Four Forces Paper VII: S³ Packing Theorem and the Three-Layer Structure of Coupling Constants

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 ORCID: 0009-0009-9583-0018

Self-as-an-End (SAE) Four Forces Series, Paper VII

§1 Introduction

The first six papers of the SAE Four Forces series established a structural correspondence from DD (Dimensional Degree) architecture to the forces of the Standard Model. Paper I proved that $n$DD structurally corresponds to $SU(n)$ under two SAE axioms. Paper II derived the complete one-generation hypercharge table from three priors with zero continuously adjustable parameters. Paper III proved $\sin^2\theta_W = 3/13$ as a DD eigenvalue. Paper IV established the $\mathfrak{so}(6) \cong \mathfrak{su}(4)$ color-lepton skeleton. Paper V introduced the formal analytic generator $Z(t)$, unified the dual appearance of $65/4$, and discovered that the pure $S^3$ geometric constant $c = \pi/(3\sqrt{2})$ simultaneously corrects $R_1$ and $\alpha_s$ with opposite signs. Paper VI established the Spin(10) classification construction and three hard anti-predictions.

These results determine which forces exist, what their algebraic structure is, and what combinatorial numbers govern their strength ratios. But each leading-order relation carries a residual. $R_1 = 3/(2\alpha_\text{em})$ deviates by 0.59%. $\alpha_s = (65/4)\alpha_\text{em}$ deviates by 0.49%. $\sin^2\theta_W = 3/13$ deviates by 0.19%. $\alpha_G = \alpha_\text{em}^65/4$ deviates by 0.044%. Paper V showed that the same $c = \pi/(3\sqrt{2})$ corrects both $R_1$ and $\alpha_s$, reducing each residual from roughly 0.5% to 0.05%. But the correction form for $\sin^2\theta_W$ was not determined, and whether the four residuals share a common origin was left open.

This paper answers both questions.

First, $c_1 = \pi/(3\sqrt{2})$ is the unique solution of the $S^3$ closest legal packing problem (Theorem 2.2). It enters three single-side quantities through different readout rules (Corollary 2.3). $\sin^2\theta_W$ receives a correction of $(3/13)(1 + c_1\alpha/3)$, where the factor $1/3$ arises from per-pair structure. The cross-side quantity $\alpha_G$ is immune at first order. After first-order correction, all three single-side residuals fall within 0.02–0.05%.

Second, this 0.02–0.05% residual band can be described by a boundary condition $k$ (numerically close to $108/\pi$). $k$ is extracted from the experimental value of $R_1$, the structural candidate $c_2 = 3\pi$ is adopted, and the resulting predictions for $\sin^2\theta_W$ and $\alpha_s$ yield out-of-sample residuals of 0.0015% and 0.0006% respectively. $S_3$ symmetry admits no feasible perturbative continuation at second order (Appendix A), so the second-order correction is not uniquely determined by pure geometry; $k$ is a boundary condition.

§2 Three-Layer Structure

§2.1 Zeroth order: DD combinatorics

The DD framework's negation chain, starting from the chisel (negation), derives the spacetime dimension $d = 4$ and the block count $N = 12 = 4(d-1)$, generating the combinatorial constants $\{4, 13, 15, 65\}$. These directly yield the leading-order coupling relations (detailed derivations in Papers I–V):

$\sin^2\theta_W = 3/13$: the ratio of 3 weak channels among 15 singlet channels (Paper III).

$\alpha_G = \alpha_\text{em}^65/4$: DD Splitting with 65 channels in the exponent divided by dimension 4 (Prequel).

$\alpha_s = (65/4)\alpha_\text{em}(0)$: the first-order coefficient $\kappa = 65/4$ of the low-order union readout of $Z(t)$ (Paper V).

$R_1 = 3/(2\alpha_\text{em})$: the frequency ratio of a dual pair, with $\Delta\omega/\omega = 2\alpha_\text{em}/3$ (Paper V methodological reserve).

