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1. Introduction

1.1 Background and scope

Paper XXVII established the upper reflection criterion: $E[(\Delta_{\text{spf}})^+ \mid H \geq h^+] > 1$ is a sufficient condition for negative composite drift at high $H$, verified numerically at $N = 10^7$ ($E[(\Delta_{\text{spf}})^+] = 1.82 > 1$). Paper XXVII also identified the two analytic inputs required for upgrading this to a theorem: a quantitative comparator margin on the even subfamily and a conditional density lower bound.

The present paper does three things: (1) extends the $n/2$ scale attenuation of Paper XXVII to a six-divisor universal family; (2) explains, via the weighted attenuation $W_2$ decomposition, why the criterion surplus remains stable as $N$ grows; (3) establishes a three-step analytic program for lower reflection (positive drift at low $H$), avoiding conditional Sathe-Selberg density estimation.

This paper does not claim to complete the analytic proof of either upper or lower reflection.

1.2 Contributions

(A) Exact identity $\delta = 1 - j$ (§2), the core algebraic tool throughout.

(B) Universal scale attenuation family (§3). Six divisors, zero exceptions, plus fixed-threshold confirmation.

(C) Weighted attenuation decomposition and even-slice surplus (§4). Pointwise exact identity $\Delta_2 = W_2 + B_2$. The even-slice $E[(\Delta_2)^+]$ decreases from 2.43 ($N = 10^4$) to 1.80 ($N = 10^9$), remaining well above 1 throughout. Via $E[(\Delta_{\text{spf}})^+] \geq d_2 \cdot E[(\Delta_2)^+]$ ($d_2 = P(\text{spf}=2 \mid C) \approx 0.66$), the even slice alone suffices to maintain the overall criterion surplus above 1 at tested scales.

(D) Three-step lower reflection program (§5). Density lower bound + unconditional tail + bulk lemma. No systematic rescue-squad phenomenon.

(E) Asymmetric quantifier structure of upper vs lower reflection (§6). $\exists$-lemma vs $\forall$-type family control.

1.3 Notation

Series conventions and Paper XXVI–XXVII notation apply. $\rho_E(n)$, $M_n$, $\delta(n)$, $S(n)$, $H(n)$, $\text{spf}(n)$ as in Paper XXVII §1.3. $\Delta_{\text{spf}}(n) = [\rho_E(n-1)+1] - [\rho_E(\text{spf}(n)) + \rho_E(n/\text{spf}(n)) + 2]$. Throughout, $E[F(n) \mid n \in C]$ denotes the finite-set empirical average over $\{n \leq N\} \cap C$.

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2. Exact Identity δ = 1 − j

Corollary. $E[\delta \mid C] = 1 - E[j \mid C]$ for any conditioning set $C$. Upper reflection ($E[\delta] < 0$) $\Longleftrightarrow$ $E[j] > 1$. Lower reflection ($E[\delta] > 0$) $\Longleftrightarrow$ $E[j] < 1$.

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3. Universal Scale Attenuation Family

3.1 Six-divisor data

For $d = 2, 3, 4, 5, 6, 7$, measured at high $H$ (Q5):

Divisor d	E[H(n−1)]	E[H(n/d)]	gap
2	3.872	3.787	+0.085
3	3.874	3.782	+0.092
4	3.875	3.781	+0.095
5	3.873	3.774	+0.099
6	3.877	3.778	+0.099
7	3.872	3.763	+0.109

All six scales confirm attenuation with zero exceptions. The gap generally increases with $d$ (from 0.085 at $d = 2$ to 0.109 at $d = 7$), reflecting the greater DP distance between $n/d$ and $n$: excursion information decays more over longer evolutionary spans.

3.2 Fixed-threshold confirmation

Numerical Observation 2. Using the fixed absolute threshold $H \geq 3.84$ (not quantile-based), the gap for $d = 2$ decreases from $+0.223$ ($N = 10^4$) to $+0.070$ ($N = 10^8$), remaining positive throughout. The trend is consistent with quantile-based results. Scale attenuation is not a quantile-sliding artifact.

