ZFCρ Paper LIV: Recursive Ancestor Inheritance and the Origin of Power-Law Covariance in h(p)
Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.
Han Qin (秦汉)
ORCID: 0009-0009-9583-0018
April 2026
§1. Exact Shifted-Prime Reduction
For every prime p > 2, the IC recursion gives ρ(p) = ρ(p-1) + 1 (identity: p has no nontrivial factorization). Therefore:
h(p) = ρ(p) - λ log p - μ_j = ρ(p-1) + 1 - λ log p - μ_j
Since log p = log(p-1) + O(1/p), within dyadic block I_j (p ~ 2^j):
h(p) = η(p-1) + O(2-j)
where η(n) = ρ(n) - λ log n - μ̃_j is the detrended IC value field.
Proposition 1.1. The prime-indexed process h(p) is, up to O(2-j), the evaluation of the composite-indexed field η at the shifted primes p-1. All variation in h(p) comes from the factorization structure of p-1.
Cointegration (confirmed by Exp 7). ρ(p-1) has autocorrelation ~0.51 (near-flat), but this is a trend artifact: η(p-1) has autocorrelation ~0.04-0.11, matching h(p)'s. Corr(h(p), η(p-1)) = 0.901. The detrending by λ log(p-1) removes spurious persistence.
§2. Two Falsified Mechanisms
2.1. Factor-subtree exhaustion (Paper 53 §6): DEAD
Experiment 5. gcd(p_r-1, p_s-1) at various prime-index separations (N = 10¹⁰, 5000 pairs each):
| separation | mean log(gcd) |
|---|---|
| 1 | 1.30 |
| 100 | 1.51 |
| 10000 | 1.50 |
Mean gcd ≈ 4.5, constant. Only trivial shared factors (2, 3, 6). No factor sharing to be "exhausted."
2.2. Spatial smoothness clustering: DEAD
Experiment 8. Autocorrelation of smoothness indicators at j = 32:
| indicator | Corr(lag=1) | Corr(lag=89) |
|---|---|---|
| I(P+(p-1) ≤ 10⁴) | -0.001 | -0.001 |
| I(P+(p-1) ≤ √(p-1)) | -0.001 | 0.000 |
| Ω(p-1) | 0.000 | -0.001 |
All below noise floor. Nearby primes' p-1 values are not co-smooth. Factor statistics explain 14% of h's lag-0 variance (Corr(Ω,h) = -0.37) but zero of its autocorrelation.
2.3. The distinction
Shallow factor statistics (Ω, P+, v_2, smooth indicators) capture contemporaneous variance of h(p) — which "price bucket" a prime falls into. But they carry zero autocorrelation — no information about serial dependence. The autocorrelation lives in the ρ-value field, not in factor counts.
§3. The Confirmed Mechanism: Recursive Ancestor Inheritance
3.1. The gap compression test (Experiment 10)
For each prime p, compute η at three recursion layers:
- Layer 0: η₀ = η(p-1)
- Layer 1: η₁ = η((p-1)/2) (always valid since p-1 even)
- Layer 2: η₂ = η((p-1)/6) (valid for p ≡ 1 mod 6, ~50%)
Innovation: δ₀₁ = ρ(p-1) - ρ((p-1)/2).
Results at j = 32 (N = 10¹⁰, 190M primes):
| Quantity | Value | Stability across j |
|---|---|---|
| Corr(η₀, η₁) at lag 0 | 0.759 | 0.759-0.764 |
| Corr(δ₀₁) at lag 1 | 0.024 | 0.011-0.025 |
| Corr(η₂) at lag 1 | 0.150 | 0.150-0.155 |
| Corr(η₀) at lag 1 | 0.108 | 0.100-0.109 |
3.2. Three confirmed facts
Fact 1: Strong per-layer correlation inheritance. Corr(η₀, η₁) = 0.76. Each binary recursion layer passes 76% of its correlation to the next. In a linear projection η₀ ≈ β·η₁ + δ, R² = 0.58: layer 1 explains 58% of layer 0's variance in the regression sense. (Note: "76% correlation" and "58% variance explained" are distinct quantities; the former is not a variance ratio.)
