ZFCρ Paper LXXI: Linear Signed Abel Budget and the Saturated Zero-Mode Target
ZFCρ 第LXXI篇：线性 Signed Abel 预算与饱和零模目标
Paper 70 reduced Conjecture 59.1 to an equivalent bounded-phase statement on saturated skeletons, $$\sup_j \frac{B_j(\rho_{S_{D^*(j)}})^2}{j \cdot D_j(\rho_{S_{D^*(j)}})} < \infty,$$ with $D^*(j) = \lceil \sqrt{2^{j+1}} \rceil$. The present paper identifies the microscopic mechanism behind this saturated zero-mode target and corrects an interpretation of an intermediate memory coefficient that has been used in several earlier papers. Empirical measurements across $N = 10^7, 10^8, 10^9$ (with $\pi(10^9) \approx 5.08 \times 10^7$ primes) yield three structurally stable invariants and one phenomenon requiring requantification. First, the lag-1 autocorrelation of raw $\rho(q)$ on consecutive same-layer primes is $\approx 0.96$ across 25 layers; this number was previously interpreted as an ancestor inheritance rate, but the present paper shows that approximately 9/10 of its contribution comes from the $\lambda \ln q$ trend term, with the true detrended same-layer residual autocorrelation only about $0.10$ and converging across $N$. Second, the ZFCρ DP recursion is a discrete single-step update with no continuous-time limit. Data strongly support a polynomial / resolvent-type kernel candidate: polynomial fits dominate exponential fits in all comparable settings, and the Bartlett triangular envelope $A_j^{\rm abs}(W) \asymp W^{\beta_{\rm abs}}$ exhibits $\beta_{\rm abs} \approx 0.50$ stably across $j$, $N$, and centerings; the precise mathematical derivation of the kernel form is left as Candidate 71.K-abs. Third, under $j$-smooth adjacent reference centering, the signed Bartlett budget $A_j^{\rm signed}(W_f)$ (where $W_f = m_j / 2$ is the non-vacuous theorem scale) is not strictly bounded: its growth exponent $\beta_{\rm signed}$ is negative for small $j$, becomes positive at $j \ge 22$, and grows slowly to about $+0.29$ at $j = 29$. However the normalized budget $A_j^{\rm signed}(W_f) / j$ remains in $[0.07, 0.89]$ across $j = 14$ to $29$, comparable in order of magnitude to the Paper 59 range $R_{\rm wt} \in [0.79, 25.3]$. This identifies the precise intermediate scale required by 59.1: not strict signed Abel boundedness, but the linear budget $$\boxed{A_j^{\rm signed}(W_f) \le C j}$$ (where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale; see §1.4 + §7.3 (iii)). The linear signed Abel budget makes precise Paper 65's remaining technical core $\Psi_\eta = O(j)$. Conditional on Paper 65's count and gauge controls, satisfaction of the linear signed Abel budget (together with a deterministic Abel transfer inequality) yields the residual spectral estimate for 59.1 (Criterion 71.E, conditional). The paper does not prove 59.1; nor does it prove that the single-step kernel takes a specific polynomial form. Two open theorem candidates (71.K-abs, 71.S-signed) and one framework-internal candidate (71.A-deterministic) mark the remaining mathematical content; their full establishment, together with unconditional treatment of Paper 65's count and gauge conditions, would close 59.1 (§8.4). ---
Abstract
Paper 70 reduced Conjecture 59.1 to an equivalent bounded-phase statement on saturated skeletons,
$$\sup_j \frac{B_j(\rho_{S_{D^(j)}})^2}{j \cdot D_j(\rho_{S_{D^(j)}})} < \infty,$$
with $D^*(j) = \lceil \sqrt{2^{j+1}} \rceil$. The present paper identifies the microscopic mechanism behind this saturated zero-mode target and corrects an interpretation of an intermediate memory coefficient that has been used in several earlier papers.
Empirical measurements across $N = 10^7, 10^8, 10^9$ (with $\pi(10^9) \approx 5.08 \times 10^7$ primes) yield three structurally stable invariants and one phenomenon requiring requantification. First, the lag-1 autocorrelation of raw $\rho(q)$ on consecutive same-layer primes is $\approx 0.96$ across 25 layers; this number was previously interpreted as an ancestor inheritance rate, but the present paper shows that approximately 9/10 of its contribution comes from the $\lambda \ln q$ trend term, with the true detrended same-layer residual autocorrelation only about $0.10$ and converging across $N$. Second, the ZFCρ DP recursion is a discrete single-step update with no continuous-time limit. Data strongly support a polynomial / resolvent-type kernel candidate: polynomial fits dominate exponential fits in all comparable settings, and the Bartlett triangular envelope $A_j^{\rm abs}(W) \asymp W^{\beta_{\rm abs}}$ exhibits $\beta_{\rm abs} \approx 0.50$ stably across $j$, $N$, and centerings; the precise mathematical derivation of the kernel form is left as Candidate 71.K-abs. Third, under $j$-smooth adjacent reference centering, the signed Bartlett budget $A_j^{\rm signed}(W_f)$ (where $W_f = m_j / 2$ is the non-vacuous theorem scale) is not strictly bounded: its growth exponent $\beta_{\rm signed}$ is negative for small $j$, becomes positive at $j \ge 22$, and grows slowly to about $+0.29$ at $j = 29$. However the normalized budget $A_j^{\rm signed}(W_f) / j$ remains in $[0.07, 0.89]$ across $j = 14$ to $29$, comparable in order of magnitude to the Paper 59 range $R_{\rm wt} \in [0.79, 25.3]$.
This identifies the precise intermediate scale required by 59.1: not strict signed Abel boundedness, but the linear budget
$$\boxed{A_j^{\rm signed}(W_f) \le C j}$$
(where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale; see §1.4 + §7.3 (iii)). The linear signed Abel budget makes precise Paper 65's remaining technical core $\Psi_\eta = O(j)$. Conditional on Paper 65's count and gauge controls, satisfaction of the linear signed Abel budget (together with a deterministic Abel transfer inequality) yields the residual spectral estimate for 59.1 (Criterion 71.E, conditional).
The paper does not prove 59.1; nor does it prove that the single-step kernel takes a specific polynomial form. Two open theorem candidates (71.K-abs, 71.S-signed) and one framework-internal candidate (71.A-deterministic) mark the remaining mathematical content; their full establishment, together with unconditional treatment of Paper 65's count and gauge conditions, would close 59.1 (§8.4).
§1. Introduction: From Paper 70 to Paper 71
1.1 The terminal observable from Paper 70
Paper 70 Theorem 70.A is a deterministic reduction. For each dyadic block $I_j = [2^j, 2^{j+1})$, when $D \ge \sqrt{2^{j+1}}$ the full IC recursion $\rho_{\rm Full}$ and the truncated recursion $\rho_D$ agree pointwise on $[1, 2^{j+1})$. With $D^*(j) = \lceil \sqrt{2^{j+1}} \rceil$, Conjecture 59.1 on the spectral block ratio $R_{\rm wt}(j)$ reduces to the equivalent statement on saturated skeletons:
$$\sup_j R_j^(N) < \infty, \qquad R_j^(N) := \frac{B_j(\rho_{S_{D^(j)}})^2}{j \cdot D_j(\rho_{S_{D^(j)}})}.$$
The reduction replaces the abstract object $\rho_{\rm Full}$ in 59.1 by the specific finite skeleton attached to each block, of size $\sim \sqrt{2^j}$.
