<\infty$ and $j_0$ with
$$B_j(\lambda)^2\ \le\ M\,j\,D_j(\lambda)\qquad\text{for all }j\ge j_0 . \tag{2.13a}$$
〔The cleared-denominator form is used (Appendix B.2): $R_{\rm wt}$ involves a division, and "$D_j\ne0$" is $\Sigma^0_1$ rather than decidable for computable reals; (2.13a) is everywhere well defined and equivalent to $R_{\rm wt}\le M$ where $D_j>0$, with the proof unchanged. It is recommended that the formal theorem version of 59.1 likewise be stated with cleared denominators, $R_{\rm wt}$ being retained only as the name of a diagnostic.〕
Proof. In four steps.
Step 1 ($jD_j\to0$, unconditional). For $p\in I_j$, Lemma 74.C.0 gives
$$|h(p)|=\bigl|\rho_{\mathcal D}(p)-\lambda\log p-\mu_{12}(p\bmod12)\bigr|=O(\log p)=O(j),$$
while $p^{-2}\le4^{-j}$ and $I_j$ contains at most $2^j$ integers (only the interval length is used; no prime counting). Hence
$$D_j(\lambda)=\sum_{p\in I_j}\frac{h(p)^2}{p^2}\ =\ O\bigl(2^j\cdot j^2\cdot4^{-j}\bigr)=O\bigl(j^2\,2^{-j}\bigr),$$
so $j\,D_j(\lambda)=O(j^3\,2^{-j})\to0$.
Step 2 ($B_j\to0$). If (2.13a) holds for $j\ge j_0$ then
$$|B_j(\lambda)|\le\sqrt{M\,j\,D_j(\lambda)}=O\bigl(j^{3/2}2^{-j/2}\bigr)\longrightarrow0 .$$
Step 3 (the difference). Suppose $\lambda_1\ne\lambda_2$ both satisfy the conclusion. Since $\mu_{12}$ is fixed and independent of $\lambda$, we have $h_{\lambda_1}(p)-h_{\lambda_2}(p)=-(\lambda_1-\lambda_2)\log p$, hence exactly
$$B_j(\lambda_1)-B_j(\lambda_2)=-(\lambda_1-\lambda_2)\sum_{p\in I_j}\frac{\log p}{p}. \tag{2.13}$$
Step 4 (contradiction). By the prime number theorem (the refinement of Mertens' first theorem with vanishing error),
$$\sum_{p\in I_j}\frac{\log p}{p}\ \longrightarrow\ \log2 . \tag{2.14}$$
By Step 2 the left side of (2.13) tends to $0$; by (2.14) the right side tends to $-(\lambda_1-\lambda_2)\log2\ne0$. Contradiction. $\square$
〔Grade T1, unconditional. Only Step 4 uses the prime number theorem (per (F2)); Steps 1–3 are elementary.〕
〔Grade C check: the deviation in (2.14) is below $2.2\times10^{-4}$ for $j=20,\ldots,23$.〕
2.4.3 The resulting decomposition
Proposition 74.C splits 59.1 in two:
$$\underbrace{\exists\lambda\ \exists C\ \forall j\ge2:\ B_j(\lambda)^2\le C\,j\,D_j(\lambda)}_{\text{a statement about }\rho_{\mathcal D}}\qquad+\qquad\underbrace{\text{“that unique }\lambda\text{ equals }\lambda_{\rm FullIC}\text{”}}_{\text{a statement about the calibration}}$$
The first is well defined once $\mu_{12}$ is fixed (uniqueness being guaranteed unconditionally by the proposition). The second is not a mathematical statement about $\rho_{\mathcal D}$.
〔$j\ge2$ rather than $j\ge1$: see the small-prime convention of §0.2.〕
〔For the standing of the second conjunct, and for the well-posedness of $\mu_{12}$ itself, see §2.5–§2.7.〕
§2.5 Unidentifiability and operational irreproducibility
2.5.1 Three items
§2.3 shows that 59.1 requires the centering constant to precision $|\varepsilon|/\sqrt{Q^{(2)}_{32}}\lesssim3.0\times10^{-5}$. This section shows in three ways that the requirement cannot currently be met:
- (a) Extrapolative unidentifiability — within the present data window and the family of models tested, the attainable precision on $\lambda$ is three orders of magnitude short, and improves very slowly with data;
- (b) Operational irreproducibility — the published specification fixes neither values, nor error bounds, nor an estimation procedure for $\mu_{12}$, so Table 1 of Paper LIX cannot be reproduced from the published text alone;
- (c) The calibration enters the observable — the position of the deep trough in the $R_{\rm wt}$ profile is controlled at least in part by the calibration cutoff, not by $\rho_{\mathcal D}$ alone.
The three differ in kind: (a) is an extrapolation limit within the present window and model family; (b) concerns the operability of this particular specification; (c) concerns the constitution of the observable itself.
〔The strength of (a) must be fixed. It is not a minimax impossibility theorem over all estimation procedures, nor has it been shown that further data can never separate the models. Only the restricted form above is supported here.〕
2.5.2 (a) Extrapolative unidentifiability
Block-optimal centers. With $\mu_{12}$ fixed, let $\lambda_j^*$ be the unique $\lambda$ with $B_j(\lambda)=0$ ($B_j$ being affine in $\lambda$, so $\lambda_j^*=B_j(0)/\sum_{I_j}\frac{\log p}{p}$). Put $I_j(M)=\{\lambda:B_j(\lambda)^2\le M\,j\,D_j(\lambda)\}$ in cleared form. This set is defined by a quadratic inequality in $\lambda$, and depending on the sign of the leading coefficient $S_j^2-MjQ_2$ it may be an interval, a half-line, or a pair of rays; throughout this paper we take the connected component containing $\lambda_j^*$ and record its half-width.
〔It may not simply be called an interval: the solution set of a general quadratic inequality can be a pair of rays.〕
By Step 1 of §2.4.2 the half-width of $I_j(M)$ has exponential factor $2^{-j/2}$, together with an at most polynomial prefactor ($j^{3/2}$ from the elementary bound). Grade C check: over $j=14,\ldots,27$, fourteen blocks, the blockwise ratio lies in $0.705$–$0.719$, against $2^{-1/2}=0.7071$.
Two propositions must first be separated. The center of the original 59.1 is already specified ($\lambda=3.856763$ and the corresponding $\mu_{12}$); "there exists some $\lambda$ making the weighted ratio bounded" is a weaker existential statement. The two must not be conflated. Put
$$\mathcal E(\vec\mu)\ :=\ \exists\lambda\in\mathbb R\ \exists C>0\ \forall j\ge2:\ B_j(\lambda,\vec\mu)^2\le C\,j\,D_j(\lambda,\vec\mu).$$
Then the trivial equivalence (a rewriting of definitions) reads
$$\mathcal E(\vec\mu)\ \iff\ \exists C,\lambda\ \ \forall j\ge2:\ \lambda\in I_j(C), \tag{2.15}$$
while the fixed-parameter version $59.1_{\rm freeze}(\lambda_0,\vec q):\ \exists C>0\ \forall j\ge2:\ B_j(\lambda_0,\vec q)^2\le C\,j\,D_j(\lambda_0,\vec q)$ relates to it by
$$59.1_{\rm freeze}(\lambda_0,\vec q)\ \Longrightarrow\ \mathcal E(\vec q),$$
the converse requiring in addition that $\lambda_{\vec q}=\lambda_0$, where $\lambda_{\vec q}$ is the center whose existence Proposition 74.C bounds to at most one.
〔Two cautions. First, the left side of (2.15) must not be written as "59.1": that would replace a fixed-$\lambda$ proposition by the weaker existential one. Second, Proposition 74.C does not entail that the unique center equals the published value.〕
Corollary 2.5.1 (necessary condition). Since $B_j(\lambda)=S_j(\lambda_j^*-\lambda)$ with $S_j=\sum_{p\in I_j}\frac{\log p}{p}$ (because $B_j(\lambda_j^*)=0$ and $B_j$ is affine in $\lambda$), if 59.1 holds for some $\lambda$ with constant $C$ then
$$|\lambda_j^*-\lambda|\ \le\ \frac{\sqrt{C\,j\,D_j(\lambda)}}{S_j}\ =\ O\bigl(j^{3/2}\,2^{-j/2}\bigr),$$
using $D_j=O(j^2 2^{-j})$ from Lemma 74.C.0 (so $jD_j=O(j^3 2^{-j})$) and $S_j\to\log2$. Rapid Cauchy convergence of $\{\lambda_j^*\}$ is thus necessary for 59.1.