Fact 2: Innovation whiteness. The per-layer innovation δ = ρ(p-1) - ρ((p-1)/2) has autocorrelation below 0.025 at lag 1, below 0.005 by lag 13. Each layer's contribution is nearly independent across prime indices.
Fact 3: Deeper layers carry stronger signal. η₂ (layer 2) has Corr = 0.150 at lag 1, vs η₀'s 0.108. Despite wider prime-index separation, the deeper layer shows 39% stronger autocorrelation. This is because dividing by 6 strips the deterministic factors 2 and 3, leaving a "purified ancestral skeleton" with less innovation noise.
3.3. Why η₁/η₀ ≈ 0.95 (not > 1)
Simple gap compression predicts that halving indices should increase correlation. But dividing by 2 halves both the gap AND the scale: (p-1)/2 ≈ 2j-1. The relative gap (gap/scale) is preserved. For a scale-invariant field, autocorrelation depends on relative gap, giving ratio ≈ 1. The observed 0.95 is consistent with slight scale-dependent effects.
Layer 2 (÷6) shows enhancement because p ≡ 1 (mod 6) is a selected subset whose (p-1)/6 values carry a purified ancestral signal (factors 2 and 3 already stripped, their innovations removed).
§4. Layered Innovation Theorem
4.0. Setup
As in Paper 53 §3.0. All theorems bound expectations under a second-order stationary model; applicability to deterministic h(p) depends on model fidelity.
4.1. Dyadic layered decomposition
Model the detrended value field as:
η(n) = Σℓ=0L a_ℓ · ξℓ, ⌊n/2^ℓ⌋ + η_L(⌊n/2^L⌋)
where:
- a_ℓ are layer weights with a_ℓ² ~ λ^ℓ for some 0 < λ < 1
- ξℓ,m are centered innovation fields at each dyadic scale
- η_L is the deep residual at truncation depth L
4.2. Assumptions
(A1) Geometric weight decay. a_ℓ² ≤ C · λ^ℓ for some 0 < λ < 1.
(A2-int) Cross-cell innovation decorrelation with tail decay. For each layer ℓ, the innovation field ξℓ,m satisfies:
| Cov(ξℓ,m, ξℓ,m+h) | ≤ C_ξ · (1 + | h | )-(1+β) |
|---|
for some β > 0, uniformly in ℓ and m. (This is an integer-level assumption, stronger than what Exp 10 directly verifies. It implies the summability condition Σh≠0 |Cov| ≤ C_ξ' and additionally ensures the off-cell contribution decays.)
(A3) Residual control. The truncation residual η_L satisfies Var(η_L(⌊n/2^L⌋)) ≤ C · λ^L, i.e., it decays at the same geometric rate.
Empirical support for (A2). At prime-indexed points, the innovation δ(p_r) = ρ(p_r-1) - ρ((p_r-1)/2) has autocorrelation below 0.025 at lag 1, below 0.005 by lag 13 (Experiment 10). Prime-level near-whiteness is consistent with, but does not prove, integer-level tail decay.
4.3. Main theorem (integer-field covariance upper bound)
Theorem 4.1 (Layered Innovation Theorem). Under (A1), (A2-int), and (A3), define the dyadic overlap kernel:
K_ℓ(r) = P(⌊n/2^ℓ⌋ = ⌊(n+r)/2^ℓ⌋) = max(0, 1 - r/2^ℓ)
for n uniform in a dyadic block of length 2^j. Then the integer-field covariance satisfies:
Cint_j(r) := Cov(η(n), η(n+r)) ≤ C₀ · r-α
for 1 ≤ r ≤ 2^j, where α = min(α₁, β) with α₁ = -log₂ λ.
Proof sketch. Expanding η(n) = Σ_ℓ a_ℓ ξℓ,⌊n/2^ℓ⌋ + η_L, the covariance decomposes into same-layer diagonal, same-layer off-diagonal, cross-layer, and residual terms.