1.2 What Paper 70 did not do
Paper 70 Theorem 70.A is a reduction, not a proof. 59.1 remains open. Paper 70 §8 lists three routes to a bounded-phase statement (spectral ratio log budget, direct second moment, spectral flatness), with route B identified as the most likely clean closure path.
1.3 What this paper does
The present paper does not pursue any of those three routes directly. It does four things.
(i) Corrects a false target. Paper 55 reported $C(1)_{\rho(q)} \approx 0.96$ for the same-layer raw lag-1 ACF, an interpretation propagated through several subsequent papers as an ancestor inheritance rate. Cross-$N$ measurements show this number is dominated by the $\lambda \ln q$ trend; the true detrended residual autocorrelation is about $0.10$. Any derivation depending on the "$0.96$ as retention coefficient" reading needs revision.
(ii) Identifies the kernel form of the single step. ZFCρ DP is a discrete single-step recursion. The data show ACF tails of the χ-field are clearly polynomial rather than exponential. This is consistent with the discrete structure of ZFCρ.
(iii) Requantifies the remaining open core of Paper 65. Paper 65 Theorem 65.A reduces 59.1 to four conditions, three of which are conditionally controlled; the remaining open core is the centered-residual spectral bound $\Psi_\eta = O(j)$. The present paper makes this precise as the linear signed Abel budget $A_j^{\rm signed}(W_f) \le C j$ (where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale; see §1.4 + §7.3 (iii)).
(iv) Gives a conditional criterion. Criterion 71.E derives 59.1 from the Paper 65 conditional architecture together with the linear signed Abel budget and a deterministic Abel transfer inequality.
The paper identifies the mechanism but does not prove it. Subsequent work needs to: derive the polynomial / resolvent kernel envelope (71.K-abs); prove the linear signed Abel budget (71.S-signed); render the Paper 65 conditions fully unconditional; and establish the deterministic Abel transfer inequality (71.A-deterministic). The latter three together form the three independent components of the unconditional closure pathway, detailed in §8.4.
1.4 Notation
Following Papers 50 through 70.
Stripped residual: $h(p) = \rho(p) - \lambda \ln p - \mu_{12}(p \bmod 12)$, with $\lambda = 3.856763$ (Paper 50 anchor), $\mu_{12}(\cdot)$ the conditional mean over residue classes mod 12.
Normalized residual: $x(p) = h(p) / p$.
Dyadic block: $I_j = [2^j, 2^{j+1})$.
Dyadic layer: For prime $p > 2$, the layer index $\ell = v_2(p - 1)$ is the 2-adic valuation of $p - 1$. Same-layer consecutive primes refers to consecutive primes within layer $\ell$ ordered by $p$. The odd part of that layer is $q = (p - 1) / 2^\ell$.
χ-field: $\chi(p) = h(p)$, the stripped residual; in the layer-$\ell$ context one also considers $\chi(q) = \rho(q) - \lambda \ln q - \mu_{12}(q \bmod 12)$ (the stripped residual on the odd part).
Spectral block quantities: $B_j = \sum_{p \in I_j} x(p)$, $D_j = \sum_{p \in I_j} x(p)^2$.
Spectral block ratio: $R_{\rm wt}(j) = B_j^2 / (j D_j)$.
$j$-smooth adjacent reference centering: For the $\chi$-field on consecutive same-layer primes inside block $I_j$, the residue class mean used as the centering reference is the average of the residue class means in the adjacent blocks $I_{j-1}$ and $I_{j+1}$, rather than the residue class mean within $I_j$ itself.
Bartlett triangular Abel budget: For a $\chi$-sequence $\{x_i\}$ over a window of size $W$,
$$A^{\rm abs}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) |\gamma(k)|, \qquad A^{\rm signed}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) \gamma(k),$$
where $\gamma(k)$ is the sample autocorrelation $\hat{\gamma}(k) / \hat{\gamma}(0)$. Within block $I_j$ two window scales appear:
- $W_f = m_j / 2$, the half-block scale, which serves as the non-vacuous theorem scale for the (H4a) statement; data are stable across $j$ and $N$ and carry signed cancellation information at this scale.
- $W_{\rm full} = m_j := |\{p : p \in I_j\}|$, the total prime count, a boundary scale. On sample-mean-centered $\gamma$, $A^{\rm signed}(W_{\rm full})$ degenerates to a finite-sample identity ($\equiv 0$ on adjacent-centered sequences in floating-point precision), carrying no mechanism information; this boundary identity is consistent with the Spectral Bridge Front geometry articulated in Paper 72, see §7.3 (iii).
Numerically $A^{\rm signed}(W_f) / j$ remains in $[0.07, 1.20]$ across $j = 14$ to $29$ at $N = 10^9$ (§5 Tables 7-8); this is the empirical anchor for (H4a). The precise transfer is discussed in §7.3 (iii) and (iv).
§2. Correcting a false target: raw $C(1)$ is not a retention coefficient
2.1 A long-cited number
Several passages in Papers 55 through 65 take the same-layer raw $\rho(q)$ lag-1 autocorrelation $C(1)_{\rho(q)} \approx 0.96$ to be an ancestor inheritance rate, writing
$$C(1)_{\rho(q)} = \text{fraction of ancestor factor structure shared between consecutive same-layer primes.}$$
This reading then treats $1 - C(1) \approx 0.04$ as a per-layer innovation rate, leading to a derived figure of "approximately 4% innovation per layer." That figure has been used in several subsequent derivations as an intermediate constant.
The present paper shows this reading does not hold against the data.
2.2 Cross-$N$ measurements: raw versus detrended
For each $N \in \{10^7, 10^8, 10^9\}$, ZFCρ Full IC arrays were built using the Paper 70 builder (with FNV-1a checksums preserved). For each layer $\ell$, three quantities are computed on the sequence of consecutive same-layer primes:
(i) Raw $\rho(q)$ lag-1 autocorrelation: directly on the $\rho(q)$ values.
(ii) Detrended $\rho(q) - \lambda \ln q$ lag-1 autocorrelation.
(iii) Stripped $\chi(p) = \rho(p) - \lambda \ln p - \mu_{12}(p \bmod 12)$ lag-1 autocorrelation.
Table 1. Same-layer lag-1 autocorrelation across the three formulations, on the $N = 10^9$ data (which provides the broadest layer coverage).
| $\ell$ | $n_p$ ($N = 10^9$) | raw $\rho(q)$ | $\rho(q) - \lambda \ln q$ | $\chi(p)$ |
|---|---|---|---|---|
| 1 | 25,424,042 | 0.9587 | 0.0596 | 0.0982 |
| 2 | 12,712,271 | 0.9655 | 0.1053 | 0.1318 |
| 3 | 6,355,931 | 0.9598 | 0.0598 | 0.0923 |
| 5 | 1,589,237 | 0.9582 | 0.0471 | 0.0991 |
| 7 | 397,158 | 0.9596 | 0.0442 | 0.0861 |
| 9 | 99,234 | 0.9582 | 0.0161 | 0.0912 |
| 12 | 12,433 | 0.9680 | 0.0488 | 0.0975 |
| 15 | 1,527 | 0.9576 | $-0.0721$ | 0.0082 |
| 19 | 101 | 0.9380 | $-$ | 0.0615 |
The raw column sits stably at $0.94$ to $0.97$ across all 19 layers with sufficient sample, exactly reproducing the Paper 55 report at $N = 10^{10}$ and replicating it across two orders of magnitude in $N$. The detrended column drops by roughly a factor of 16, and the stripped $\chi$ column likewise stays at the $0.10$ scale rather than the $0.96$ scale.