〔Improvable to $O(j2^{-j/2})$ by invoking the Chebyshev bound $m_j=O(2^j/j)$, which gives $D_j=O(j2^{-j})$. The elementary form is retained here (using only that $I_j$ contains at most $2^j$ integers), so the conclusion stated is the weaker one.〕
〔The necessary condition alone does not yield the converse. $R_{\rm wt}=S_j^2(\lambda_j^*-\lambda)^2/(jD_j(\lambda))$ is controlled by the convergence rate only in its numerator; no lower bound on $D_j$ is supplied. With the typical scale $jD_j\asymp\sigma^2 2^{-j}/(2\log2)$ and the rate $O(j^{3/2}2^{-j/2})$ given by the corollary, one obtains $R_{\rm wt}=O(j^3 2^{-j})/O(2^{-j})=O(j^3)$, not yet enough to conclude boundedness. Establishing the converse requires a further nondegeneracy lower bound on the second moment. A limit on phrasing: the above shows only that the converse cannot be derived; it is not a refutation of the converse, no actual counterexample having been constructed for $\rho_{\mathcal D}$.〕
Unidentifiability of the extrapolation. The measured $\lambda_j^*$ drifts slowly with $j$. Fitting three equally plausible drift models to the same data and extrapolating to $j\to\infty$:
Table 2.1 · Extrapolations of three drift models (grade C)
| Data range |
$A+B/j$ |
$A+B/j+C/j^2$ |
$A+(B+C\log j)/j$ |
Spread |
| $j=14\ldots22$ ($N\sim8\times10^6$) | 3.8906 | 3.8425 | 3.7938 | 0.097 |
| $j=19\ldots27$ ($N\sim2.7\times10^8$) | 3.8708 | 3.8450 | 3.8191 | 0.052 |
The last two models differ by only $17\%$ in residual sum of squares and cannot be separated on this data: over $j=19\ldots27$, $\log j$ varies by only $11.9\%$ while $1/j$ varies by $42\%$, so the two terms are nearly collinear.
> Conclusion: the attainable extrapolated precision on $\lambda$ is about $\pm0.05$ (absolute), while the requirement of §2.3 converted through (2.12b) is about $|\varepsilon|\lesssim2.3\times10^{-5}$ — a gap of roughly $3.3$ orders of magnitude. Moreover a $32$-fold increase in data only halves the spread.
〔The factor depends on the conditional conversion (2.12b); see the limits stated there.〕
〔Limit. Table 2.1 was computed from the self-built table with externally supplied $\mu_{12}$ ratios, not in the calibration of Paper L. What may be cited is the conclusion about model discriminability and the $2^{-j/2}$ half-width law; none of the individual values of $\lambda_\infty$ in the table may be cited.〕
The nature of the gap. $\lambda$ and the drift model are entangled in the data. Simply enlarging computations of the same kind has so far shown no capacity to close a three-order gap ($32$-fold data halving the spread); absent theoretical control of the second-order term in $\rho_{\mathcal D}(p)$, finite-range extrapolation lacks adequate warrant.
〔It must not be written as "more data will not help" — that would be an assertion about all future computation, and is not proved here.〕
2.5.3 (b) Operational irreproducibility, and sensitivity to the truncation
Paper L §2.2 defines $\mu_{12}(r)$ as an $X\to\infty$ class-mean limit (see (2.16) in §2.6.1) and supplies neither the four values, nor error bounds, nor a designated estimation procedure. Using the most natural finite-$X$ plug-in estimate induced by that definition — the class mean at some truncation $X$ — the result is highly sensitive to the truncation point.
Table 2.2 · Sensitivity to the truncation (grade C)
| $j$ |
14 |
18 |
22 |
24 |
26 |
27 |
| This paper ($X=3\times10^8$) | 8.32 | 16.10 | 17.04 | 7.71 | 0.44 | 7.84 |
| Paper LIX Table 1 ($N=10^{10}$) | 8.73 | 18.12 | 24.32 | 18.07 | 4.21 | 0.79 |
| Ratio | 0.95 | 0.89 | 0.70 | 0.43 | 0.105 | 9.92 |
Conditions: the same $\rho_{\mathcal D}$ table (§0.3, 33/33); the same block primes ($m_j$ exact for $j=14\ldots27$, 14/14); $\mu_{12}$ taken as the plug-in value of the definition of Paper L §2.2 at $X=3\times10^8$.
Two differences, not one. The principal controlled change is the calibration cutoff; but by Appendix A.1.1 the second pass of Paper LIX uses the unrounded $\lambda_{10^{10}}^{\rm OLS}$, and that value is not published, so only the displayed value $3.856763$ is available here. There is therefore a second, irremovable substitution by rounding.
> Table 2.2 is accordingly a demonstration of truncation sensitivity, not a controlled experiment in which a single variable is changed. 〔This does not weaken the conclusion; it strengthens the point that the published text does not suffice for exact reproduction.〕
Accurate statement.
> Under the most natural finite-$X$ plug-in estimate the resulting $R_{\rm wt}$ profile is highly sensitive to the truncation point; since the published text supplies neither the four limit values, nor error bounds, nor a designated estimation procedure, Table 1 cannot be reproduced operationally from the published text alone.
Four limits that must be stated.
- No claim is made that the values of Paper LIX are in error. Table 2.2 concerns reproducibility, not correctness.
- What changes is the estimate, not the limit itself (if it exists). It must not be written that "changing only the truncation changed the $\mu_{12}$ in the definition".
- It must not be inferred that different $\vec q$ give different truth values. Table 2.2 demonstrates different finite-block profiles; two infinitary propositions may both be true or both false, and no difference in truth value is proved here.
- The mismatch pattern is consistent with the mechanism of §2.3 but does not constitute its proof: the ratio deviates monotonically in $j$ and inverts near a zero of $B_j$, in agreement with the $2^j$ amplification of (2.10), but this is a consistency observation, not an independent verification.
〔A discrepancy once unlocated, now closed. By (2.16) at $X=3\times10^8$ this paper obtains $\sum_r\mu_{12}(r)=-1.03$, whereas Paper L reports about $+4.94$. By the upstream clarification of Appendix A.1.3, $4.94$ is in the $c=\rho_{\mathcal D}+2$ convention: each class mean shifts by $2$ and the sum by $4\times2=8$; the $\rho$-version sum at $p\le2000$ upstream is $-2.991$, and $-2.991+8=5.01$. The discrepancy is a convention difference; there is no substantive disagreement.〕
2.5.4 (c) The calibration enters the observable: the position of the deep trough
By Appendix A.1.2, $\mu_{12}(r)$ is the arithmetic mean of $\eta$ over class $r$ for $p\le N$. Hence, exactly and class by class,
$$\sum_{\substack{p\le N\\ p\equiv r\,(12)}}h(p)=0\qquad(r=1,5,7,11). \tag{2.19}$$
This is not an approximation but an identity of the construction: four exact linear constraints spanning all blocks. Moreover $2^{33}=8.59\times10^9<10^{10}$, so every block with $j\le32$ lies inside the calibration range.
> Consequently $R_{\rm wt}(j)$ is controlled at least in part by the calibration protocol; it is not a function of $\rho_{\mathcal D}$ and the block index alone.
〔This corresponds to the version split of §2.7: the proposition actually tested is $59.1_{\rm calib}(N)$, and this section shows that its observable profile depends on $N$.〕
A discriminating test. Take ten fixed blocks $j=14,\ldots,23$ and vary only the calibration cutoff $N$ — that is, refit $\lambda$ and $\mu_{12}$ by the two-pass procedure of A.1.1–A.1.2 at different $N$.
Table 2.5 · The same blocks under four calibration cutoffs (grade C)
| Cutoff $N$ |
$j{=}14$ |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
Trough |
| $2^{24}$ | 0.57 | 1.11 | 0.01 | 0.05 | 0.01 | 0.21 | 0.48 | 0.23 | 0.13 | 0.71 | $j{=}16/18$ |
| $2^{26}$ | 1.47 | 2.49 | 0.46 | 0.72 | 0.28 | 0.02 | 0.06 | 0.12 | 0.67 | 0.38 | $j{=}19$ |
| $2^{27}$ | 2.06 | 3.39 | 0.97 | 1.42 | 0.84 | 0.28 | 0.02 | 0.00 | 0.37 | 0.47 | $j{=}21$ |
| $2^{28}$ | 2.88 | 4.63 | 1.84 | 2.63 | 1.98 | 1.13 | 0.52 | 0.31 | 0.02 | 0.16 | $j{=}22$ |
> The same $\rho_{\mathcal D}$, the same primes, the same blocks — only the fitting range differs, and the trough migrates monotonically. In this experiment the trough position is therefore not determined by $\rho_{\mathcal D}$ and the block index alone; it is controlled at least in part by the calibration protocol.
Comparison with the published profile.
|
Top block |
Trough |
Distance |
| Here, $N=2^{24}$ | 23 | 16 | 7 |
| Here, $N=2^{26}$ | 25 | 19 | 6 |
| Here, $N=2^{27}$ | 26 | 21 | 5 |
| Here, $N=2^{28}$ | 27 | 22 | 5 |
| Paper LIX Table 1 ($N=10^{10}$) | 32 | 27 | 5 |
Mechanism. Let $T_j=\sum_{p\in I_j}h(p)$ be the unweighted block sum. By (2.19), $\sum_jT_j=0$ exactly; and since prime counts grow like $2^j$, the compensation falls almost entirely on the topmost blocks. Grade C measurement at $N=2^{28}$: $T_j$ is negative for $j=14\ldots21$, turns positive over $j=22\ldots26$ rising to $+7424$, and is pulled back to $-16877$ at $j=27$.
