ZFCρ Thermodynamics Paper IV

Toward a First-Principles Derivation of the Tsallis q Parameter: the Interpolation Family, a Reparametrization Lemma, and the Channel-Averaged Shielding Conjecture

Han Qin

SAE Research · 2026

Abstract

Tsallis nonextensive statistical mechanics introduced a parameter q to characterize deviations from Boltzmann-Gibbs statistics. Over thirty years, q has been widely fitted in experiment (QGP q ≈ 1.15, cosmic rays q ≈ 1.2, solar wind q ≈ 1.7, gravitational systems q ≈ 2.0) but has never been derived from first principles.

This paper identifies the exact interpolation family to which q belongs. The Tsallis q-exponential at q = 1 + 1/K takes the form

e_q(-x) = (1 + x/K)^{-K}

whose K → ∞ limit is the mathematical definition of the exponential function:

lim_{K→∞} (1 + x/K)^{-K} = e^{-x}

The Boltzmann distribution e^{-βE} is thus the infinite-order limit of this interpolation family. At q = 2 (K = 1), one recovers the DD resolvent (1+x)⁻¹, the natural kernel of the ZFCρ DP recursion. The parameter K admits a conjectural physical interpretation as the effective feedback order.

An iterated discrete shielding derivation shows that K layers of equal-weight single-layer resolvent screening naturally produce the q-exponential. An extremal inequality (Theorem 2) proves that equal-weight allocation yields the strongest shielding under fixed total coupling; unequal allocations increase the effective q. A supplementary lemma shows that the second-order effective q equals 1 + Σλ² (Herfindahl index of layer weights), linking q directly to the concentration of shielding across layers.

A reparametrization lemma proves that neither the second-order Taylor coefficient nor the tail exponent is invariant under general reparametrizations. Corollary: any regime diagnostic must first fix a canonical control variable.

A channel-averaged shielding conjecture is proposed: in an f/r dual-channel system,

q = Ω_eff / n_ch

where Ω_eff is the effective shell depth (the physical generalization of the ZFCρ prime-factor count Ω) and n_ch is the channel count. For dual channels (n_ch = 2), q = Ω_eff/2. The conjecture has exact endpoint anchoring (Ω_eff = 2 ↔ q = 1, Ω_eff = 4 ↔ q = 2), a structural explanation for q ≥ 1 (Ω_eff ≥ n_ch), and admits a multi-channel generalization Ω_eff = Σq_i. The first-principles content of this paper lies in the interpolation family and the iterated shielding theorems; q = Ω_eff/n_ch remains a structurally motivated conjecture. Predictions and compatibility checks are given, the hardest being: independently extract K from any nonequilibrium system's autocorrelation function, then predict q = 1 + 1/K.

§1 The Problem: Where Does Tsallis q Come From?

1.1 Thirty Years of Free Parameters

In 1988, Tsallis proposed a generalization of Boltzmann-Gibbs statistics [1]. The central object is the q-exponential

$$e_q(u) = [1 + (1-q)u]_+^{1/(1-q)}$$

which recovers the standard exponential when q = 1 and generates power-law tails when q ≠ 1.

Over three decades, q-exponentials have been successfully fitted across physical systems. Tsallis distributions fitted to ALICE and CMS data [2,3] yield q ≈ 1.11–1.15 for hadron transverse momentum spectra [16]. The solar-wind q-triplet gives q_stat ≈ 1.71 ± 0.07 [4]. Cosmic-ray spectra yield q ≈ 1.2 [5]. Gravitational systems approach q ≈ 2 [6]. Applications of Tsallis entropy to black-hole thermodynamics continue through 2025 [7,8].

Yet q remains a free parameter. Each system is fitted independently, and no microscopic mechanism has predicted which q a given system should exhibit.

This is the central criticism of Tsallis statistics. The Boltzmann distribution e^{-βE} has no free parameter—given temperature T, the distribution is fully determined. Tsallis adds a parameter q, gaining fitting power but losing explanatory power.

1.2 Three Layers of Objects

Before entering the derivation, three layers of objects must be distinguished. These are frequently conflated in the literature.

