§1 Introduction

1.1 The Problem

Paper 44 (DOI: 10.5281/zenodo.19247859) introduced the Reset-Slack decomposition

\(A(m) = \psi(\Delta M^-(m)) - U^-(m)\)

where \(\psi(x) = -\min(x,1)\) and \(U^-(m) = (1-\Delta M^-(m))^+ - J(m) \geq 0\) is the reset slack. Paper 44 proved the Reset-Slack Reduction theorem \(\mathrm{Var}(A) \leq 2\cdot\mathrm{Var}(\Delta M^-) + 2\cdot E[U^2]\), reducing \(\mathrm{Var}(A)\) to two independent inputs.

Paper 44 described U as "the amount by which DP min falls short of its theoretical ceiling" — an efficiency loss. The S/U overlap analysis found Jaccard(S>0, U>0) ≈ 0: U > 0 does not come from current-step selection errors (S > 0), but from "construct's historical accumulation." But what is "construct's historical accumulation" in mathematical terms?

This paper answers that question.

1.2 A Priori Proposition

In the SAE framework, the remainder is the structural residue of chisel-construct operations, not random noise. If U is the remainder's realization in the DP recurrence, SAE requires: (1) U should not come from current-step selection errors, (2) U should come from the system's historical state, (3) U's transmission should be exact — no more, no less.

If U were mainly due to current-step selection errors (overlapping with S/T), or could only be statistically correlated rather than pointwise recovered, the SAE "historical carrier" thesis would fail. This paper's data and theorems rule out these failure modes.

1.3 Notation

As in Paper 44 §1.2. Additional notation:

• \(R(m) := M(m) - \rho_E(m) \geq 0\) — predecessor defect reservoir
• \(B(m) := 1 - \Delta M^-(m) = 1 - M(m) + M(m-1)\) — signed capacity
• \(C(m) := B(m)^+ = \max(B(m), 0)\) — release capacity
• For brevity, \(U := U^-\) throughout.

Note: All quantities (\(\rho_E, M, J, U, R, B\)) take integer values.

1.4 Main Results

(1) Capacity-Allocation Law (Theorem 1). \(U(m) = \min(C(m), R(m-1))\), \(J(m) = \max(C(m) - R(m-1), 0)\). The local dynamics of DP is capacity allocation: current capacity first repays the predecessor's historical debt, surplus converts to jump.

(2) Reservoir Dynamics (Theorem 2). \(R(m) = \max(R(m-1) - B(m), 0)\). The reservoir follows a Lindley-type recurrence; B > 0 discharges, B < 0 recharges.

(3) Three-sector unification. R2 (shielded) = empty reservoir; reset front = full repayment; true interior = truncation. Paper 44's three-layer structure consists of three special cases of the Capacity-Allocation Law.

(4) A posteriori evidence. Five data groups confirm U is a historical quantity, not a current selection error.

(5) Reduction of Conjecture 1. \(U \leq R(m-1) \Rightarrow E[U^2] \leq E[R^2]\). If \(E[R^2 \mid \mathrm{exposed}, \Omega=k]\) is bounded (data ≈ 2.4, k-independent), Conjecture 1 holds.

§2 A Posteriori Evidence: U Is a Historical Quantity

2.1 S/U Overlap Near Zero

Paper 44 auxiliary experiment. Jaccard(S>0, U>0) < 0.005 for k ≥ 4. Over 93% of U > 0 events occur at integers where M = M_spf (SPF split is globally optimal) — the chisel made the correct selection but U is still nonzero. U > 0 is not a selection error.

2.2 Predecessor Ω Strongly Biased toward Low Ω

When U > 0, Ω(m-1) = 2 accounts for 68–79%; when U = 0, only 16–33%.

Ω(m-1)	U>0 share (k=4)	U=0 share (k=4)	U>0 share (k=8)	U=0 share (k=8)
2	67.6%	15.9%	78.8%	32.8%
3	25.7%	29.7%	18.6%	36.9%
4	5.4%	25.1%	2.3%	19.2%
≥5	1.3%	29.3%	0.3%	11.1%

Low-Ω integers are inefficient dissipators (P(J>0) ~ 30%), frequently taking the successor path, accumulating R > 0. High-Ω integers are efficient dissipators (P(J>0) > 77%), almost always jumping, R = 0. All system imbalance originates from dissipation-deficient low-complexity regions.