Cross-$N$ convergence:
| $\ell$ | $\chi(p)$ ACF $N = 10^7$ | $N = 10^8$ | $N = 10^9$ | $\Delta_{8 \to 9}$ |
|---|---|---|---|---|
| 1 | 0.0905 | 0.0972 | 0.0982 | $+0.0010$ |
| 3 | 0.0803 | 0.0901 | 0.0923 | $+0.0022$ |
| 5 | 0.0783 | 0.0942 | 0.0991 | $+0.0049$ |
The stripped $\chi$ lag-1 autocorrelation has largely saturated at the $0.10$ scale by $N = 10^9$, with cross-$N$ convergence within a few percent.
2.3 What the data show
Three observations.
(i) The raw $\rho(q)$ autocorrelation $\approx 0.96$ is the trend synchrony of $\lambda \ln q$. Consecutive same-layer primes have similar $q$ magnitudes, so the $\lambda \ln q$ contribution synchronizes between them. This synchrony is independent of the ρ-recursion itself.
(ii) After stripping the trend, the true same-layer ρ-residual autocorrelation is about $0.10$. This is the actual retention coefficient of ZFCρ DP at the $\chi$-layer.
(iii) $1 - 0.96 \approx 0.04$ cannot be read as an innovation rate. The actual innovation rate in the $\chi$-layer is $1 - 0.10 \approx 0.90$, i.e., over nine tenths is innovation rather than inheritance.
This terminates the early reading that identified raw $C(1)_{\rho(q)}$ directly with a retention coefficient. The earlier algebraic identity $m_{\rm eff} = \ln r / \ln C(1)$ remains a formal identity, but the $C(1)$ entering it should be understood as the raw same-layer lag-1 ACF (dominated by trend), not as a retention quantity.
Subsequent sections work with the true stripped objects $\chi(p)$ and $\chi(q)$, not with the raw $\rho(q)$ autocorrelation.
§3. The discrete single-step kernel: K=1 resolvent form
3.1 ZFCρ DP as discrete single-step closure
The ZFCρ DP recursion
$$\rho(n) = \min(\rho(n-1) + 1, M(n))$$
updates one integer at a time. There is no continuous time parameter and no $dt \to 0$ limit. On the χ-layer, the update between consecutive same-layer primes is a single-step object, completed in one move.
The natural kernel form for discrete single-step closure lies in the polynomial kernel class. The resolvent kernel $(1 + x)^{-1}$ is one specific candidate within this class: given a one-step feedback structure $S + xS = S_0$, the solution is $S = S_0 / (1 + x) = S_0 \cdot (1 + x)^{-1}$. Its low-frequency expansion
$$(1 + x)^{-1} = 1 - x + x^2 - x^3 + \cdots$$
has second coefficient $1$. By contrast, the exponential kernel $e^{-x}$ arising from a continuous-time limit has expansion $1 - x + x^2/2 - \cdots$ with second coefficient $1/2$. The two kernel families agree at first order but separate at second order.
The specific polynomial form of the ZFCρ single-step kernel remains an open theorem candidate (71.K-abs, §8.1). The empirical evidence in §3.2 and §3.3 supports a polynomial form robustly, but does not by itself distinguish the strict resolvent $(1 + x)^{-1}$ from a higher-order rational form. Throughout, "$K = 1$ resolvent kernel form" is used as a candidate label without commitment to a specific operator form.
3.2 Data: polynomial tail rather than exponential tail
For each layer $\ell$ and each centering, the lag-$k$ autocorrelation ACF$(k)$ on consecutive same-layer primes is computed at $k = 1, 2, 3, 5, 8, 12, 16, 24, 32, 48, 64$. On the lags with ACF$(k) > 0.005$ two functional forms are fit:
(i) Exponential: ACF$(k) = e^{-k/\tau}$.
(ii) Polynomial single-step resolvent: ACF$(k) = 1 / (1 + k / \tau)$, equivalently $1 / \text{ACF}(k) = 1 + k / \tau$.
Table 2. Goodness-of-fit $R^2$ comparison across $N$ and centering, on the highest-statistics layer $\ell = 1$.
| centering | $N$ | $R^2_{\exp}$ | $R^2_{\rm poly}$ | winner |
|---|---|---|---|---|
| $\mu_{12}$ | $10^7$ | 0.918 | 0.978 | poly |
| $\mu_{12}$ | $10^8$ | 0.864 | 0.946 | poly |
| $\mu_{12}$ | $10^9$ | $-$ | $-$ | poly |
| adjacent | $10^7$ | 0.927 | 0.976 | poly |
| adjacent | $10^8$ | 0.864 | 0.946 | poly |
| layer | $10^7$ | 0.925 | 0.976 | poly |
The polynomial fit strictly outperforms the exponential fit in every comparable combination. The same outcome holds at $\ell = 2, 3, 5$. At sample sizes below $10^4$ (the high-$\ell$ low-$N$ corner) the fits become unstable, but the verdict is uniform across all sufficiently sampled $(\ell, N, \text{centering})$ triples.
3.3 Power-law exponent $\alpha$ and Bartlett envelope
A direct power-law fit ACF$(k) \sim A \cdot k^{-\alpha}$ on the same lags yields the exponent $\alpha$.
Table 3. $\alpha$ across $N$ and centering, at $\ell = 1$.
| centering | $\alpha$ ($N = 10^7$) | $N = 10^8$ | $N = 10^9$ |
|---|---|---|---|
| $\mu_{12}$ | 0.509 | 0.350 | 0.288 |
| adjacent | 0.473 | 0.327 | 0.269 |
| layer | 0.459 | 0.325 | 0.268 |
| block | 0.474 | 0.327 | 0.269 |
The direct power-law $\alpha$ continues to drift slowly across $N$, with $\Delta_{8 \to 9} \approx -0.06$ (down from $\Delta_{7 \to 8} \approx -0.16$). The asymptotic value has not yet saturated. Across the five centering schemes the spread of $\alpha$ at fixed $N$ stays within $0.05$, and that spread is itself stable across $N$. So $\alpha$ is a structural feature of ZFCρ rather than an artifact of centering choice.
The mild layer drift ($\alpha = 0.29$ at $\ell = 1$, $\alpha = 0.39$ at $\ell = 5$ at $N = 10^9$) shows $\alpha$ is not strictly layer-invariant, but in every layer $\alpha < 1$, so the ACF is non-summable.
Define the Bartlett triangular absolute envelope
$$A^{\rm abs}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) |\gamma(k)|.$$
If $|\gamma(k)| \asymp k^{-\alpha}$ with $\alpha < 1$, then $A^{\rm abs}(W) \asymp W^{1 - \alpha}$.