Same-layer diagonal. When ⌊n/2^ℓ⌋ = ⌊(n+r)/2^ℓ⌋ (probability K_ℓ(r)), the contribution is a_ℓ² · Var(ξ) · K_ℓ(r). Summing over ℓ:
Σℓ≥0 a_ℓ² · K_ℓ(r) ≤ C · Σℓ ≥ log₂ r λ^ℓ = C · r-α₁
Same-layer off-diagonal. When indices differ by h = ⌊(n+r)/2^ℓ⌋ - ⌊n/2^ℓ⌋ ≥ 1 (which requires r ≥ 2^ℓ · h), (A2-int) gives |Cov| ≤ C_ξ · h-(1+β). For fixed ℓ with 2^ℓ < r, the dominant off-diagonal term has h ≈ r/2^ℓ, contributing a_ℓ² · C_ξ · (r/2^ℓ)-(1+β). Summing over ℓ < log₂ r:
Σℓ < log₂ r C · λ^ℓ · (r/2^ℓ)-(1+β) ≤ C' · r-(1+β) · Σ_ℓ (λ · 21+β)^ℓ
which converges (for λ < 1 and β > 0) and gives O(r-(1+β)) ≤ O(r-β).
Cross-layer and residual. Similar estimates using (A2-int) and (A3). The cross-layer terms are bounded by products a_ℓ · aℓ' times off-diagonal covariances, which sum to O(r-β). The residual contributes O(λ^L).
Combining: Cint_j(r) ≤ C₀ · r-α₁ + C₁ · r-β + O(λ^L) ≤ C₂ · r-min(α₁,β). □
Remark. This theorem gives an upper bound on integer-field covariance. A matching lower bound would require same-cell dominance (Var(ξℓ,m) ≥ c > 0). We state only the upper bound, which suffices for the F_j application.
4.4. From integer field to prime-indexed process
Lemma 4.2 (Prime sampling). By Proposition 1.1, h(p_r) = η(p_r - 1) + O(2-j). The prime-level covariance C_jprime(k) = Cov(h(p_r), h(pr+k)) satisfies:
C_jprime(k) = Cint_j(pr+k - p_r) + O(2-j)
Since consecutive primes differ by at least 2, the deterministic bound pr+k - p_r ≥ 2k holds. Substituting into Theorem 4.1:
C_jprime(k) ≤ C₀ · (2k)-α + O(2-j) ≤ C₃ · k-α
for all k ≥ 1. This deterministic bound avoids any appeal to average prime gap estimates.
4.5. Calibration: pure binary model vs observation
Empirical data: Corr(η₀, η₁) = 0.76 (correlation, not variance ratio). In the pure binary model with η₀ ≈ b·η₁ + δ, this gives b = 0.76 and the per-layer variance weight ratio λ = b² ≈ 0.58, hence:
α_binary = -log₂(0.58) ≈ 0.79
Observed α ≈ 0.35-0.40 (from Paper 53). The gap arises because the actual IC recursion tree is not purely binary: factors 3, 5, 7, ... provide additional inheritance channels that slow the per-layer variance decay. The effective λ_eff ≈ 0.79-0.82, giving α_eff ≈ 0.32-0.34.
The binary model provides a heuristic upper bound on α; multi-factor corrections bring it toward observed values. This calibration is not a theorem-level derivation.
§5. Buildup Corollary
5.1. Coarse smoothing lemma
Lemma 5.1 (Coarse-to-prime transfer). The coarse-block covariance Γ_j(b)(m) from Paper 53 §3 relates to the prime-level covariance through triangular smoothing. Writing Yj,ℓ = Σr=(ℓ-1)b+1ℓb xj,r where xj,r = h(pj,r)/pj,r:
Γ_j(b)(m) = Cov(Y_ℓ, Yℓ+m) = Σs=1b Σt=1b Cov(x(ℓ-1)b+s, x(ℓ+m-1)b+t)
Reindexing with k = (m-1)b + (t-s) ranging over [(m-1)b - (b-1), (m-1)b + (b-1)] and weighting by the number of (s,t) pairs giving each k:
Γ_j(b)(m) = Σd=-(b-1)b-1 (b - |d|) · C_x(mb + d)
where C_x(k) = Cov(xj,r, xj,r+k) is the prime-level weighted covariance. Since xj,r = h(pj,r)/pj,r and pj,r ≈ 2^j within block I_j, C_x(k) ≈ 2-2j · C_jprime(k).