Kernel layer. The fundamental weight function: Boltzmann writes e^{-βE}, resolvent writes (1+x)⁻¹, Tsallis writes e_q(-x). This is the raw weight assigned to microstates.

Occupation layer. The statistical distribution function. For bosons, the Bose-Einstein distribution 1/(e^{βE}−1) uses e^{βE} at the kernel layer. Replacing the kernel modifies the occupation layer.

Observed flux layer. The spectrum actually measured by a distant observer. It is the occupation layer after propagation effects (e.g. greybody factors for black holes).

This paper operates exclusively at the kernel layer. Modifications at the occupation and flux layers are deferred to subsequent work.

1.3 Contributions

This paper makes five contributions:

(1) Exact interpolation family: q = 1+1/K connects Boltzmann (K → ∞) and DD resolvent (K = 1) in a single-parameter family.

(2) Iterated shielding derivation: K layers of equal-weight discrete screening naturally produce the q-exponential. An extremal inequality proves equal weight is the strongest shielding. A supplementary lemma reveals q_eff^{(2)} = 1 + Σλ², linking q to layer-weight concentration.

(3) Reparametrization lemma: Neither the second-order Taylor coefficient nor the tail exponent is reparametrization-invariant.

(4) Channel-averaged shielding conjecture: q = Ω_eff/n_ch links Tsallis q to the ZFCρ prime-factor structure through channel counting.

(5) Physical interpretation of experimental q values and falsifiable predictions.

§2 The Exact Interpolation Family

2.1 q = 1 + 1/K

Setting q = 1 + 1/K (K > 0) in the Tsallis q-exponential definition yields

$$e_q(-x) = [1 + (q-1)x]^{1/(1-q)} = \left(1 + \frac{x}{K}\right)^{-K}$$

This is an exact algebraic identity, free of approximation.

2.2 Boltzmann as Infinite-Order Limit

$$\lim_{K \to \infty} \left(1 + \frac{x}{K}\right)^{-K} = e^{-x}$$

This is not an approximation. It is the mathematical definition of the exponential function itself.

Within the interpolation family considered here, the Boltzmann distribution e^{-βE} corresponds to the K → ∞ continuous limit—a system whose single-step feedback has been subdivided into infinitely many micro-steps, each contributing vanishing feedback, the macroscopic result being smooth exponential decay.

The DD resolvent (1+x)⁻¹ corresponds to K = 1: feedback completes in a single step with no subdivision. This is the natural kernel of the ZFCρ DP recursion X_{n+1} = min(X_n + A_n, B_n).

2.3 Physical Meaning of K

The precise mathematical identity of K is the shape parameter of the q-exponential family.

The physical interpretation of K is a conjecture, not a theorem: K is the effective feedback order—the number of coherent feedback steps the system sustains before decorrelation.

K → ∞ should not be read as "infinite memory" (semantically counterintuitive since Boltzmann is memoryless). The more accurate reading is "infinite subdivision": a single feedback step is partitioned into infinitely many micro-steps, each with vanishing coupling. The calculus continuous limit is the mathematical realization of this operation.

K = 1 is "no subdivision": feedback completes in one step without intermediaries.

Intermediate values K ∈ (1, ∞) correspond to finite subdivision. Smaller K means coarser, more direct single-step feedback; larger K means finer, smoother feedback.

2.4 Connection to m_eff

Thermo II [11] defined the effective layer count m_eff, extracted from η ≈ 1 − C(1)^{m_eff}. Thermo III [12] provided preliminary support for this structure in the Brusselator.

K and m_eff may measure the same physical quantity through different extraction routes. Their precise relationship is not yet established. In ZFCρ, m_eff = 5.47, yielding q = 1.18 if K = m_eff. In the Brusselator, m_eff = 11.3, yielding q = 1.09. These values fall within the range of observed experimental q values, but the identification K ≈ m_eff remains conjecture.