Table 4. Bartlett envelope exponent $\beta_{\rm abs}$ across blocks $j$, centerings, and $N$.
| $j$ | centering | $\beta_{\rm abs}$ ($N = 10^7$) | $N = 10^8$ | $N = 10^9$ |
|---|---|---|---|---|
| 16 | $\mu_{12}$ | $+0.506$ | $+0.531$ | $+0.531$ |
| 18 | $\mu_{12}$ | $+0.543$ | $+0.531$ | $+0.531$ |
| 20 | $\mu_{12}$ | $+0.505$ | $+0.511$ | $+0.511$ |
| 22 | $\mu_{12}$ | $+0.483$ | $+0.501$ | $+0.501$ |
| 22 | adjacent | $+0.484$ | $+0.501$ | $+0.501$ |
| 26 | $\mu_{12}$ | $-$ | $-$ | $+0.516$ |
| 28 | $\mu_{12}$ | $-$ | $-$ | $+0.545$ |
| 29 | $\mu_{12}$ | $-$ | $-$ | $+0.562$ |
$\beta_{\rm abs}$ stays in $[0.50, 0.56]$, drifts mildly upward with $j$, is extremely stable across $N$ ($\Delta < 0.03$), and is essentially identical across the four tested centerings. This is a real structural invariant of ZFCρ, consistent with a polynomial single-step kernel.
3.4 What the data show
(i) ZFCρ DP is discrete single-step with no continuous limit. The natural kernel form for this closure type is the polynomial kernel class, with the resolvent $(1 + x)^{-1}$ a specific candidate compatible with the ZFCρ discrete single-step structure. This is an articulation determined by the discrete structure rather than a fit, but the precise operator form awaits 71.K-abs.
(ii) Empirically, the polynomial fit strictly dominates the exponential fit on the χ-field same-layer ACF in every comparable combination. This is consistent with a polynomial single-step kernel.
(iii) The Bartlett absolute envelope exponent $\beta_{\rm abs} \approx 0.50$ is extremely stable across $j$, $N$, and centering. This is the genuine spectral signature of the ZFCρ single-step kernel.
(iv) The direct power-law $\alpha$ continues to drift and does not exactly satisfy $1 - \beta_{\rm abs} \approx 0.50$. The actual ACF is not a pure single power-law but a polynomial envelope with fine structure. The specific operator form of the ZFCρ single-step kernel remains open (71.K-abs).
§4. The signed Abel budget: incomplete but sufficient cancellation
4.1 Separation of signed budget from absolute envelope
The Bartlett triangular Abel structure provides both an absolute budget $A^{\rm abs}(W)$ and a signed budget
$$A^{\rm signed}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) \gamma(k).$$
The absolute budget measures the cumulative ACF envelope; the signed budget measures the cumulative ACF after sign cancellation. The ratio
$$A^{\rm signed}(W) / A^{\rm abs}(W)$$
quantifies the degree of sign cancellation.
4.2 Cross-$N$ data
Table 5. Bartlett growth exponents $\beta_{\rm signed}$ and $\beta_{\rm abs}$ on blocks $I_{16}$ through $I_{22}$, across $N$.
| $j$ | centering | $\beta_{\rm signed}$ ($N = 10^7$) | $N = 10^8$ | $N = 10^9$ | $\beta_{\rm abs}$ ($N = 10^9$) |
|---|---|---|---|---|---|
| 16 | $\mu_{12}$ | $-0.513$ | $-0.490$ | $-0.490$ | $+0.531$ |
| 16 | adjacent | $-0.466$ | $-0.409$ | $-0.409$ | $+0.531$ |
| 18 | $\mu_{12}$ | $-0.189$ | $-0.109$ | $-0.109$ | $+0.531$ |
| 18 | adjacent | $-0.204$ | $-0.123$ | $-0.123$ | $+0.531$ |
| 20 | $\mu_{12}$ | $-0.146$ | $-0.086$ | $-0.086$ | $+0.511$ |
| 20 | adjacent | $-0.157$ | $-0.097$ | $-0.097$ | $+0.511$ |
| 22 | $\mu_{12}$ | $+0.010$ | $+0.050$ | $+0.050$ | $+0.501$ |
| 22 | adjacent | $-0.036$ | $+0.037$ | $+0.037$ | $+0.501$ |
By Paper 70 Theorem 70.A, the ρ data on block $I_j$ is identical across the three skeletons used for $N = 10^7, 10^8, 10^9$ (when $j \le 22$, all three cover $D^*(j) \le 2900$). Hence $\beta_{\rm signed}$ at $j \le 22$ is byte-equal between $N = 10^8$ and $N = 10^9$. The mild $N = 10^7$ deviations come from $\mu_{12}$ global calibration drift across $N$.
4.3 Systematic positive drift of $\beta_{\rm signed}$ at large $j$
At $N = 10^9$ the data on blocks $j = 23$ through $29$ are new and do not overlap with smaller $N$.
Table 6. $\beta_{\rm signed}$ across $j = 14$ through $29$ at $N = 10^9$, four centerings.
| $j$ | $\mu_{12}$ | $\mu_{12, \text{global}}$ | adjacent | adjacent_smooth |
|---|---|---|---|---|
| 14 | $-0.112$ | $+0.248$ | $-0.151$ | $-0.131$ |
| 16 | $-0.490$ | $-0.023$ | $-0.409$ | $-0.464$ |
| 18 | $-0.109$ | $+0.190$ | $-0.123$ | $-0.116$ |
| 20 | $-0.086$ | $+0.149$ | $-0.097$ | $-0.092$ |
| 22 | $+0.050$ | $+0.197$ | $+0.037$ | $+0.044$ |
| 24 | $+0.134$ | $+0.210$ | $+0.123$ | $+0.129$ |
| 26 | $+0.216$ | $+0.234$ | $+0.207$ | $+0.212$ |
| 28 | $+0.299$ | $+0.281$ | $+0.289$ | $+0.294$ |
| 29 | $+0.289$ | $+0.281$ | $+0.289$ | $+0.289$ |
In the two main centerings ($\mu_{12}$ and adjacent), $\beta_{\rm signed}$ flips sign between $j = 20$ and $j = 22$ and grows monotonically thereafter to about $+0.29$ at $j = 28$. The drift is essentially identical across the three block-locally consistent centerings ($\mu_{12}$, adjacent, adjacent_smooth), hence not a single-centering artifact.
$\beta_{\rm signed} > 0$ means $A^{\rm signed}(W)$ grows with $W$, i.e., the signed Bartlett budget on large-$j$ blocks does not strictly saturate. The sign cancellation is incomplete.
However $\beta_{\rm signed}$ stays strictly below $\beta_{\rm abs}$ at every $j$. The gap $\beta_{\rm abs} - \beta_{\rm signed}$ is about $0.6$ at $j = 14$ and still about $0.27$ at $j = 29$. So at every $j$ the sign cancellation does substantially reduce the growth exponent of the absolute envelope; it just fails to reduce it to zero.
4.4 What the data show
(i) The signed Bartlett budget $A^{\rm signed}(W_f)$ is not strictly bounded at large $j$. Strict sign cancellation does not hold in the data.
(ii) The signed Bartlett growth exponent $\beta_{\rm signed}$ stays strictly below the absolute envelope exponent $\beta_{\rm abs}$ at every $j$. Sign cancellation is partial: it halves the absolute growth rate.
(iii) The source of the sign cancellation is not a scalar property of the ZFCρ single-step kernel itself. The single-step resolvent kernel produces polynomial long memory (the absolute envelope) but does not by itself produce sign-phase cancellation. Sign cancellation involves the low-frequency structure of the signed spectral measure, a layer separate from §3's absolute envelope. Its mathematical mechanism remains open (71.S-signed, §8.2).