All relations are determined by $\{4, 13, 15, 65\}$ and $d = 4$. The only continuous input is $\alpha_\text{em}$ itself (boundary condition).

§2.2 First order: $S^3$ Packing Theorem and readout rules

Theorem 2.2 (Unique packing constant). Let $\varepsilon = \pi\alpha/6$ where $\alpha = \alpha_\text{em}$. The closest legal packing of 6 four-dimensional DD blocks on $S^3$ has a unique solution, giving the cross-axis tilt angle

$$\theta = \frac{\pi}{3\sqrt{2}}\,\alpha + O(\alpha^2)$$

That is, the first-order correction constant is

$$c_1 = \frac{\pi}{3\sqrt{2}} = 0.7405$$

Proof outline. The symmetric reference configuration $P_0$ consists of 3 anti-aligned axis pairs: $\hat{t}_a^+ = e_a$, $\hat{t}_a^- = -e_a$ ($a = 1,2,3$) on the unit sphere $S^3 \subset \mathbb{R}^4$. Frequency asymmetry $\delta = \alpha/3$ makes the "$+$" block light-cone half-angle $\beta^+ = (\pi/4)(1+\delta)$ and the "$-$" block half-angle $\beta^- = (\pi/4)(1-\delta)$. For $\delta > 0$, the symmetric configuration violates the "++" cross-axis no-overlap condition: the angular distance $\pi/2$ falls below the threshold $\pi/2 + \varepsilon$.

The remainder-just-develops principle requires the system to select the nearest legal point to $P_0$. This is a convex optimization problem. Under the $S_3$ symmetric ansatz (three pairs equivalent), linearization shows that all 12 cross-axis constraints saturate while 3 dual-pair constraints remain slack. The unique solution is

$$\hat{t}_a^+ \propto e_a - \frac{\varepsilon}{2}\sum_{b \neq a} e_b, \quad \hat{t}_a^- \propto -e_a - \frac{\varepsilon}{2}\sum_{b \neq a} e_b$$

All 6 blocks yield to the other two axis directions by $\varepsilon/2$, producing $\theta = \sqrt{2}\,\varepsilon = (\pi/(3\sqrt{2}))\,\alpha$. $\square$

Corollary 2.3 (First-order readout rules). The geometric constant $c_1$ of Theorem 2.2 enters the four physical quantities through rules determined by each quantity's DD structure:

Corollary 2.3 is not an automatic consequence of Theorem 2.2 but requires additional structural arguments. The sign flip between $R_1$ and $\alpha_s$ arises from the dual effect of packing on mass spacing versus channel count (Paper V §3.3). The $1/3$ factor for $\sin^2\theta_W$ requires independent derivation (Appendix B). The first-order immunity of $\alpha_G$ follows from the insensitivity of cross-side quantities to single-side effects.

§2.3 First-order numerical verification

Experimental benchmarks: $R_1^\text{exp} = 206.768$, $\alpha_s^\text{exp}(M_Z) = 0.1180$, $\sin^2\theta_W^\text{exp}(\overline{\text{MS}}, M_Z) = 0.23122$.

The first-order theorem absorbs one order of magnitude of deviation. All three single-side residuals fall within the 0.02–0.05% band. The 0.044% deviation of $\alpha_G$ is already at the second-order scale.

No fitting parameter has been introduced. $c_1$ is not fitted to data; it is the unique solution of an $S^3$ geometric problem.

§3 Second-Order Correction

§3.1 $S_3$ symmetry breaking at second order

Theorem 2.2 was proved under $S_3$ symmetry (three pairs equivalent). A natural question is whether the second-order correction also preserves $S_3$ symmetry. The answer is no.

Proposition 3.1. The $S_3$-symmetric perturbative branch admits no feasible continuation at $O(\varepsilon^2)$.