§3 Iterated Shielding Derivation

3.1 Discrete Shielding Model

Let S₀ > 0 be the initial signal, x ≥ 0 the total shielding amplitude (control parameter), and K a positive integer (number of shielding layers). Given K nonnegative couplings c₁, ..., c_K ≥ 0, define the signal after K layers of discrete shielding as

$$S_K(\mathbf{c}) = S_0 \prod_{i=1}^{K} (1 + c_i)^{-1}$$

subject to the total coupling constraint Σc_i = x.

Physical picture: each prime factor contributes one layer of multiplicative screening. The signal is attenuated by (1+c_i)⁻¹ at each layer. The total coupling x is fixed (corresponding to the total control parameter, e.g. βE). In Boltzmann statistics, x = βE has a clear physical meaning (ratio of energy to thermal-bath temperature). In the SAE/ZFCρ framework, x can be understood as the total unresolved structural tension facing the current state—the system absorbs it by distributing it across K shielding layers. The precise ontological identity of x is an open problem (§10).

3.2 Theorem 1 (Discrete Shielding Iteration)

(a) Equal-weight allocation. If c_i = x/K for i = 1, ..., K, then

$$S_K^{\text{eq}}(x) = S_0 \left(1 + \frac{x}{K}\right)^{-K} = S_0 \cdot e_q(-x), \quad q = 1 + \frac{1}{K}$$

Proof: Direct substitution. ∎

(b) Boltzmann limit.

$$\lim_{K \to \infty} \left(1 + \frac{x}{K}\right)^{-K} = e^{-x}$$

Proof: Take logarithms; the standard limit lim_{K→∞} K·log(1+x/K) = x applies. ∎

(c) Uniqueness. Let a function family F_K satisfy: (i) it is generated by K identical single-layer resolvent iterations, F_K(x) = S₀·[1+γ_K(x)]^{-K}; (ii) the total coupling is normalized by exact additive constraint K·γ_K(x) = x. Then F_K(x) = S₀·(1+x/K)^{-K} is the unique such family.

Proof: From K·γ_K(x) = x, obtain γ_K(x) = x/K. Substitute. ∎

Note: Uniqueness requires the exact additive normalization Σc_i = x. If only first-order normalization is imposed (K·γ_K(x) = x + O(x²)), uniqueness fails.

3.3 Theorem 2 (Extremal Inequality Under Fixed Total Coupling)

Let c_i ≥ 0 with Σc_i = x. Then

$$\left(1 + \frac{x}{K}\right)^{-K} \leq \prod_{i=1}^{K}(1+c_i)^{-1} \leq (1+x)^{-1}$$

Left equality holds iff c₁ = ... = c_K = x/K (equal weight). Right equality holds iff some c_j = x with all others zero (full concentration).

Proof (left bound). Since log(1+t) is concave on [0,∞), Jensen's inequality gives

(1/K)·Σlog(1+c_i) ≤ log(1 + (1/K)·Σc_i) = log(1+x/K)

Multiply by K, exponentiate, and invert to obtain the bound. Equality in Jensen requires all c_i equal. ∎

Proof (right bound). Expand the product: ∏(1+c_i) = 1 + Σc_i + Σ_{i

Physical interpretation: Equal-weight allocation produces the strongest shielding (smallest surviving signal) for a given total coupling. The more concentrated the coupling across fewer layers, the weaker the shielding.

3.4 Corollary (Existence and Uniqueness of K_eff)

Fix x > 0 and define f_ν(x) = (1+x/ν)^{-ν} for ν > 0. Then f_ν(x) is strictly decreasing in ν.

Therefore, for any c_i ≥ 0 with Σc_i = x, there exists a unique K_eff ∈ [1, K] such that

$$\prod_{i=1}^{K}(1+c_i)^{-1} = \left(1 + \frac{x}{K_{\text{eff}}}\right)^{-K_{\text{eff}}}$$

and one may define

$$q_{\text{eff}} = 1 + \frac{1}{K_{\text{eff}}} \in \left[1 + \frac{1}{K},\; 2\right]$$

Any unequal-weight system can be uniquely re-encoded as an equal-weight system with a smaller effective order. Larger q strictly corresponds to smaller K_eff.

