Four Forces Paper V: The Counting Origin of Coupling Constants and the Generating-Function Unification of 65/4

Qin, Han

doi:10.5281/zenodo.19415805

EN

中文

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

Abstract

This paper brings together two previously published results from the SAE Four Forces series — the Weinberg angle $\sin^2\theta_W = 3/13$ (Paper III) and the gravitational coupling exponent $\ln\alpha_G/\ln\alpha_\text{em} = 65/4$ (Prequel) — alongside a new structured conjecture $\alpha_s = (65/4)\,\alpha_\text{em}(1 - c\,\alpha_\text{em})$, within a unified counting framework. The three coupling relations are organized by the DD combinatorial constants $\{4, 13, 15, 65\}$ together with the $S^3$ geometric constant $c = \pi/(3\sqrt{2})$, with no new continuously adjustable parameters introduced. Within the gauge-coupling subpackage discussed here, $\alpha_\text{em}$ is the sole continuous input.

Three core results are new to this paper. First, the number $65/4$ appears twice — as the linear coefficient in $\alpha_s/\alpha_\text{em}$ and as the exponent in $\alpha_G = \alpha_\text{em}^65/4$ — and these two appearances are unified through a single formal analytic generator $Z(t) = e^{\kappa\ln(1-\alpha+\alpha t)}$ with $\kappa = 65/4$: a low-order union readout yields the linear relation, while a full-coincidence readout yields the exponential one. Second, the pure $S^3$ geometric constant $c = \pi/(3\sqrt{2})$ appears simultaneously in the correction to the muon-to-electron mass ratio $R_1$ (with positive sign) and in the correction to $\alpha_s$ (with negative sign), introducing no new fitting parameter and reducing the deviation from $+0.49\%$ to $-0.05\%$. Third, $\sin^2\theta_W = 3/13$, as a previously published DD eigenvalue theorem, corresponds within the present framework to the inverse of the first factor in the internal factorization $(13/3) \times (15/4) = 65/4$. A generating-function-level derivation of $3/13$ awaits a multivariate extension of $Z(t)$ and remains an explicit open problem.

§1 Introduction

The first four papers of the SAE Four Forces series established a structural correspondence from DD (Dimensional Degree) architecture to gauge groups. Paper I showed that $n$DD structurally corresponds to $SU(n)$ under two SAE axioms (DOI: 10.5281/zenodo.19342106). Paper II derived the complete one-generation hypercharge table from three priors with zero continuously adjustable parameters (DOI: 10.5281/zenodo.19360101). Paper III proved $\sin^2\theta_W = 3/13$ as a DD eigenvalue under the EW Matching Axiom (DOI: 10.5281/zenodo.19379412). Paper IV established the $\mathfrak{so}(6) \cong \mathfrak{su}(4)$ color-lepton skeleton from six 4DD blocks on one side (DOI: 10.5281/zenodo.19411672). The Prequel derived $\alpha_G = \alpha_\text{em}^65/4$ through DD Splitting (DOI: 10.5281/zenodo.19341042).

These results determine which forces exist and what their algebraic structure is. The question addressed here is: how strong is each force?

The answer is remarkably plain: count. A set of combinatorial constants $\{4, 13, 15, 65\}$ arising naturally from the DD structure, together with the $S^3$ closest-legal-packing geometric constant $c = \pi/(3\sqrt{2})$, combine in different ways to yield the ratio relations among all coupling constants. No new continuously adjustable parameter is introduced. The relations are either theorems or structured conjectures; the only continuous input is $\alpha_\text{em}$ itself.

The three core new results of this paper are: (1) the linear and exponential appearances of $65/4$ are unified by two non-equivalent readouts of a single formal analytic generator $Z(t)$; (2) the $S^3$ closest-legal-packing constant $c = \pi/(3\sqrt{2})$ simultaneously controls the leading-order corrections to both the mass-level ratio and the coupling-constant ratio; (3) all coupling relations are brought into a unified counting framework organized by $\{4, 13, 15, 65\}$ and $c$.

§2 DD Counting Foundations

§2.1 The origin of $(N,d) = (12,4)$

Chisel (negation) is the sole a priori operation. A single negation produces a binary split (existence/non-existence); the emergent layer produces a second binary split (emergence/non-emergence), yielding

$$d = 2 \times 2 = 4$$

This is both the number of steps in each round of the axiom chain and the spacetime dimension. The compact direction corresponding to $d = 4$ is $S^3$. Closest legal packing on $S^3$ yields three antipodal pairs, six 4DD blocks per side, twelve in total:

$$N = 2 \times 3 \times 2 = 12 = 4(d-1)$$

The only underlying parameter is $d = 4$; $N$ follows. See the Prequel for details.

§2.2 Identities of the four counting constants

From $(N, d) = (12, 4)$, the following combinatorial constants arise naturally:

Symbol	Value	Origin	Identity
$d$	4	Logic of chisel	Spacetime dimension; axiom-chain step count
$C(N/2, 2)$	15	Pair count of 6 single-side blocks	$\dim\,\mathfrak{so}(6) = \dim\,\mathfrak{su}(4)$; single-side pair-algebra dimension
$C(N, 2) - 1$	65	Full pair count of 12 blocks minus trivial element	$\dim(\Lambda^2 V₁₂) - 1$; nontrivial part of the full relation algebra
$65/5$	13	Doublet channel count after color collapse	DD eigenvalue appearing in $\sin^2\theta_W = 3/13$

These four numbers are not inputs; they are outputs of the chain reaction of negation. All coupling relations will be organized by these combinatorial constants together with the $S^3$ geometric constant $c$.