Four limits (this result is strong and they must be written next to it).
- No claim that the whole oscillation is a calibration artefact. Only the trough is located here. The period-8 structure of Table 1 of Paper LIX and the secondary peak at $j=30$ are not explained.
- No claim that the $j=32$ value of Paper LIX is an artefact. The distance $5$ is extrapolated from four cutoffs; the measured sequence is $7,6,5,5$, not constant.
- No claim that 59.1 is false. An observable whose profile is affected by the calibration protocol may nevertheless be bounded.
- No variance decomposition or exclusive causal attribution has been carried out. This test cannot quantify what fraction of the whole $R_{\rm wt}$ profile derives from the calibration, nor can it exclude that the trough is jointly determined by block-internal arithmetic structure and the calibration protocol.
〔Relation to §2.5.2: that section is a limit on estimation precision (binding on any procedure within the stated window and model family); this one concerns the observable itself — the $R_{\rm wt}$ profile contains a feature positioned by the calibration cutoff. The two are independent.〕
§2.6 Uniqueness of the class-mean gauge
2.6.1 The definition in Paper L
Paper L §2.2 defines, for $r\in\{1,5,7,11\}$,
$$\mu_{12}(r)\ :=\ \lim_{X\to\infty}\ \frac{1}{\pi_r(X)}\sum_{\substack{p\le X\\ p\equiv r\ (12)}}\eta(p),\qquad \eta(p)=\rho_{\mathcal D}(p)-\lambda\log p . \tag{2.16}$$
The existence of these four limits is not proved there. This section examines what their existence constrains about $\lambda$.
2.6.2 The uniqueness theorem
Notation. For $r\in\{1,5,7,11\}$, $\lambda\in\mathbb R$ and $X\ge13$ (so that $\pi_r(X)>0$ in all four classes), set
$$M_r(X;\lambda)=\frac{1}{\pi_r(X)}\sum_{\substack{p\le X\\ p\equiv r\,(12)}}\bigl(\rho_{\mathcal D}(p)-\lambda\log p\bigr),\qquad L_r(X)=\frac{1}{\pi_r(X)}\sum_{\substack{p\le X\\ p\equiv r\,(12)}}\log p .$$
Proposition 74.D. For $\lambda_1\ne\lambda_2$, exactly
$$M_r(X;\lambda_1)-M_r(X;\lambda_2)=-(\lambda_1-\lambda_2)\,L_r(X), \tag{2.17}$$
and by the prime number theorem for arithmetic progressions $L_r(X)\sim\log X\to\infty$. Therefore:
> For each $r$, at most one $\lambda$ can make $M_r(X;\lambda)$ converge to a finite real. In particular at most one common $\lambda$ can make all four $\mu_{12}(r)$ of (2.16) simultaneously well posed.
Proof. (2.17) is immediate on subtracting the definitions (the $\rho_{\mathcal D}(p)$ terms cancel). For $L_r$: by the prime number theorem in arithmetic progressions, $\sum_{p\le X,\,p\equiv r}\log p\sim X/\varphi(12)$ while $\pi_r(X)\sim X/(\varphi(12)\log X)$, so $L_r(X)\sim\log X$. Now suppose $\lambda_1\ne\lambda_2$ and both $M_r(X;\lambda_1)$ and $M_r(X;\lambda_2)$ converge to finite reals. Then so does their difference; but by (2.17) that difference equals $-(\lambda_1-\lambda_2)L_r(X)$, whose absolute value tends to $+\infty$. Contradiction. $\square$
〔Grade T1, unconditional. External result invoked: the prime number theorem for arithmetic progressions (per (F2)).〕
Summary sentence.
> Well-posedness of $\mu_{12}$ forces the centering slope to be unique; but uniqueness of the centering slope does not by itself deliver $\mu_{12}$.
2.6.3 The direction not claimed here
Suppose a prime-side slope $\lambda_*$ exists in some sense, say
$$\frac{1}{L_r(X)}\cdot\frac{1}{\pi_r(X)}\sum_{\substack{p\le X\\ p\equiv r}}\rho_{\mathcal D}(p)\ \longrightarrow\ \lambda_* . \tag{2.18}$$
Convergence of $M_r(X;\lambda_*)$ does not follow from (2.18). All that (2.18) yields is
$$M_r(X;\lambda_*)=o\bigl(L_r(X)\bigr)=o(\log X),$$
and $o(\log X)$ entails neither $O(1)$ nor convergence.
An abstract counterexample. Suppose
$$\frac{1}{\pi_r(X)}\sum_{p\le X,\ p\equiv r}\rho_{\mathcal D}(p)\ =\ \lambda_*L_r(X)+\sqrt{L_r(X)} .$$
Then (2.18) holds, while $M_r(X;\lambda_*)=\sqrt{L_r(X)}\to+\infty$: the centered class mean diverges.
〔This is not a claim that $\rho_{\mathcal D}$ has that shape. It exhibits that, in the present logic, existence of a slope does not entail existence of the intercept limit. Hence:
> "$\lambda$ is exactly the prime-side slope" is a necessary condition for the existence of $\mu_{12}$, not a proved sufficient one.〕
2.6.4 Class-by-class drift (grade C)
The empirical counterpart of Proposition 74.D: at $\lambda=3.856763$, do the four $M_r(X;\lambda)$ exhibit nonzero drift?
Table 2.3 · Regression slope of $M_r(X;\lambda)$ against $\log X$ ($\lambda=3.856763$; $X$ doubling from $10^6$ to $1.28\times10^8$; equally weighted OLS on the last six points)
| Class $r$ |
1 |
5 |
7 |
11 |
Sum |
| Slope | $+0.0207$ | $+0.0164$ | $+0.0142$ | $+0.0101$ | $+0.0614$ |
| Implied $\lambda$ ($=3.856763+{}$slope) | 3.87749 | 3.87317 | 3.87095 | 3.86682 | — |
Three readings.
- Class-by-class reporting is required. Reporting only the sum $\bar\mu(X)=\sum_r\mu_r(X)$ would allow drifts in different classes to cancel, and a small slope in the sum would then be misread as approximate convergence of all four limits. The measurement shows the four to be of the same sign, of the same order, without cancellation — but that was unknown before the table was produced. Note also that inverting the summed slope for $\lambda$ requires an explicit division by $4$ ($0.0614/4=0.0153$, giving $3.87211$).
- At $\lambda=3.856763$ all four $M_r$ exhibit same-sign positive drift within the present window.
〔The reading must be sharply downgraded. Even if $M_r(X;\lambda_*)\to\mu_r$ holds, no finite window's regression slope is required to vanish — convergence requires only that suitable moving-window slopes eventually tend to zero. Table 2.3 therefore supports "within the present finite window all four classes still drift with the same sign"; it cannot by itself refute existence of the limits, nor can $0.0614/4$ be read unconditionally as the true $\lambda_*-\lambda$.
Conditional estimate: if one further assumes that the drift in the present window is dominated by slope mismatch $\lambda_*-\lambda$, the residual second-order term being negligible over that window, then the summed slope gives $\lambda_*-\lambda\approx0.0614/4=0.01535$. This is a finite-window estimate under a model assumption, not an unconditional consequence of Proposition 74.D.〕
- The four slopes are not equal ($0.0207$ against $0.0101$), and the implied $\lambda$ values spread by $0.0107$. No inference is drawn here that the four classes have different asymptotic slopes: that spread is far below the identification window of §2.5.2 ($\pm0.05$), so "the four share one $\lambda_*$" is neither established nor refuted. This is the empirical counterpart of the situation described in §2.6.3 — even granted a common slope, whether the four $M_r$ converge simultaneously remains an independent question.
Comparison with §2.3. Under the model condition above, the measured deviation converted through (2.12b) exceeds the requirement by about $2.8$ orders of magnitude ($0.0153$ against $2.3\times10^{-5}$).
〔Doubly conditional: first, $\lambda_*-\lambda\approx0.0153$ is conditional on the drift being dominated by slope mismatch; second, the absolute conversion is conditional on the extrapolation of $\sqrt{Q^{(2)}_{32}}$ in (2.12b).〕
〔It is not appropriate to derive a further precision requirement from the $\bar\mu\approx4.94$ reported in Paper L: that value is in the $c=\rho_{\mathcal D}+2$ convention (Appendix A.1.3), and the reading "stable to the reported precision" is too loose.〕
§2.7 Five versions of Conjecture 59.1
§2.5–§2.6 show that of the two constants on which 59.1 depends, $\mu_{12}$ is defined by (2.16) as four limits whose existence is unproved, and that their existence in turn forces $\lambda$ to be unique (Proposition 74.D). The single word "59.1" therefore does not name a single proposition in the current literature. This section separates it into five and locates each.