3.5 Supplementary Lemma (Second-Order Effective q)

Let c_i = λ_i·x with λ_i ≥ 0 and Σλ_i = 1. Then as x → 0,

$$\prod_{i=1}^{K}(1+\lambda_i x)^{-1} = 1 - x + \frac{1+\sum_i \lambda_i^2}{2} x^2 + O(x^3)$$

Matching with the Tsallis q-exponential expansion e_q(-x) = 1 − x + (q/2)x² + O(x³) gives

$$q_{\text{eff}}^{(2)} = 1 + \sum_{i=1}^{K} \lambda_i^2$$

The layer-weight concentration (Herfindahl index) Σλ² directly determines the local effective q.

Value of this formula: one need not prove equal sharing to obtain q. The locally measured q is the shielding-weight concentration. Equal sharing is merely the most uniform endpoint of this spectrum.

3.6 Status Summary

§4 Reparametrization Lemma

4.1 Second-Order Coefficient Is Not Invariant

Under the standard parametrization (z = x):

$$e^{-x} = 1 - x + \frac{x^2}{2} - \cdots$$

$$(1+x)^{-1} = 1 - x + x^2 - \cdots$$

The second-order coefficients differ by a factor of 2, noted in the literature but not elevated to a systematic diagnostic [9].

However, under reparametrization z = f(x) with f(x) = x + a₂x² + O(x³),

$$\frac{1}{1+f(x)} = 1 - x + (1-a_2)x^2 + O(x^3)$$

Choosing a₂ = 1/2 makes the second-order coefficient exactly 1/2—identical to Boltzmann.

Proposition 1. The second-order Taylor coefficient is not invariant under general reparametrizations z = f(x).

4.2 Tail Exponent Is Not Invariant

For the resolvent kernel R_K(z) = (1+z/K)^{-K}, when z = f(x) with f(x) ~ x^ν (large x),

$$R_K(f(x)) \sim x^{-K\nu}$$

The tail exponent reads Kν, not K.

Proposition 2. The tail exponent is not invariant under general nonlinear reparametrizations.

4.3 Corollary

The combined implication of Propositions 1 and 2:

Any kernel-shape regime diagnostic—whether second-order coefficient or tail exponent—must first fix the canonical control variable before extraction and comparison.

The second-order coefficient is too local (probing only the x → 0 neighborhood). The q-family is a more appropriate global object, but q extraction also requires a fixed canonical variable. The factor-of-2 diagnostic retains value as a gauge-fixed convenience under the canonical parametrization z = βE, but its status is "gauge-fixed convenience" rather than "reparametrization-invariant diagnostic."

§5 Phenomenological Anchors

5.1 Experimental q-Value Table

Heuristic reading of K: in the QGP, K ≈ 7 suggests roughly 7 steps of coherent feedback depth. For the solar wind, K ≈ 3 suggests ~3 steps of local self-organization. For the black hole, K = 1 is the simplest one-step feedback. These readings depend on the physical interpretation of K (§2.3), which is itself conjectural.

5.2 Conditions of Use

The q values above come from independent fitting by different experimental groups with non-uniform protocols. LHC q values are fitted from power-law tails of particle transverse-momentum spectra. The solar-wind q_stat is fitted from proton density distributions. Comparability is limited.

A single system may have multiple q values. The solar-wind q-triplet yields q_stat ≈ 1.71, q_rel ≈ 3.8, q_sen ≈ −0.6 [4]. This paper treats only the stationary distribution kernel q (i.e., q_stat).

The table should therefore be viewed as phenomenological anchors, not evidence for precise mapping. Its value lies in the observation that all known experimental q_stat values fall within the Boltzmann–resolvent corridor q ∈ [1, 2], consistent with K ∈ [1, ∞).

§6 Channel-Averaged Shielding Conjecture

6.1 Statement

Conjecture (channel-averaged shielding law). In an f/r dual-channel system, the Tsallis q parameter equals the per-channel effective shell depth:

$$q = \frac{\Omega_{\text{eff}}}{n_{\text{ch}}}$$

where Ω_eff is the effective shell depth (the physical generalization of the ZFCρ prime-factor count Ω(n)) and n_ch is the channel count. For dual channels (f/r decomposition), n_ch = 2.