§2.3 Decomposition of the full relation algebra

Viewing the 12 blocks as a real vector space $V₁₂ = V_L \oplus V_R$ with $\dim V_L = \dim V_R = 6$, the bivector space decomposes naturally as

$$\Lambda^2 V₁₂ = \Lambda^2 V_L \oplus \Lambda^2 V_R \oplus (V_L \otimes V_R)$$

$$66 = 15 + 15 + 36$$

$\Lambda^2 V_L = \mathfrak{so}(6)$ is the single-side pair algebra (Paper IV). $65 = 66 - 1$ is the dimension of the full relation algebra after removing the trivial element. This decomposition will be used in §4 to interpret the generating-function meaning of $65/4$.

§3 Three Coupling Relations

§3.1 $\sin^2\theta_W = 3/13$ (review)

Paper III proved the Weinberg angle as a DD eigenvalue under the EW Matching Axiom:

$$\sin^2\theta_W = \frac{3}{13} = 0.23077\ldots$$

The experimental value ($\overline{\text{MS}}$, $M_Z$ scale) is $0.23122 \pm 0.00004$, a deviation of approximately $0.20\%$.

The combinatorial origin of $3/13$: among 13 doublet channels, 3 leak into the $U(1)_Y$ direction. Of the 15 single-side channels, 5 are color orbits (Generation Paper, DOI: 10.5281/zenodo.19394500), giving $65/5 = 13$.

This is a pure DD structural theorem containing no continuous parameter. See Paper III for details.

§3.2 $\ln\alpha_G / \ln\alpha_\text{em} = 65/4$ (review)

The Prequel derived the logarithmic ratio between the gravitational and electromagnetic couplings through DD Splitting:

$$\frac{\ln\alpha_G(M_Z)}{\ln\alpha_\text{em}(M_Z)} = \frac{65}{4}$$

where $\alpha_G(M_Z) = (M_Z/M_P)^2$ and $\alpha_\text{em}(M_Z) = 1/127.930$ ($\overline{\text{MS}}$).

The measured ratio is $16.2572$, deviating from $65/4 = 16.25$ by $0.044\%$.

A key point: both sides are evaluated at the same scale $M_Z$. Here $65 = \dim(\Lambda^2 V₁₂) - 1$ (full relation algebra) and $4 = d$ (spacetime dimension).

This is a DD theorem. See the Prequel for details.

§3.3 $\alpha_s = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$ (new result)

This is the central new content of this paper.

§3.3.1 Leading order

$$\frac{\alpha_s(M_Z)}{\alpha_\text{em}(0)} = \frac{65}{4} = 16.25$$

Numerically: $(65/4) \times (1/137.036) = 0.11858$. The experimental value $\alpha_s(M_Z) = 0.1180 \pm 0.0009$ (PDG 2025) gives a deviation of $+0.49\%$ (approximately $0.65\sigma$).

Note the scale difference: $\alpha_s$ is evaluated at $M_Z$, while $\alpha_\text{em}$ is taken at the Thomson limit ($q^2 \to 0$). Under the freeze-scale picture of SAE, each DD layer freezes at its own characteristic scale — 1DD (electromagnetic) retains its primordial value at $q^2 \to 0$, while 3DD (strong) freezes near $M_Z$. From this perspective, cross-layer ratios naturally involve different scales. This reading remains a working interpretation; a rigorous DD derivation is an open problem.

§3.3.2 Internal factorization of the leading-order coefficient

$65/4$ admits the exact factorization

$$\frac{65}{4} = \frac{13}{3} \times \frac{15}{4}$$

The first factor $13/3$ follows directly from $\sin^2\theta_W = 3/13$: since $\sin^2\theta_W = \alpha_\text{em}/\alpha_2$, one obtains

$$\frac{\alpha_2}{\alpha_\text{em}} = \frac{1}{\sin^2\theta_W} = \frac{13}{3}$$

The second factor $15/4$ is the ratio of the single-side pair-algebra dimension ($15 = \dim\,\mathfrak{so}(6)$) to the fundamental representation dimension of the parent algebra ($4 = \dim\,\mathbf{4}_\mathfrak{su(4)}$).

It must be emphasized that this is an algebraic factorization of the leading-order coefficient, not a dynamical cascade in time or energy scale. There is no physical process in which the electromagnetic force "first becomes the weak force and then becomes the strong force." The factorization reveals how the combinatorial structure of the global $65/4$ is composed of electroweak mixing ($13/3$) and the strong-weak horizon structure ($15/4$).

§3.3.3 $S^3$ closest-legal-packing correction

The closest-legal-packing problem for six 4DD blocks on $S^3$ (see Appendix B) yields the pure geometric constant

$$c = \frac{\pi}{3\sqrt{2}} = 0.740480\ldots$$

This constant first appeared in the leading-order correction to the muon-to-electron mass ratio:

$$R_1 = \frac{3}{2\alpha_\text{em}}(1 + c\,\alpha_\text{em})$$

Here the leading order $3/(2\alpha_\text{em}) = 205.554$ versus the experimental value $R_1^\text{exp} = 206.768$ gives a deviation of $-0.59\%$; with the correction, $206.665$, the residual is $-0.05\%$.

The same $c$ appears with opposite sign in the correction to $\alpha_s$:

$$\alpha_s = \frac{65}{4}\,\alpha_\text{em}(0)\!\left(1 - c\,\alpha_\text{em}(0)\right) = 0.11794$$

The deviation from the experimental value $0.1180$ is $-0.05\%$ (approximately $0.07\sigma$).