Table 2.4 · Five versions of 59.1
| Version |
Definition |
Logical status |
| $59.1_{\rm pub}$ | As displayed in Paper LIX §4.1, with $\mu_{12}$ per (2.16) | Partially defined: presupposes the four limits, which the text does not prove |
| $59.1_{\rm pub}^{\exists}$ | $\bigl[\text{all four limits of (2.16) exist}\bigr]\wedge\bigl[R_{\rm wt}\text{ bounded}\bigr]$ | A complete proposition, of higher quantifier complexity than the frozen versions |
| $59.1_{\rm calib}(N)$ | $\lambda=\lambda_N^{\rm OLS}$ and $\mu_{12}$ both computed by the two-pass procedure of A.1.1–A.1.2 at cutoff $N$, the second pass using the unrounded OLS value | Completely determined and effectively evaluable. Both $\lambda_N^{\rm OLS}$ and $\mu_{12}(r)$ are computable reals with a computable modulus, not rationals (the former a quotient of finite sums of $\log p$, the latter containing $\sum\log p$). The figure $3.856763$ is its displayed value, not its definition. This is the version actually tested by Table 1 of Paper LIX |
| $59.1_{\rm freeze}(\vec q)$ | $\lambda=3856763/10^6$ exactly, $\mu_{12}$ frozen to a given $\vec q\in\mathbb Q^4$ | Complete and effectively evaluable; $\vec q$ is a rational truncation of the $59.1_{\rm calib}$ values, hence a stipulation made by this paper |
| $59.1_*$ | Defined with the true asymptotic center $(\lambda_*,\mu_*)$ | That center is not yet defined |
2.7.1 Six notes
First, $59.1_{\rm freeze}$ is a stipulation of this paper, not a reading of the source. Paper L writes $\mu_{12}$ as a limit, not as decimals; interpreting the published seven-figure $3.856763$ as the exact rational $3856763/10^6$ is a stipulation made here.
Second, failure to publish four decimals produces operational irreproducibility, not necessarily semantic indeterminacy. If the four limits of (2.16) exist they are uniquely determined by the definition, even if no reader knows their decimal expansions. Thus $59.1_{\rm pub}^{\exists}$ is a determinate proposition; what is missing is operability (§2.5.3), not semantics.
Third, $59.1_{\rm calib}(N)$ is the version this series has actually studied, and it is indexed by $N$. By the two-pass procedure both $\lambda$ and $\mu_{12}$ depend on the cutoff $N$; and §2.5.4 shows that the position of the deep trough in the $R_{\rm wt}$ profile migrates with $N$. So "59.1", as the object under test, is a family $\{59.1_{\rm calib}(N)\}$, not a single proposition.
Fourth, rational freezing is a truncation and must carry enough digits. Since $\mu_{12}(r)=\frac{1}{\pi_r(N)}\bigl(\sum\rho_{\mathcal D}(p)-\lambda\sum\log p\bigr)$ with $\sum\log p$ transcendental, the value is not rational. Taking $\vec q\in\mathbb Q^4$ is a rounding, and by the sensitivity of §2.3.4 rounding can change the proposition: a uniform shift $c$ in $\mu_{12}$ moves the block-optimal $\lambda$ by about $c/(j\log2)$, and the half-width at $j=32$ is $7.2\times10^{-5}$, so one needs
$$|c|\ \lesssim\ 1.6\times10^{-3}.$$
To leave adequate margin, publication of at least four decimal places is recommended here.
〔Two limits. First, the display above is an order-of-magnitude estimate — the usual rounding error at three decimals is at most $5\times10^{-4}$, already below $1.6\times10^{-3}$, so the estimate does not show four to be the minimum necessary precision. Second, and more importantly, however many digits are given, the rationally truncated version and the original computable-real version remain literally different propositions. One may say that the finite-block error falls inside the permitted window; one may not say that the two "become the same proposition".〕
Fifth, this paper uses the parametrized notation $59.1_{\rm freeze}(\vec q)$ throughout. A concrete instance awaits the upstream values (Appendix A). None of the other conclusions here depends on that instance.
Sixth, $59.1_{\rm freeze}(\vec q)$ is directly attackable. Its $\lambda$ is frozen by definition and requires no estimation, so the unidentifiability of §2.5 does not block attacks on it. What it faces is a different question: whether it carries natural mathematical meaning — whether the frozen $\vec q$ corresponds to any intrinsic structure of $\rho_{\mathcal D}$, or is merely a set of numbers. This paper does not adjudicate that; see §3.
2.7.2 Summary of this part
The deliveries of §2 fall into two groups.
Group one: unconditional results with no dependence on $\lambda,\mu_{12}$ whatever (§2.1). Theorem 74.A (global lower bound), Theorem 74.A′ (equality exactly at $4^k$), Corollary 74.A.1 ($\liminf=5/\log4$), Corollary 74.A.2 (capacity constant $\gamma=\log4/5$) with Corollary 74.A.2′ ($A(5k)=4^k$), and Corollaries 74.A.3–74.A.4 (closed form on powers of two and the increment alternation). These are produced by the recursion (0.1) itself, without any centering constant, prime sampling, or spectral apparatus. No originality is claimed for them (§2.1.0).
Group two: the specification audit (§2.2–§2.7). Corollary 74.B (the only possible value of the full-$n$ limit); Proposition 74.0 (the exact identity) and the precision requirement (2.12) it yields; Proposition 74.C (with $\mu_{12}$ fixed, at most one $\lambda$); Proposition 74.D (well-posed $\mu_{12}$ forces a unique slope); and the three empirical limits of §2.5.
The relation between the two groups. Group two shows that the current specification of 59.1 demands a centering far beyond the resolution of accessible data; group one shows that even so, the recursion itself carries exact, unconditional, statable arithmetic content. This paper takes the latter to be the more reliable footing at the present stage.
§3 Handoff
3.0 The character of this part
This paper is the stage closure and specification audit of ZFCρ Papers L–LXXIII. It takes up no successor line of attack. The function of this part is to classify the open questions and record where they go, so that later work need not reread the preceding seventy-three papers in order to pick them up.
3.1 What this paper closes
| # |
Closed |
On the basis of |
| 1 | Global lower bound and liminf for $\rho_{\mathcal D}$ | Theorem 74.A, Corollary 74.A.1 |
| 2 | Complete characterization of the equality set | Theorem 74.A′ |
| 3 | Capacity constant of the min-plus weighted grammar, and exact capacity points | Corollaries 74.A.2, 74.A.2′ |
| 4 | Closed form and increments of $\rho_{\mathcal D}$ on powers of two | Corollaries 74.A.3–74.A.4 |
| 5 | The only possible value of the full-$n$ limit | Corollary 74.B |
| 6 | Uniqueness of $\lambda$ with $\mu_{12}$ fixed | Proposition 74.C |
| 7 | The constraint imposed on the centering slope by well-posedness of $\mu_{12}$ | Proposition 74.D |
| 8 | Factorization of $R_{\rm wt}$ into an arithmetic and a statistical factor | Proposition 74.0 |
| 9 | That "59.1" is not a single proposition in the literature | §2.7, five versions |
| 10 | The two values in Appendix A of Paper L | §0.4 (none of the three models yields the reported values) |
〔Items 1–5 and 8 depend on no centering constant; items 6, 7, 9 are results of the specification audit.〕
3.2 What remains open, and in what class
Class A: shown here to be currently inexecutable
| Question |
Obstruction |
| Any statistical route presupposing the unknown true asymptotic center $(\lambda_*,\mu_*)$ | Currently inexecutable given the published data and existing theory: attainable extrapolated precision $\pm0.05$ against a required normalized $3.0\times10^{-5}$, about $3.3$ orders through the conditional conversion (2.12b) (§2.5.2). Not "the object is unestimable in principle" |
| Operational reproduction of Table 1 of Paper LIX | Neither the four $\mu_{12}$ values, nor error bounds, nor an estimation procedure is published (§2.5.3) |
〔What is sealed here is current executability, not correctness. Should $\lambda_*$ be determined theoretically, class A revives in full.〕
Class B: well defined and directly attackable; not attacked here
$59.1_{\rm freeze}(\vec q)$: both $\lambda$ and $\mu_{12}$ are frozen by definition and require no estimation, so §2.5.2 does not block it. What it faces is a different question — whether the frozen $\vec q$ corresponds to any intrinsic structure of $\rho_{\mathcal D}$, or is merely a set of numbers. Not adjudicated here.
Class C: not treated here; handed off unchanged
- Existence of the prime-subsequence limit $\lim_p\rho_{\mathcal D}(p)/\log p$ (the method of Theorem 74.A is unavailable at primes);
- Whether the four limits of (2.16) exist (Proposition 74.D gives only a necessary condition);
- The $20\%$ contribution of $\mu_{12}$ to $F_W$ (P73 §11.1 item 5; not closed);
- $c_0,c_t$ and the ratio $9.87$; the two-axis $0.30$; the non-stationarity of the ACF (Table 1.2, items 1–3);
- The mechanisms indicated by the exploratory observations of §4.2.