2.3 Excursion Length = 2

U > 0 almost always occurs at exactly the second non-jump step: P(len=2) > 81% for k ≥ 4. P(len=1) = 0 (J(m-1) = 0 is a necessary condition for U > 0). Remainder transfer is instantaneous — a single non-jump step suffices; no long-excursion accumulation is needed.

2.4 R(m-1) Concentrates at 1

Distribution of R(m-1) given U > 0:

R(m-1)	Share (k=4)	Share (k=8)
1	65.1%	70.3%
2	25.5%	23.3%
3	7.6%	5.5%
4	1.5%	0.8%

\(E[R \mid U>0] \approx 1.34\)–\(1.46\). \(\mathrm{Var}[R \mid U>0] \approx 0.4\)–\(0.5\). R = 1 (minimum debt: multiplicative target exceeds successor by exactly 1) is the dominant mode.

2.5 U and R(m-1): Front-Dominated Correspondence

On the front sector (J(m-1) = 0, J(m) > 0, comprising 95%+ of U > 0), U = R(m-1) pointwise exactly (corollary of Theorem 1). On the rare true interior (J(m-1) = 0, J(m) = 0, <5%), U = C < R(m-1) (truncation state). The empirical distributions of U and R(m-1) nearly coincide; deviation comes from rare interior truncation.

§3 Capacity-Allocation Law

3.1 Definitions

• Signed capacity: \(B(m) := 1 - M(m) + M(m-1)\)
• Release capacity: \(C(m) := B(m)^+ = \max(B(m), 0)\)
• Predecessor reservoir: \(R(m) := M(m) - \rho_E(m) \geq 0\)

\(C(m)\) is the current step's release capacity. \(R(m-1)\) is the predecessor's historical debt.

3.2 Theorem 1 (Capacity-Allocation Law)

3.3 Theorem 2 (Reservoir Dynamics)

3.4 Three-Sector Unification (Discharge Regime, B > 0)

Sector	Condition	R(m-1) vs C	U	J	R(m)
Shielded	R(m-1) = 0	0 ≤ C	0	C	0
Reset front	0 < R(m-1) < C	debt < capacity	R(m-1)	C − R(m-1)	0
True interior	R(m-1) ≥ C > 0	debt ≥ capacity	C	0	R(m-1) − C

When B(m) ≤ 0, C(m) = 0, so U = J = 0. The dynamics is given by Theorem 2's recharge/neutral recurrence: R(m) = R(m-1) + |B(m)| (recharge) or R(m) = R(m-1) (neutral).

Paper 44's R2 (U = 0 on the shielded sector) is the R(m-1) = 0 special case. Paper 44's predecessor-state decomposition (\(E[U^2] = (1-w_\mathrm{shielded}) \times D(k)\)) is the macroscopic statistical consequence of Theorem 1.

3.5 Charge-Discharge Dynamics

\(B(m) = 1 - M(m) + M(m-1)\) is the signed capacity:

• B > 0 (discharge, 66%): Multiplicative target becomes cheaper. Capacity first repays reservoir, surplus becomes jump. R(m) = max(R(m-1) − B, 0) — reservoir decreases or clears.
• B < 0 (recharge, 13%): Multiplicative target becomes more expensive. Reservoir increases: R(m) = R(m-1) + |B|.
• B = 0 (neutral, 21%): Reservoir unchanged: R(m) = R(m-1).

Complete dynamics: the system carries a reservoir R ≥ 0 (historical debt). At each step, the signed capacity B determines discharge or recharge. Jump occurs only when discharging and capacity exceeds debt.

§4 Remainder Transfer Law

4.1 Precise Statement

This is not a conservation law (some total remaining constant) but a pointwise transfer law — the predecessor's debt is exactly transmitted, no more, no less. Macroscopically it manifests as conservation: all slack (U) has a source (predecessor reservoir), all reservoir has a destination (cleared by discharge, or carried forward).

