1. Introduction

1.1 Background and scope

Paper XXVI established the exact decomposition $\rho_E(N) = \pi(N) + S_{\text{comp}}(N)$, confirmed that primes are the sole unconditional source of positive growth, and reduced (L3) to a composite-drift correction theorem. The present paper attacks the first step of that correction: negative drift at high $H$.

When $\rho_E$ is elevated relative to $\ln n$ ($H(n-1)$ above a threshold $h^{+}$), composite increments are, on average, negative — the system is pulled back. This is a one-sided result: it controls the upper reflection ($\rho_E$ cannot remain indefinitely above its equilibrium) but does not directly control the lower reflection (positive drift at low $H$ arises from a different mechanism).

This paper does not claim to close (L3). Lower reflection, the passage from two-sided reflection to pred gap $> \tau_k$, and the connection to Foster–Lyapunov stationarity theory are deferred.

1.2 Contributions

• (A) Algebraic framework for the spf comparator (§2): saving lower bound and negative drift sufficient condition.
• (B) Complete deconfounding of height-drift monotonicity (§3): 75 data points across (spf, $\Omega$, gap-position) triple stratification, zero exceptions.
• (C) Numerical confirmation of the sufficient condition (§4): $E[(\Delta_{\text{spf}})^{+} \mid H \geq 3.84] = 1.82 > 1$; whole-population aggregation with even subfamily alone exceeding 1.
• (D) Direct measurement of scale attenuation (§5): $H(n/2)$ anchors at $h_0 \approx 3.79$, attenuation ratio negative.
• (E) Precise characterization of the remaining analytic gap (§5–6): under the spf-comparator route, analytic closure requires a quantitative comparator margin on the even subfamily and a conditional density lower bound. Scale attenuation serves as the mechanistic lemma for the margin estimate.

1.3 Notation

Series conventions and Paper XXVI notation apply. $\rho_E(n)$: Erdős additive complexity, $\rho_E(1) = 0$. $M_n = \min_{ab=n,\, a,b\ge 2}(\rho_E(a)+\rho_E(b)+2)$; $M_p = +\infty$ for primes. $\delta(n) = \rho_E(n) - \rho_E(n-1)$. $S(n) = \rho_E(n-1)+1-\rho_E(n)$ (saving; $S(n) \geq 0$ holds identically). $H(n) = \rho_E(n)/\ln n$ (height ratio). $\text{spf}(n)$: smallest prime factor of $n$.

Throughout, $E[F(n) \mid n \in C]$ denotes the finite-set empirical average: $E_N[F(n) \mid n \in C] = \tfrac{1}{|C \cap [1,N]|}\sum_{n \in C,\, n \le N} F(n)$.

2. Algebraic Framework: the spf Comparator

2.1 Saving lower bound

For any composite $n$ (so that $M_n$ is finite), let $p = \text{spf}(n)$, $q = n/p$. By definition of $M_n$: $$M_n \leq \rho_E(p) + \rho_E(q) + 2.$$ Define the spf comparator gap: $$\Delta_{\text{spf}}(n) = [\rho_E(n-1)+1] - [\rho_E(p) + \rho_E(q) + 2].$$

2.2 Negative drift sufficient condition

Since $\delta(n) = 1 - S(n)$:

3. Complete Deconfounding of Height-Drift Monotonicity

3.1 Raw phenomenon

$E[\delta \mid \text{composite}]$ decreases monotonically from $+0.70$ (Q1, low $H$) to $-1.16$ (Q5, high $H$) across height quintiles.

3.2 Layer 1: (spf, $\Omega$) stratification

Across 28 tested (spf, $\Omega$) combinations (spf $\in \{2,3,5,7\}$, $\Omega \in \{2,\ldots,8\}$), $E[\delta \mid H \text{ quintile}]$ is strictly monotonically decreasing. Zero exceptions.

3.3 Layer 2: gap-position stratification

Composites are classified by position within inter-prime gaps: first, middle, last, solo.

Observation 1. In middle-of-gap composites (excluding first/last position effects), $E[\delta]$ remains strictly monotonically decreasing with $H$:

3.4 Layer 3: triple stratification

In middle-of-gap composites with spf $= 2$, further stratified by $\Omega$ ($\Omega = 2$ to $16$):

Observation 2. $15 \times 5 = 75$ data points, all strictly monotonically decreasing, zero exceptions.

Representative data (middle-of-gap, spf $= 2$):

spf $= 3$ triple stratification is likewise fully monotonic (complete data released with scripts).

