ZFCρ Paper LXVII: Scale-Dependent Erosion Rate and the Intrinsic Spectral Bridge — Discovery of the C_eff Trajectory
ZFCρ 第LXVII篇：尺度依赖侵蚀率与内生谱桥——C_eff轨迹的发现

§1. Status After Paper 66

Paper 66 discovered the two-regime Ψ_η(W) structure: Regime 1 (power-law growth, W^{0.7}) and Regime 2 (crash, 5–40×). Power-law extrapolation overshoots reality by 100–10,000×. Thirteen hypotheses killed. Surviving mechanism: gradual chiseling, but insufficient alone for 59.1.

§2. Framework 5a Killed (Kills 14–15)

Kill 14: Simple Schur complement bridge reduces Ψ by only 10–30% (need 70%+). Peak at W = 10% m (not 0.5%).

Kill 15: Non-dyadic blocks [3·2^j, 4·2^j] show same peak-then-crash. Dyadic geometry not essential.

These kills close the class of simple externally imposed bridge/boundary mechanisms. Any surviving spectral bridge must be intrinsic and scale-dependent.

§3. C_eff Definition and Log-Budget Identity

Definition. C_eff(W₁ → W₂) = 1 − [log Ψ(W₂) − log Ψ(W₁)] / [log W₂ − log W₁].

C_eff < 1: Ψ growing (Regime 1). C_eff > 1: Ψ shrinking (Regime 2). C_eff = 1: critical summability threshold.

Proposition 67.A (Log-Budget Identity, exact). For any scale chain W₀ < W₁ < ⋯ < W_R = m_j:

log Ψ(m_j) − log Ψ(W₀) = Σ_r (1 − C_{j,r}) · Δ_r

where Δ_r = log(W_{r+1}/W_r). This is a direct telescoping identity from the definition.

Corollary 67.B (Cumulative Erosion Criterion). Let W₀ be a fixed starting scale (e.g., W₀ = 100 or 500, not varying with j). If Ψ(W₀) = O(j) and the cumulative budget Σ(1 − C_{j,r})·Δ_r ≤ B for some constant B independent of j, then Ψ(m_j) ≤ Ψ(W₀)·e^B = O(j), and Paper 65's conditional theorem yields the L² analogue of 59.1. More generally, if budget ≤ B·log j, then Ψ(m_j) = O(j^{B+1}) — a polynomial bound but not automatically O(j). For 59.1, budget = O(1) is required.

Note: when C_eff < 1 (Regime 1), each term (1 − C_r)·Δ_r is positive, driving Ψ growth. Once C_eff crosses 1.0 (Regime 2), the terms become negative, actively subtracting from the budget. It is this eventually increasing property of C_eff that locks the total budget from above.

Scope caveat: the bounded budget claim rests on j = 22–32 data. Whether the budget remains bounded for j > 32 cannot be determined empirically; this is a core Paper 68 target.

§4. C_eff Trajectory: All j = 22–32

Representative trajectory (j = 32, m = 190M):

C_eff rises from ~0.28 (shallow trough at W ≈ 500–1K) through 0.43, 0.66, 0.76, crossing 0.99 at W ≈ 2M (1.1% of m), then reaching 2.89 at W = 5–10M.

All 11 dyadic blocks (j = 22–32) show the same pattern: eventually increasing C_eff crossing 1.0. Crossing at W/m_j ≈ 1–3%.

Note: C_eff is not strictly monotone — a shallow trough at the smallest scales (W < 1K) precedes the sustained rise. "Eventually increasing" is the precise characterization.

§5. Block-Relative Collapse and Non-Dyadic Confirmation

Observation 67.C (Universal C_eff Trajectory). Across all j = 22–32, C_eff(W) starts near 0.3, exhibits a shallow trough, then eventually increases past 1.0. The crossing occurs at W/m_j of order 10⁻² (range 0.4%–3.7%).