4.2 SAE Interpretation of Charge-Discharge

• Discharge (B>0): The chisel finds a cheaper multiplicative path. Capacity first repays historical debt, surplus becomes jump. If debt is zero (R(m-1) = 0, shielded), all becomes jump — R2, the chisel-remainder coincidence state.
• Recharge (B<0): The chisel finds no better path than successor. Reservoir increases — the construct accumulates new historical debt.
• Neutral (B=0): Historical state transmitted unchanged.

The DP recurrence is not a memoryless greedy algorithm — it carries history (reservoir), releasing under favorable conditions, accumulating under unfavorable ones. The Le Chatelier mechanism's microscopic realization: high-frequency discharge (66%) continually clears the reservoir, low-frequency recharge (13%) occasionally accumulates, but accumulation amounts (R = 1 dominates) are minimal and typically cleared by the next discharge.

4.3 Dissipator Picture

Low-Ω integers (Ω ≤ 3) are inefficient dissipators: P(J>0) ~ 30–57%, frequently recharging. High-Ω integers (Ω ≥ 4) are efficient dissipators: P(J>0) > 77%, frequently discharging.

68–79% of U > 0 predecessors have Ω = 2 — all system imbalance originates from dissipation-deficient low-complexity regions.

§5 Impact on the H′ Closure Chain

5.1 Reduction of Conjecture 1

By Theorem 1, \(U(m) = \min(C(m), R(m-1)) \leq R(m-1)\). Therefore

\(E[U^2 \mid \Omega(m)=k] \leq E[R(m-1)^2 \mid \Omega(m)=k]\)

Exact sector decomposition:

\(E[U^2 \mid \Omega=k] = P(\text{front} \mid \Omega=k) \cdot E[R_\mathrm{prev}^2 \mid \text{front}, k] + P(\text{interior} \mid \Omega=k) \cdot E[C^2 \mid \text{interior}, k]\)

Since front comprises 95%+ of the exposed sector, numerically:

\(E[U^2 \mid \Omega=k] \approx P(\text{exposed} \mid \Omega=k) \cdot E[R_\mathrm{prev}^2 \mid \text{exposed}, k]\)

Data: \(E[R^2 \mid \text{exposed}, \Omega=k] \approx 2.4\) (nearly k-independent).

5.2 Relationship to Paper 44's Attack Hierarchy

Paper 45's value is not in replacing Route A. Its value lies in the ontological explanation: it shows why Conjecture 1 should hold — U is strictly bounded by the reservoir, a historical quantity, whose second moment concerns a single integer's ρ_E-to-M relationship, far simpler than controlling the difference ΔM⁻.

§6 Numerical Evidence

6.1 Theorem 1 Verification

8,670,831 samples, zero failures. Across all Ω shells (k = 2 to 14).

Sector	Samples	Pass rate
Shielded (R(m-1) = 0)	5,800,087	100.000000%
Reset front (0 < R < C)	2,565,983	100.000000%
True interior (R ≥ C)	304,761	100.000000%

6.2 Theorem 2 Verification

8,670,842 samples, zero failures. Across all Ω shells. R(m) ≥ 0 verification: zero violations. \(R(m) = M(m) - \rho_E(m) \geq 0\) holds exactly for all m ≤ 10⁷.

6.3 B(m) Distribution

B(m)	Share	Meaning
B > 0	66.4%	Discharge (multiplicative target becomes cheaper)
B = 0	20.6%	Neutral (multiplicative target unchanged)
B < 0	13.0%	Recharge (multiplicative target becomes more expensive)

§7 SAE Interpretation

7.1 Deepening the Pre-Dimensional Pattern

Paper 44's transfer coefficient τ(k) ∈ [0.14, 0.25] suggested a pre-dimensional pattern. Paper 45 provides a deeper foundation: the weak correlation between selection output (A) and structural input (Δf) is not because "the remainder passively absorbs fluctuation" — but because the remainder is exactly transmitted between steps. The remainder is not created to absorb fluctuation; it is the precise continuation of the previous step's cost differential.

7.2 Deepening R2

Paper 44 interpreted R2 (J(m-1) > 0 ⟹ U(m) = 0) as "chisel-remainder coincidence." Paper 45 gives a more precise statement: R2 does not say "the remainder is zero after a jump" — it says "the jump emptied the reservoir, so the next step has no historical debt to repay." The microscopic mechanism of chisel-remainder coincidence is reservoir clearing.