Appendix to class C: a conditional refutation criterion for H′ (registered, not developed)
The object of this paper is 59.1; H′ lies downstream and is not treated here. But one structure was registered during preparation which is independent of 59.1 and of every centering constant, and which supplies a route to refuting H′; it is handed off here.
Probability space (must be explicit). Let $\mathcal C_N=\{2\le n\le N: n\ \text{composite}\}$, and let $\mathbb P_N,\mathbb E_N$ denote the uniform probability and expectation on $\mathcal C_N$. All conditional quantities below refer to that measure.
Let the jump quantity be
$$J(n):=\rho_{\mathcal D}(n-1)+1-\rho_{\mathcal D}(n)\ \ge\ 0$$
(nonnegativity from the successor candidate). Telescoping gives exactly
$$\sum_{n=2}^{N}J(n)=N-1-\rho_{\mathcal D}(N)=N+O(\log N), \tag{3.1}$$
the last step by Lemma 74.C.0. By (0.2), $J(p)=0$ identically at primes, so the whole mass sits on the composites:
$$\mathbb E_N J=\frac{N-1-\rho_{\mathcal D}(N)}{N-1-\pi(N)}\longrightarrow1. \tag{3.2}$$
Put $q_N:=\mathbb P_N(J>0)$. By Hölder's inequality (applied to $J=J\cdot\mathbf1_{J>0}$ with exponents $p$ and $p/(p-1)$),
$$\mathbb E_N[J]\ \le\ \bigl(\mathbb E_N[J^p]\bigr)^{1/p}q_N^{1-1/p}\quad\Longrightarrow\quad \mathbb E_N[J^p]\ \ge\ \frac{\mathbb E_N[J]^p}{q_N^{p-1}}\ \sim\ q_N^{1-p}. \tag{3.3}$$
> Contrapositive: if some $L^{1+\varepsilon}$ moment of $J_N$ can be shown uniformly bounded (in particular if jump depth admits a uniform upper bound), then H′ is false.
Attribution (to be observed strictly)
| Content |
Source |
| The telescoping structure of (3.1); density–depth reciprocity | Paper XII |
| $H'\iff q_N\to0$ | Paper XXX |
| The first-moment tension of (3.2), and that the active-jump conditional mean must diverge | Paper XXX, on the identity of Paper XII |
| The finite-scale observation that the active-jump mean is about $1.68$ | Paper XXX |
| The Hölder family of higher-moment lower bounds (3.3) | External review (Appendix C) |
| The uniform $L^{1+\varepsilon}$ refutation criterion | External review (Appendix C) |
A single unified attribution sentence. The telescoping structure of (3.1) originates in Paper XII; Paper XXX applies it to the composite conditioning of H′, establishes $H'\iff q_N\to0$ and the first-moment tension, and reports the finite-scale behaviour of the active-jump conditional mean. This paper restates all of it self-containedly in unified notation. What external review added is the Hölder-type higher-moment lower bound together with its contrapositive: if there is $\varepsilon>0$ with $\sup_N\mathbb E_N[J^{1+\varepsilon}]<\infty$, then H′ is false.
〔Grade: (3.1), (3.2) are T1 (elementary; sources as tabulated); (3.3) and its contrapositive are T1 (added by external review, restated here). Grade C checks: (3.1) is exact at $N=10^5,10^6,10^7,10^8$; among the primes $p\le10^8$ the number with $J(p)\ne0$ is $0$.〕
A locating observation (grade C). By (3.2), H′ (that is, $q_N\to0$) necessarily requires $\mathbb E_N[J\mid J>0]=\mathbb E_N[J]/q_N\to\infty$. Measured:
| $N$ |
$10^5$ |
$10^6$ |
$10^7$ |
$10^8$ |
| $\mathbb E_N[J\mid\text{composite}]$ | 1.1056 | 1.0851 | 1.0712 | 1.0611 |
| $q_N$ | 0.6503 | 0.6415 | 0.6350 | 0.6302 |
| $\mathbb E_N[J\mid J>0]$ | 1.7002 | 1.6917 | 1.6870 | 1.6837 |
〔The reading must be strict. H′ is an asymptotic statement; finite data neither support nor refute it. What the table records is that within the range examined, $N\le10^8$, no growth of the active-jump conditional mean required by H′ has been observed — $\mathbb E_N[J\mid J>0]$ decreases monotonically across four orders of magnitude, and $q_N$ falls by only about $0.005$ per decade. Whether that transition has already begun at finer scales, or will occur at larger ones, finite data cannot decide. This table agrees with the corresponding observation of Paper XXX and is not a new finding here.〕
〔Why registered here. (3.1)–(3.3) depend on neither $\lambda$, nor $\mu_{12}$, nor any calibration, and are therefore unaffected by the specification problems of §2; and the uniform $L^{1+\varepsilon}$ criterion added by external review is complementary to existing lines of attack. This paper registers only; it does not develop.〕
3.3 A specification problem: three complexity models
It emerged during preparation that the series has in fact used three different complexity models at different stages, without systematic distinction in the literature.
Table 3.1 · The three models
| Symbol |
Constant origin |
Constructors |
Costs |
Appears in |
| $\rho_{\mathcal H}$ | $\rho(0)=0$ | $S,+,\times$ | $1,1,2$ | The history-term model of the early papers |
| $\rho_{\mathcal H^\times}$ | $\rho(0)=0$ | $S,\times$ | $1,2$ | The addition-free form |
| $\rho_{\mathcal D}$ | $\rho(1)=0$ | $S,\times$ | $1,2$ | From Paper XXX onward, including this paper |
The three are not related by additive constants. Grade C check: $\rho_{\mathcal H}-\rho_{\mathcal D}$ over $n=1,\ldots,20$ runs $1,1,1,1,1,1,1,2,2,\ldots$ — it takes different values at two small integers, hence is not a constant shift. That alone suffices to require separate notation for the three models.
〔A further grade C observation: $\rho_{\mathcal H}(4^m)=6m-2$ and $\rho_{\mathcal D}(4^m)=5m-2$ agree pointwise for $m=1,\ldots,5$; were both to hold for all $m$ the difference would be $m$. No such infinitary statement is claimed here: the corresponding theorem for $\rho_{\mathcal H}$ requires a separate proof (its addition branch prevents direct transfer of the induction of Theorem 74.A), and this paper studies only $\rho_{\mathcal D}$.〕
A proof requiring attention. The early papers reduce $\rho_{\mathcal H}$ to $\rho_{\mathcal H^\times}$ by an "addition domination lemma". The existing proof contains a step of the form
$$X\le A+B,\quad B\ge C\ \Longrightarrow\ X\le A+C,$$
which is invalid (a larger upper bound cannot be replaced by a smaller one).
> Registered here: addition elimination should stand as the open conjecture $\rho_{\mathcal H}=\rho_{\mathcal H^\times}$, not as a proved lemma.
Grade C check: no counterexample for $n\le3000$. Stronger empirical evidence already exists in the antecedent literature: Observation 3.3 of Dogra (2018) reports $c_{\{1,S,+,\times\}}(n)=c_{\{1,S,\times\}}(n)$ for all $n\le4.5\times10^6$ (the charging convention differs from the P32 convention for $\rho_{\mathcal H}$, so this is not the same proposition, but the direction agrees). That paper likewise offers only a computational observation, with no proof.
Effect on this paper: none. All results here concern $\rho_{\mathcal D}$, which is defined directly by (0.1) and whose well-definedness depends on no reduction. But the notation of this paper must follow Table 3.1: what earlier literature called $\rho_{\rm Full}$ is $\rho_{\mathcal D}$ here, Full having referred to a complete factor skeleton — a different axis from complete history grammar, a distinction not previously drawn.
Handoff. Further formalization of the three models may, if wanted, be issued as a technical specification or errata document. This paper promises no numbering, form, or timetable for it (see the delimitation in §5.6).
3.4 Where things go
This paper hands off classes A, B and C of §3.2 together with the specification problem of §3.3. It designates no scheme for taking them up and claims no direction more likely to succeed than another.
Two remarks bearing directly on the conclusions here, for whoever takes them up:
- The revival condition for class A is explicit: existence of $\lambda_*$ together with a computable modulus of convergence. Both are theoretical questions; enlarging computations of the same kind has so far shown no capacity to close the gap (§2.5.2).
- $\rho_{\mathcal D}$ itself carries unmined elementary content. The results of §2.1 use only $\rho_{\mathcal D}(4)=3$ and a page of induction, and had not appeared in this series (their statements, however, are already in Dogra (2018); §2.1.0). For the methodological import of that fact see §5.5.
