1. Introduction

1.1 Background and scope

The ZFCρ series establishes a general theorem: any formalization $C(U)$ necessarily produces a nonempty remainder $\rho$. Within this framework, $D(N) \to 1$ — the convergence of the Lindley defect density for the Erdős additive complexity function — is a central quantitative target.

The proof chain for $D(N) \to 1$ involves three conditional layers (L1), (L2), (L3). Layer (L3) — the requirement that $E[\text{pred gap}] > \tau_k$ under shell conditioning $\Omega(m) = k$, $D_m = 0$ — is the remaining open link in the dyadic/predecessor-gap branch. Paper XXV reduced (L3) to Cesàro control of the dyadic shell constant $c_\text{dyad}$ and the predecessor gap. The present paper does not claim to close (L3); rather, it builds the structural tools needed for that closure and gives a precise characterization of the remaining gap.

1.2 Contributions

Six results:

• (A) Zero-jump lemma (§2). $D_n > 0$ implies $j(n) = 0$. This short algebraic fact significantly simplifies subsequent analysis.
• (B) Additive decomposition of the predecessor gap (§3). $\text{pred gap}(m) = R_I + N_{0,I} - E_I$, providing an exact accounting framework.
• (C) Prime source term and composite correction (§4). The exact algebraic identity $\rho_E(N) = \pi(N) + S_{\text{comp}}(N)$. The prime contribution is deterministic +1 at each prime; the composite correction is numerically negative globally and in every sampled dyadic interval.
• (D) Asymptotic uniformity under shell conditioning (§5). Four step rates and the excess rate vary by less than 0.05 percentage points across shells $\Omega = 5\text{–}12$, suggesting that a shell-uniform proof route may be feasible.
• (E) Interior drift adjudication (§6). In extra-clean windows ($D_{m-1} = 0$ and $D_{2m-1} = 0$), $E[\text{pred gap}]$ remains above 2.5. Boundary spillover modulates the signal by approximately $\pm 0.6$ but is not the dominant source of the positive mean.
• (F) Precise characterization of the remaining gap (§8). The $O(1/m)$ precision barrier is identified as intrinsic. A composite-drift correction theorem is formulated as the exact target for closing (L3).

1.3 Relation to existing literature

The prime/composite increment separation has no precedent in the integer complexity literature. The closest external prototype is Arias de Reyna's prime-recursive surrogate $L(n)$: $L(1) = 1$, $L(p) = 1 + L(p-1)$ for primes, $L(ab) = L(a) + L(b)$. Arias proved $\|n\| \le L(n) \le (3/\log 2)\log n$, and explicitly noted the fundamental difference between the complete additivity of $L$ and the non-additivity of true complexity.

The contribution of the present paper may be viewed as the first systematic characterization of a similar separation for true complexity (which is non-additive). The transition from $L(n)$ to true complexity requires controlling the "good factorization" effect of non-additivity — precisely the composite-drift correction problem characterized in §8.

Known external obstacles include: non-additivity of true complexity ($\|mn\| < \|m\| + \|n\| + 2$ is possible); the $(3+\varepsilon)$-type lower bound for the global constant remains unproven (Cordwell et al. 2018); extrema are controlled by sparse smooth families (Altman 2012); probabilistic methods have known pitfalls (Shriver / Steinerberger).

1.4 Notation

We follow series conventions. $\rho_E(n)$: Erdős additive complexity, $\rho_E(1) = 0$. $M_n = \min_{ab=n,\, a,b\ge 2}(\rho_E(a)+\rho_E(b)+2)$; for primes $p$, $M_p = +\infty$ (no nontrivial factorization). Recurrence: $\rho_E(n) = \min(\rho_E(n-1)+1, M_n)$. For composites: $G(n) = \rho_E(n-1)+1-M_n$, $j(n) = \max(G(n),0)$, $D_n = M_n - \rho_E(n)$. For primes, the recurrence degenerates to $\rho_E(p) = \rho_E(p-1)+1$; for uniformity of counting, we set $G(p) = j(p) = D_p = 0$ (see Convention 2.0). $\pi(x)$: prime counting function. $\Omega(n)$: number of prime factors of $n$ counted with multiplicity.

2. The Zero-Jump Lemma

Corollary 1.1. $j(n) > 0$ implies $D_n = 0$. Jumps occur only where the defect vanishes.

Numerical verification. At $N = 10^7$ there are 1,279,213 maximal runs (excursions) of $D > 0$. In every excursion, $\Sigma j = 0$ exactly. The contrapositive holds without exception in the computed range.