Important caveat: this is an effective equal-depth approximation. In real systems, f and r channels may have highly asymmetric shielding depths—for example, Thermo III [12] reports η_{r←f} ≈ 0.192 versus η_{f←r} ≈ 0.002 in the Brusselator. The equal-depth approximation q = Ω_eff/n_ch averages out this asymmetry. A more precise treatment uses the multi-channel generalization Ω_eff = Σq_i (§6.6), allowing different channels to have different q_i. The macroscopically measured q is typically captured by the dominant channel (the one carrying the largest variance).

Combining with q = 1 + 1/K gives

$$K = \frac{n_{\text{ch}}}{\Omega_{\text{eff}} - n_{\text{ch}}}$$

6.2 Endpoint Anchoring

Both endpoints are definitional within their respective frameworks:

Endpoint alignment is not a fitting result. Ω = 2 is the smallest shell in ZFCρ with two prime factors. q = 1 is the definition of Boltzmann statistics. Their correspondence follows from the combinatorial constraint that at least n_ch prime factors are needed to distribute one per channel.

6.3 Structural Reason for Division by n_ch

The Ω_eff prime factors are distributed across n_ch channels. Each channel receives Ω_eff/n_ch shielding layers. q counts per-channel shielding depth, not total system shielding.

Structural reason for q ≥ 1 (within the channel-averaged shielding model): Ω_eff ≥ n_ch. At least n_ch prime factors are needed to assign one to each channel. Within this model, configurations with fewer than n_ch factors do not constitute effective dissipative systems.

Exclusion of q < 1 within this framework: From q = 1+1/K, if q < 1 then K = 1/(q−1) < 0. Negative K would mean negative iterated shielding layers, which has no physical meaning in the discrete iteration model (each shielding layer represents one actual feedback operation). Stationary-state feedback kernels with q < 1 are therefore excluded within this framework. The q < 1 branch in Tsallis statistics (corresponding to compact-support distributions) involves a different physical mechanism outside the scope of the layered resolvent model.

6.4 q Not Limited to [1, 2] (Mathematical Extension / Higher-Order Conjectural Regime)

Ω_eff can exceed 2n_ch. In ZFCρ, Ω has no upper bound. A second phase-transition structure exists between Ω = 7 and 8 (qualitative change in single-layer closure behavior; ZFCρ Paper 42 [15] data: Ω = 7 closes but extremely slowly, Ω = 8 may never close). This corresponds to q > 2, K < 1.

This regime is a mathematical extension of the channel-averaged shielding law and currently lacks independent empirical support. The physical meaning of K < 1 (if any) may be "feedback intensity exceeding first order"—single-step feedback producing multiple cascading effects. This interpretation belongs to a higher-order conjectural regime, not on the same evidence tier as the empirically common q ∈ [1, 2] corridor.

6.5 Preliminary Remarks on the q-Triplet

The solar-wind q-triplet yields three q values: q_stat ≈ 1.71, q_rel ≈ 3.8, q_sen ≈ −0.6 [4]. As noted in §5.2, this paper treats only q_stat.

A preliminary observation worth recording: if different observables (stationary distribution, relaxation dynamics, sensitivity) probe different depths of a system's feedback structure, then it is natural for the same system to yield different q values from different observables—each reflects a different effective K.

This observation is currently speculative. Mapping each q to specific physical layers requires independent arguments beyond this paper's scope. We merely record the numerical coincidence that q_rel ≈ 3.8 maps to Ω_eff ≈ 7.6, which falls in the vicinity of the ZFCρ second phase-transition zone (Ω = 7–8). Whether this coincidence has structural significance is an open question.

6.6 Multi-Channel Generalization

A more general form (proposed by 公西华):

$$\Omega_{\text{eff}} = \sum_{i=1}^{n_{\text{ch}}} q_i$$

The equal-depth approximation (all q_i equal) recovers q = Ω_eff/n_ch. Unequal depths allow different channels to carry different shielding loads.

This generalization matters because it resolves the question of Ω_eff > 4: one need not push a single dual-channel q above 2. The more natural route is channel proliferation. Complex systems grow in complexity not by deepening the same channel but by spawning new functional channels.