§4 Registered Exploratory Observations
4.0 The character of this part
This part registers a set of computations carried out during preparation. They do not constitute a route selection by this paper, and this paper takes up none of them. The purpose of registering is to make the observations directly usable by later work without redoing them.
4.1 Statement of temporal discipline
Everything below was executed before it was stated. None of it is pre-registered, and none of it carries confirmatory standing. Computing conditions: the self-built $\rho_{\mathcal D}$ table (§0.3, 33/33); items (i)–(iii) over $j=18\ldots27$, items (iv)–(vi) over $j=22\ldots27$; $N=3\times10^8$.
4.2 The observations (grade C)
(i) The mod-12 class-mean ordering $E[\rho_{\mathcal D}(p)\mid p\equiv1]10/10 blocks, with a block range of about $1.06$.
〔The ordering survives removal of $\lambda\log p$: the class-to-class spread of $\lambda E[\log p\mid r]$ amounts to $0.03\%$–$1.24\%$ of the principal effect (below $0.2\%$ for $j\ge22$). This is measured, not asserted.〕
(ii) Exact congruence facts (for $p>3$):
| $p\bmod24$ |
1 |
5 |
7 |
11 |
13 |
17 |
19 |
23 |
| $v_2(p-1)$ | $\ge3$ | $=2$ | $=1$ | $=1$ | $=2$ | $\ge3$ | $=1$ | $=1$ |
| $v_3(p-1)$ | $\ge1$ | $=0$ | $\ge1$ | $=0$ | $\ge1$ | $=0$ | $\ge1$ | $=0$ |
〔The columns for $p\equiv1,17$ give only a lower bound ($17\equiv41\pmod{24}$ while $v_2(16)=4\ne3=v_2(40)$).〕
Two negative controls, both same-signed in 10/10 blocks: $[11]-[23]=-0.1936$ (the two classes have $(v_2,v_3)=(1,0)$ exactly fixed, so this is a clean counterexample to "$(v_2,v_3)$ is a complete state variable"); and $[7]-[19]=-0.0515$ (both have $v_2=1$ but different exact $v_3$ distributions, so this refutes only "determined by $v_2$ alone").
(iii) The depth does not attenuate. Amplitudes: mod-12 range $1.0645$; mean of the four mod-24 differences $0.1778$; mean of the eight mod-48 differences $0.2343$. Depth three exceeds depth two, so the reading "$\mu_{12}$ is the depth-one truncation of a rapidly attenuating structure" does not hold. Grouped: at depth three the two differences that refine $v_2$ average $0.583$, the six that do not average $0.118$.
(iv) The currently dominant first-order state variable. After stratifying by the exact values of $(v_2,v_3)$, the residual mod-12 range averages $0.2501$ against $1.0645$ unstratified — a fall of about $76.5\%$ in that residual-range statistic.
〔Three limits: (a) this is not variance explained; (b) it must not be read as "the correct state variable" — about $23\%$ of the statistic remains, and (ii) already shows structure persisting after $(v_2,v_3)$ is exactly fixed; (c) it may be called only a strong first-order explanatory variable among the low-dimensional variables tested so far.〕
(v) Parity alternation (restricted to $v_3\ge1$; see the confounding statement). The marginal effect of $E[\rho_{\mathcal D}(p)\mid v_2]$ alternates with the parity of $v_2$: entering an even $v_2$ saves $0.53$–$0.72$, entering an odd $v_2$ costs $0.14$–$0.24$; same-signed in 8/8 steps. The residual mod-12 range likewise alternates: $0.4155$ at even $v_2$ against $0.1793$ at odd (12/12 cells).
Confounding statement (must be written). Write $p=2^{v_2}3^{v_3}m+1$ with $\gcd(m,6)=1$. When $v_3=0$ one has $p\equiv(-1)^{v_2}m+1\pmod3$, so coprimality of $p$ to $3$ forces the parity of $v_2$ to be determined by $m\bmod3$: $m\equiv1$ forces $v_2$ even, $m\equiv2$ forces it odd. Hence the four odd-part classes cannot all be populated when $v_3=0$, and the parity effect is inseparable there. The results above are restricted to $v_3\ge1$.
Source-side mechanism. The increment $c_\Delta$ of Corollary 74.A.4 (alternating $2/3$ from $m\ge2$) supplies an exact, mechanistically compatible source-side explanation of the observed shape.
〔It must not be written as a "prediction": the observation came first, the closed form second. And the transfer from the source-side closed form to prime conditional means remains unproved.〕
(vi) The second difference $\mathcal A$. For $j$, $a\ge1$, $b\ge1$ and odd-part class $r$ put
$$M_{j,a,b,r}=E\Bigl[\rho_{\mathcal D}(p)\ \Big|\ p\in I_j,\ v_2(p-1)=a,\ v_3(p-1)=b,\ \tfrac{p-1}{2^a3^b}\equiv r\ (12)\Bigr]$$
$$\mathcal A_{j,a,b,r}=M_{j,a,b,r}-2M_{j,a+1,b,r}+M_{j,a+2,b,r} \tag{4.1}$$
Over $j=24\ldots27$ ($b=1$) and $j=25\ldots27$ ($b=2$), with $a=1\ldots5$ and $r=1$ — thirty-five cells in all — the sign pattern $+,-,+,-,+$ is correct in all thirty-five. The mean $|\mathcal A|$ is about $0.75$ at $b=1$ and about $0.65$ at $b=2$: each stable, but unequal.
4.3 A three-layer placement of $\mathcal A$
| Layer |
Content |
Status |
| Unconditional fact | The definition (4.1) does not use $\lambda$ | Holds |
| Model condition | Any term affine in $a$ vanishes in a second difference | Holds (algebra) |
| Unproved interface | Whether $E[c(q)\mid j,a,b,r]$ (with $q=(p-1)/2^a$, odd) is approximately affine in $a$ | Unproved |
Full-transfer baseline. If the split is purely 2-primary and the remainder conditional mean is locally affine in $a$, then $|\mathcal A|=1$. The measured $\approx0.75$ is therefore "about $75\%$ of that model baseline".
〔Two prohibitions: it must not be written as "$75\%$ of the theoretical upper bound" ($1$ is not read off from the candidate-construction inequality alone; it needs the second condition); and the shortfall must not be attributed directly to "the 2-primary split not always being optimal" (it may equally arise from $\Delta_a^2E[c(q)]\ne0$, from conditional distributions, from channel switching, or from finite-block bias).〕
4.4 A pre-registered prediction, handed off
The following is the only quantitative prediction in this part that its author has not executed. This paper does not undertake to test it; it is registered here for later use.
P74-F (finite-block prediction). Cells: $j\in\{28,\ldots,32\}$; $b=1$; $r=1$; $a\in\{1,\ldots,5\}$; minimum sample size $400$ per cell, cells below that recorded as empty. Let $N_{\rm eff}$ be the number of non-empty cells and $e$ the number of sign errors ($\mathcal A=0$ counts as an error).
| Outcome |
Condition |
| Pass | $e\le1$ and $\overline{ | \mathcal A | }_j\ge0.55$ in every block |
| Inconclusive | $e=2$; or exactly one block with $0.50\le\overline{ | \mathcal A | }_j<0.55$; or $N_{\rm eff}\le20$ |
| Trigger | $e\ge3$; or any block with $\overline{ | \mathcal A | }_j<0.50$; or an equally weighted OLS slope of the five block means against $j$ of $\le-0.03$ per block |
What it does not test. This item does not refute "$|\mathcal A|\not\to0$". Any finite prefix is compatible with both alternatives.
〔The publication record guarantees only that this is a proposal and not a backfill. Neither its trigger nor its passage constitutes an adjudication of any research route — this paper selects none.〕
4.5 Limits
First, the shape of the observations in this part was revised three times during preparation, each revision after inspecting the same body of exploratory data. That fact lowers their confirmatory standing. An earlier draft proposed a "first delivery" on the basis of that shape; the proposal is withdrawn along with the change in this part's standing, while the observations themselves are retained.
Second, the source-side mechanism of (v) explains only half. The alternation of $c_\Delta$ is a theorem (Corollary 74.A.4); the transfer from that closed form to prime conditional means is unproved, and six definitional gaps remain untreated (definition of the event, counting of tied optima, node scope, the averaging convention, tightness of the upper bound, and $\Delta_a^2E[c(q)]$).
Third, the recursive residual structure of (iv) persists: the $[11]-[23]$ of (ii) remains nonzero after $(v_2,v_3)$ is exactly fixed.
Fourth, this part does not explain the $20\%$ contribution of $\mu_{12}$ to $F_W$ (class means are a first-moment quantity, the $F_W$ contribution a spectral one). P73 §11.1 item 5 is not closed.
Fifth, the term "Pratt tree" is not used. A Pratt certificate is a primality certificate built around a prime; a $\rho$-optimal construction certificate may split composite factors, may use successor edges, and may select a different tree. Until a formal mapping is established the two must not be identified.