3.5 Conclusion

Within all tested (spf, $\Omega$, gap-position) strata, height-drift monotonicity is preserved without exception. The effect of $H(n-1)$ on composite drift cannot be explained by confounding from these arithmetic covariates.

4. Numerical Confirmation of the Sufficient Condition

4.1 $E[(\Delta_{\text{spf}})^{+}]$ by height quintile

Observation 3.

Crossover at $H \approx 3.807$. Every fine-bin above this point exceeds 1, with no oscillation.

4.2 Whole-population aggregation

Decompose $E[(\Delta_{\text{spf}})^{+} \mid C_N(h^{+})] = \sum_p d_p \cdot a_p$ where $d_p = P(\text{spf}=p \mid C_N(h^{+}))$, $a_p = E[(\Delta_{\text{spf}})^{+} \mid \text{spf}=p, C_N(h^{+})]$.

Observation 4 (Q5).

The even subfamily alone contributes $d_2 \cdot a_2 = 1.269 > 1$. The conditional density $d_2 \approx 0.663$ exceeds the unconditional density ($\approx 0.5$) under high-$H$ truncation, providing additional margin.

4.3 Numerical closure

Taking $h^{+} = 3.84$: $$E[\delta \mid \text{composite},\; H \geq 3.84] \leq 1 - 1.823 = -0.823.$$ Measured: $E[\delta] = -1.16$ (gap due to non-spf splits achieving better savings; one-sided bound is not tight). Analytic closure requires the margin and density inputs specified in §5.5.

5. Scale Attenuation

5.1 Motivation

For even $n$ (spf $= 2$): $\Delta_2 = \rho_E(n-1) - \rho_E(n/2) - 2$. The sufficient condition holds at high $H$ because $\rho_E(n/2)$ does not track $\rho_E(n-1)$ — scale attenuation.

5.2 Direct measurement

Observation 5 (spf $= 2$).

$E[H(n/2)]$ anchors at $\approx 3.79$, independent of $H(n-1)$. Fine-bin data (high $H$, spf $= 2$):

As $H(n-1)$ increases from 3.83 to 4.03, $E[H(n/2)]$ decreases from 3.79 to 3.69.

5.3 Attenuation ratio

With $h_0 = 3.80$: $\text{excess\_parent} = H(n-1) - h_0$, $\text{excess\_child} = H(n/2) - h_0$.

Observation 6. In Q4/Q5 (spf $= 2$):

5.4 Mechanism

$\rho_E(n/2)$ is computed by the DP at position $n/2$, before $\rho_E(n-1)$. The height excursion accumulates during the $\approx n/2$ steps from $n/2$ to $n-1$; the value at $n/2$ reflects the pre-excursion state.

5.5 Analytic target

Under the current spf-comparator route, upgrading the numerical closure of §4 to analytic closure requires the following inputs.

A concrete sufficient route (even-dominance version). Define $C_{2,N}(h^{+}) = C_N(h^{+}) \cap \{\text{spf}(n) = 2\}$, and let $d_2(h^{+}, N) = P(\text{spf}(n)=2 \mid C_N(h^{+}))$, $a_2(h^{+}, N) = E[(\Delta_2)^{+} \mid C_{2,N}(h^{+})]$. If one can prove:

• (i) Quantitative comparator margin: $a_2(h^{+}, N) \geq A_0$ for all large $N$.
• (ii) Even density lower bound: $d_2(h^{+}, N) \geq d_0$.
• (iii) $d_0 \cdot A_0 > 1 + c$.

Then $E[(\Delta_{\text{spf}})^{+} \mid C_N(h^{+})] \geq d_2 \cdot a_2 \geq d_0 \cdot A_0 > 1 + c$, yielding the main target.

Scale attenuation as a mechanistic lemma for (i). If one can quantify the attenuation as $$E[\rho_E(n/2) \mid C_{2,N}(h^{+})] \leq h^{+} \cdot E[\ln(n-1) \mid C_{2,N}(h^{+})] - (3 + \eta)$$ for some $\eta > 0$, then $a_2 \geq 1 + \eta$ follows.

Numerical status. At $N = 10^7$: $d_2 \approx 0.663$, $a_2 \approx 1.91$, $d_2 \cdot a_2 \approx 1.27 > 1$. Analytic proof is the central open problem of the current route in this series.

6. Asymmetry Between High and Low Ends

6.1 One-sided nature of the spf comparator

Propositions 1–2 provide a sufficient condition for upper reflection. The low end (positive drift at low $H$) cannot be derived from the same mechanism.

When $H$ is low, $(\Delta_{\text{spf}})^{+} = 0$ means only that the spf split loses to the additive path — not that all splits lose. Since $M_n$ minimizes over all factorizations, a more balanced split may still win. Thus $(\Delta_{\text{spf}})^{+} = 0$ does not imply $S = 0$ or $\delta = +1$.