Observation 67.D (Block-Relative Collapse). When plotted against W/m_j, C_eff trajectories from all j collapse to a common shape. Against absolute W, crossing shifts over two orders of magnitude (10K to 5M) — no collapse. The screening length scales with block size, not with any absolute prime count.

Non-Dyadic Persistence. Non-dyadic blocks [3·2^j, 4·2^j] for j = 25, 27, 29 reproduce the same C_eff trajectory shape and crossing. The non-dyadic crossing is delayed by roughly a factor of 2–3 relative to dyadic (e.g., dyadic j=25 crosses at W/m ≈ 1%, non-dyadic at W/m ≈ 5%), likely due to lower reference accuracy. The fundamental pattern is preserved.

§6. Cumulative Erosion Budget

Budgets across j = 22–32 range from 0.5 to 3.4, with no systematic growth. Compatible with O(1) and O(log j) — current data cannot distinguish.

For Corollary 67.B: only O(1) bounded budget directly yields Ψ_full = O(j) (59.1 satisfied). O(log j) budget only gives O(j^{B+1}) polynomial bound, which does not automatically satisfy 59.1. Current empirical evidence: if budget ≤ 3.5 for all j (O(1) scenario), then Ψ_full = O(j). Whether this bound persists for j > 32 is a Paper 68 target.

Budget-predicted log(Ψ_full) matches actual within ±0.5 in log scale.

§7. Small-W C_eff Drift

Observation 67.E. C_eff(100→200) decreases from 0.55 (j = 22) to 0.31 (j = 32). The IC field becomes locally smoother at larger j — neighboring primes' construction landscapes share more structure, producing weaker short-lag decorrelation. This recovers Paper 66's α_fit drift in the C_eff framework.

§8. Via Negativa → Via Rho Transition

This is the first round with zero kills after 15 consecutive kills across Papers 62–67. Via Negativa completed the systematic clearing of the search space; C_eff is the unique surviving quantitative object that passes all empirical tests.

Paper 67 marks the transition from elimination to construction. The C_eff trajectory is the primary surviving quantitative object that passes all empirical tests, providing a target that any eventual IC-recursion derivation must reproduce.

Epistemic caution: first all-confirm removes the natural self-correction of kills. Every claim in this paper carries explicit epistemic status labels (identity / conditional / empirical observation).

§9. Paper 68 Target

Derive from IC recursion ρ(n) = min(ρ(n−1)+1, min_{ab=n}[ρ(a)+ρ(b)+2]):

(a) Why C_eff eventually increases

(b) Why crossing occurs at W/m_j = O(1)

(c) Cumulative budget bound

Candidate frameworks: influence-cone overlap divergence, near-optimal factorization multiplicity, min-plus corrector theory.

§10. Status Map

Content	Status
Proposition 67.A: log-budget identity	Exact identity
Corollary 67.B: budget bounded → Ψ = O(j)	Conditional theorem
Observation 67.C: C_eff eventually increasing, crossing 1	Empirically confirmed
Observation 67.D: block-relative collapse W/m_j ≈ 1–3%	Empirically confirmed
Observation 67.E: small-W drift	Empirically confirmed
Non-dyadic persistence	Empirically confirmed
Budget bounded (0.5–3.4)	Empirically confirmed
15 hypotheses killed	Empirical/analytic
C_eff increase from IC recursion	Open (Paper 68)
59.1 unconditional	Open (Paper 68–69)

§1. Paper 66后的状态

1.1. 两阶段谱结构

Paper 66（DOI: 10.5281/zenodo.19701479）发现Ψ_η(W)先按power-law增长（Regime 1: W^{1-C}, C ≈ 0.3），在W ≈ 0.5% m_j处到达peak，然后crash 5-40倍（Regime 2）。Power-law外推超出实测100-10000倍。

1.2. 13 kills和surviving mechanism

Paper 66通过13次假说否决（坐标幻象/均值artifact/状态空间shortcut/screening模型/祖先shortcut）排除了所有macro-cancellation和shortcut mechanisms。Surviving bare mechanism是min操作的累计微侵蚀（gradual chiseling），但Paper 66的conditional theorem 66.A（ε_k ~ C/k → power-law）alone不给59.1——需要additional mechanism控制full-block Ψ。