3.3 Divisor-dependent inequality family

The data supports a family of band inequalities: for each divisor $d$,

$$E[H(n/d) \mid H(n-1) \geq h^{+},\; d \mid n,\; n \leq N] \leq h^{+} - \eta_d^{+}(N)$$ $$E[H(n/d) \mid H(n-1) \leq h^{-},\; d \mid n,\; n \leq N] \geq h^{-} + \eta_d^{-}(N)$$

where $\eta_d$ depends on $d$ and $N$, and all divisors share an approximate fixed point $h_0 \approx 3.80$ ($N = 10^7$).

3.4 Physical interpretation

$\rho_E(n/d)$ is computed by the DP at position $n/d$, before $\rho_E(n-1)$. The evolution from $n/d$ to $n-1$ spans $n(1-1/d)$ steps. Height excursions ($\rho_E(n-1)$ deviating above its mean) accumulate during these steps; the value at $n/d$ reflects the pre-excursion state. Larger $d$ means $n/d$ is further from $n-1$, so excursion information decays more.

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4. Weighted Attenuation W₂ and Stability of the Criterion Surplus

4.1 Identification of the key object

Paper XXVII focused on the raw attenuation gap $H(n-1) - H(n/2)$, which decays with $N$. However, the object truly relevant to the criterion surplus is the logarithmically weighted attenuation.

For even composites $n$ (i.e., $\text{spf}(n) = 2$), with $\rho_E(2) = 1$, the exact identity is:

$$\Delta_2(n) = \rho_E(n-1) - \rho_E(n/2) - 2$$

Rewriting in terms of $H$ (exact version):

$$\Delta_2(n) = \underbrace{(H(n-1)-H(n/2)) \cdot \ln(n/2)}_{W_2(n) \text{ (weighted attenuation)}} + \underbrace{H(n-1) \cdot \ln\!\left(\frac{2(n-1)}{n}\right) - 2}_{B_2(n) \text{ (baseline term)}}$$

For large $n$, $\ln(2(n-1)/n) = \ln(2-2/n) \to \ln 2$, so $B_2(n) \to H(n-1)\cdot\ln 2 - 2$. For finite $n$, $\ln(2-2/n) < \ln 2$, hence $B_2(n) < H(n-1)\cdot\ln 2 - 2$ pointwise. Define the asymptotic baseline target $b(h^+) = h^+\cdot\ln 2 - 2$; for finite $N$, the precise conditional expectation $E[\ln(2-2/n) \mid C]$ should be used (see §4.5), rather than $b(h^+)$ itself as a ready-made lower bound. At tested scales $N \geq 10^4$, $E[\ln(2-2/n) \mid C]$ is extremely close to $\ln 2$ (numerically < 0.001 difference), though this is an empirical observation.

4.2 N-stability of W₂

The $W_2$ column below is the empirical mean of pointwise $W_2(n) = (H(n-1)-H(n/2))\cdot\ln(n/2)$ over the conditioning set $C_{2,N}(h^+)$ (not the product of $E[\text{raw gap}] \times E[\ln(n/2)]$). The $B_2$ column is likewise the empirical mean of pointwise $B_2(n)$.

Numerical Observation 3.

N	raw gap	E[ln(n/2)]	W₂	B₂	E[(Δ₂)⁺]
10⁴	0.223	7.8	1.73	0.70	2.43
10⁵	0.150	10.0	1.47	0.69	2.16
10⁶	0.108	12.2	1.31	0.69	2.00
10⁷	0.085	14.5	1.22	0.68	1.90
10⁸	0.070	16.8	1.16	0.68	1.84
10⁹	0.059	19.1	1.11	0.68	1.80

$W_2$ decreases slowly (1.73 → 1.11), with diminishing rate of decrease (Δ = −0.26, −0.16, −0.09, −0.06, −0.05). Observed $E[B_2] \approx 0.68$, stable (slightly above the asymptotic lower-bound target $b(h^+) \approx 0.66$).