2.1 Structure of D > 0 excursions

Excursions are extremely short: 95.63% have length 1, 4.36% length 2, 0.01% length 3. By Lemma 1, each step within an excursion contributes $\rho$-increment +1 (pure additive repair).

3. Additive Decomposition of the Predecessor Gap

3.1 Step classification

Every step $n \ge 2$ belongs to exactly one of four classes:

• (R) Repair: $D_n > 0$. By Lemma 1, $j(n) = 0$, $\rho$-increment $= +1$.
• (N₀) Free zero: $D_n = 0$, $j(n) = 0$. For composite $n$: $M_n = \rho_E(n)$ and $M_n \ge \rho_E(n-1)+1$, so $\rho$-increment $= +1$. Primes placed in this class by convention; likewise contribute $\rho$-increment $= +1$.
• (J₁) Unit jump: $D_n = 0$, $j(n) = 1$. Here $M_n = \rho_E(n-1)$, $\rho$-increment $= 0$.
• (J≥2) Large jump: $D_n = 0$, $j(n) \ge 2$. Here $M_n \le \rho_E(n-1)-1$, $\rho$-increment $= 1 - j(n) \le -1$.

The four classes form a complete partition: $R + N_0 + J_1 + J_{\ge 2} = N - 1$.

3.2 Decomposition theorem

For the interval $I = [m, 2m-1]$, define $R_I, N_{0,I}, J_{1,I}, J_{\ge 2,I}$ as counts of each class in $I$, and $E_I = \sum_{n \in I,\, D_n=0,\, j(n)\ge 2}(j(n)-1)$.

3.3 Global identity

For the full range $[2, N]$: $$\rho_E(N) = R_{[2,N]} + N_{0,[2,N]} - E_{[2,N]}.$$

3.4 Numerical verification

40,000 samples ($\Omega = 5\text{–}12$, 5,000 each): zero mismatches. Global check: $\Sigma j = 9{,}999{,}941 = (N-1) - \rho_E(N)$. ✓

4. Prime Source Term and Composite Correction

4.1 Deterministic increment at primes

The contribution of primes to $\rho_E$ is a deterministic, irreversible ascent: +1, without exception, independent of the current value of $\rho_E$.

4.2 Global prime/composite decomposition

Decompose the telescoping sum by primality:

4.2a Structural remark

At each $n$, the recurrence stages a competition between two paths: the additive path (+1, universal, structure-independent) and the multiplicative path ($M_n$, depending on the factorization of $n$). Composites admit at least one nontrivial factorization, so $M_n$ is finite and the multiplicative path may win. Primes have no nontrivial factorization; $M_p = +\infty$ and the additive path wins unconditionally.

Consequently, the +1 increment at primes is unconditional and forced — independent of the current $\rho_E$ value and the local structure around $n$. In this sense, primes are the sole unconditional source of positive increment in the recurrence; the true asymptotic difficulty lies in understanding the precise behavior of the composite correction.

4.3 Numerical observations

Observation 1. At $N = 10^7$: $\pi(N) = 664{,}579$; $\rho_E(N) = 58$; $S_{\text{comp}} = -664{,}521$. Near-perfect cancellation:

Observation 2 ($N$-dependence of composite rate). $\bar\delta_{\text{comp}}(N)$ tends to $0^-$:

Asymptotically, $\bar\delta_{\text{comp}} \approx -\pi(N)/(N-1) \approx -1/\ln N \to 0^-$, since $\rho_E(N) = O(\log N)$ is negligible relative to $\pi(N) \sim N/\ln N$.

4.4 Dyadic prime/composite decomposition

For $I = [m, 2m-1]$: $$\text{pred gap}(m) = [\pi(2m-1)-\pi(m-1)] + S_{\text{comp}}(I).$$

Observation 3. Across all tested shells ($\Omega = 5\text{–}12$) and window types: $P(S_{\text{comp}}(I) < 0) = 1.0000$.

4.5 Relation to Arias de Reyna's L(n)

Arias de Reyna's surrogate $L(n)$ is completely additive: $L(ab) = L(a) + L(b)$. In $L$ there is no "composite consumption" — multiplicative structure creates no complexity loss. True complexity differs by non-additivity: $\|mn\| < \|m\| + \|n\|$ is possible. These "good factorizations" are the microscopic source of composite consumption. The present paper provides the first quantitative separation of this non-additive effect via the exact identity of Proposition 4.

4.6 Limitations

Proposition 4 is a global identity. It confirms $S_{\text{comp}}(N) = \rho_E(N) - \pi(N)$ but does not give an independent estimate of $S_{\text{comp}}$ on local intervals or under shell conditioning. The shell-conditioned asymptotics of the composite drift is the central open problem of this series, deferred to subsequent work.