1.3. Paper 67的目标

理解Regime 2的crash mechanism：什么使Ψ(W)在大W处停止增长并下降？

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§2. Framework 5a Killed（Kills 14-15）

2.1. Kill 14：Simple Schur complement bridge太弱

Power-law kernel K₀(k) + 单一zero-mode projection（Schur complement K_bridge = K₀ - K₀a(a'K₀a)⁻¹a'K₀）只减少Ψ 10-30%。Peak位置在W = 10% m（不是0.5% m）。需要5-10倍更强的correction。

单一零模投影不足以解释crash。

2.2. Kill 15：Non-dyadic blocks同样crash

[3·2^25, 4·2^25]（m = 1.8M）和[3·2^27, 4·2^27]（m = 6.7M）的sub-block Ψ同样show peak-then-crash pattern。

Crash不依赖dyadic block geometry。机制内在于IC field。

2.3. Simple externally imposed mechanisms的终结

Kills 14-15共同排除了"simple externally imposed bridge/boundary"作为crash mechanism：bridge projection（Kill 14）和dyadic boundary geometry（Kill 15）均fail。Paper 66的Kill 8另行排除了naive Thermo hierarchy作为structural mechanism。综合来看，任何surviving的spectral bridge必须是intrinsic且scale-dependent的——不是外加的single constraint。

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§3. C_eff定义与Log-Budget Identity

3.1. 定义

设Ψ_j(W)是j-block内以size W sub-blocks测得的方差放大因子。定义尺度区间[W₁, W₂]上的effective erosion exponent：

C_eff(W₁ → W₂) = 1 - [log Ψ(W₂) - log Ψ(W₁)] / [log W₂ - log W₁]

当C_eff < 1时Ψ(W)在增长（ACF正贡献主导）。当C_eff > 1时Ψ(W)在下降（ACF负/零贡献使variance增速低于W线性）。C_eff = 1是临界阈值：Σk^{-C}在C = 1处从divergent变成convergent。

3.2. Proposition 67.A（Log-Budget Identity）

Proposition（精确恒等式）。 对任意尺度链W₀ < W₁ < ... < W_R = m_j，设Δ_r = log(W_{r+1}/W_r)。则：

log Ψ_j(m_j) - log Ψ_j(W₀) = Σ_{r=0}^{R-1} (1 - C_{j,r})·Δ_r

其中C_{j,r} = C_eff(W_r → W_{r+1})。

推导。 直接从C_eff定义展开telescoping sum。每个term (1 - C_{j,r})·Δ_r = log Ψ(W_{r+1}) - log Ψ(W_r)。求和得结论。

状态： Exact identity from definition。不是条件定理——是数学恒等式。

3.3. Corollary 67.B（Cumulative Erosion Criterion）

Corollary。 设W₀为固定起始尺度（例如W₀ = 100或500，不随j变化）。若Ψ_j(W₀) = O(j)，且cumulative erosion budget满足

Σ_r (1 - C_{j,r})·Δ_r ≤ B

for some constant B independent of j，则Ψ_j(m_j) ≤ Ψ_j(W₀)·e^B = O(j)。

推导。 从Proposition 67.A: Ψ(m_j) = Ψ(W₀)·exp(budget)。若budget ≤ B则Ψ(m_j) ≤ Ψ(W₀)·e^B。

对59.1的含义： 若Corollary 67.B成立（budget = O(1)），结合Paper 65条件定理（59.1 ⟸ gauge + count + Ψ_η = O(j)）即得59.1的L²类比。

更一般：若budget ≤ B·log j，则Ψ_j(m_j) ≤ Ψ_j(W₀)·j^B = O(j^{B+1})。这只给多项式上界，不自动给O(j)。59.1严格要求budget = O(1)（即B·log j with B → 0）或等价地budget bounded by constant。