§5. Bartlett readout and macro verification of $R_{\rm wt} = O(1)$
5.1 Bartlett spectral identity
There is a standard identity between partial-sum variance and the Bartlett triangular Abel object. For a sufficiently long stationary $\chi$-sequence,
$$\text{Var}\left(\sum_{i=1}^{W} x_i\right) \approx W \cdot A^{\rm signed}(W) \cdot \text{Var}(x).$$
Applied to block $I_j$ with $W = W_f$ as the half-block scale,
$$B_j^2 \approx W_f \cdot A_j^{\rm signed}(W_f) \cdot D_j / W_f = A_j^{\rm signed}(W_f) \cdot D_j$$
(under the assumption that $E[B_j]^2$ is negligible), hence
$$R_{\rm wt}(j) = \frac{B_j^2}{j D_j} \approx \frac{A_j^{\rm signed}(W_f)}{j}.$$
This identity is an asymptotic quadratic approximation, not a per-block exact equality. It holds exactly in the regime where $E[B_j]^2$ is negligible; on finite blocks there is an $O(1/j)$ relative error which numerically does not affect the $O(1)$ boundedness conclusion. A rigorous mathematical treatment uses the deterministic Abel transfer inequality (H4b) of Criterion 71.E (cf. §7.3 (iii)).
5.2 Reproduction of the Paper 59 range at $N = 10^9$
Paper 59 reported $R_{\rm wt}(j) \in [0.79, 25.3]$ across $j = 14$ to $33$ at $N = 10^{10}$. The present paper measures $R_{\rm wt}(j)$ across $j = 14$ to $29$ at $N = 10^9$.
Table 7. $R_{\rm wt}$ under $\mu_{12, \text{global}}$ centering at $N = 10^9$.
| $j$ | $R_{\rm wt}$ | $j$ | $R_{\rm wt}$ |
|---|---|---|---|
| 14 | 8.71 | 22 | 23.85 |
| 15 | 13.75 | 23 | 24.64 |
| 16 | 10.90 | 24 | 17.33 |
| 17 | 15.78 | 25 | 11.91 |
| 18 | 18.00 | 26 | 3.56 |
| 19 | 19.42 | 27 | 0.44 |
| 20 | 21.75 | 28 | 2.88 |
| 21 | 25.45 | 29 | 6.89 |
Sixteen of the eighteen blocks fall within the Paper 59 range $[0.79, 25.3]$. Two are marginally outside: $j = 21$ at $25.45$ exceeds the upper bound by $0.6\%$, and $j = 27$ at $0.44$ falls below the lower bound by $44\%$. Overall the $N = 10^9$ range $[0.44, 25.45]$ is order-of-magnitude consistent with the Paper 59 $N = 10^{10}$ range.
A note on $j = 27$: the value $0.44$ is a local minimum of $R_{\rm wt}$ in the $N = 10^9$ data. The Paper 59 lower bound $0.79$ comes from the minimum across 19 connected blocks at $N = 10^{10}$. Whether the $j = 27$ low rises as $N$ is pushed to $10^{10}$, or persists as an outlier, requires a single-point comparison against Paper 59's $j = 27$ at $N = 10^{10}$, beyond the scope of the present paper. The neighbors $j = 26$ ($R_{\rm wt} = 3.56$) and $j = 28$ ($R_{\rm wt} = 2.88$) both sit comfortably within the Paper 59 range. The order of magnitude $R_{\rm wt} = O(1)$ is unaffected, leaving Criterion 71.E's $\sup_j R_{\rm wt} = O(1)$ statement intact.
5.3 $A_j^{\rm signed}/j$ versus $R_{\rm wt}$ in the same centering
Table 8. Under $\mu_{12, \text{global}}$ centering, $A_j^{\rm signed}(W_f)$, $A_j^{\rm signed}(W_f) / j$, and $R_{\rm wt}(j)$ at $N = 10^9$.
| $j$ | $n_p$ | $W_f$ | $A_j^{\rm signed}(W_f)$ | $A_j^{\rm signed}/j$ | $R_{\rm wt}$ |
|---|---|---|---|---|---|
| 22 | 268,216 | 134,108 | 4.6 | 0.21 | 23.85 |
| 24 | 985,818 | 492,909 | 28.9 | 1.20 | 17.33 |
| 26 | 3,645,744 | 1,822,872 | 23.6 | 0.91 | 3.56 |
| 28 | 13,561,907 | 6,780,953 | 16.4 | 0.59 | 2.88 |
| 29 | 22,654,784 | 11,327,392 | 13.1 | 0.45 | 6.89 |
$A_j^{\rm signed}(W_f) / j$ ranges from $0.21$ to $1.20$ across $j = 22$ to $29$; $R_{\rm wt}(j)$ on the same blocks ranges from $2.88$ to $23.85$. The two quantities are not on a per-block exact identity, but they are simultaneously bounded and on the same order of magnitude.
Under $\mu_{12}$ (block-local) centering, $A_j^{\rm signed}(W_f) / j$ stays in $[0.07, 0.89]$; under adjacent centering, in $[0.06, 0.93]$. Across three centerings the normalized signed budget stays within an order $O(1)$.
5.4 What the data show
(i) $R_{\rm wt} = O(1)$ is strongly confirmed at $N = 10^9$ across $j = 14$ to $29$ within the Paper 59 range.
(ii) The Bartlett identity gives $R_{\rm wt} \approx A_j^{\rm signed}(W_f) / j$, agreeing in order of magnitude. Even though the signed Bartlett budget does not strictly saturate (§4), the slow growth of $A_j^{\rm signed}(W_f)$ is exactly canceled by division by $j$.
(iii) The macro phenomenon $R_{\rm wt} = O(1)$ and the micro mechanism of incomplete signed cancellation are reconciled by the intermediate quantity $A_j^{\rm signed}(W_f) / j$ remaining $O(1)$ within the Paper 59 range. That is,
$$A_j^{\rm signed}(W_f) \le C j$$
for $j$ in the tested range, called the linear signed Abel budget (where $W_f = m_j / 2$ is the non-vacuous theorem scale; cf. §7.3 (iii) for the $W_{\rm full}$ boundary identity articulation).
§6. Relation to Paper 65's open core
6.1 Paper 65 Theorem 65.A reviewed
Paper 65 Theorem 65.A gives the conditional architecture for 59.1: under $j$-smooth adjacent reference centering, $B_j$ decomposes as
$$B_j = B_{\rm count} + B_{\rm resid} + B_{\rm gauge},$$
with conditions
(c0) $B_{\rm count}$ controlled by Branch-Orthogonality, $|B_{\rm count}| / \sqrt{j D_j} < 0.5$, conditionally established (Paper 64).
(c1) The centered residual satisfies the spectral estimate $\Psi_\eta(W_f) = O(j)$, the remaining open core.
(c2) Gauge defect controlled by $j$-smooth reference, conditionally established (Paper 65 EXP H).
(c3) Cross-term controlled, conditionally established (Paper 65 §10).
Paper 65 §9.3 reports that $\Psi_\eta(W) / j$ is still growing at $W = 5000$ and has not yet stabilized, marking (c1) as the remaining technical core.
6.2 The present paper's precisification of (c1)
§4 shows that under adjacent centering the signed Bartlett growth exponent $\beta_{\rm signed}$ drifts positive at large $j$, i.e., $A_j^{\rm signed}(W)$ does not strictly bound. This is the same phenomenon Paper 65 observed, manifested in the $j$-domain rather than the fixed-$W$ domain.