Low-$H$ positive drift ($E[\delta] = +0.70$ at Q1) arises from a different mechanism: at low $H$, jump probability is low ($P(j \geq 1) = 26\%$), and the additive path ($\delta = +1$) dominates. Formalizing this requires different tools (e.g., an upper bound on jump probability at low $H$), deferred to Paper XXVIII.

6.2 Division of labor

7. Auxiliary Structures

7.1 Stability of $h_0$

The height-drift crossover $h_0(N)$:

$h_0$ increases monotonically with decreasing increments, suggesting convergence to $h_0^\infty \approx 3.82$–$3.85$.

7.2 Imbalance of winning factorizations

Median balance ratio of winning factorizations: $0.0007$. At high $\Omega$ it rises to $0.68$. The spf comparator is most effective at low-to-moderate $\Omega$; high-$\Omega$ composites (where more balanced splits dominate) are exponentially sparse by Sathe–Selberg weights.

7.3 Drift structure within inter-prime gaps

The strong negative drift at the first position reflects the "prime pushes $\rho_E$ up → next composite pulls back" step-by-step mechanism.

7.4 Technical note: $H(n)$ and Markov property

$H(n)$ is a deterministic arithmetic sequence, not a Markov chain. The full application of Foster–Lyapunov stability theory requires defining a suitable randomized or augmented-state process; Paper XXI's Lindley isomorphism provides the interface. Within this paper, the upper reflection criterion is a conditional-expectation sign control statement and does not require Markov structure.

8. Discussion

8.1 Physical picture

At high $H$, the spf split references $\rho_E(n/\text{spf}(n))$, which was computed before the current height excursion built up. The excursion makes the additive path expensive relative to the spf split, so jump probability increases and drift turns negative. This time-asymmetry effect — the DP evaluates factors at smaller scales before the excursion accumulates — is the microscopic mechanism behind upper reflection.

8.2 Relation to Paper XXVI

Paper XXVI established that composites are net consumers of $\rho_E$. The present paper explains why: numerically, $\rho_E$ spends substantial mass above $h_0$, where composite drift is negative. The system does not collapse because primes inject deterministic $+1$ increments, pushing $\rho_E$ back toward $h_0$.

8.3 Open problems

1. Even subfamily comparator margin and density lower bound (§5.5): central remaining problem for analytic closure under the spf-comparator route.
2. Lower reflection theorem: positive drift at low $H$ via jump probability bounds (Paper XXVIII).
3. From two-sided reflection to pred gap $> \tau_k$: dyadic-scale prime-composite balance.
4. Precise asymptotics of $h_0(N)$: convergence rate and relation to PNT.
5. Chebyshev bound as a corollary: if upper reflection generalizes to $S_{\text{comp}}(N) = -\Theta(N/\ln N)$, the Chebyshev bound $\pi(N) = \Theta(N/\ln N)$ follows directly.

8.4 Acknowledgments

ChatGPT 5.4 Pro identified the correct sufficient condition $E[(\Delta_{\text{spf}})^{+}] > 1$ (§2.2), the one-sided nature of the spf comparator (§6), scale attenuation as the core bottleneck (§5), and the necessity of subgroup aggregation (§4.2). Gemini suggested Foster–Lyapunov over strict monotonicity. Grok ensured consistency with the preceding 26 papers. Claude wrote all numerical computation scripts (Blocks 1–5) and drafted working notes v1–v3. The final text was completed independently by the author; all mathematical judgments are the author's responsibility.

9. Data Sources and Reproducibility

All numerical results are based on full DP computations ($N = 10^7$, SPF sieve + multiplicative sieve). Sanity check: $\rho_E(10^7) = 58$.

References

1. ZFCρ Papers I–XXVI. H. Qin. Paper I: 10.5281/zenodo.18914682. Paper XXI: 10.5281/zenodo.19037934. Paper XXV: 10.5281/zenodo.19054726. Paper XXVI: 10.5281/zenodo.19059834.
2. J. Arias de Reyna. Complexity of natural numbers and arithmetic compact coding. Preprint.
3. K. Cordwell, S. Epstein, A. Hemmady, S. J. Miller, E. Steiner (2018). On the number of 1's needed to represent n. J. Number Theory, 189:17–34.
4. H. Altman (2014). Integer complexity, addition chains, and well-ordering. PhD thesis, Rutgers University.
5. S. P. Meyn, R. L. Tweedie (1993). Markov chains and stochastic stability. Springer-Verlag, London.