Corollary: for a three-channel system (n_ch = 3), the mapping becomes q = Ω_eff/3.

6.7 Epistemic Status of Linearity

The linear map q = Ω_eff/n_ch is the simplest form satisfying endpoint anchoring. Nonlinear maps (e.g. q = (Ω_eff/n_ch)^α) satisfy the endpoints for α = 1 only if both endpoints must equal 1 and 2 respectively. More generally, any monotone continuous function with q(2) = 1 and q(4) = 2 is a candidate.

Linearity is chosen by Ockham's razor: no empirical data currently require nonlinear correction. Linearity itself is not derived from endpoint anchoring—it is a simplicity assumption, corresponding physically to the equal-depth approximation (§6.1). Should future data systematically deviate from linear predictions, introducing nonlinear corrections is the natural next step.

6.8 Status Summary

§7 Physical Location of the Phase-Transition Window

7.1 Preliminary Correspondence Between Ω-Window and q-Window

The ZFCρ five-indicator phase-transition window lies at Ω ∈ [2.75, 4.01]. Under q = Ω_eff/2, this maps to q ∈ [1.375, 2.005]. This window covers some mid-to-high q systems (solar wind, gravitational systems, black holes) but not all known q_stat systems—QGP (q ≈ 1.15) and cosmic rays (q ≈ 1.2) lie below the window.

7.2 Second Phase-Transition Zone

ZFCρ exhibits a second phase transition between Ω = 7 and 8 (qualitative change in single-layer closure). q = 3.5–4. The solar-wind q_rel ≈ 3.8 maps to Ω ≈ 7.6, falling in this zone—but q_rel measures relaxation dynamics rather than stationary distribution. Different q types may probe different Ω layers.

7.3 Complete h(Ω) Behavior (N = 10^10 Confirmed)

At N = 10^10, h(Ω) decreases monotonically through Ω = 29 with no upturn. |Δh| converges to ~0.10 for Ω ≥ 12. The M̄ slope crosses −1.000 at Ω = 11, with high-Ω values approaching zero from below.

Interpretation: Multiplicative suppression of additive accumulation continues but enters diminishing marginal returns at high Ω. Each additional prime-factor layer contributes a roughly constant marginal |Δh| ≈ 0.10—neither increasing nor decreasing.

§8 Black-Hole Crossover Conjecture (Interpretive Layer)

The following content belongs to the interpretive layer and does not constitute exact results or conjectures of this paper.

8.1 T_H(M_P)/T_P = 1/(8π)

The Hawking temperature of a Planck-mass black hole is T_H = T_P/(8π). This is an exact algebraic result (verified by substituting M_P = √(ℏc/G) into T_H = ℏc³/(8πGMk_B)).

At T_H, a typical Hawking photon (E ~ k_BT_H) has βE = 1. Boltzmann gives e⁻¹ ≈ 0.368; resolvent gives (1+1)⁻¹ = 0.5; deviation 36%.

8.2 Crossover Conjecture

Under certain observer-dependent and probe-dependent coarse-grainings, the effective thermal kernel may crossover from a memory-bearing form (small K, resolvent-like) to a Gibbs-like form (large K, Boltzmann-like). The vicinity of a black-hole horizon may be the most prominent physical scenario for such a crossover.

This is not a claim that "the event horizon is an observer-independent thermodynamic phase surface." It is an observer-/extraction-protocol-dependent crossover conjecture.

8.3 Complementarity with Island Formula

The DD resolvent framework is highly compatible with the island formula developed during 2019–2026 [14]. The island formula provides a computational tool (how to compute radiation entropy so the Page curve holds). The DD framework provides a mechanistic interpretation (why the effective kernel takes different forms in different regions). The two are complementary, not contradictory.

8.4 Emergence of Temperature (Discussion Level)

Within this framework, temperature T may be viewed as the Boltzmann-side coarse-grained parameter—sufficient when the effective kernel lies very close to the q = 1 (K → ∞) end. k_B serves as the coefficient translating between feedback-order language and temperature language.