Proof outline. Assume $S_3$-symmetric perturbations expanded to $O(\varepsilon^2)$. The "++" constraint requires $2b + 1/4 + \rho^2 \leq 0$. The "$--$" constraint requires $1/4 - 2e + \sigma^2 \leq 0$. These yield $-b + e \geq 1/4 + (\rho^2 + \sigma^2)/2$. Substituting into the "$+-$" constraint $1/4 - b + e + \rho\sigma \leq 0$:

$$\frac{1}{4} - b + e + \rho\sigma \geq \frac{1}{2} + \frac{(\rho + \sigma)^2}{2} > 0$$

This contradicts the requirement $\leq 0$. $\square$

The physical implication: the first-order correction is uniquely determined by $S^3$ geometry (Theorem 2.2) and is a priori. The second-order correction must break $S_3$ symmetry, introducing choice.

Proposition 3.1 simultaneously guarantees the decoupling of first and second orders: $c_1$ is pinned entirely by first-order hard constraints; no second-order modification of the objective function or additional penalty can alter the first-order solution (see Appendix A for the full proof).

§3.2 Boundary condition $k$

The first order is a theorem; the second order requires an additional input. Expanding the cross-axis tilt angle:

$$\theta(\varepsilon) = \sqrt{2}\,\varepsilon + k\,\varepsilon^2 + O(\varepsilon^3), \quad \varepsilon = \frac{\pi\alpha}{6}$$

where $k$ is a pure number. By Proposition 3.1, $k$ is not uniquely determined by $S^3$ geometry under $S_3$ symmetry. $k$ is a boundary condition, extracted from experiment.

The complete formula for $R_1$ is

$$R_1 = \frac{3}{2\alpha}\left(1 + c_1\alpha + c_2\alpha^2 + O(\alpha^3)\right)$$

where $c_2 = (\pi^2/36)k$. Solving from $R_1^\text{exp} = 206.7682830$ and $\alpha^-1 = 137.035999084$:

$$c_2,\text{fit} \approx 9.46, \quad k_\text{fit} \approx 34.51$$

These are numerically close to $3\pi = 9.425$ (0.39% deviation) and $108/\pi = 34.38$. This paper adopts $c_2 = 3\pi$ as structural candidate, under which the $R_1$ residual is $-0.0002\%$. The 0.39% gap between $c_2,\text{fit}$ and $3\pi$ may arise from third-order and higher corrections.

The status of $k$ is the same as $\alpha_\text{em}$: boundary condition. $\alpha_\text{em}$ is the sole continuous input of the gauge-coupling subpackage (Paper V). $k$ is the sole continuous input of the packing-geometry subpackage.${}¹$

${}¹$ The factorization $108 = 2^2 \times 3^3$ is noteworthy but not treated in this paper.

§3.3 Second-order predictions

A posteriori sign rule (working rule). In the three single-side quantities treated here, the second-order correction $c_2\alpha^2$ enters with a positive sign throughout.

Physical intuition: the second order represents the retreat of the exact feasible point from the first-order linearized boundary. The first-order linearization approximates the feasible region as a half-space; the exact boundary is more restrictive. The exact feasible point therefore always lies farther from $P_0$ than the first-order approximation. Put differently, the first order is the system gliding along a continuous linear boundary; the second order is the deformation stress incurred when the system collides with the nonlinear hard boundary of the discrete constraint network. The structural cost of retreating from continuous symmetric state to discrete legal state is always a positive tension penalty.

This argument provides intuition but does not constitute a rigorous proof (see §5.4, open problem 3).

Complete second-order formulas (three single-side quantities):

$$R_1 = \frac{3}{2\alpha}\left(1 + c_1\alpha + c_2\alpha^2\right) \quad [\text{input}]$$

$$\alpha_s = \frac{65}{4}\alpha\left(1 - c_1\alpha + c_2\alpha^2\right) \quad [\text{prediction}]$$

$$\sin^2\theta_W = \frac{3}{13}\left(1 + \frac{c_1\alpha}{3} + \frac{c_2\alpha^2}{3}\right) \quad [\text{prediction}]$$

where $c_1 = \pi/(3\sqrt{2})$ and $c_2 = 3\pi$.