Decomposition verification: $E[W_2] + E[B_2] = E[\Delta_2] = 1.9027$, $E[(\Delta_2)^+] = 1.9049$ ($N = 10^7$). The inequality $E[(\Delta_2)^+] \geq E[\Delta_2]$ holds (the gap 0.0022 arises from a small fraction of $\Delta_2 < 0$ contributions), confirming that $\Delta_2$ is overwhelmingly positive under high-$H$ conditioning.

4.3 Why the even-slice surplus does not vanish

The raw gap decays at approximately $C/\ln N$. However, multiplied by $\ln(n/2) \approx \ln N$, $W_2 \approx C\cdot\ln N / \ln N = C$ — compatible with a constant. The diminishing rate of $W_2$ decline is consistent with this.

The asymptotic baseline lower bound $b(h^+) = h^+\cdot\ln 2 - 2 > 0$ holds whenever $h^+ > 2/\ln 2 \approx 2.885$ (known: $h^+ = 3.84 \gg 2.885$). This positive floor derives purely from arithmetic: $\ln(n-1) - \ln(n/2) \to \ln 2$.

Using the exact lower bound $E[(\Delta_2)^+] \geq E[W_2] + E[B_2]$: even if $E[W_2] \to 0$ (requiring raw gap decay faster than $1/\ln N$, contradicted by data), $E[B_2]$ provides a positive contribution at tested scales (approximately 0.68, with asymptotic lower-bound target $b(h^+) \approx 0.66$; the observed value is slightly higher because $E[H(n-1) \mid C] > h^+$ on the conditioning set) — but this alone is insufficient to exceed 1. The data supports $E[W_2]$ not tending to zero, and possibly approaching a positive constant (of order 1), keeping the even-slice $E[(\Delta_2)^+]$ stable at approximately 1.8 at tested scales.

From even-slice to overall criterion surplus. By the aggregation framework of Paper XXVII:

$$E[(\Delta_{\text{spf}})^{+} \mid C_N(h^{+})] \geq d_2(C) \cdot E[(\Delta_2)^{+} \mid C_{2,N}(h^{+})]$$

where $d_2(C) = P(\text{spf}(n) = 2 \mid C_N(h^+))$, $C_{2,N}(h^+) = C_N(h^+) \cap \{\text{spf}(n) = 2\}$. Numerically $d_2 \approx 0.66$. Hence the overall surplus $\geq 0.66 \times 1.80 \approx 1.19 > 1$ ($N = 10^9$).

4.4 H sub-bin decomposition

Numerical Observation 4.

H range	W₂	B₂	E[(Δ₂)⁺]
[3.84, 3.89]	1.00	0.68	1.68
[3.89, 3.94]	1.95	0.71	2.66
[3.94, 4.00]	3.10	0.74	3.84
[4.00, 4.05]	4.35	0.78	5.12

$W_2$ is monotonically increasing across the four tested $H$ sub-bins. The higher $H$ is, the stronger the weighted attenuation and the larger the criterion surplus.

4.5 Analytic target for upper reflection

Main target. Prove that there exist $h^+ > 0$, $c > 0$ such that for all sufficiently large $N$:

$$E[(\Delta_{\text{spf}})^{+} \mid C_N(h^{+})] > 1 + c.$$

Even-dominance sufficient route. Define $d_2(C) = P(\text{spf}(n) = 2 \mid C_N(h^+))$. By

$$E[(\Delta_{\text{spf}})^{+} \mid C_N(h^{+})] \geq d_2(C) \cdot E[(\Delta_2)^{+} \mid C_{2,N}(h^{+})]$$

and Proposition 2's exact lower bound $E[(\Delta_2)^+] \geq E[W_2] + E[B_2]$:

• $E[B_2 \mid C] \geq h^+ \cdot E[\ln(2-2/n) \mid C] - 2$ (exact, from $H(n-1) \geq h^+$ pointwise). For large $N$, $E[\ln(2-2/n) \mid C] \to \ln 2$, so $E[B_2]$ asymptotically approaches $b(h^+) = h^+\cdot\ln 2 - 2 \approx 0.66$ (asymptotic positive lower-bound target). At tested scales, $E[B_2] \approx 0.68$, slightly above $b(h^+)$ because $E[H(n-1) \mid C] \approx 3.87 > h^+ = 3.84$.
• Required: $E[W_2(N; h^+)] \geq W_0 > 0$ for all large $N$.
• Required: $d_2(C) \geq d_0 > 0$ (even conditional density has a positive lower bound).
• Combined: $d_0 \cdot (W_0 + b(h^+)) > 1$ (numerically $d_2 \approx 0.66$, $W_2 + B_2 \approx 1.80$, product $\approx 1.19 > 1$, $N = 10^9$).

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5. Lower Reflection: Numerical Evidence and Three-Step Analytic Program

5.1 Jump statistics

Numerical Observation 5.

H bin	P(j=0)	E[j]	E[δ]
Q1 (low)	73.9%	0.30	+0.70
Q3	33.2%	0.94	+0.06
Q5 (high)	5.7%	2.16	−1.16

5.2 No systematic rescue-squad phenomenon

Numerical Observation 6. Among jumping composites ($j \geq 1$) at low $H$ ($H \leq 3.74$):

• The optimal split (the factorization $n = a\cdot b$ achieving $M_n$) is not the spf split in 73.8% of cases, but winners are highly dispersed (the most frequent small factor 4 accounts for only 13.9%).
• $j = 1$ accounts for 87.3%; large jumps are virtually absent.
• The imbalance ratio $\min(a,b)/\max(a,b)$ of the winning split has median 0.0001.

Numerical Observation 7 (Sathe-Selberg suppression of high-Ω rescue squads). $P(j \geq 1 \mid H \leq 3.74, \Omega = k)$:

Ω	P(j≥1)	density among low-H composites
2	6.7%	28.8%
4	31.8%	20.9%
8	80.9%	0.9%
12	98.1%	0.04%

High-$\Omega$ numbers jump with near certainty even at low $H$, but Sathe-Selberg weights render their density negligible.

5.3 Three-step framework

Target. $E[j \mid \text{composite}, H(n-1) \leq h^-] < 1 - c^-$, i.e., $E[\delta \mid H \leq h^-] > c^- > 0$.

Step 1: Density lower bound. $\alpha(N) = P(H(n-1) \leq h^- \mid \text{composite}, n \leq N) \geq \alpha_0 > 0$.

Numerical Observation 8.

N	P(H ≤ 3.74)
10⁵	0.425
10⁶	0.288
10⁷	0.181
10⁸	0.106
10⁹	0.064

$\alpha$ is declining (as $h_0(N)$ rises), but $h^- = 3.74$ lies approximately 0.09 below $h_0^\infty \approx 3.83$. The bound $\alpha > 0$ is not a lightweight input: even if the height distribution centers near $h_0^\infty$, variance collapse could cause the mass at the fixed threshold 3.74 to vanish. Step 1 is therefore one of the core unresolved theorems of the lower reflection route, requiring stationary theory for the height process (Paper XXI's Lindley isomorphism may provide an interface).

Step 2: Unconditional tail control. This step avoids direct estimation of conditional Sathe-Selberg densities. By the definition of conditional expectation and $\mathbf{1}_{H\leq h^-} \leq 1$:

$$E[j \cdot \mathbf{1}_{\Omega > K} \mid H \leq h^{-}] = \frac{E[j \cdot \mathbf{1}_{\Omega > K} \cdot \mathbf{1}_{H \leq h^{-}}]}{P(H \leq h^{-})} \leq \frac{E[j \cdot \mathbf{1}_{\Omega > K}]}{\alpha}$$

Numerical Observation 9 ($N = 10^9$, fixed $K$).

K	E[j·1_{Ω>K}] (unconditional)	bound / α
8	0.081	1.267
9	0.043	0.671
10	0.022	0.348

At $K = 10$, the tail bound $\approx 0.35$ ($N = 10^9$).