7.3 Precise Microscopic Mechanism of Le Chatelier

Paper 44's Le Chatelier picture: \(E[U^2] = (1-w_\mathrm{shielded}) \times D(k)\), frequency suppression and decoherence intensity counterbalancing.

Paper 45 makes D(k) precise. The exact expression is \(D(k) = E[\min(C, R_\mathrm{prev})^2 \mid J(m-1)=0, \Omega(m)=k]\). Since front dominates (95%+), numerically \(D(k) \approx E[R^2 \mid \text{front}, k]\). The front's E[R²] slowly declines from ~2.7 to ~2.3 — not "selection bias purification" (Paper 44's formulation), but the reservoir's second moment gently declining within the front sector.

§8 Open Questions

1. k-dependence of \(E[R(m-1)^2 \mid J(m-1)=0, \Omega(m)=k]\). Data says ≈ 2.4, k-independent. Proof requires current-to-predecessor fiber control.

2. Structural explanation of R = 1 dominance (65–70%). R = 1 means M(m) = ρ_E(m) + 1 — minimum debt mode.

3. Reservoir Dynamics and Paper 21's Lindley recurrence. \(R(m) = \max(R(m-1) - B(m), 0)\) is Lindley-type. What is its stationary distribution?

4. Generalization of the Capacity-Allocation Law. Does it hold for all min/max recurrence systems?

§9 Retrospective on Paper 44

Paper 44's Le Chatelier picture and three-layer narrative do not require modification. Paper 45 provides the precise microscopic mechanism:

Data Sources

Scripts: su_overlap_detail.c, u_diag.c, cap_alloc_verify.c, reservoir_dynamics.c (C, gcc -O2). Data: ρ_E via DP min for n ≤ 10⁷+1.

References

[1] H. Qin. ZFCρ Paper XLIV (Why DP min produces local smoothness). DOI: 10.5281/zenodo.19247859.

[2] H. Qin. ZFCρ Paper XLII (H' conditional closure). DOI: 10.5281/zenodo.19226607.

[3] H. Qin. ZFCρ Paper XLIII (Self-correction at N=10¹⁰). DOI: 10.5281/zenodo.19240183.

[4] H. Qin. ZFCρ Paper XVIII (Anti-correlation engine). DOI: 10.5281/zenodo.19024385.

[5] H. Qin. ZFCρ Paper XX (Unconditional B-bound). DOI: 10.5281/zenodo.19027892.

[6] H. Qin. ZFCρ Paper XXI (Queue isomorphism). DOI: 10.5281/zenodo.19037934.

Acknowledgments

ChatGPT (Gongxihua): Exact min/max formulation of the Capacity-Allocation Law (upgrading from U = −D(m-1) to U = C ∧ R(m-1)). Proposal of Reservoir Dynamics R(m) = (R(m-1) − B(m))⁺. Two review rounds: proof gap correction (Case 2 inequality direction), exact/approximate layer separation, recharge dynamics completion. SPF Skeleton Reduction framework for Conjecture 2 attack.

Claude (Zilu): All numerical experiments (S/U overlap, U > 0 diagnostics, Capacity-Allocation exact verification, Reservoir Dynamics verification). Text drafting, working notes v1–v3, initial algebraic proof sketch (two cases). First discovery of the U = −D(m-1) empirical regularity.

Claude (Thermodynamic thread): SAE positioning of the Remainder Transfer Law (pointwise transfer, not conservation). Dissipator picture (low Ω = inefficient dissipators). Correction suggestion for Paper 44's "decoherence" narrative. R2 = reservoir clearing microscopic interpretation.

Gemini (Zixia): Case 1 proof tightening (M(m) ≤ ρ_E(m-1) ⟹ C ≥ 1+R > R). "Fundamental Theorem" positioning suggestion.

Grok (Zigong): Series consistency audit, proof completeness check, Conjecture 1 statement tightening, references completion.

Han Qin (author): Directional decisions (U's ontology as Paper 45 main line), SAE a priori–a posteriori structure design, "Remainder Transfer Law" naming, Paper 45/46 scope decision. Final text independently completed by the author.