§1 Introduction

1.1 Background

Paper XXXVI (DOI: 10.5281/zenodo.19153002): 1/φ(m) joint profile, explanation rate plateau (19–24%), cofactor test (~80% global path effect), Route C repositioned to directly prove \(\mu_{\mathrm{aff}} - \mu_{\mathrm{gen}} = O(1)\).

1.2 Notation

• A = \(E[\rho(pq-1)]\) (\(q\) prime), B = \(E[\rho(pr-1)]\) (\(r\) random odd, \(\mu_{\mathrm{aff}}\)), C = \(E[\rho(\text{random } n)]\) (\(\mu_{\mathrm{gen}}\))
• \(\pi_{\mathrm{aff}} = P(\text{first-move=mult} \mid pr-1)\), \(\pi_{\mathrm{gen}} = P(\text{first-move=mult} \mid \text{random})\)
• \(\mu_{\mathrm{aff}}^{\mathrm{mult}}\), \(\mu_{\mathrm{gen}}^{\mathrm{mult}}\) = conditional means given first-move=mult
• Similarly \(\mu_{\mathrm{aff}}^{\mathrm{add}}\), \(\mu_{\mathrm{gen}}^{\mathrm{add}}\)

1.3 Goal

Achieve quantitative causal decomposition of B-C erosion: which term drives it?

§2 First-Step Asymmetry: A p-Independent Constant

\(\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}}\) fluctuates between +0.10 and +0.17 with no systematic trend. First-step asymmetry does not drive B-C erosion.

§3 Exact Three-Term Decomposition

3.1 Formula

$$B\text{-}C = E_p + C_p^{\mathrm{mult}} + C_p^{\mathrm{add}}$$

where:

• \(E_p = (\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}}) \times (\mu_{\mathrm{gen}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\) — entry effect
• \(C_{\mathrm{mult}} = \pi_{\mathrm{aff}} \times (\mu_{\mathrm{aff}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{mult}})\) — mult branch continuation
• \(C_{\mathrm{add}} = (1 - \pi_{\mathrm{aff}}) \times (\mu_{\mathrm{aff}}^{\mathrm{add}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\) — add branch continuation

3.2 Data

Exact to machine precision (residual = 0.0000 at all \(p\)).

3.3 Which Term Drives Erosion?

§4 Continuation Bias Reversal

4.1 Child ρ Difference Conditioned on first-move=mult

4.2 Interpretation

After multiplicative splitting, shifted children go from advantage (−0.59) to disadvantage (+0.96). Continuation bias is not just decaying — it is reversing. This is the deepest microscopic driver of B-C erosion: within the mult branch, shifted integers' continuation advantage reverses into a penalty.

4.3 Consistency with §3

§3 shows \(C_{\mathrm{mult}}\) shrinks from −1.39 to −0.53 (still negative). §4 shows child diff|mult has already turned positive. No contradiction: \(C_{\mathrm{mult}} = \pi_{\mathrm{aff}} \times (\mu_{\mathrm{aff}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{mult}})\) is the weighted overall mult-branch bias (including first-step saving), while child diff|mult is the continuation-only component after removing the first step. The overall mult branch is still favorable (first-step saving persists), but the continuation part has reversed.

§5 Rich First-Step Features Break the 20% Ceiling

The previous ~20% ceiling was a low-dimensional artifact. Rich features (7 dimensions: \(v_2\), \(v_3\), is_mult, sf_bin, saving_bin, child_size_bin, n_options; 355 levels) explain 45–51%. Gain decays with \(p\) (+45% to +6%): at large \(p\), rich and simple features converge.

§6 Memory Retention Rates

6.1 Unconditional Child Retention

6.2 Reconciling Unconditional (83–94%) and Mult-Conditioned (31% to −146%) Retention

Define child-level conditional means: \(\nu_{\mathrm{aff}}^{\mathrm{mult}} = E[\rho(\text{child}) \mid pr-1, \text{first-move=mult}]\), \(\nu_{\mathrm{gen}}^{\mathrm{mult}} = E[\rho(\text{child}) \mid \text{random}, \text{first-move=mult}]\). Let \(\delta_{\mathrm{mult}} = \nu_{\mathrm{aff}}^{\mathrm{mult}} - \nu_{\mathrm{gen}}^{\mathrm{mult}}\), \(\delta_{\mathrm{add}} = \nu_{\mathrm{aff}}^{\mathrm{add}} - \nu_{\mathrm{gen}}^{\mathrm{add}}\). The total child bias decomposes as:

$$D_{\mathrm{child}} = (\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}})(\nu_{\mathrm{gen}}^{\mathrm{mult}} - \nu_{\mathrm{gen}}^{\mathrm{add}}) + \pi_{\mathrm{aff}} \cdot \delta_{\mathrm{mult}} + (1-\pi_{\mathrm{aff}}) \cdot \delta_{\mathrm{add}}$$