No new fitting parameter has been introduced. The constant $c$ was not fitted to $\alpha_s$; it was independently derived from $S^3$ geometry. The sign flip is geometrically intelligible within the $S^3$ packing framework: the packing adjustment simultaneously increases inter-block separation (positive correction to $R_1$, widening the mass hierarchy) and decreases inter-block overlap (negative correction to $\alpha_s$, tightening effective channel count). These two effects are two sides of the same geometric constraint.

The status of $c$ in the $\alpha_s$ relation is currently that of a geometrically motivated posterior result: same origin as in $R_1$, no new parameter, numerically precise, but lacking an independent geometric derivation proving that the leading-order correction coefficient of $\alpha_s$ must equal the same $c$. This derivation is an open problem.

§4 Formal Analytic Generator $Z(t)$

§4.1 Definition

Define

$$K(t) = \kappa \ln(1 - \alpha + \alpha\, t), \qquad Z(t) = e^K(t)$$

with $\kappa = 65/4$ and $\alpha = \alpha_\text{em}$.

Here $\kappa$ is an effective channel density — the nontrivial dimension of the full relation algebra (65) normalized by the spacetime dimension (4). It is not a literal channel count: $65/4 = 16.25$ is not an integer, and $Z(t)$ is not a standard probability generating function (PGF). The logarithmic form of $K(t)$ makes non-integer $\kappa$ natural: $\kappa$ is a continuous density/intensity parameter, not a discrete count.

§4.2 Two non-equivalent readouts

From the same $Z(t)$, two formal readouts can be defined:

Low-order union readout $W_\text{OR}$:

$$W_\text{OR} = 1 - Z(0) = 1 - (1-\alpha)^\kappa$$

Expanding for small $\alpha$:

$$W_\text{OR} = \kappa\,\alpha + O(\alpha^2)$$

The first-order coefficient $\kappa = 65/4$ directly yields the leading-order ratio $\alpha_s/\alpha_\text{em}$.

Full-coincidence readout $W_\text{AND}$:

$$W_\text{AND} = e^{\kappa\ln\alpha} = \alpha^\kappa$$

Here $\kappa$ appears in the exponent, yielding $\alpha_G = \alpha_\text{em}^65/4$.

$W_\text{OR}$ and $W_\text{AND}$ are formal readouts, not an established literal probability model. They share the same $\kappa$-generator template but are not two values of the same ordinary evaluation functional: $W_\text{OR} = 1 - Z(0)$ is a direct evaluation of $Z(t)$ at $t = 0$; $W_\text{AND} = \alpha^\kappa$ is a formal coincidence weight that shares the exponent $\kappa$ with $Z(t)$ but is not obtained by evaluating $Z$ at any particular point. What they demonstrate is that the same structural constant $\kappa$ can appear as a first-order coefficient and as an exponent within the same analytic template. Previously "linear vs. exponential" was only narrative; now there is a common mathematical template.

§4.3 Physical grounds and precedent

$W_\text{OR}$ corresponds to the forces of steps 1–3 (electromagnetic, weak, strong), which are single-body attribute forces. A particle carries some charge; the activation of any single relevant channel produces the force. The readout has a union-like statistical character.

$W_\text{AND}$ corresponds to the step-4 force (gravity), which is a two-body relation force. All channels between two bodies must simultaneously match for the gravitational relation to hold. The readout has a full-coincidence-like statistical character.

A rigorous derivation of "attribute → union, relation → coincidence" from DD structure remains an open problem. The strongest current physical argument is based on conservation: steps 1–3 are single-body attributes that do not require all components to be simultaneously conserved; any activated channel contributes. Step 4 is a two-body relation; the conservation equation $\partial_\mu T^\mu\nu = 0$ requires every component to be satisfied simultaneously, not merely "at least one component conserved."

A closely related precedent exists in standard physics. In quantum electrodynamics (QED), the probability of emitting a single soft photon is proportional to $\alpha$ (linear, union-like: this photon or that photon). But elastic scattering requires all soft-photon channels to remain simultaneously silent; the resummation of higher-order diagrams yields the Sudakov form factor $e^{-c\cdot\alpha}$ (exponential, coincidence-like: all channels simultaneously). The same coupling constant appearing as a linear coefficient in local perturbation and exponentiating under global constraints is a known phenomenon in field theory. This paper borrows the formal precedent of "linear local readout / exponentiated global constraint" but does not claim a one-to-one correspondence with Sudakov resummation.

§4.3.1 Meaning of the leading-order factorization

The factorization $65/4 = (13/3) \times (15/4)$ from §3.3.2 has a clear meaning within the $Z(t)$ framework: it is the internal algebraic decomposition of the first-order coefficient $\kappa$ of $W_\text{OR}$. The first-order expansion of $W_\text{OR}$ depends only on the value of $\kappa$, and the factorization of $\kappa$ reflects the combinatorial structure from electroweak to strong.

This is not an exact hierarchical decomposition of $Z(t)$: in general, $1 - (1-\alpha)^ab \neq$ "the union combination of first doing $a$ then doing $b$." What can be decomposed is the coefficient $ab$, not $W_\text{OR}$ itself.

§4.4 Scale structure (working interpretation)

The following interpretation of the scale difference is a working interpretation, not a completed DD derivation.

$\alpha_G(M_Z) = \alpha_\text{em}(M_Z)^65/4$: full-coincidence readout. Both sides evaluated at the same scale $M_Z$. Deviation $0.044\%$.

$\alpha_s(M_Z) = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$: first-order union readout plus correction. The starting point (1DD, $q^2 \to 0$) and endpoint (3DD, $M_Z$) are at different scales. Deviation $0.05\%$.