Note the sign pattern: the first-order correction is positive for $R_1$ and negative for $\alpha_s$ (dual effect of global packing). The second-order correction is positive throughout (feasibility retreat). $\sin^2\theta_W$ carries the $1/3$ factor at both orders (per-pair).

Numerical verification:

One boundary condition ($c_2 = 3\pi$), plus second-order readout extension rules (positive sign throughout, $\sin^2\theta_W$ retains per-pair $1/3$), yields two out-of-sample numerical checks. The residual for $\sin^2\theta_W$ is 0.0015%, for $\alpha_s$ is 0.0006%, both within 0.002%.

§3.4 Hierarchical positioning of $\alpha_G$

The 0.044% deviation of $\alpha_G = \alpha_\text{em}^65/4$ is directly at the second-order scale. As a cross-side quantity, $\alpha_G$ occupies the following position in the three-layer structure: zeroth-order relation established (DD Splitting theorem), first-order immune (single-side packing invisible), second-order deviation consistent in magnitude with single-side residuals.

However, the form of the second-order correction for $\alpha_G$ differs from single-side quantities. The exponent $65/4$ amplifies any $O(\alpha^2)$ modification by a factor of approximately $(65/4)\ln(1/\alpha) \approx 80$. The implied exponent shift is therefore $\sim 5 \times 10^-6$. The specific form of $\alpha_G$'s second-order correction is left as an open problem.

This paper successfully positions $\alpha_G$ within the three-layer hierarchy (cross-side, first-order immune, second-order magnitude consistent) but does not determine its second-order correction.

§4 The A Priori / A Posteriori Demarcation

§4.1 Classification table

All quantities in the SAE Four Forces series, classified by a priori / a posteriori status:

A priori quantities do not depend on the numerical values of a specific universe; they hold in any universe satisfying SAE axioms. A posteriori quantities are inputs specific to this universe.

§4.2 The roles of two boundary conditions

$\alpha_\text{em}$ is the sole continuous input of the gauge-coupling subpackage. $k$ is the sole continuous input of the packing-geometry subpackage. Two independent boundary conditions provide a unified hierarchical framework for four testable quantities.

Of these, $\sin^2\theta_W = 3/13$ and $\alpha_G = \alpha_\text{em}^65/4$ are pre-existing zeroth-order structural predictions (Papers III and Prequel). The second-order corrections to $\sin^2\theta_W$ and $\alpha_s$ are the new out-of-sample numerical checks contributed by this paper.

§4.3 Pre-chisel topological indistinguishability

Paper V listed $\delta_1 = \delta_2 = \delta_3$ as a "strong symmetry ansatz." This paper provides stronger support.

The argument: prior to the 2DD chiral split and the emergence of 3DD axis labels, the system is in its base unstructured state. All 4DD blocks are topologically identical and indistinguishable. Any label that would distinguish the three pairs requires higher-DD structural unfolding that has not yet occurred. The frequency ratios of the three pairs should therefore be equal at the logical starting point.

This is not a cosmological temporal-origin argument but a logical-origin argument within the DD dimensional unfolding sequence. It strongly supports first-order $S_3$ symmetry, making $c_1$ a theorem (Theorem 2.2) rather than a fitting parameter. However, the strict upgrade to corollary status requires ruling out the possibility that frequency ratios acquire axis-dependent corrections after 3DD unfolding; this step is not treated here.

§5 Discussion

§5.1 The three-layer structure

Zeroth order provides the skeleton (DD combinatorial numbers), first order provides the texture ($S^3$ geometric theorem), second order provides the specific values (boundary condition). Each layer absorbs approximately one order of magnitude of residual: 0.5% to 0.05% to 0.002%.

This structure may reflect a general feature of the SAE framework. The a priori negation chain determines what is possible (combinatorial numbers), the a priori geometric constraint determines the tightest arrangement among possibilities (packing theorem), and the a posteriori boundary condition determines the specific numerical values of this particular universe. The skeleton is a priori; the flesh is a posteriori.