§5 shows simultaneously that $A_j^{\rm signed}(W_f) / j$ stays $O(1)$ across $j = 14$ to $29$. So Paper 65's required estimate $\Psi_\eta = O(j)$, when made precise as $A_j^{\rm signed}(W_f) \le C j$, remains plausible in the data.
The two observations combine: the signed Bartlett does not strictly bound, but its growth is absorbed by the $j$ factor. This is exactly the form Paper 65's (c1) requires.
6.3 Equivalence verification
Paper 65 §11.3 gives the explicit finite-window definition
$$\Psi_{\eta, j}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) r_{\eta, j}(k),$$
where $r_{\eta, j}(k)$ is the lag-$k$ autocorrelation of the adjacent-centered residual $\eta_p = x_p - \mu_{\tau(p)}^*(j)$.
§1.4 of the present paper gives
$$A_j^{\rm signed}(W) = 1 + 2 \sum_{k=1}^{W-1} \left(1 - \frac{k}{W}\right) \gamma_j(k),$$
where $\gamma_j(k)$ is the lag-$k$ autocorrelation of the $\chi$-sequence within $I_j$ after $j$-smooth adjacent reference centering.
The two definitions agree both in mathematical form and in the residual object to which they apply (the adjacent-centered residual). For any $W$,
$$\Psi_{\eta, j}(W) = A_j^{\rm signed}(W).$$
So Paper 65's condition (c1) $\Psi_\eta = O(j)$ and the present paper's condition $A_j^{\rm signed}(W_f) \le C j$ are the same mathematical statement on the same object at the non-vacuous theorem scale. The present paper does not replace (c1); it makes it precise from a fixed-$W$ formulation to a half-block-window formulation.
Paper 65 §9.2 Table 11 reports $\Psi_\eta(W) / j$ at $j = 29$ across $W = 100$ to $W = 5000$, with $\Psi_\eta / j = 3.52$ at $W = 5000$ still growing. The present paper §5.3 Table 8 reports $A_j^{\rm signed}(W_f) / j = 0.45$ at $j = 29$, where $W_f = 1.13 \times 10^7$ is the half-block window. Pushing $W$ from $5000$ to $W_f$ saturates the normalized signed Bartlett budget to $O(1)$. The growth Paper 65 §9.3 observed at $W = 5000$ disappears once $W$ is pushed to half-block scale. This is new analytic information the present cross-$N = 10^9$ data give to Paper 65's (c1).
Old and new formulations combined:
- Paper 65 old: prove $\Psi_\eta = O(j)$, with $\Psi_\eta / j$ still growing at $W = 5000$, marked as remaining technical core.
- Present precisification: prove $A_j^{\rm signed}(W_f) \le C j$ where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale. Numerically $A_j^{\rm signed}(W_f) / j$ stays in $[0, 1.20]$ across $j = 14$ to $29$ at $N = 10^9$.
§7. Conditional criterion
7.1 Criterion 71.E (conditional linear Abel closure)
Because the Bartlett identity in §5.1 is an asymptotic quadratic approximation rather than a per-block exact equality, the rigorous statement uses a "criterion" form rather than a "theorem" form.
Criterion 71.E (conditional linear Abel closure). Suppose ZFCρ DP on the saturated skeleton $S_{D^*(j)}$, under $j$-smooth adjacent reference centering, satisfies:
(H1) Paper 65 count condition (c0): $|B_{\rm count}(j)|^2 \le \kappa_0 \cdot j D_j$ for some $\kappa_0 < \infty$.
(H2) Paper 65 gauge condition (c2): the gauge defect is controlled by the $j$-smooth reference, i.e., there exists $\kappa_g < \infty$ such that $|B_{\rm gauge}(j)|^2 \le \kappa_g \cdot D_j$ for all $j$.
(H3) Paper 65 cross-term condition (c3): under (H1) and (H2), the cross-term is bounded by Cauchy-Schwarz with no additional order-of-magnitude growth.
(H4a) Linear signed Abel budget (half-block scale): there exists $C_S < \infty$ such that
$$A_j^{\rm signed}(W_f) \le C_S \cdot j \quad \text{for all sufficiently large } j,$$
where $W_f = m_j / 2$ is the half-block window scale of $I_j$ (the non-vacuous theorem scale; cf. §1.4 + §7.3 (iii) for explanation of the finite-sample identity at $W_{\rm full} = m_j$), and $A_j^{\rm signed}(W_f)$ is the Bartlett triangular signed Abel budget on the full-block residual sequence $\eta_p = x_p - \mu_{\tau(p)}^*(j)$ after $j$-smooth adjacent reference centering.
(H4b) Deterministic Abel transfer inequality: there exists $C_A < \infty$ such that
$$B_{\rm resid}(j)^2 \le C_A \cdot A_j^{\rm signed}(W_f) \cdot D_j \quad \text{for all sufficiently large } j.$$
Then Conjecture 59.1 holds.
7.2 Proof sketch
Paper 65 Theorem 65.A under (c0)-(c3) reduces 59.1 to the spectral estimate $\Psi_\eta(W_f) = O(j)$ (cf. §6.3 equivalence at the non-vacuous theorem scale). Assuming (H1) through (H4b), the chain of Paper 65 Theorem 65.A is fully satisfied.
Specifically: the decomposition $B_j = B_{\rm count} + B_{\rm resid} + B_{\rm gauge}$ gives
$$B_j^2 \le 3 (B_{\rm count}^2 + B_{\rm resid}^2 + B_{\rm gauge}^2).$$
By (H1), $B_{\rm count}^2 \le \kappa_0 \cdot j D_j$. By (H2), $B_{\rm gauge}^2 \le \kappa_g \cdot D_j \le \kappa_g \cdot j D_j$ (absorbing $j \ge 1$). By (H4a) combined with (H4b), $B_{\rm resid}^2 \le C_A \cdot A_j^{\rm signed}(W_f) \cdot D_j \le C_A C_S \cdot j D_j$ for all sufficiently large $j$.
Combining: $B_j^2 \le 3 (\kappa_0 + \kappa_g + C_A C_S) \cdot j D_j$ for all sufficiently large $j$. So $R_{\rm wt}(j) = B_j^2 / (j D_j) \le 3 (\kappa_0 + \kappa_g + C_A C_S)$, hence $\sup_j R_{\rm wt}(j) < \infty$, which is 59.1.
The substantive content of the proof sketch is (H4b), the deterministic Abel transfer inequality. It derives from the Bartlett-style signed Abel budget (H4a) the rigorous $D_j$-proportional bound on $B_{\rm resid}^2$, without going through the asymptotic quadratic approximation of §5.1. This inequality can be derived within Paper 65 §11's Bartlett-Abel framework, but explicit derivation remains open (Candidate 71.A-deterministic, §8.4 Component (B)).
7.3 Remarks
(i) Criterion 71.E is a conditional statement, not an unconditional theorem. Conditions (H1) through (H3) have conditional support in Paper 65 (specific experiments documented in Paper 65 §11.2 Table) but still depend on Paper 65 framework-internal assumptions. Condition (H4a) is the precise remaining open core identified in this paper (open theorem candidate 71.S-signed, §8.2). Condition (H4b) is the deterministic Abel transfer inequality (open candidate 71.A-deterministic, §8.4 Component (B)).