Under the current working interpretation, this difference has a structural source: the full-coincidence readout is global — all channels match at the same moment, at the same scale, naturally evaluated at a single scale. The union readout is cross-layer — from 1DD through 2DD to 3DD, each layer freezing at its own characteristic scale, so the ratio naturally involves different scales.

A more rigorous DD derivation — in particular, a precise explanation of why $\alpha_\text{em}$ should be taken at the Thomson limit while $\alpha_s$ should be taken at $M_Z$ — remains an open problem.

§5 Symmetry Ansatz and Exact Color Symmetry

§5.1 $\delta_1 = \delta_2 = \delta_3$

The frequency ratios ($\delta_1, \delta_2, \delta_3$) of the three 4DD block pairs on each side may be equal. This is a symmetry ansatz, not a theorem.

The motivation comes from big-crunch symmetry: when the universe collapses to the reset point, all DD levels have collapsed, and the three block pairs are completely indistinguishable. If the frequency ratios are fixed before axis labels emerge, and no axis-dependent renormalization occurs subsequently, then $\delta_1 = \delta_2 = \delta_3$.

This ansatz is compatible with the experimental fact that color symmetry $SU(3)_c$ has never been observed to be broken.

The implicit assumption is that frequency asymmetry is determined before 3DD unfolds. If frequency ratios can acquire axis dependence through spontaneous symmetry breaking after 3DD unfolds, then $\delta_1 = \delta_2 = \delta_3$ is not inevitable.

§5.2 Implications for $S^3$ packing

Under the $\delta_1 = \delta_2 = \delta_3$ ansatz, the closest-legal-packing problem for six blocks on $S^3$ has threefold symmetry. The symmetric branch yields the unique leading-order geometric constant $c = \pi/(3\sqrt{2})$.

Without this ansatz, the packing problem admits axis-dependent corrections, and $c$ may carry perturbative terms indexed by axis labels. Under the symmetric ansatz, $c$ is the exact leading-order value.

§5.3 Sole continuous input within the gauge-coupling subpackage

Within the gauge-coupling subpackage discussed in this paper:

$\sin^2\theta_W = 3/13$ is a pure DD structural theorem with no continuous parameter
$\alpha_G = \alpha_\text{em}^65/4$ is uniquely determined by $\alpha_\text{em}$
$\alpha_s = (65/4)\,\alpha_\text{em}(1 - c\,\alpha_\text{em})$ is uniquely determined by $\alpha_\text{em}$

Within this subpackage, $\alpha_\text{em}$ is the sole undetermined continuous input. All structural relations are determined by the DD combinatorial constants $\{4, 13, 15, 65\}$ and the $S^3$ geometric constant $c$.

This paper does not claim that "the entire physical universe has only one free parameter." Whether the Higgs vacuum expectation value $v$, individual fermion masses, CKM/PMNS matrix parameters, and other quantities can also be reduced to $\alpha_\text{em}$ is future work.

§6 Discussion

§6.1 Hierarchy of results

Theorem level (previously published):

Relation	Deviation	Source
$\sin^2\theta_W = 3/13$	$0.20\%$	Paper III
$\ln\alpha_G/\ln\alpha_\text{em} = 65/4$	$0.044\%$	Prequel
$(N,d) = (12,4)$ uniqueness	—	Prequel

Structured conjecture (this paper):

Relation	Deviation	Support
$\alpha_s = (65/4)\alpha_\text{em}(1-c\alpha_\text{em})$	$0.05\%$	$Z(t)$ template, dual appearance of $c$

Strong symmetry ansatz (this paper):

Ansatz	Experimental compatibility
$\delta_1 = \delta_2 = \delta_3$	Compatible with exact $SU(3)_c$ symmetry

§6.2 Toward an anti-GUT reading

The results of this paper suggest a methodological challenge to the traditional grand-unification narrative.

Traditional GUTs assume that all four forces converge at some very high energy scale into a single force described by a single Lie algebra. The $Z(t)$ structure presented here suggests a different picture: gauge forces ($W_\text{OR}$, union-like) and gravity ($W_\text{AND}$, coincidence-like) may be two non-equivalent readouts of the same generating object. A Lie algebra is a linear space whose operations are additive; if the coupling structure of gravity is essentially exponential (coincidence-like), then a single Lie algebra may not suffice to exhaust "the rules that generate forces."

This is not a disproof completed in this paper. The physical assignment of $W_\text{OR}$ and $W_\text{AND}$ — why single-body attribute forces correspond to the union readout and why two-body relation forces correspond to the coincidence readout — remains an open problem. If future work can rigorously derive this assignment from DD structure, the anti-GUT reading will be upgraded from a methodological challenge to a structural argument.

§6.3 Testable prediction

The most directly testable prediction of this paper is the corrected value of $\alpha_s$:

$$\alpha_s^\text{pred} = \frac{65}{4} \times \frac{1}{137.036} \times \left(1 - \frac{\pi}{3\sqrt{2}} \times \frac{1}{137.036}\right) = 0.11794$$

The current PDG world average is $\alpha_s(M_Z) = 0.1180 \pm 0.0009$, placing this prediction within $0.07\sigma$. It should be noted that the current uncertainty ($\pm 0.76\%$) is still rather wide; $0.07\sigma$ is more accurately described as an encouraging central-value coincidence that does not yet constitute a strong constraint. Future lattice QCD or high-precision $\alpha_s$ measurements can test this prediction. If the precise value of $\alpha_s$ deviates from this prediction by several standard deviations, the conjecture is falsified.