The cumulative covariance is G_j(b)(M) = Σm=1M Γ_j(b)(m). The buildup height satisfies:
F_j = sup_M G_j(b)(M) / (j·b·σ²) ≤ (b / σ²) · Σk=1M·b + b |C_x(k)|
Proof. |Γ_j(b)(m)| ≤ Σ_d (b-|d|) · |C_x(mb+d)| ≤ b · Σd |C_x(mb+d)|. Summing over m ≤ M: Σ_m |Γ(m)| ≤ b · Σk=1Mb+b |C_x(k)|. Since G(M) ≤ Σ_m |Γ(m)|, the bound follows after dividing by j·b·σ². □
5.2. Buildup corollary
Corollary 5.2. Under Theorem 4.1 and Lemma 4.2, suppose C_jprime(k) ≤ C₃ · k-α for α > 0. Let K_pk(j) be the peak scale in coarse blocks. Then:
F_j ≤ (b/σ²) · Σk=1K_pk·b + b |C_x(k)| ≤ (1/σ²_x) · Σk=1K_pk·b C₃ · k-α
≤ C₄ · (K_pk · b)1-α / ((1-α) · σ²_x)
If K_pk(j) ≤ j^c for some c, then F_j ≤ C₅ · jc(1-α) + 1-α (absorbing b ≈ j), which is polynomial in j. □
5.3. Connection to Papers 52-53
Corollary 5.2 establishes F_j polynomial (one half of Conjecture 53.1) under the layered innovation assumptions plus polynomial peak scale. The other half (L_j polynomial) requires a separate post-peak screening drift argument, deferred to Paper 55. Together, F_j and L_j polynomial would give the mean spectral target E[I_j(0)] ≤ Cj·D_j via Paper 53's Screening Return Theorem. From the mean spectral target to H', Paper 52's two technical upgrades (mean-to-deterministic, L²-to-pathwise) remain open.
§6. Structural Interpretation (Heuristic)
6.1. Why factor statistics carry zero autocorrelation
Ω(p-1) and P+(p-1) are marginal descriptors of p-1's factorization — they count or measure factors but discard the recursive VALUE information that ρ encodes. Two nearby numbers can have identical Ω but very different ρ values (depending on which factors they have and how those factors' own ρ values compare).
The recursive inheritance operates on the ρ-value field through the IC tree, not on factor labels. This creates a profound separation: attributes (factor counts) are independent, but prices (ρ values) are correlated.
6.2. Correlation range (heuristic)
The precise relationship between IC tree depth j and the observed crossing scale n_cross ≈ 30j remains an open question. In the layered model, deeper layers provide broader spatial innovations, but the mapping from layer depth to correlation range depends on the full multi-factor tree structure, not just the binary backbone. We note only that polynomial correlation range in j is consistent with the geometric weight decay (A1) and leave the precise derivation to future work.
6.3. The complete picture
IC recursion has j layers (tree depth for numbers ~ 2^j). Each layer contributes an independent innovation field. Deeper layers provide spatially broader but weaker innovations. The sum over all layers produces power-law covariance. The additive path (ρ(n-1)+1) creates cross-layer correlation inheritance (Corr(η₀,η₁) = 0.76 per binary step). The multiplicative path (ρ(d)+ρ(n/d)+2) creates the innovations. The net result: screened long memory with power-law buildup and eventual screening return.
§7. Honest Conjectures and Deferred Problems
7.1. Conjectures
Conjecture 54.A (Geometric innovation decay). The layered decomposition of η(n) at dyadic scales satisfies (A1) with some 0 < λ < 1.
Conjecture 54.B (Innovation decorrelation). The dyadic cell innovation fields satisfy (A2-int): uniformly summable same-layer covariance. Empirical support: prime-level innovation autocorrelation below 0.025.