Unconditional child retention (83–94%) includes the composition advantage (shifted walks mult more often). Mult-conditioned child retention (31% to −146%) isolates the within-branch continuation effect (\(\delta_{\mathrm{mult}}\)). Total child bias is sustained by entry composition, while within-mult continuation has already reversed. The two forces are in competition.

§7 Matched-Scale \(\Delta_{\mathrm{penalty}}\)

\(\Delta_{\mathrm{penalty}}\) is nearly constant (0.698–0.705). \((B\text{-}C)/\Delta\) zero-crossing comes entirely from B-C's one-sided erosion.

§8 Per-Step Saving and Path-Memory (Auxiliary)

8.1 Per-Step ρ-Saving Difference

Per-step \(\rho\)-saving difference ~0.07 (pr-1 vs random, conditional on mult). As a dimensional heuristic, B-C of −1 to −2 corresponds to roughly 15–25 steps of accumulation, but this is not a precise dynamical conclusion.

8.2 Path-Memory Decay (Auxiliary)

Preliminary decay signal but noisy (500 samples, PATH_MAX=10⁷). Auxiliary evidence only.

§9 Complete Picture

9.1 Quantitative Causal Chain (First-Order Decomposition Closed)

(a) Entry \(E_p\) (5–14% of B-C, nearly constant): \(\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}} \approx 0.14\). Does not drive erosion. Contributes 5%.

(b) Mult continuation \(C_{\mathrm{mult}}\) (59–70%, drives 72% of erosion): Shifted integers starting with mult are still easier to compress (\(C_{\mathrm{mult}} < 0\)), but the advantage shrinks −1.39 to −0.53. At child level, continuation-only part has already reversed (−0.59 to +0.96).

(c) Add continuation \(C_{\mathrm{add}}\) (21–29%, drives 23% of erosion): Similar to \(C_{\mathrm{mult}}\) but smaller.

9.2 Route C Status

9.3 Route C Step 2b Precise Target

Define \(G(X;p) := \mu_{\mathrm{aff}}(X;p) - \mu_{\mathrm{gen}}(X)\).

Current data: \(G\) erodes through zero one-sidedly while \(\Delta\) is constant. First-order causal structure fully identified. Quantitative decomposition complete.

§10 Discussion

10.1 Core Contributions

(a) Exact three-term decomposition (machine precision). (b) \(C_{\mathrm{mult}}\) drives 72% of erosion. (c) Continuation bias reversal. (d) Rich features break 20% ceiling (45–51%). (e) Unified interpretation of retention rates. (f) Entry is constant background; continuation is the erosion engine.

10.2 Progress Table

10.3 Open Problems

(1) Can \(C_{\mathrm{mult}}\)'s decay rate be rigorized? (2) What is the algebraic mechanism of continuation reversal? (3) Does \(\Delta_{\mathrm{penalty}}\) grow at \(10^{11}\)–\(10^{12}\)? (4) Prove \(G = O(1)\) via contraction of the continuation bias operator.

§11 Data Sources

References

[1] ZFCρ Papers I–XXXVI. Paper XXXVI DOI: 10.5281/zenodo.19153002. Paper XXXV DOI: 10.5281/zenodo.19143732. Paper XXXIV DOI: 10.5281/zenodo.19140015.

Acknowledgments

Claude (Zilu) wrote all numerical scripts, drafted working notes v1–v2 and the formal text, discovered first-move p-invariance and the rich-feature ceiling breakthrough. ChatGPT (Gongxihua) proposed the exact three-term decomposition formula \(B\text{-}C = E_p + C_{\mathrm{mult}} + C_{\mathrm{add}}\), corrected v1's "bounded memory" overclaim, suggested the rich feature experiment and entry/continuation causal decomposition. Gemini (Zixia) contributed the geometric decay model (\(\gamma \approx 0.20\)) and the interpretation "memory did not reverse; the penalty became visible." Grok (Zigong) confirmed Lindley consistency and identified the matched-scale Δ flatness as consistent with Paper XXXIV. The final text was independently completed by the author.