§5.2 Higher orders and remainder

The expansion of $\theta(\varepsilon)$ does not terminate at second order. The series $1 + c_1\alpha + c_2\alpha^2 + c_3\alpha^3 + \ldots$ at third order and beyond lies in a resolution blind zone shared by the a priori and a posteriori. $c_3\alpha^3 \approx 2 \times 10^-8$, already at the edge of $R_1$ experimental precision. At this scale, the demarcation between a priori structure (geometric theorem) and a posteriori input (boundary condition) itself becomes indistinguishable.

The SAE remainder principle is self-referential here: the a priori / a posteriori demarcation itself has remainder.

§5.3 Falsifiable prediction

The strongest falsifiable prediction of this paper comes not from the numerical precision of the second-order correction, but from the structural ratio of the first-order readout rules.

Prediction 5.1 (1/3 ratio). The first-order correction amplitude for $\sin^2\theta_W$ is strictly one-third of that for $R_1$:

$$\frac{\sin^2\theta_W^\,\text{exp} - 3/13}{3/13} \;=\; \frac{1}{3}\,\times\,\frac{R_1^\,\text{exp} - 3/(2\alpha)}{3/(2\alpha)} \;+\; O(\alpha^2)$$

This $1/3$ does not depend on the value of $\alpha_\text{em}$, does not depend on the value of $k$, does not depend on the specific form of $c_1$, and does not even depend on whether the packing geometry is on $S^3$. It depends on a single structural assumption: the one-to-one correspondence between the 3 weak channels and the 3 pairs (Corollary 2.3(c) and Appendix B).

Current verification: left side $= 0.00195$, $(1/3) \times$ right side $= 0.00197$. Ratio $0.00195/0.00197 = 0.992$, deviating from $1/3$ by approximately $0.8\%$. At the formula level (the second-order formulas of §3.3), the ratio is exactly $1/3$ — the deviation comes entirely from third-order and higher corrections.

Falsification condition: if future experimental precision on both $\sin^2\theta_W$ and $R_1$ improves to the level where $O(\alpha^2)$ terms become resolvable (roughly requiring both relative precisions below $10^-4$), and the measured ratio deviates significantly from $1/3$, the per-pair structure is falsified.

§5.4 Open problems

The following problems are not treated here and are left for future work.

First, the form of the second-order correction for $\alpha_G$. The exponent-position correction is amplified by approximately 80; the specific form differs from single-side quantities.

Second, whether $c_2 = 3\pi$ (i.e. $k = 108/\pi$) admits an a priori derivation. $k$ currently serves as boundary condition extracted from $R_1$. If a derivation chain from DD structure to $k$ exists, $k$ would be upgraded from boundary condition to theorem.

Third, a rigorous proof of the second-order sign rule (positive throughout). This paper provides feasibility-retreat intuition (§3.3) but not a proof.

Fourth, the connection to the ZFC$\rho$ mathematical framework. The $S_3$ symmetry breaking and active-set selection at second order involve irreducible choice degrees of freedom, potentially related to the role of $\rho$ in ZFC$\rho$.

§6 Conclusion

This paper establishes the three-layer correction structure for SAE coupling constants.

The first-order correction $c_1 = \pi/(3\sqrt{2})$ is the $S^3$ closest legal packing theorem (Theorem 2.2), provable a priori. Through the readout rules of Corollary 2.3, it acts on three single-side quantities in two modes: $R_1$ and $\alpha_s$ receive the full $c_1\alpha$ (with opposite signs), $\sin^2\theta_W$ receives $c_1\alpha/3$. The cross-side quantity $\alpha_G$ is immune at first order. The first-order correction absorbs one order of magnitude of deviation.