5. Asymptotic Uniformity Under Shell Conditioning

5.1 Step rates and excess rate

Observation 4 (Asymptotic uniformity). Under $\Omega(m) = k$, $D_m = 0$ ($N = 10^7$, 5,000 samples per shell):

Maximum cross-shell span: $R/m \sim 0.00034$, $N_0/m \sim 0.00041$. $J_{\ge 2}/m$ is identical to displayed precision (0.29195).

5.2 Prime density uniformity

Prime density in $[m, 2m-1]$ also shows no shell dependence (deviations < 0.7%, no systematic trend).

5.3 Interpretation

Under $D_m = 0$, the interval's internal statistics are insensitive to the factor structure of $m$. This is consistent with Paper XXI's Lindley isomorphism: $D_m = 0$ numerically behaves as a quasi-regeneration point, after which the system carries no significant memory of $m$'s factorization in its bulk statistics.

The bulk rate uniformity does not extend to pred gap itself, which still depends on $k$ (ranging from $\approx 2.84$ at $\Omega=5$ to $\approx 2.69$ at $\Omega=12$). This $O(0.15)$ variation suggests that shell dependence enters only through higher-order corrections.

6. Interior Drift and Boundary Effects

6.1 Window classification

Under $D_m = 0$, $\Omega(m) = k$:

• Clean: $D_{2m-1} = 0$ (no right spillover)
• Extra-clean: $D_{m-1} = 0$ and $D_{2m-1} = 0$ (no spillover on either side)
• Dirty: $D_{2m-1} > 0$ (right boundary in excursion)

6.2 Adjudication

Observation 5. Extra-clean windows retain full signal:

6.3 Boundary modulation

Observation 6 ($\Omega = 8$):

Right spillover raises pred gap by $\approx +0.6$; left spillover lowers it by $\approx -0.25$. The baseline (Type A) already exceeds $\tau_k \approx 2.50$.

7. Auxiliary Structures

7.1 Alpha decay law

Observation 7. $E[j \mid D=0, \Omega=k]$ increases roughly linearly with $k$ (from 0.794 at $k=3$ to 3.605 at $k=12$).

Observation 8. The slope $\alpha(N)$ decays as $1/\ln\ln N$: $\alpha \cdot \ln\ln N \approx 0.87$ across five orders of magnitude ($N = 10^5$ to $10^7$, variation < 2%).

7.2 Negative correlation compression

Observation 9. $\text{corr}(q_I, \bar j_{0,I})$ ranges from $-0.998$ ($\Omega=5$) to $-0.99999$ ($\Omega=12$). The product $q \cdot \bar j_0$ has its fluctuation compressed by factors of 15 to 150 relative to the marginals.

7.3 Direct measurement of c_local

Observation 10. $c_{\text{local}} > 3.6$ for all shells ($\Omega = 3\text{–}12$), with margin over $\tau_k/\ln 2 \approx 3.607$ ranging from $+0.26$ to $+0.49$.

8. Precise Characterization of the Remaining Gap

8.1 Six-digit near-perfect cancellation

From full-precision statistics: $$(R+N_0)/m \approx 0.40732, \quad E/m \approx 0.40731.$$ The difference, $\text{pred gap}/m \approx 2.8/m$, is $O(10^{-6})$ for $m \sim 10^6$. This precision requirement is an intrinsic feature of the problem, not a deficiency of method.

8.2 The O(1/m) barrier

Every proof strategy explored converges to the same requirement: local precise asymptotics of the composite correction term.

8.3 Composite-drift correction theorem

8.4 Known external obstacles

(i) Non-additivity of true complexity. (ii) The $(3+\varepsilon)$ lower bound is unproven (Cordwell et al. 2018). (iii) Extrema controlled by sparse smooth families (Altman 2012). (iv) Known flaws in probabilistic approaches (Shriver / Steinerberger).

8.5 Tools provided by this paper

(a) Lemma 1 restricts jumps to $D=0$ positions. (b) The additive decomposition $R+N_0-E$ gives an exact accounting framework. (c) The prime/composite decomposition reduces the problem to the composite correction. (d) Shell uniformity suggests a single-argument proof may suffice. (e) Interior drift adjudication focuses attention on the interval interior. (f) $c_{\text{local}} > 3.6$ provides numerical safety margin of $O(0.1)$.