Free photon gas sits at the complete-mixing limit (K = ∞, q = 1). In this limit, Boltzmann description is exact and temperature is exact—this is why the Planck spectrum and CMB measurements are so precise.

The stronger claim that "temperature is an emergent quantity" currently lacks mathematical closure. Upgrading it requires at minimum an explicit map from (K, η, ...) to β. Until then, it remains a discussion-level interpretive framework.

§9 Predictions and Compatibility Checks

Predictions

P1 (Conditional prediction: predict q from K). If a system's effective kernel is indeed controlled by the same K that governs both the layered resolvent shape and the observable memory depth, then K extracted independently from the f-channel autocorrelation function should agree with Tsallis q = 1 + 1/K fitted from the distribution tail. This is the paper's hardest prediction and can be immediately cross-validated on existing data.

P2 (Three-channel system). If a system with n_ch = 3 exists (three symmetric drive/restore channels), its q = Ω_eff/3 rather than Ω_eff/2. For the same Ω_eff, a three-channel system has lower q—closer to Boltzmann. This is a direct corollary of the channel-averaged shielding conjecture, testable once a three-channel system is identified.

Compatibility Checks (Gauge-Fixed / Compatibility Level)

C1 (Second-order coefficient). Under the canonical parametrization (z = βE), the second-order Taylor coefficient of (1+x/K)^{-K} is (K+1)/(2K). For K ≈ 7 (QGP), this gives ≈0.571; for pure Boltzmann, 0.500. The 14% difference is a gauge-fixed convenience check: it depends on the choice of canonical variable (§4) and is not a reparametrization-invariant diagnostic. Consistency would support the framework if future experiments resolve second-order kernel corrections.

C2 (Brusselator q-value compatibility). Thermo III [12] extracted m_eff = 11.3 from the Brusselator via η and C(1). m_eff is a DP-level mechanism parameter, not automatically identifiable with Tsallis kernel parameter K. If K ≈ m_eff holds approximately, the Brusselator's steady-state fluctuation distribution should approach q = 1 + 1/11.3 = 1.09. This is a compatibility check testing two assumptions simultaneously (K ≈ m_eff and layered-resolvent applicability).

C3 (Hawking radiation kernel-level inference). At the kernel level, a black hole with effective K = 1 (q = 2) would have its Hawking radiation kernel biased toward the (1+βE)⁻¹ power-law form rather than e^{-βE}. The deviation at βE ~ 1 is approximately 36%. Note: from kernel to observed flux requires occupation-layer corrections (Bose-Einstein/Fermi-Dirac) and greybody factors. This inference is discussed within the §8.2 crossover conjecture framework and is not presented as an independent prediction.

§10 Open Problems and Claim Boundaries

10.1 Status Map

10.2 Propositions Beyond This Paper's Scope

The following propositions may hold within the SAE framework, but this paper's evidence does not support them at the claim level. They are marked as interpretive or long-term open:

• "The event horizon is a Boltzmann/resolvent physical phase surface" — currently only an observer-dependent crossover conjecture
• "The information paradox is resolved by kernel switching" — currently only compatibility / interpretive layer
• "The numerical coincidence Ω = 4.01 and βE = 4.01 constitutes structural identity" — currently coincidence-level + indirect support from q = Ω_eff/n_ch
• "Temperature is not fundamental but emergent" — direction supported, but lacking an explicit (K, η) → β map
• "q = Ω_eff/n_ch" — structurally motivated conjecture, not theorem

10.3 Open Problems

• For a given class of physical systems, prove that the effective thermal kernel can be written as a layered resolvent product (Assumption A1) and determine the statistical structure of layer weights {λ_i}. Under full equal-sharing, the exact relation q = Ω_eff/n_ch follows; under general unequal weighting, the local effective parameter satisfies q_eff^{(2)} = 1 + Σλ².
• The precise relationship between K and m_eff. Is K = m_eff? Or K = f(m_eff)?
• Whether the "canonical control variable" is βE or something more fundamental.
• The physical domain of applicability of Ω_eff = Σq_i (multi-channel budget generalization).
• An analytic explanation for |Δh| converging to ~0.10 at high Ω.

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