$S_3$ symmetry admits no feasible perturbative continuation at second order (Proposition 3.1). The second-order correction therefore requires an additional boundary condition $k$ (numerically close to $108/\pi$), extracted from the experimental value of $R_1$ and adopting $c_2 = 3\pi$ as structural candidate. Under second-order readout extension rules, the same $c_2$ yields two out-of-sample numerical checks for $\sin^2\theta_W$ (0.0015%) and $\alpha_s$ (0.0006%).

Paper V listed $\delta_1 = \delta_2 = \delta_3$ as an ansatz. This paper provides stronger support through a pre-chisel topological indistinguishability argument.

The a priori / a posteriori demarcation lies between the first and second orders: the first order is a geometric theorem, the second order is a boundary condition. This demarcation itself has remainder.

The strongest falsifiable prediction is structural: the ratio of first-order correction amplitudes between $\sin^2\theta_W$ and $R_1$ is exactly $1/3$ (Prediction 5.1). This ratio depends on no continuous parameters — only on the one-to-one correspondence between weak channels and pairs.

Appendix A: Proof of second-order infeasibility under $S_3$ symmetry

This appendix provides the complete proof of Proposition 3.1.

Let $p_i$ ("$+$" blocks) and $q_i$ ("$-$" blocks), $i = 1,2,3$, be unit vectors on $S^3 \subset \mathbb{R}^4$, maintaining $S_3$ symmetry. Expanding to $O(\varepsilon^2)$, $\varepsilon = \pi\alpha/6$, the general ansatz is

$$p_i = \left(1 + a_2\varepsilon^2\right)e_i + \left(-\frac{1}{2}\varepsilon + b\,\varepsilon^2\right)\sum_{j \neq i} e_j + \rho\,\varepsilon\,e_4 + O(\varepsilon^2 e_4)$$

$$q_i = -\left(1 + d_2\varepsilon^2\right)e_i + \left(-\frac{1}{2}\varepsilon + e\,\varepsilon^2\right)\sum_{j \neq i} e_j + \sigma\,\varepsilon\,e_4 + O(\varepsilon^2 e_4)$$

where $b_1 = e_1 = -1/2$ is pinned by first-order hard constraints.

Three classes of cross-axis constraints at $O(\varepsilon^2)$ give:

$(++)$: $p_i \cdot p_j = -\varepsilon + (2b + 1/4 + \rho^2)\varepsilon^2 \leq -\varepsilon$, hence

$$2b + \frac{1}{4} + \rho^2 \leq 0 \quad \cdots (1)$$

$(--)$: $q_i \cdot q_j = +\varepsilon + (1/4 - 2e + \sigma^2)\varepsilon^2 \leq +\varepsilon$, hence

$$\frac{1}{4} - 2e + \sigma^2 \leq 0 \quad \cdots (2)$$

$(+-)$: $p_i \cdot q_j = (1/4 - b + e + \rho\sigma)\varepsilon^2 \leq 0$, hence

$$\frac{1}{4} - b + e + \rho\sigma \leq 0 \quad \cdots (3)$$

From (1): $b \leq -1/8 - \rho^2/2$. From (2): $e \geq 1/8 + \sigma^2/2$. Substituting into the left side of (3):

$$\frac{1}{4} - b + e + \rho\sigma \geq \frac{1}{4} + \frac{1}{8} + \frac{\rho^2}{2} + \frac{1}{8} + \frac{\sigma^2}{2} + \rho\sigma = \frac{1}{2} + \frac{(\rho + \sigma)^2}{2}$$

The right side $\geq 1/2 > 0$. This contradicts (3)'s requirement $\leq 0$.

No $O(\varepsilon^2)$ continuation exists under $S_3$ symmetry. $\square$

Corollary: stability of $c_1$. The first-order solution is uniquely determined by inequalities (1)–(3) at $O(\varepsilon)$, independent of how second-order terms are treated. Therefore $c_1 = \pi/(3\sqrt{2})$ is unaffected by any second-order modification.