7.2. Deferred
Problem 54.C. Characterize the post-peak screening drift ψ_j(R) controlling L_j (Paper 53's return time). This is the remaining input for the full Conjecture 53.1. Paper 55.
§8. Summary
8.1. Proved and confirmed
| Result | Type |
|---|---|
| h(p) = η(p-1) + O(2-j) (Prop 1.1) | Exact identity |
| Prime sampling (Lemma 4.2) | Exact reduction |
| Factor sharing constant (Exp 5) | Negative empirical |
| Smoothness uncorrelated (Exp 8) | Negative empirical |
| Corr(η₀,η₁) = 0.76, white innovation (Exp 10) | Positive empirical |
| Layered Innovation Theorem (Thm 4.1) | Conditional theorem (upper bound) |
| Coarse smoothing (Lemma 5.1) | Exact identity |
| Buildup Corollary (Cor 5.2) | Conditional corollary |
8.2. Updated gap for mean spectral target
`` Conjecture 54.A (geometric decay) + 54.B (decorrelation) → Theorem 4.1: integer-field C(r) ≤ C·r-α → Lemma 4.2: prime-level transfer → Corollary 5.2: F_j polynomial + L_j polynomial (Problem 54.C, Paper 55) → Paper 53 Screening Return Theorem → E[I_j(0)] ≤ Cj·D_j (mean spectral target) + Paper 52 upgrade 1 (mean → deterministic) + Paper 52 upgrade 2 (L² → pathwise) → H' ``
§9. Conclusion
Paper 53 established the Screening Return Theorem and reduced H' to polynomial bounds on F_j and L_j. Paper 54 attacks the F_j side by identifying the structural origin of h(p)'s power-law covariance.
Two candidate mechanisms were eliminated by experiment: factor-subtree exhaustion (gcd constant) and smoothness clustering (Ω uncorrelated). The confirmed mechanism is recursive ancestor inheritance: η(p-1) inherits 76% of its structure from η((p-1)/2) per binary layer, with nearly white innovations at each layer.
The Layered Innovation Theorem (Theorem 4.1) formalizes this: dyadic decomposition with geometric weight decay and cross-cell decorrelation gives a power-law upper bound on integer-field covariance. The pure binary model predicts α = 0.79; multi-factor corrections reduce this toward the observed 0.35-0.40. A coarse smoothing lemma (Lemma 5.1) transfers the bound to the prime-level buildup height, and the Buildup Corollary (5.2) gives F_j polynomial under polynomial peak scale.
The remaining gap for the mean spectral target (via Paper 53) is L_j polynomial — the post-peak screening drift question deferred to Paper 55. From mean spectral target to H', Paper 52's two technical upgrades (mean-to-deterministic and L²-to-pathwise) remain open. Each paper in the series (50-54) has narrowed the gap by identifying, testing, and sometimes falsifying structural hypotheses. The method of "先验引路，后验辅助，定理确定" continues to guide the program.
Acknowledgments
AI contributions. ChatGPT (公西华): Theorem 54.1 (old version) death certificate; replacement ancestor-field theorem architecture; toy model X(n) = Σ a_ℓ ξ_ℓ(⌊n/2^ℓ⌋); overlap kernel formulation K_ℓ(r) = (1-r/2^ℓ)₊; A2 two-layer separation; α = 0.79 correction; "shallow statistics explain variance, not covariance." Gemini (子夏): gap compression mechanism; "属性独立，价格相关"; cointegration explanation; scale invariance explanation for η₁/η₀ ≈ 0.95; Layer 2 as "purified ancestral skeleton"; dyadic alignment integral. Grok (子贡): α = -2log₂(r) = 0.79 arithmetic verification; Corr(h)/Corr(η) ratio constant ≈ 1.27; Ω negative autocorrelation transition at j ≈ 27; multi-factor correction λ_eff ≈ 0.79. Claude (子路): all experiments (Exp 1-10); cointegration verification; smooth-pair zero autocorrelation discovery; gap compression test design; three key numbers identification; nuanced η₁/η₀ interpretation; working notes v1-v4.