§5 Conclusion and Disposition
5.1 What is established unconditionally
Table 5.1 · T1 deliveries
| Item |
Statement |
Frozen premises |
| Theorem 74.A | $c(n)\ge\alpha\log n$, $\alpha=5/\log4$ | P32 convention; $n\ge1$ |
| Theorem 74.A′ | Equality exactly at $4^k$ | $k\ge1$ |
| Corollary 74.A.1 | $c(4^k)=5k$; $\liminf\rho_{\mathcal D}/\log n=5/\log4$ | $k\ge1$ |
| Corollary 74.A.2 | Capacity constant $\gamma=\log4/5=1/\alpha$ | $K,L\ge2$; eventually superadditive Fekete |
| Corollary 74.A.2′ | $A(5k)=4^k$ | $k\ge1$ |
| Corollary 74.A.3 | $c(2^m)=\lceil5m/2\rceil$ | $m\ge1$ (three cases) |
| Corollary 74.A.4 | Increments $1/2/3$ ($m=1$ exceptional) | as above |
| Corollary 74.B | If the full-$n$ limit exists it equals $5/\log4$ | existence assumed |
| Proposition 74.0 | Arithmetic/statistical factorization of $R_{\rm wt}$ | $D_j\ne0$ |
| Lemma 74.C.0 | $\rho_{\mathcal D}(n)=O(\log n)$ | elementary |
| Proposition 74.C | With $\mu_{12}$ fixed, at most one $\lambda$ | PNT |
| Proposition 74.D | Well-posed $\mu_{12}$ forces a unique slope | PNT for arithmetic progressions |
| (0.2) | $\rho_{\mathcal D}(p)=\rho_{\mathcal D}(p-1)+1$ | P32 convention only |
Statement of originality (§2.1.0). By (0.0), $\rho_{\mathcal D}$ differs from the shortest-representation complexity of Dogra (2018) by the constant $1$, and that paper already reports the lower bound and the equality case. No originality is claimed for any of Theorem 74.A through Corollary 74.A.4; what is offered is an independent proof in the min-plus normalization. Nor is originality claimed for the closed form $c(2^m)$ — failure to find it in a search is not evidence of priority.
Four items deserve separate mention (for their mathematical content, not as claims of originality).
Theorem 74.A′ characterizes the equality set completely as $\{4^k\}_{k\ge1}$; together with $A(5k)=4^k$ this closes the first group: a sharp global lower bound, the complete equality set, exact capacity points, and a closed form on powers of two, forming one theorem package.
Corollaries 74.A.3–74.A.4 give the complete closed form of $\rho_{\mathcal D}$ on powers of two. The $2/3$ alternation of the increments has a direct mechanical reading: $2^{\text{even}}$ divides into whole copies of $4$, and $c(4)/2=2.5$ is the least cost per doubling; $2^{\text{odd}}$ necessarily leaves an isolated $2$, at cost $3$.
Corollary 74.A.2 (capacity constant) is thus far the only real invariant of this series produced by the recursion's own definition without any fitted constant. Its existence has an independent source (superadditivity of $A(K)$ plus Fekete); its exact value comes from the sharp lower bound, so it is the capacity-dual formulation of one result rather than an independent proof. With Corollary 74.A.2′ the capacity points at $K=5k$ are exact.
Propositions 74.C and 74.D reach one class of conclusion from two directions: the two centering constants of 59.1 cannot be specified independently. The former constrains $\lambda$ given $\mu_{12}$; the latter constrains $\lambda$ from the definition of $\mu_{12}$. Both are unconditional.
5.2 What the specification audit shows
The current specification of 59.1 demands a centering far beyond the resolution of accessible data.
- By §2.3.4, inverting the published measurements gives a required normalized precision $|\varepsilon|/\sqrt{Q^{(2)}_{32}}\lesssim3.0\times10^{-5}$;
- By §2.5.2, the attainable extrapolated precision on $\lambda$ within the present window and model family is about $\pm0.05$, and three equally plausible drift models cannot be separated on that data;
- By §2.6, $\mu_{12}$ under the definition of Paper L §2.2 is four limits of unproved existence, whose well-posedness forces $\lambda$ to be unique — and that unique $\lambda$ is exactly the unidentifiable one;
- By §2.5.3, the published text supplies neither the four $\mu_{12}$ values, nor error bounds, nor a designated estimation procedure, so its numerical table cannot be reproduced operationally from the published text alone;
- By §2.5.4, the position of the deep trough in the $R_{\rm wt}$ profile is controlled at least in part by the calibration cutoff.
The gap is not a matter of computing power. $\lambda$ and the drift model are entangled in the data; separating them requires knowing the second-order term of $\rho_{\mathcal D}(p)$, which is a theoretical question.
The resulting classification (§2.7): $59.1_{\rm pub}$ is partially defined; $59.1_{\rm pub}^{\exists}$ is complete but of higher quantifier complexity; $59.1_{\rm calib}(N)$ is completely determined but indexed by $N$; $59.1_{\rm freeze}(\vec q)$ is its rational truncation; $59.1_*$ has no center yet defined. The five are not the same proposition, and the previous literature drew no such distinction.
5.3 What remains open
- Existence of the prime-subsequence limit $\lim_p\rho_{\mathcal D}(p)/\log p$. This is precisely the limit on which the definitions of $\eta$ and $\mu_{12}$ depend. The method of Theorem 74.A is unavailable: its factor step degenerates to a vacuous truth at primes. Compare the still-separated liminf and limsup for integer complexity $\|n\|$.
- Whether the four limits of (2.16) exist. Proposition 74.D gives only "if they exist, $\lambda$ is unique"; the converse is unproved (the counterexample of §2.6.3 shows that existence of a slope does not entail existence of the intercept limit).
- The $20\%$ contribution of $\mu_{12}$ to $F_W$ (P73 §11.1 item 5). This paper finds only a candidate origin for the class-mean structure; the item is not closed.
- The three unexplained empirical objects left by P50–P73: $c_0,c_t$ and the ratio $9.87$; the two-axis $0.30$; the non-stationarity of the ACF. Not treated here.
- The transfer problem of §4. Attackable as an empirical programme; not attackable as a foreseeable route to a theorem (§4.3).
5.4 Disposition
First, a clarification request to the upstream literature (Appendix A): one item only — the exact values of the four $\mu_{12}$ of Paper L §2.2 at $N=10^{10}$. The definition of $\lambda$ and the procedure for computing $\mu_{12}$ are settled by A.1.1–A.1.2. That item affects exact reproduction and the freezing of an instance, not the location of any earlier discrepancy (the roughly six-unit difference in $\bar\mu$ is explained by the $c=\rho_{\mathcal D}+2$ convention; A.1.3). No other conclusion here depends on it.
Second, an upstream erratum is registered (§0.4): in the P32 column of Appendix A of Paper L, $\rho_{\mathcal D}(100)$ should be $15$ (reported $19$) and $\rho_{\mathcal D}(1000)$ should be $24$ (reported $28$). The discrepancy is not explained by a difference in factor cost.
Third, instrument verification is established as a standing check (§0.3, Appendix D): the anchor check and the recursion-invariant check, recommended unconditionally after any new table is built. The latter is the test that refuted the 1860 truncation at the time.
Fourth, no route in §4 is undertaken. That part is registered as exploratory; its single pre-registered item, P74-F, is a finite-block prediction whose trigger does not entail failure of any route (§4.4).
Fifth, $59.1_{\rm freeze}$ is not sealed. It is precisely defined, directly attackable, and unobstructed by unidentifiability. This paper simply does not choose it — a budgetary judgement, not a logical conclusion.
5.5 Methodological register
First, four occurrences of one failure mode (§1.5, Table 1.3): two transcription errors and one omitted instrument check within the manuscript, plus one omitted literature check at the stage level. Neither transcription error was a computational error — the underlying output was correct, and the fault lay in the step from computed output to tabulated statement, where the data were read as supporting a cleaner assertion than they did. These are the third through sixth instances of the failure mode behind causes of death 1 and 2.
Supplement: the fourth is at the stage level. From the adoption of the $\mathcal D$ model onward the direct antecedent went unidentified, that function differing from Dogra's $d$ by the constant $1$. Instrument verification and literature verification are two directions of one activity — the first asks is what I am computing the thing this series has been computing?, the second is the thing this series has been computing something already computed elsewhere? This series established the first and not the second. It is recommended that any new object undergo one literature verification before its second paper, and that this be placed alongside instrument verification as a standing check.
Second, the difference in detection mechanism deserves separate record. The first two (causes of death 1 and 2) each consumed a full paper's work; the first three of Table 1.3 were caught during the preparation of this one, the fourth by external search. The difference lies not in the errors but in whether external review existed and whether the self-check was performed.
Third, this paper introduces grade C (certified computation / finite empirical result). Under the three grades fixed by the methodology papers, a finite-sample computation is neither a mathematical fact (T1), nor a reframing of established results (T2), nor an internal identification proposal (T3) — the three leave no slot for it. Forcing it into T2 would hollow out the criterion "mathematical content not new, organization new"; forcing it into T1 would hollow out "has a proof". Hence a separate grade.