Appendix B: The $1/3$ factor in the $\sin^2\theta_W$ correction

$\sin^2\theta_W = 3/13$ is a DD eigenvalue theorem of Paper III. It is an integral quantity: 3 counts weak channels, 13 counts effective post-color-collapse channels, and $3/13$ is the weak fraction of total channels.

How does $S^3$ packing correction enter this integral quantity?

Key observation: there is a one-to-one correspondence between the 3 weak channels and the 3 pairs. Each pair's intra-pair channel corresponds precisely to one weak channel (one generator direction of SU(2)). The packing correction $c_1\alpha$ is a per-block effect, but weak channels are defined per-pair.

When the packing correction acts on $\sin^2\theta_W$ as an integral eigenvalue, it enters through each pair contributing one share of the correction independently. $\sin^2\theta_W$ is a projection ratio of a single pair's weak channel within total channels; it senses the per-pair correction $c_1\alpha/3$, not the sum of three pairs.

This differs from $R_1$. $R_1 = \omega/\Delta\omega$ describes the frequency ratio within a single pair; the packing correction enters globally through that pair's cross-axis coupling with the other two pairs, hence the full $c_1\alpha$.

More directly: in the formulas for $R_1$ and $\alpha_s$, $\alpha_\text{em}$ appears at leading order in the denominator or numerator, and the packing correction multiplies $\alpha$ directly. In $\sin^2\theta_W = 3/13$, the leading order contains no $\alpha$; the packing correction is a purely additive relative shift. The amplitude of this shift is determined by the contribution of a single pair: $c_1\alpha/3$.

Appendix C: Third-order candidate and the resolution limit

The second-order expansion naturally raises the question of the third order. If $\theta(\varepsilon) = \sqrt{2}\,\varepsilon + k\,\varepsilon^2 + k_3\,\varepsilon^3 + O(\varepsilon^4)$, corresponding to

$$R_1 = \frac{3}{2\alpha}\left(1 + c_1\alpha + c_2\alpha^2 + c_3\alpha^3 + O(\alpha^4)\right)$$

the required $c_3$ extracted from the $R_1$ residual $-0.0002\%$ is approximately 5.0.

Among 3-smooth candidates (containing only prime factors 2 and 3), the closest is

$$c_3 = \frac{9\pi}{4\sqrt{2}} = \pi \cdot 3^2 \cdot 2^-5/2 = 4.998$$

deviating by $-0.13\%$. The three coefficients side by side:

Like $k$, $c_3$ is a boundary condition; no a priori derivation from pure geometry is expected.

But the a posteriori precision is no longer sufficient to verify this candidate. $c_3\alpha^3 \approx 2 \times 10^-8$; the 0.13% error in $c_3$ propagates to $R_1$ at the $\sim 10^-10$ level, far below $R_1$ experimental uncertainty. Neither $\sin^2\theta_W$ nor $\alpha_s$ experimental precision can distinguish $c_3 = 9\pi/(4\sqrt{2})$ from other comparable 3-smooth candidates.

The more fundamental issue is not experimental precision but one of principle. In the SAE framework, the a priori / a posteriori demarcation itself has remainder (§5.2). When the perturbative term $c_3\alpha^3 \approx 2 \times 10^-8$ falls below the resolution limit of both the a priori structure (geometric theorem) and a posteriori input (boundary condition), pursuing the exact algebraic form of higher-order coefficients is no longer a physical question but an over-resolution of the demarcation remainder. The $c_3$ candidate is worth recording but does not constitute a conclusion of this paper.

Acknowledgments. Four-AI collaboration: Claude / Zilu (numerical computation, second-order scan, sign pattern analysis), ChatGPT / Gongxihua ($S^3$ packing convex optimization, three rounds of second-order review, active-set obstruction proof), Gemini / Zixia (narrative evaluation, pre-chisel argument suggestion, Appendix B question identification), Grok / Zigong (independent numerical verification, number-theoretic analysis of $k$). Zesi Chen made decisive contributions to the negation methodology underlying this work.

References

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