注意累积budget的物理含义：当C_eff < 1时（Regime 1），每一项(1 - C_r)·Δ_r为正，为budget做正贡献，驱动Ψ增长。一旦C_eff穿越临界阈值1.0（Regime 2），(1 - C_r)变为负值，budget开始被扣除。正是C_eff的eventually increasing性质使得Regime 2的negative budget抵消Regime 1的positive budget，锁死总budget的上限。

Scope caveat： Bounded budget claim目前依赖j=22-32范围内的empirical evidence。对j > 32, budget是否继续bounded无法从当前data确定。Asymptotic budget behavior是Paper 68 target的core sub-question之一。

状态： Conditional on budget bound。Budget的bound是Paper 68 target。

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§4. C_eff轨迹：全j=22-32精细测量

4.1. 半decade分辨率

W grid: 100, 200, 500, 1K, 2K, 5K, 10K, 20K, 50K, 100K, 200K, 500K, 1M, 2M, 5M, 10M, 20M, 50M（对每个j，只要K ≥ 4）。

4.2. Representative trajectory: j = 32（m = 190M）

Scale range	C_eff	Regime
100→200	0.31
200→500	0.29	Shallow trough
500→1K	0.28	← minimum
1K→2K	0.29
2K→5K	0.30
5K→10K	0.34	Regime 1:
10K→20K	0.37	power-law
20K→50K	0.43	growth
50K→100K	0.38
100K→200K	0.43
200K→500K	0.66
500K→1M	0.76	Approaching
1M→2M	0.99	← crossing
2M→5M	1.46	Regime 2:
5M→10M	2.89	crash
10M→20M	1.57

C_eff从shallow trough（~0.28 at lag 500-1K）开始，经持续上升穿过C = 1 at W ≈ 2M = 1.1% of m_j，然后继续增大至2.89。

注意：前几个scale的C_eff有浅trough（0.31 → 0.29 → 0.28），不是严格单调。但从trough之后持续上升——"eventually increasing"是precise characterization。Initial trough的mechanism是open sub-question，可能与small-W drift（Observation 67.E）相关。

4.3. 穿越1.0在所有j确认

j	m_j	Approx crossing W	W/m_j
22	268K	~10K	3.7%
23	514K	~5-10K	1-2%
24	986K	~5-10K	0.5-1%
25	1.9M	~20-50K	1-3%
26	3.6M	~20-50K	0.5-1.4%
27	7.0M	~50-100K	0.7-1.4%
28	13.6M	~50-100K	0.4-0.7%
29	26M	~200-500K	0.8-2%
30	51M	~500K-1M	1-2%
31	98M	~1-2M	1-2%
32	190M	~2-5M	1-3%

全部11个j在W/m_j量级10^{-2}处穿越（实测范围约0.4%-3.7%）。

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§5. Block-Relative Collapse与Non-Dyadic确认

5.1. Observation 67.C（Universal C_eff Trajectory）

对j = 22-32的所有dyadic blocks，C_eff(W)从短尺度~0.3起步，经shallow trough后eventually increasing，穿过critical threshold 1.0。穿越点在W/m_j量级10^{-2}（0.4%-3.7%）。

5.2. Observation 67.D（Block-Relative Collapse）

以W/m_j为横轴绘制C_eff轨迹时，所有j=22-32的曲线坍缩到共同形状。穿越C = 1的位置集中在W/m_j ≈ 0.4%-3.7%带。

以absolute W为横轴时，穿越位置从10K（j=22）到2-5M（j=32）跨越两个数量级——不collapse。

含义： crash的尺度与block size成比例。Screening length不是absolute number of primes，而是block size的固定fraction。

5.2. Non-dyadic confirmation

Non-dyadic [3·2^25, 4·2^25]和[3·2^27, 4·2^27]和[3·2^29, 4·2^29]的C_eff轨迹与对应dyadic blocks定性一致：从~0.3起步，eventually increase，穿越1.0，crash。