9. Discussion

9.1 The remainder and incompressibility

The prime/composite decomposition reveals a fundamental picture of complexity growth. The multiplicative path, via factorization, borrows known complexities of smaller numbers — a compression mechanism that bypasses the step-by-step +1 accumulation. Composites, having at least one nontrivial factorization, can access multiplicative compression. The data shows this compression is extraordinarily efficient: the aggregate composite contribution is negative (§4.3), meaning composites absorb more complexity than they accumulate through additive steps. Multiplicative structure, as a whole, is a net absorber of complexity.

Primes have no nontrivial factorization. Multiplicative compression is unavailable to them. They are the only positions in the system that must be reached entirely through the additive path — blind spots of the compression mechanism. The +1 increment at primes is unconditional, forced, deterministic.

This resonates directly with the central thesis of the ZFCρ series: formalization (here, multiplicative decomposition) is a compression tool; the remainder is what survives compression. Primes, as positions unreachable by multiplicative compression, are the microscopic carriers of that remainder.

9.2 The heat sink analogy

The six-digit cancellation of §8.1 admits a thermodynamic reading. Multiplicative structure acts as a highly efficient heat sink, absorbing nearly all complexity growth. Primes are the heat sink's blind spots: at each prime, the system is forced to accept an irreversible +1 increment. The growth of $\rho_E$ is the residual heat that the sink fails to cover.

The sink is highly efficient ($|S_{\text{comp}}| \sim \pi(N) \sim N/\ln N$) but imperfect (residual $= \rho_E(N) = O(\log N)$). The predecessor gap $\approx 2.8$ is this residual projected onto the dyadic scale.

9.3 Open problems

1. The composite-drift correction theorem (§8.3).
2. Local asymptotics of $S_{\text{comp}}(I)$ under shell and $D_m = 0$ conditioning.
3. Analytic proof of shell uniformity (§5). Can it be derived from the Lindley regeneration structure of Paper XXI?
4. Analytic form of the alpha decay law (§7.1). Can $\alpha \cdot \ln\ln N \approx 0.87$ be derived from Paper XXIII's 80/3 law or Sathe-Selberg weights?
5. If $\lim \rho_E(N)/\ln N$ exists (denote it $c_{\text{main}}$), what is its precise value and relation to PNT? Note $c_{\text{main}}$ (if it exists) differs from $c^* = \liminf \rho_E(n)/\ln n$.

9.4 Acknowledgments

The data analysis and structural exploration in this paper benefited from multi-model AI collaboration (Claude, ChatGPT, Gemini, Grok). ChatGPT proposed the additive decomposition $R + N_0 - E$ as the proof object (§3) and the composite-drift correction as the precise target (§8.3). Gemini identified the importance of the alpha decay law and the non-constancy of the composite consumption rate ($\approx -1/\ln N$). Grok's sustained review ensured consistency with the preceding 25 papers. Claude wrote all numerical computation scripts and drafted working notes v1–v4. The final text was completed independently by the author; all mathematical judgments are the author's responsibility.

10. Data Sources and Reproducibility

All numerical results are based on full DP computations ($N = 10^7$, SPF sieve + multiplicative sieve, $O(N \log N)$). Scripts will be released with the paper.

All sanity checks pass: $\rho_E(100)=15$, $\rho_E(10^3)=24$, $\rho_E(10^4)=32$, $\rho_E(10^5)=41$, $\rho_E(10^6)=49$, $\rho_E(10^7)=58$. $\Sigma j = (N-1) - \rho_E(N)$ holds exactly.

References

1. ZFCρ Papers I–XXV. H. Qin. DOIs: Paper I (10.5281/zenodo.18914682), Paper XXI (10.5281/zenodo.19037934), Paper XXII (10.5281/zenodo.19039953), Paper XXIII (10.5281/zenodo.19041689), Paper XXIV (10.5281/zenodo.19044696), Paper XXV (10.5281/zenodo.19054726).
2. J. Arias de Reyna. Complexity of natural numbers and arithmetic compact coding. Unpublished manuscript / preprint.
3. K. Cordwell, S. Epstein, A. Hemmady, S. J. Miller, E. Steiner. On the number of 1's needed to represent n. J. Number Theory, 189:17–34, 2018.
4. H. Altman. Integer complexity, addition chains, and well-ordering. PhD thesis, Rutgers University, 2014.
5. C. E. Shriver. On integer complexity and the Markov chain approach. Preprint.
6. P. Erdős. Some remarks on number theory. Riveon Lematematika, 9:45–48, 1953.
7. R. K. Guy. Every number is expressible as the sum of how many polygonal numbers? Amer. Math. Monthly, 101:169–172, 1994.