Fourth, the shape of §4 was revised three times, each time after inspecting the same body of exploratory data (§4.5). That fact lowers its confirmatory standing, and it is registered accordingly as a post hoc candidate regularity. The number of revisions is not a calibrated reliability measure; what it discharges is disclosure of the research path.
Fifth, a reusable conclusion. The technique behind every T1 delivery of this paper was available throughout the preceding seventy-three papers. The induction of Theorem 74.A needs only $\alpha\log\frac54<1$ and additivity of the factor candidates; its base case needs only $\rho_{\mathcal D}(4)=3$, a three-line computation. The three cases of Corollary 74.A.3 are two lines each. Proposition 74.D is a subtraction plus the prime number theorem for arithmetic progressions. None required a tool this series did not already have.
They went unfound not because they were difficult, but because the attention of twenty-four papers was on the statistical structure of the residual and never returned to the recursion's own definition.
> The reusable form. When a line of research repeatedly relocates the same obstruction (§1.2, the frontier loop), the return is more likely to lie in re-examining the definition of the object than in searching for a better tool. Both groups of results here, and the specification problem discovered in §3.3, arose that way.
〔Grade T3 (a methodological register; not admissible as a premise of any mathematical argument).〕
5.6 Delimiting the stage closure
> ZFCρ closes a stage at Paper LXXIV.
Three components of that phrase must be written out, so that it is read neither as a promise nor as a termination.
First, no sequel is promised. This paper does not announce Paper LXXV, sets no timetable for any open question, and incurs no obligation toward whoever takes them up by virtue of having handed them off. The handoff list of §3.2 is not to be read as an outline for a sequel.
Second, no termination is declared. Should a result appear that makes one of the questions of §3.2 attackable again — for instance the existence of $\lambda_*$ together with a computable modulus of convergence, which would revive class A in full — the series may be continued upon it. A closure settles the accounts of the current stage; it is not a verdict on the object.
Third, neither continuation nor discontinuation alters the standing of this paper. Its T1 deliveries (Table 5.1) are produced by the recursion (0.1) itself and do not depend on anyone taking up the handoff; the same holds for the specification audit of §2. The completeness of this paper is not conditional on a sequel.
〔Why not "termination". Termination would be a stronger judgement than the evidence supports. What is established here is that routes depending on the unknown true center $59.1_*$ are currently inexecutable and that the specification requires respecification ($59.1_{\rm freeze}$ remains directly attackable; §2.7, §3.2) — not that the object is without content. The theorem package of §2.1 shows precisely that the latter is false.〕
What is closed and what is not. This paper closes the stage of P50–P73, in which 59.1 served as the current target of closure. It does not declare $\rho_{\mathcal D}$, or the weighted construction-cost problem generally, to be exhausted.
5.7 One closing sentence
Papers L through LXXIII, twenty-four in all, placed the whole of their effort on a target whose specification had not been fixed. This paper shows that the current form of that target demands a centering far beyond the resolution of accessible data; and that the same recursion carries exact, unconditional, statable arithmetic content — the package formed by Theorems 74.A, 74.A′ and Corollaries 74.A.1–74.A.4, 74.A.2′ — which those twenty-four papers never touched (though its statements were already in Dogra (2018); §2.1.0).
Delivery Index
| Part |
Content |
Load-bearing |
| §0 | Object, notation, 33 anchor checks, one upstream erratum | Yes (premises) |
| §1 | Closing ledger: one unconditional load, a frontier loop, five causes of death, four unexplained objects, four occurrences of one failure mode | Yes (register) |
| §2 | Theorem 74.A package; specification audit (74.0, 74.B, 74.C, 74.D); five versions of 59.1 | Sole principal load |
| §3 | Handoff: what is closed, the A/B/C classification, the three-model specification problem, a conditional refutation criterion for H′ | No |
| §4 | Registered exploratory observations; three-layer placement of $\mathcal A$; the handed-off pre-registration P74-F | No |
| §5 | Conclusion and disposition | — |
| Appendices A–D | Upstream clarifications; arithmetical complexity placement; external contributions; verification, scripts and specifications | — |
What This Paper Does Not Claim
Collected here, per the registration discipline of this series, are the items declared throughout the text as not to be concluded from this paper.
- That any cause of death in Table 1.1 is in error or requires re-examination
- That the technical content of Papers LXXI–LXXIII is in error
- That the $20\%$ contribution of $\mu_{12}$ to $F_W$ has been explained
- The correctness of the standard-IC column of Appendix A of Paper L
- That $\lim_p\rho_{\mathcal D}(p)/\log p$ exists
- Any specific standing for $\lambda_{\rm FullIC}$
- Numerical identity between $\lambda-h_0$ and the prime penalty
- That the three-row reading of (2.9) is a consequence of the identity
- That the numerical values of Paper LIX are in error
- That different $\vec q$ give different truth values
- That $\mu_{12}$ exists if and only if $\lambda=\lambda_*$ (sufficiency unproved; §2.6.3)
- That the four classes have different asymptotic slopes
- That $59.1_{\rm freeze}(\vec q)$ is unattackable
- That Table 2.2 is a controlled experiment under one specification
- That 59.1 has been refuted, or that its subject is indeterminate
- That the existence of $\lambda$ has been proved
- That the origin of $\mu_{12}$ has been identified
- That P73 §11.1 item 5 is closed
- That $p\bmod24$ determines $v_2(p-1)$
- That the source-side $2/3$ alternation has been transferred to a law for prime conditional means
- That $|\mathcal A|=1$ is a theoretical upper bound
- That the shape of $\mathcal A$ has undergone independent testing
- That prime-side structure was predicted (observation preceded the closed form; it is a source-side mechanistic explanation)
- That addition elimination is false (only that its existing proof fails)
- That the distinction between $\rho_{\mathcal H}$ and $\rho_{\mathcal D}$ affects any conclusion here
- That any successor route has been selected
- That the observations of §4.2 constitute evidence for any direction
- That "agreement among several parties" constitutes grounds for any conclusion
- That routes depending on the unknown center have been refuted (only their current executability is sealed)
- That the route selected here is more likely to succeed (§4.5 gives reasons, not probabilities)
References
- A. S. Dogra, Minimal Representations of Natural Numbers Under a Set of Operators, arXiv:1801.01360 (2018). — Shortest symbolic representations over $\{1,S,\times\}$; its complexity $d$ satisfies $d=\rho_{\mathcal D}+1$ (equation (0.0)). The antecedent for every result in §2.1.
- A. P. Matos, Kolmogorov complexity in multiplicative arithmetic, Technical Report, DCC-FCUP (2015). — Cited by Dogra as the source of an equivalent formulation of the equality characterization. Not consulted here.
- E. K. Gnang, M. Radziwiłł, C. Sanna, Counting Arithmetical Formulas, European J. Combin. 47 (2015), 40–53. — Counting of arithmetical formulas; Dogra's Theorem 1.4 generalizes it. Directly relevant to the history-fibre direction handed off in §3.3.
- A. S. Dogra, Optimal Presentations of Mathematical Objects, arXiv:1812.00972 (2018).
- H. M. Bahig, On a generalization of addition chains: Addition–multiplication chains, Discrete Math. (2008). — The addition–multiplication chain model. Key difference from the object here: chain models permit reuse of intermediate results (a DAG), whereas the factor candidates here form an expression tree and a subexpression used twice must be paid for twice. Example: $1\to2\to4\to16$ takes three steps in a chain model, while $\rho_{\mathcal D}(16)=8$.
- Mahler–Popken integer complexity $\|n\|$ (operator set $\{1,+,\times\}$, counting leaves). Not equivalent to $\rho_{\mathcal D}$: arbitrary $a+b$ is permitted and multiplication carries no separate charge.
- P. Koiran, $\tau$-complexity and arithmetic circuit lower bounds. — The circuit version, permitting reuse.
- R. Lipton, D. Dobkin, Complexity measures and hierarchies for the evaluation of integers and polynomials (1976). — An early source for the idea that constructing constants itself carries cost.
〔Positioning. Items 5–8 are different models from $\rho_{\mathcal D}$ (reuse permitted or not; arbitrary addition or not; leaves counted or operators counted) and are listed to locate the object of this paper within the existing taxonomy, not as direct antecedents. Only item 1 differs from the object here by a constant; items 2 and 3 are respectively the source of the equality characterization and an antecedent for the history-fibre direction.〕
Appendices A–D follow the Chinese authoritative version and are reproduced there in full: A, upstream clarifications and residual requests; B, arithmetical complexity placement ($\Sigma^0_2$ placement, the family $59.1^\flat(C)$, Proposition 74.3, and $59.1^\sharp$); C, registration of external contributions; D, the 33-item anchor table, reference implementations, checksums, and machine-readable attachments.