Step 3: Bulk lemma. Required (with $K = K(N) \to \infty$ in the asymptotic version):

$$E[j \mid H \leq h^{-},\; \Omega \leq K(N)] < 1 - c_0$$

Since $K(N) \to \infty$, the split family size within the bulk grows with $N$, making control harder than in the fixed-$K$ version.

From $E[j \mid H \leq h^-] = E[j\cdot\mathbf{1}_{\Omega\leq K} \mid H \leq h^-] + E[j\cdot\mathbf{1}_{\Omega>K} \mid H \leq h^-]$, the second term is controlled by Step 2.

Note: $E[j\cdot\mathbf{1}_{\Omega\leq K} \mid H \leq h^-] = P(\Omega \leq K \mid H \leq h^-) \cdot E[j \mid H \leq h^-, \Omega \leq K]$.

Empirical verification (not part of the proof): $E[j \mid H \leq 3.74] = 0.17$ ($N = 10^9$). With $P(\Omega \leq 9 \mid H \leq h^-) \approx 0.99$, $E[j \mid H \leq h^-, \Omega \leq 9] \leq 0.17/0.99 \leq 0.18$. Well below 1.

Analytic route: The proof of the bulk lemma must be independent of empirical means, established via the symmetric face of scale attenuation: at low $H$ with low $\Omega$, $M_n$ is generically above the additive path (§5.5(iii)).

5.4 Advantages and costs of the three-step approach

The three-step approach avoids direct estimation of $P(\Omega > K \mid H \leq h^-)$ (conditional Sathe-Selberg), but at the cost of introducing two new independent bottlenecks: the positive lower bound on $\alpha(N)$ (Step 1) and the controllability of the $j$-weighted unconditional $\Omega$-tail under $K = K(N) \to \infty$ (Step 2). This remains a valuable repositioning — converting a conditional density problem into two separately attackable inputs — but should not be described as having "bypassed" the problem entirely.

5.5 Analytic targets for lower reflection

The three steps each require:

(i) Density lower bound $\alpha > 0$: Requires stationary theory for the height process, ruling out variance collapse. This is the core unresolved input of the lower reflection route. Paper XXI's Lindley isomorphism may provide an interface.

(ii) Unconditional tail ($K = K(N) \to \infty$): Must prove $E[j\cdot\mathbf{1}_{\Omega>K(N)}] / \alpha(N) \to 0$ with $K(N) \to \infty$ growing sufficiently fast. Standard Sathe-Selberg weights control $P(\Omega > K)$, but $j$-weighting adds difficulty.

(iii) Bulk lemma ($K = K(N) \to \infty$): $E[j \mid H \leq h^-, \Omega \leq K(N)] < 1 - c_0$. Since $K(N) \to \infty$, the split family size grows with $N$, increasing the control difficulty. A strong sufficient route is: at low $H$ with low $\Omega$, $M_n$ is generically above the additive path (all competitive splits are too expensive). This can be reduced to the symmetric face of scale attenuation: when the parent is below the mean, children do not follow (they stay near the mean), so $M_n > \text{additive}$.

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6. Asymmetric Quantifier Structure of Upper and Lower Reflection

6.1 Same mechanism, different proof structures

Scale attenuation drives both directions:

• High $H$ → children do not follow parent's elevation → a good split exists (spf) → jump → negative drift.
• Low $H$ → children do not follow parent's depression → all splits are expensive → no jump → positive drift.

The proof-level asymmetry:

Upper reflection ($\exists$-lemma). Only one effective comparator (spf) needs to yield $E[(\Delta_{\text{spf}})^+] > 1$. A single-witness lower bound. Paper XXVII established this route via the spf comparator.

Lower reflection ($\forall$-type family control). The target quantity is $E[j \mid H \leq h^-] < 1$. Since $j$ is determined by the best split, the proof manifests as a best-split family control problem: a strong and convenient sufficient route is that all competitive splits in the bulk are too expensive, with the sparse tail controlled separately. The three-step framework decomposes this family control into "finite bulk (few split options, all expensive) + sparse tail (high $\Omega$, many options but negligible density)."