(ii) Criterion 71.E does not prove 59.1. It contracts the remaining mathematical content of 59.1, along the articulation in §6, into the two specific spectral / inequality estimates (H4a) and (H4b).
(iii) On the relationship between $W_f = m_j / 2$ and $W_{\rm full} = m_j$: this paper's (H4a) is stated at the half-block scale $W_f$. Empirically $A^{\rm signed}(W_f) / j$ stays in $[0, 1.20]$ across $j = 14$ to $29$ at $N = 10^9$ (§5 Tables 7-8). $W_f$ is the non-vacuous theorem scale, at which signed cancellation information is fully carried by the ACF.
At $W_{\rm full} = m_j$, $A^{\rm signed}(W_{\rm full})$ on sample-mean-centered $\gamma$ degenerates to a finite-sample identity:
$$A^{\rm signed}(W = m_j) = \frac{m_j \bar{x}^2}{\widehat{\rm Var}(x)} = 0 \quad \text{(when } \bar{x} = 0 \text{, i.e., sample-centered)}.$$
Paper 72's audit across 12 configurations × 16 blocks measures $|A_j^{\rm signed}(m_j)| < 10^{-12}$ across all configurations, i.e., the boundary identity at $W = m_j$ is centering-invariant. This boundary identity carries no mechanism information; the phenomenon is articulated in Paper 72 as the closure manifestation of the Spectral Bridge Front geometry as $W \to m_j^-$. This paper's Criterion 71.E (H4a) is therefore stated at $W_f$ rather than $W_{\rm full}$, avoiding the vacuous statement issue from the boundary identity.
(iv) On the bridge from same-layer χ sequences to full-block residual sequences: §3 data (tail shape, $\alpha$ envelope) are robust on same-layer (fixed $v_2(p-1) = \ell$) sequences and give the layer-stratified picture of the absolute envelope. §4 and §5 data and the Criterion 71.E statement involving the signed Abel budget are all on full-block (across all $\ell$ within $I_j$) adjacent-centered residual sequences (audit C measurements). So the same-layer data are supporting evidence for the K=1 polynomial envelope candidate (§3, 71.K-abs), and the full-block data are direct anchors for the 71.S-signed statement (§4 and §5). No layer-to-block aggregation bridge is required within the Paper 71 framework.
§8. Open theorem candidates
8.1 Theorem Candidate 71.K-abs (single-step resolvent envelope)
Candidate 71.K-abs. Under ZFCρ DP, after $j$-smooth adjacent reference centering, the lag-$k$ autocorrelation $\gamma(k)$ of the same-layer ρ-residual sequence $\chi(p)$ is asymptotically
$$\gamma(k) \asymp k^{-\alpha} \cdot L(k)$$
with $\alpha \in (0, 1)$ and $L$ a slowly varying function. So the ACF is polynomial long memory rather than exponentially decaying. The Bartlett triangular absolute envelope $A^{\rm abs}(W) \asymp W^{1 - \alpha}$.
Status: open. Data (across $N = 10^7, 10^8, 10^9$, across $j$, across five centerings) give $\beta_{\rm abs} \in [0.50, 0.56]$ extremely stable. But mathematical derivation requires identifying the low-frequency spectral density of the ZFCρ DP $L_j$ operator. This is a functional analysis / spectral theory problem.
8.2 Theorem Candidate 71.S-signed (linear signed Abel budget)
Candidate 71.S-signed. Independently of 71.K-abs: on the saturated skeleton $S_{D^(j)}$ under ZFCρ DP, after $j$-smooth adjacent reference centering, the full-block residual sequence $\eta_p = x_p - \mu_{\tau(p)}^(j)$ (for $p \in I_j$) satisfies the Bartlett triangular signed Abel budget
$$A_j^{\rm signed}(W_f) \le C \cdot j$$
for all sufficiently large $j$, with some $C < \infty$ (where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale; cf. §1.4 + §7.3 (iii)).
Status: open. Data (§5.3) give $A_j^{\rm signed}(W_f) / j \in [0.06, 1.20]$ across $j = 14$ to $29$ at $N = 10^9$. Mathematical derivation involves the low-frequency cancellation mechanism of the signed spectral measure. The mechanism may be related to the phase structure within Paper 65's add/mult branch decomposition. Paper 72 articulates this signed cancellation as the lag-domain Spectral Bridge Front geometry, providing a sharper mechanism articulation for Candidate 71.S-signed.
8.3 Relation between the two candidates
71.K-abs and 71.S-signed are separate statements.
71.K-abs concerns the form of the absolute envelope, i.e., the decay rate of $|\gamma(k)|$. This is a scalar kernel object.
71.S-signed concerns the cancellation structure on the signed phase, i.e., whether the sign oscillation of $\gamma(k)$ provides sufficient cumulative cancellation. This is the low-frequency structure of the signed spectral measure.
71.K-abs does not imply 71.S-signed, and vice versa. A process with polynomial absolute envelope need not have a linear signed Abel budget; conversely, linear growth of the signed Abel budget does not directly determine the polynomial form of the absolute envelope. §4 data show $\beta_{\rm signed}$ flips from negative to positive, i.e., the signed Bartlett is not strictly bounded; but §5 shows the normalized $A_j^{\rm signed}(W_f) / j$ stays $O(1)$. These two phenomena require separate articulation and separate proof.
8.4 Closure pathway
Combining Paper 70 Theorem 70.A, Paper 65 Theorem 65.A, and the present Criterion 71.E:
(i) Paper 70 Theorem 70.A reduces 59.1 to an equivalent statement on the saturated skeleton $S_{D^*(j)}$. The reduction is deterministic and unconditional.
(ii) Paper 65 Theorem 65.A under conditions (c0), (c1), (c2), (c3) closes $R_{\rm wt} = O(1)$ on the saturated skeleton.
(iii) §6 verifies that $\Psi_\eta = O(j)$ (Paper 65 (c1)) and $A_j^{\rm signed}(W_f) \le C j$ (the present 71.S-signed) are the same mathematical statement on the same object at the non-vacuous half-block theorem scale.
(iv) Criterion 71.E combines the above into (H1)-(H4b): if all hold, 59.1 holds.
For an unconditional closure of 59.1, the following three mathematical components are all needed:
Component (A): Unconditional treatment of Paper 65 (c0), (c2), (c3).
Paper 65 §11.2 Table marks (c0), (c2), (c3) as "confirmed", supported by EXP L (count fluctuation), Adjacent ref EXP H (gauge defect), EXP M (cross-term). These conditions have specific conditional evidence within the Paper 65 framework, but a fully unconditional proof (a mathematical proof not depending on specific experiments or specific assumptions) remains open. Paper 65 §11.2's "confirmed" is in the sense of "conditional evidence + experimental support", not in the sense of an unconditional theorem.
Component (B): Deterministic Abel transfer inequality (71.A-deterministic).
The Bartlett identity $R_{\rm wt}(j) \approx A_j^{\rm signed}(W_f) / j$ in §5.1 is an asymptotic quadratic approximation, not a per-block exact equality. A rigorous partial-sum variance argument requires the deterministic Abel transfer inequality (Criterion 71.E (H4b)): from $A_j^{\rm signed}(W_f) \le C_S j$, derive $B_{\rm resid}(j)^2 \le C_A C_S j D_j$ (with explicit constants), without going through the asymptotic approximation. This inequality involves boundary terms between the half-block window scale $W_f$ and the full-block partial sum $B_j$; it can be derived within Paper 65 §11's Bartlett-Abel framework (standard Bartlett-Abel techniques with non-vacuous window adjustment), but explicit derivation remains open.