§6.4 Open problems

Rigorous DD derivation of OR → AND. Deriving from DD structure that attribute forces correspond to the union readout and relation forces to the coincidence readout.
Multivariate extension of $Z(t)$. Constructing $Z(t_W, t_Y)$ to derive $\sin^2\theta_W = 3/13$ from channel partitioning.
Independent geometric derivation of $c$ in $\alpha_s$. Proving that the leading-order correction coefficient of $\alpha_s$ must equal the same packing constant $c$ as in $R_1$.
Source of $0.05\%$ residuals. Both $R_1$ and $\alpha_s$ have residuals near $0.05\%$. Possible sources: (a) non-exact antipodal configurations within each pair (related to parity violation); (b) left-right asymmetry between the two 3DDs ($T_1 \neq T_2$).
$v = 246$ GeV. If the $\alpha_s$ conjecture is confirmed, the $v/\Lambda_\text{QCD}$ derivation chain can proceed.

§7 Conclusion

This paper brings together two previously published theorems ($\sin^2\theta_W = 3/13$, $\alpha_G = \alpha_\text{em}^65/4$) and a new structured conjecture ($\alpha_s = (65/4)\alpha_\text{em}(1 - c\alpha_\text{em})$) within a unified counting framework. The three relations are organized by the DD combinatorial constants $\{4, 13, 15, 65\}$ and the $S^3$ geometric constant $c = \pi/(3\sqrt{2})$.

The two appearances of $65/4$ — as a linear coefficient in $\alpha_s/\alpha_\text{em}$ and as an exponent in $\alpha_G$ — are unified by two non-equivalent readouts of the formal analytic generator $Z(t)$. The pure $S^3$ geometric constant $c = \pi/(3\sqrt{2})$ simultaneously controls the leading-order corrections to both the mass-level ratio and the coupling-constant ratio, with no new fitting parameter.

The complete physical dictionary of $Z(t)$ — deriving from DD structure the precise assignment of union and coincidence readouts — as well as the generating-function-level derivation of $3/13$, are explicit next steps.

Appendix A: Numerical Verification Table

Using $\alpha_\text{em}(0)^-1 = 137.035999177$ (CODATA 2022), $\alpha_\text{em}(M_Z)^-1 = 127.930$ ($\overline{\text{MS}}$, PDG), $\alpha_s(M_Z) = 0.1180 \pm 0.0009$ (PDG 2025), $c = \pi/(3\sqrt{2}) = 0.740480$.

$\alpha_s$ predictions

Formula	Prediction	Deviation	Deviation ($\sigma$)
$(65/4)\,\alpha_\text{em}(0)$	$0.11858$	$+0.49\%$	$+0.65$
$(65/4)\,\alpha_\text{em}(0)(1-c\,\alpha_\text{em}(0))$	$0.11794$	$-0.05\%$	$-0.07$
$(65/4)\,\alpha_\text{em}(M_Z)$	$0.12702$	$+7.65\%$	$+10.0$
$(65/4)\,\alpha_\text{em}(M_Z)(1-c\,\alpha_\text{em}(M_Z))$	$0.12633$	$+7.06\%$	$+9.3$

Best fit: $\alpha_\text{em}(0)$ with $c$ correction, deviation $0.07\sigma$.

$\alpha_G$ logarithmic ratio

Quantity	Value
$\alpha_G(M_Z) = (M_Z/M_P)^2$	$5.579 \times 10^-35$
$\ln\alpha_G(M_Z)/\ln\alpha_\text{em}(M_Z)$	$16.2572$
$65/4$	$16.2500$
Deviation	$0.044\%$

$\sin^2\theta_W$

Quantity	Value
$3/13$	$0.23077$
Experimental ($\overline{\text{MS}}$, $M_Z$)	$0.23122 \pm 0.00004$
Deviation	$0.20\%$

Dual appearance of $c$

Quantity	Leading-order deviation	Corrected deviation	Sign of correction
$R_1 = m_\mu/m_e$	$-0.59\%$	$-0.05\%$	$+c$
$\alpha_s/\alpha_\text{em}$	$+0.49\%$	$-0.05\%$	$-c$

Appendix B: $S^3$ Closest Legal Packing

Six 4DD blocks on $S^3$ form three pairs, each pair arranged antipodally along one axis. Under the $\delta_1 = \delta_2 = \delta_3$ symmetry ansatz, the closest-legal-packing problem has threefold symmetry. The geometric constraint imposed by each block's time arrow on the other blocks yields the nearest legal configuration, with leading-order displacement

$$c = \frac{\pi}{3\sqrt{2}} = 0.740480\ldots$$

As a heuristic reading, the factor structure of $c$ may reflect the curvature of $S^3$ ($\pi$), the threefold symmetry ($3$), and the diagonal factor between orthogonal directions on $S^3$ ($\sqrt{2}$), but this decomposition is interpretive rather than rigorously derived.

In $R_1 = m_\mu/m_e$, $c$ appears with positive sign (packing widens mass-level separation):

$$R_1 = \frac{3}{2\alpha_\text{em}}(1 + c\,\alpha_\text{em}) = 206.665 \quad\text{vs}\quad 206.768\text{ (experiment)}$$

In $\alpha_s/\alpha_\text{em}$, $c$ appears with negative sign (packing tightens effective channels):

$$\alpha_s = \frac{65}{4}\,\alpha_\text{em}(1 - c\,\alpha_\text{em}) = 0.11794 \quad\text{vs}\quad 0.1180\text{ (experiment)}$$

Sign flip: the same $S^3$ packing adjustment simultaneously increases separation-type quantities ($+c$) and decreases overlap-type quantities ($-c$).

SAE Four Forces Series, Paper V.