Non-dyadic的穿越点relatively于dyadic延迟约factor 2-3（例如dyadic j=25在W/m ≈ 1%穿越，non-dyadic j=25在W/m ≈ 5%穿越）。这一延迟likely来自non-dyadic blocks使用的enclosing dyadic reference accuracy较低。但fundamental pattern（eventually increasing C_eff, crossing 1.0, crash）完全保持。

含义（强化）： C_eff的scale-dependent erosion是IC field的intrinsic property，不依赖于dyadic block boundaries。

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§6. 累计侵蚀Budget

6.1. 测量结果

j	Budget = Σ(1-C_r)·Δ_r	log j
22	0.53	3.09
23	1.43	3.14
24	0.50	3.18
25	1.26	3.22
26	1.50	3.26
27	1.92	3.30
28	1.47	3.33
29	1.30	3.37
30	3.17	3.40
31	1.54	3.43
32	3.35	3.47

6.2. 解读

Budgets range 0.5 to 3.4。不systematic grow with j。Compatible with O(1)和O(log j)——当前data不能区分这两种scenario。

对Corollary 67.B：只有O(1) bounded budget直接给出Ψ_full = O(j)（59.1 satisfied）。O(log j)的budget只给出O(j^{B+1})的多项式上界，不自动满足59.1。

当前empirical evidence：若budget ≤ B = 3.5 for all j（O(1) scenario），则Ψ_full ≤ Ψ(W₀)·e^{3.5} ≈ 33·Ψ(W₀) = O(j)。但j > 32时budget是否继续bounded无法从当前data确定。

Budget bounded是core empirical claim。Specific growth rate（O(1) vs O(log j)）是Paper 68 target。

6.3. Budget prediction vs actual

j	Predicted log Ψ_full	Actual log Ψ_full	Diff
28	3.56	3.57	0.01
29	3.39	3.10	0.29
30	5.26	4.80	0.46
31	3.62	3.34	0.28

Budget identity prediction matches actual within ±0.5 in log scale。Mismatch from coarse W grid discretization。

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§7. Small-W C_eff Drift

7.1. Observation 67.E（Small-W Drift）

j	C_eff(100→200)
22	0.55
25	0.41
27	0.36
29	0.33
31	0.32
32	0.31

短尺度erosion rate从0.55（j=22）降到0.31（j=32）。

7.2. 解读

这是Paper 66的α_fit drift在C_eff framework中的精确表达。物理解释：较大j上的IC field在局部更smooth——邻近primes的construction landscapes共享更多structure，short-lag decorrelation更慢。

这是second-order effect，不影响main discovery（C_eff crossing 1.0）。但它影响peak height和crossing位置的j-dependence。

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§8. Via Negativa → Via Rho

8.1. 第一轮零kill

Papers 62-67总共16轮实验。前15轮每轮至少kill一个hypothesis（总计15 kills）。本轮（Paper 67 EXP A/B/C）是第一轮所有predictions confirmed, zero kills。

8.2. 方法论意义

Via Negativa的15次kill完成了搜索空间的systematic clearing。剩余空间的primary surviving quantitative object——C_eff trajectory——通过所有current tests。

Paper 67 marks transition from pure Via Negativa（eliminating shortcuts）to integrated Via Negativa + Via Rho（identifying empirical anchor for construction）。C_eff trajectory提供quantitative target：any eventual mechanism must reproduce this measured trajectory。

8.3. Epistemic caution

First all-confirm round removes the natural self-correction of kills。Via Negativa的失败hypothesis自己暴露；Via Rho的confirmations需要额外rigor。Paper 67的每个claim都require precision in epistemic status labeling（identity vs conditional vs empirical observation）。

"All predictions confirmed"可能意味着(a) found right mechanism或(b) predictions loose enough to be confirmed by多种mechanisms。本文通过specific quantitative predictions（C_eff crossing at 1-3% of m_j, non-dyadic persistence, budget bounded）对(b)的空间进行了限制。