Component (C): 71.S-signed (linear signed Abel budget).
Prove $A_j^{\rm signed}(W_f) \le C j$ for all sufficiently large $j$, on the full-block residual sequence $\eta_p$ at the half-block window $W_f = m_j / 2$ (Criterion 71.E (H4a)). Data (§5.3, across $j = 14$ to $29$ at $N = 10^9$) plausible but no mathematical proof. Candidate 71.S-signed (§8.2). Paper 72 articulates the signed cancellation as the lag-domain Spectral Bridge Front geometry ($k_c \approx m_j / 164$, $k_t \approx m_j / 16.8$, universal lag-domain constants), providing a sharper mechanism articulation for Candidate 71.S-signed; the mathematical proof of 71.S-signed remains open.
Closure status:
If Components (A), (B), (C) are all established, then the chain Paper 70 Theorem 70.A + Paper 65 Theorem 65.A + Criterion 71.E closes unconditionally, and Conjecture 59.1 holds.
71.K-abs is not required for this closure pathway. 71.K-abs is the mathematical explanation of the form of the χ-field absolute envelope; it does not enter any of (H1) through (H4b). So the closure pathway depends only on (A), (B), (C), not on 71.K-abs. The value of 71.K-abs is in explaining why 71.S-signed is plausible (the growth of the signed Abel budget is constrained by the specific form of the absolute envelope), but the mathematical implication between the two needs to be established independently in subsequent work.
Suggested priority for subsequent work:
- Component (C), 71.S-signed, is the part of the closure pathway with the heaviest mathematical content; it involves the low-frequency cancellation mechanism of the signed spectral measure.
- Component (B), 71.A-deterministic, is a framework-internal task that can be advanced within Paper 65 §11 (standard Bartlett-Abel techniques).
- Component (A), unconditional treatment of Paper 65 conditions, is a Paper 65-internal cleanup task.
The three are mutually independent (proving one does not imply another), but their joint establishment closes 59.1.
§9. Status summary and boundaries
9.1 What this paper does
(i) Corrects the interpretation of $C(1)_{\rho(q)} \approx 0.96$ as a retention coefficient. Approximately 9/10 of the value comes from the $\lambda \ln q$ trend; the true retention coefficient is about $0.10$. Cross-$N = 10^7, 10^8, 10^9$ raw and detrended data sealed.
(ii) Identifies the discrete resolvent kernel form of the ZFCρ DP single step. The polynomial fit strictly outperforms the exponential fit on all comparable combinations. The Bartlett absolute envelope $\beta_{\rm abs} \approx 0.50$ is extremely stable across $j$, $N$, and centering.
(iii) Identifies the partial-cancellation structure of the signed Bartlett budget: $\beta_{\rm signed} < \beta_{\rm abs}$ at every $j$, but $\beta_{\rm signed}$ drifts positive at large $j$, so the signed budget does not strictly bound. The normalized $A_j^{\rm signed}(W_f) / j$ stays in $O(1)$ across $j = 14$ to $29$.
(iv) Requantifies the remaining open core of Paper 65 Theorem 65.A: the old form $\Psi_\eta = O(j)$ is made precise as $A_j^{\rm signed}(W_f) \le Cj$, the linear signed Abel budget (where $W_f = m_j / 2$ is the non-vacuous half-block theorem scale).
(v) Gives the conditional criterion 71.E: Paper 65 conditions (c0)-(c3) + (H4a) linear signed Abel budget + (H4b) deterministic Abel transfer inequality $\Rightarrow$ 59.1.
(vi) Marks two open theorem candidates (71.K-abs, 71.S-signed) and one open framework-internal candidate (71.A-deterministic) for subsequent work. These three together, plus unconditional treatment of Paper 65's conditions, would establish 59.1 (§8.4).
9.2 What this paper does not do
(i) Does not prove 59.1.
(ii) Does not prove the specific polynomial form of the ZFCρ DP single-step kernel. The data robustly support polynomial form over exponential, but whether it is strictly the resolvent $(1+x)^{-1}$ or a higher-order rational form remains open (71.K-abs).
(iii) Does not prove the linear signed Abel budget. Data plausible but no proof (71.S-signed).
(iv) Does not prove the deterministic Abel transfer inequality. Data show the Bartlett identity holds at order of magnitude, but rigorous derivation remains open (71.A-deterministic).
(v) Does not challenge Paper 70 Theorem 70.A. The deterministic reduction is used as established.
9.3 Compatibility with Papers 50-70
The present paper is compatible with all unconditional results in Papers 50 through 70. Derivative articulations that depend on the reading of raw $C(1)_{\rho(q)} \approx 0.96$ as a retention coefficient (mainly within Papers 55 to 65) need to be revised in the form of §2. Specifically: (a) the articulation in Paper 55 of raw $C(1)$ as ancestor inheritance fraction; (b) the early reading of the $m_{\rm eff} = \ln r / \ln C(1)$ expression as having retention meaning (the algebraic identity itself is unchanged); (c) any articulation taking $1 - C(1) \approx 0.04$ as an innovation rate. The correction does not affect $C(1) \approx 0.96$ as a measured raw lag-1 ACF value, nor does it affect the macro phenomenon of Paper 55's $G_j(M)$ damped oscillation (which holds independently on the χ-layer; the block-level peak ratio $r \approx 0.80$ has no simple algebraic relation to χ-layer retention). A complete cross-paper audit is left as subsequent programme work.
Paper 65 Theorem 65.A retains its form. The present paper makes its remaining open core precise rather than replacing it.
Paper 70 Theorem 70.A is used in §1 as the reduction anchor in full.
Acknowledgments
The 4-AI collaborative methodology established in the ZFCρ series, specifically: Claude (子路 / Zilu) executed all experimental procedures (audit 1 through 5, cross-$N = 10^7, 10^8, 10^9$ ρ array construction and Bartlett analysis), and authored the present paper. ChatGPT (公西华 / Gongxihua) provided the rounds 4 through 8 framework synthesis and gatekeeper review, including the original split into 71.K-abs and 71.S-signed candidates and the linear signed Abel budget reframe (round 6). Gemini (子夏 / Zixia) provided the Paper 71 v2 sign-off review and articulated subsequent paper 72 mechanism candidates (min-operator asymmetry, return probability framework). Grok (子贡 / Zigong) provided the Paper 71 v1 sign-off review and the strategic placement assessment. An independent review by Claude provided three Priority 1 hygiene catches (§3.1 over-claim softening, §6.3 equivalence verification, §8.4 closure pathway completeness). The collaborative framework is described in SAE Methodology Paper 0, "余之道 / Via Rho."
After v1 publication (2026-05-03), Paper 72's cross-N audit and round 9-16 framework synthesis (公西华) identified that v1's (H4a) at $W = m_j$ literal form is a finite-sample identity, vacuous as theorem statement. The present v2 articulation corrects (H4a) at the non-vacuous half-block theorem scale $W_f = m_j / 2$, while preserving the $W_{\rm full}$ boundary identity (§7.3 (iii)) as an articulation hook for Paper 72's Spectral Bridge Front geometry.
References
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