Citations to earlier papers in this series: Prequel DOI: 10.5281/zenodo.19341042; Paper I DOI: 10.5281/zenodo.19342106; Paper II DOI: 10.5281/zenodo.19360101; Paper III DOI: 10.5281/zenodo.19379412; Paper IV DOI: 10.5281/zenodo.19411672; Generation Paper DOI: 10.5281/zenodo.19394500.

写作声明：本文由秦汉独立写作。所有智识决策、框架设计与编辑判断均由作者本人作出。

秦汉 (Han Qin)

ORCID: 0009-0009-9583-0018

摘要

本文将SAE四力系列的已发表结果——Weinberg角 $\sin^2\theta_W = 3/13$（篇III）和引力耦合指数 $\ln\alpha_G/\ln\alpha_\text{em} = 65/4$（前置篇）——与一条新的structured conjecture $\alpha_s = (65/4)\,\alpha_\text{em}(1 - c\,\alpha_\text{em})$ 收入同一个计数框架。三条耦合关系由DD组合数 $\{4, 13, 15, 65\}$ 与 $S^3$ 几何常数 $c = \pi/(3\sqrt{2})$ 共同组织，不引入新的连续拟合参数；在当前gauge-coupling子包内，$\alpha_\text{em}$ 是唯一的连续输入。

本文的核心新结果有三个。第一，$65/4$ 在 $\alpha_s/\alpha_\text{em}$ 中作为线性系数，在 $\alpha_G$ 中作为指数，两次出现可由同一个formal analytic generator $Z(t) = e^{\kappa\ln(1-\alpha+\alpha t)}$（$\kappa = 65/4$）的两种非等价读出统一：低阶union读出给出线性关系，full coincidence读出给出指数关系。第二，纯 $S^3$ 几何常数 $c = \pi/(3\sqrt{2})$ 同时出现在muon-electron质量比 $R_1$ 的修正（符号为正）和 $\alpha_s$ 的修正（符号为负）中，未引入新的拟合参数，将 $\alpha_s$ 的偏差从 $+0.49\%$ 降至 $-0.05\%$。第三，$\sin^2\theta_W = 3/13$ 作为已发表的DD本征值定理，在本文框架中对应 $65/4$ 的内部因式分解 $(13/3) \times (15/4)$ 的第一个因子的倒数。$3/13$ 的生成函数级读出仍待 $Z(t)$ 的多变量扩展，是明确的开放问题。

§1 引言

SAE四力系列的前四篇建立了从DD（Dimensional Degree）结构到规范群的对应关系。篇I证明nDD在两条SAE公理下结构对应SU(n)（DOI: 10.5281/zenodo.19342106）。篇II从三条先验出发导出一代费米子的完整超荷表，零连续自由参数（DOI: 10.5281/zenodo.19360101）。篇III在EW Matching Axiom下推出 $\sin^2\theta_W = 3/13$ 为DD本征值（DOI: 10.5281/zenodo.19379412）。篇IV从单侧6个4DD block出发建立 $\mathfrak{so}(6) \cong \mathfrak{su}(4)$ 的color-lepton骨架（DOI: 10.5281/zenodo.19411672）。前置篇通过DD Splitting推出 $\alpha_G = \alpha_\text{em}^65/4$（DOI: 10.5281/zenodo.19341042）。

这些结果已经确定了"有哪些力"以及"力的代数结构是什么"。本篇的问题是：每条力有多强？

答案的核心思路异常简朴：数数。 从DD结构中自然产生的一组组合数 $\{4, 13, 15, 65\}$，加上 $S^3$ closest legal packing给出的几何常数 $c = \pi/(3\sqrt{2})$，以不同的方式组合，给出全部耦合常数的比值关系。不需要新的连续拟合参数。结构是定理或structured conjecture，唯一的连续输入是 $\alpha_\text{em}$ 本身。

本篇的三个核心新结果是：（1）$65/4$ 的线性和指数两次出现由同一个formal analytic generator $Z(t)$ 的两种读出统一；（2）$S^3$ closest legal packing几何常数 $c = \pi/(3\sqrt{2})$ 同时控制质量层级比和耦合常数比的一阶修正；（3）全部耦合关系收入 $\{4, 13, 15, 65\}$ 与 $c$ 的统一计数框架。

§2 DD计数基础

§2.1 $(N,d) = (12,4)$ 的来源

凿（否定）是唯一的先验操作。否定一次产生二分（有/无），涌现层再产生一个二分（涌现/不涌现），给出

$$d = 2 \times 2 = 4$$

这是每一轮公理链的步数，也是时空维度。$d = 4$ 对应的紧致方向是 $S^3$。$S^3$ 上的closest legal packing给出三对对径方向，每侧六个4DD block，两侧共十二个：

$$N = 2 \times 3 \times 2 = 12 = 4(d-1)$$

底层只有一个参数 $d = 4$，$N$ 是它的推论。详见前置篇。

§2.2 四个计数常数的身份

从 $(N, d) = (12, 4)$ 出发，以下组合数自然产生：

符号	数值	来源	身份
$d$	4	凿的逻辑	时空维度，公理链步数
$C(N/2, 2)$	15	单侧6个block的pair计数	$\dim\,\mathfrak{so}(6) = \dim\,\mathfrak{su}(4)$，单侧pair代数维数
$C(N, 2) - 1$	65	12个block的全pair计数减去平凡元素	$\dim(\Lambda^2 V₁₂) - 1$，全关系代数的非平凡部分
$65/5$	13	色坍缩后的doublet通道数	DD本征值，出现在 $\sin^2\theta_W = 3/13$ 中