6.2 Division of labor: Papers XXVII–XXVIII

Result	Tool	Paper
Upper reflection criterion E[(Δ_spf)⁺] > 1	spf comparator (∃-lemma)	XXVII
Universal attenuation family	six divisors + fixed threshold	XXVIII
W₂ decomposition + surplus stability	weighted attenuation	XXVIII
Three-step lower reflection program	density + tail + bulk	XXVIII
Analytic closure of upper reflection	W₂ asymptotics + baseline	open
Analytic closure of lower reflection	α > 0 + bulk lemma	open
Two-sided reflection → pred gap > τ_k	occupancy + dyadic control	subsequent

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7. Discussion

7.1 Physical meaning of weighted attenuation

$W_2 = (H(n-1)-H(n/2))\cdot\ln(n/2)$ measures the cumulative effect of the height excursion across the DP evolution from $n/2$ to $n-1$. The raw gap decays with $N$ (as $H(n/2)$ approaches $h_0$), while the DP step count grows simultaneously ($\ln(n/2)$ increases); their product $W_2$ is compatible with a constant magnitude. This parallels "decay rate × time scale" balancing in dissipative systems, though the asymptotic behavior remains to be analytically confirmed.

7.2 Long-range connection to Chebyshev bounds

Paper XXVI established $\rho_E(N) = \pi(N) + S_{\text{comp}}(N)$. If universal scale attenuation could ultimately be extended to magnitude control on $S_{\text{comp}}$ ($S_{\text{comp}} = -\Theta(N/\ln N)$), the Chebyshev bound $\pi(N) = \Theta(N/\ln N)$ would follow as a corollary. The $W_2$ decomposition of the present paper provides preliminary support for this long-range target, though the gap remains very large.

7.3 Open problems

(1) Asymptotic behavior of $W_2$. The data is compatible with a positive plateau but does not yet discriminate against slower decay. An analytic proof requires the precise decay rate of the raw gap.

(2) Density lower bound $\alpha > 0$. Stationary theory for the height process, ruling out variance collapse.

(3) Bulk lemma. Uniform upper bound on all competitive splits at low $H$ with low $\Omega$, under $K(N) \to \infty$.

(4) From two-sided reflection to pred gap $> \tau_k$. Occupancy control + dyadic-scale balance.

7.4 Acknowledgments

ChatGPT 5.4 Pro identified weighted attenuation $W_2$ as the correct surplus object (§4), proposed the three-step lower reflection framework avoiding conditional Sathe-Selberg (§5), the $\exists$ vs $\forall$ quantifier precision (§6), and the fixed-$K$ asymptotic issue requiring $K(N) \to \infty$ (§5.3). Gemini proved that gap $\propto 1/\ln N$ multiplied by $\ln(n/p)$ yields a constant-order contribution (§4.3) and identified the conditional Sathe-Selberg measure trap (§5.4). Grok ensured consistency with the preceding 27 papers. Claude wrote all numerical computation scripts and drafted working notes v1–v3. The final text was completed independently by the author; all mathematical judgments are the author's responsibility.

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8. Data Sources and Reproducibility

Script	Measurement	Section
p28_block123.py	Six-divisor attenuation + lower reflection mechanism + N-stability	§3, §5.1
p28_block456.py	Criterion surplus N-stability + fixed threshold + low-H winning splits	§3.2, §4.2, §5.2
p28_block78.py	W₂ + density lower bound + tail bound	§4, §5.3

All scripts pass sanity checks: $\rho_E(10^7) = 58$, $\rho_E(10^8) = 66$, $\rho_E(10^9) = 74$.

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References

1. ZFCρ Papers I–XXVII. H. Qin. Paper XXVI DOI: 10.5281/zenodo.19059834. Paper XXVII DOI: 10.5281/zenodo.19062684.
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.