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§9. Paper 68 Target

9.1. Paper 67完成了什么

1. 引入C_eff作为统一两个regime的定量对象
2. 给出精确的log-budget identity（Proposition 67.A）
3. 建立conditional criterion（Corollary 67.B）
4. 全j=22-32 comprehensive C_eff trajectory
5. Non-dyadic和curve collapse confirmation
6. 15 kills的Via Negativa到Via Rho transition

9.2. Paper 67没有完成什么

1. 没有从IC axiom推导C_eff的上升rate
2. 没有从IC axiom推导1-3% crossing ratio
3. 没有unconditional proof of 59.1
4. 没有确定budget是O(1)还是O(log j)

9.3. Paper 68 target

从IC递推ρ(n) = min(ρ(n-1)+1, min_{ab=n}[ρ(a)+ρ(b)+2])推导：

(a) C_eff(W)的eventually increasing behavior

(b) 穿越C = 1的scale W的order（W = O(m_j)？）

(c) Cumulative budget的bound

候选框架：

• Influence-cone overlap divergence（IC tree共享节点概率随lag衰减）
• Near-optimal factorization multiplicity（更大lag → 更多独立factorization events → stronger decorrelation）
• Min-plus corrector theory（ρ as calibrated corrector field with bounded low-frequency power）

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§10. Status Map

内容	层级
Proposition 67.A: log-budget identity	Exact identity
Corollary 67.B: budget bounded → Ψ = O(j)	Conditional theorem
Observation 67.C: C_eff eventually increasing, crossing 1.0 at all j	Empirically confirmed
Observation 67.D: block-relative collapse W/m_j ≈ 1-3%	Empirically confirmed
Observation 67.E: small-W drift C_eff(small) decreasing with j	Empirically confirmed
Non-dyadic persistence	Empirically confirmed
Cumulative budget bounded (0.5-3.4)	Empirically confirmed
15 hypotheses killed (Papers 62-67)	Empirical/analytic
C_eff increase rate from IC recursion	Open (Paper 68)
Budget specific growth rate (O(1) vs O(log j))	Open (Paper 68)
59.1 unconditional proof	Open (Paper 68-69)

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Methods

A. 数据

IC function ρ(n)预计算到N = 10¹⁰（int16 binary），包含455,052,511个素数。

B. Centered residual

x_p = [ρ(p-1) + 1 - λ·ln p - μ_12(p mod 12)] / p（raw residual）。η_p = x_p - μ_{τ(p)}^{adj}(j)（adjacent-centered residual）。Adjacent reference uses j±1 blocks' branch-specific means。

C. Sub-block Ψ estimation

Block I_j divided into K = m/W complete sub-blocks。S_b = Σ_{primes in b} η_p。Ψ(W) = Var_b(S_b) / (W·σ²_η)。Standard error optimistic baseline assuming sub-block independence; strict confidence requires block-jackknife or K_eff correction。Confidence tiers: K ≥ 30 (high), 10 ≤ K < 30 (medium), K < 10 (exploratory)。

D. C_eff computation

C_eff(W₁ → W₂) = 1 - [log Ψ(W₂) - log Ψ(W₁)] / [log W₂ - log W₁]。Cumulative budget = Σ(1-C_r)·Δ_r where Δ_r = log(W_{r+1}/W_r)。

E. Non-dyadic blocks

Blocks [3·2^j, 4·2^j) for j = 25, 27, 29。Reference from enclosing dyadic block j+1's adjacent neighbors。

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参考文献

[1] H. Qin, "ZFCρ Paper LXVI," Zenodo (2026). DOI: 10.5281/zenodo.19701479.

[2] H. Qin, "ZFCρ Paper LXV," Zenodo (2026). DOI: 10.5281/zenodo.19687343.

[3] H. Qin, "ZFCρ Paper LXIV," Zenodo (2026). DOI: 10.5281/zenodo.19674521.

[4] H. Qin, "ZFCρ Paper LX," Zenodo (2026). DOI: 10.5281/zenodo.19480563.

[5] H. Qin, "SAE Methodology Paper 00 (Via Rho)," Zenodo (2026). DOI: 10.5281/zenodo.19657439.