这四个数不是输入，是否定链式反应的输出。全部耦合关系将由这些组合数与 $S^3$ 几何常数 $c$ 共同组织。

§2.3 全关系代数的分解

将12个block视为实向量空间 $V₁₂ = V_L \oplus V_R$，$\dim V_L = \dim V_R = 6$，其bivector空间自然分解为

$$\Lambda^2 V₁₂ = \Lambda^2 V_L \oplus \Lambda^2 V_R \oplus (V_L \otimes V_R)$$

$$66 = 15 + 15 + 36$$

$\Lambda^2 V_L = \mathfrak{so}(6)$ 是单侧pair代数（篇IV）。$65 = 66 - 1$ 是去掉平凡元素后的全关系代数维数。这个分解将在 §4 中用于解释 $65/4$ 的生成函数含义。

§3 三条耦合关系

§3.1 $\sin^2\theta_W = 3/13$（回顾）

篇III在EW Matching Axiom下推出Weinberg角为DD本征值：

$$\sin^2\theta_W = \frac{3}{13} = 0.23077\ldots$$

实验值（$\overline{\text{MS}}$, $M_Z$标度）为 $0.23122 \pm 0.00004$，偏差约 $0.20\%$。

$3/13$ 的组合来源：13个doublet通道中有3个泄漏到 $U(1)_Y$ 方向。15个单侧通道中有5个色轨道（世代篇, DOI: 10.5281/zenodo.19394500），$65/5 = 13$。

此结果是纯DD结构定理，不含连续参数。详见篇III。

§3.2 $\ln\alpha_G / \ln\alpha_\text{em} = 65/4$（回顾）

前置篇通过DD Splitting推出引力耦合与电磁耦合的对数比值：

$$\frac{\ln\alpha_G(M_Z)}{\ln\alpha_\text{em}(M_Z)} = \frac{65}{4}$$

其中 $\alpha_G(M_Z) = (M_Z/M_P)^2$，$\alpha_\text{em}(M_Z) = 1/127.930$（$\overline{\text{MS}}$）。

实测比值 $16.2572$，与 $65/4 = 16.25$ 的偏差为 $0.044\%$。

关键：两侧在同一能标 $M_Z$ 评估。$65 = \dim(\Lambda^2 V₁₂) - 1$（全关系代数），$4 = d$（时空维度）。

此结果是DD定理。详见前置篇。

§3.3 $\alpha_s = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$（新结果）

这是本篇的核心新内容。

§3.3.1 Leading order

$$\frac{\alpha_s(M_Z)}{\alpha_\text{em}(0)} = \frac{65}{4} = 16.25$$

数值：$(65/4) \times (1/137.036) = 0.11858$。实验值 $\alpha_s(M_Z) = 0.1180 \pm 0.0009$（PDG 2025），偏差 $+0.49\%$（约 $0.65\sigma$）。

注意能标的差异：$\alpha_s$ 在 $M_Z$ 处评估，$\alpha_\text{em}$ 取Thomson极限（$q^2 \to 0$）。在SAE的冻结能标图像下，各DD层冻结在各自的特征能标——1DD（电磁）在 $q^2 \to 0$ 处保持原生值，3DD（强力）在 $M_Z$ 附近冻结。由此观之，跨层比值天然涉及不同能标。此读法目前仍是working interpretation，严格的DD推导是开放问题。

§3.3.2 Leading-order系数的内部因式分解

$65/4$ 可以精确分解为

$$\frac{65}{4} = \frac{13}{3} \times \frac{15}{4}$$

第一个因子 $13/3$ 直接来自 $\sin^2\theta_W = 3/13$：由 $\sin^2\theta_W = \alpha_\text{em}/\alpha_2$，得

$$\frac{\alpha_2}{\alpha_\text{em}} = \frac{1}{\sin^2\theta_W} = \frac{13}{3}$$

第二个因子 $15/4$ 是单侧pair代数维数（$15 = \dim\,\mathfrak{so}(6)$）除以母代数fundamental表示维数（$4 = \dim\,\mathbf{4}_\mathfrak{su(4)}$）。

需要强调：这是leading-order一阶系数的代数因式分解，不是时间或能量尺度上的动力学级联演化。$\alpha_\text{em}$、$\alpha_2$、$\alpha_s$ 之间不存在"先变成弱力再变成强力"的过程。这个分解展示的是全局 $65/4$ 的组合结构如何由电弱混合（$13/3$）和强-弱视界结构（$15/4$）的乘积构成。

§3.3.3 $S^3$ Closest Legal Packing修正

$S^3$ 上六个4DD block的closest legal packing问题（见附录B）给出纯几何常数

$$c = \frac{\pi}{3\sqrt{2}} = 0.740480\ldots$$

此常数最初出现在muon-electron质量比的一阶修正中：

$$R_1 = \frac{3}{2\alpha_\text{em}}(1 + c\,\alpha_\text{em})$$

其中leading order $3/(2\alpha_\text{em}) = 205.554$ vs 实验值 $R_1^\text{exp} = 206.768$，偏差 $-0.59\%$；修正后 $206.665$，残差 $-0.05\%$。

同一个 $c$ 以相反的符号出现在 $\alpha_s$ 的修正中：

$$\alpha_s = \frac{65}{4}\,\alpha_\text{em}(0)\!\left(1 - c\,\alpha_\text{em}(0)\right) = 0.11794$$

相对实验值 $0.1180$ 的偏差为 $-0.05\%$（约 $0.07\sigma$）。

这里没有新增拟合参数。$c$ 不是为 $\alpha_s$ 拟合的，是从 $S^3$ 几何独立求解的。符号翻转在 $S^3$ packing的框架下可以理解：packing调整同时增大block间的间距（$R_1$ 的正修正，撑开质量层级）和减小block间的重叠（$\alpha_s$ 的负修正，收紧有效通道数）。这两个效应是同一个几何约束的两面。

目前 $c$ 在 $\alpha_s$ 中出现的地位是"几何上有动机的posterior结果"：来源和 $R_1$ 相同，未新增参数，数值精确，但尚缺独立的几何推导证明 $\alpha_s$ 的一阶修正系数必须等于同一个 $c$。此推导是开放问题。

§4 Formal Analytic Generator $Z(t)$

§4.1 定义

定义

$$K(t) = \kappa \ln(1 - \alpha + \alpha\, t), \qquad Z(t) = e^K(t)$$

其中 $\kappa = 65/4$，$\alpha = \alpha_\text{em}$。

$\kappa$ 是effective channel density，即全关系代数的非平凡维数（65）在时空维度（4）上的归一化密度。它不是literal channel count——$65/4 = 16.25$ 不是整数，$Z(t)$ 不是标准的概率生成函数（PGF）。$K(t)$ 的对数形式使非整数 $\kappa$ 显得自然：$\kappa$ 是连续的密度/强度参数，不是离散个数。

§4.2 两种非等价读出

从同一个 $Z(t)$，可以定义两个formal readout：

低阶union读出 $W_\text{OR}$：

$$W_\text{OR} = 1 - Z(0) = 1 - (1-\alpha)^\kappa$$

在小 $\alpha$ 下展开：

$$W_\text{OR} = \kappa\,\alpha + O(\alpha^2)$$

一阶系数 $\kappa = 65/4$ 直接给出 $\alpha_s/\alpha_\text{em}$ 的leading order。

Full coincidence读出 $W_\text{AND}$：

$$W_\text{AND} = e^{\kappa\ln\alpha} = \alpha^\kappa$$

$\kappa$ 出现在指数位置，给出 $\alpha_G = \alpha_\text{em}^65/4$。

$W_\text{OR}$ 和 $W_\text{AND}$ 是formal readouts，不是已建立的字面概率模型。两者共享同一个 $\kappa$-generator模板，但不是同一个ordinary evaluation functional的两个值：$W_\text{OR} = 1 - Z(0)$ 是 $Z(t)$ 在 $t=0$ 处的直接evaluation；$W_\text{AND} = \alpha^\kappa$ 是formal coincidence weight，它与 $Z(t)$ 共享指数 $\kappa$，但不是通过对 $Z$ 在某个点取值而得到的。它们展示的是：同一个结构常数 $\kappa$ 在同一个解析模板中既能以一阶系数出现，又能以指数出现。 以前"线性 vs 指数"只是叙事；现在有了共同的数学模板。

§4.3 物理根据与Precedent

$W_\text{OR}$ 对应步1-3的力（电磁、弱、强），它们是单体属性力。一个粒子带有某种荷，任何一个相关通道激活就产生力。读出的统计模式是union-like的。

$W_\text{AND}$ 对应步4的力（引力），它是双体关系力。两个主体的全部通道同时match才构成引力关系。读出的统计模式是full-coincidence-like的。

从DD结构严格推出"属性→union，关系→coincidence"仍是开放问题。目前最强的物理依据是守恒论证：步1-3是单体属性，不要求全分量守恒，任一通道有贡献即可；步4是双体关系，守恒方程 $\partial_\mu T^\mu\nu = 0$ 要求每个分量同时满足，不是"至少一个分量守恒就行"。

标准物理中存在密切相关的precedent。在量子电动力学（QED）中，单个软光子的发射概率正比于 $\alpha$（线性，union-like：这个光子或那个光子）。但弹性散射要求全部软光子通道同时保持沉默，高阶图的重组给出Sudakov form factor $e^{-c\cdot\alpha}$（指数，coincidence-like：所有通道同时）。同一个耦合常数在局域微扰下做线性系数，在全局约束下指数化，是场论中已知的现象。本文借用"线性局域读出/指数化全局约束"的形式先例，但不声称与Sudakov resummation有一一对应。

§4.3.1 Leading-order因式分解的含义

§3.3.2的因式分解 $65/4 = (13/3) \times (15/4)$ 在 $Z(t)$ 的框架下有清晰的含义：它是 $W_\text{OR}$ 的一阶系数 $\kappa$ 的内部代数分解。$W_\text{OR}$ 的一阶展开只取决于 $\kappa$ 的值，而 $\kappa$ 的因式分解反映了全关系代数从电弱到强的组合结构。

这不是 $Z(t)$ 的精确层级分解：一般地，$1 - (1-\alpha)^ab \neq$ "先做 $a$ 再做 $b$ 的union组合"。可以分解的是系数 $ab$，不是 $W_\text{OR}$ 本身。

§4.4 Scale结构（working interpretation）

以下对scale差异的解释是当前的working interpretation，不是已完成的DD推导。

$\alpha_G(M_Z) = \alpha_\text{em}(M_Z)^65/4$：full coincidence读出。两侧在同一能标 $M_Z$ 评估。偏差 $0.044\%$。

$\alpha_s(M_Z) = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$：union读出的一阶系数加修正。起点（1DD, $q^2 \to 0$）和终点（3DD, $M_Z$）在不同能标。偏差 $0.05\%$。

在当前的working interpretation下，这个差异有结构性的来源：full coincidence读出是全局的，全部通道在同一时刻同一能标match，自然在单一能标评估。Union读出是跨层的，从1DD经2DD到3DD，每一层冻结在各自的特征能标，比值天然涉及不同能标。

更严格的DD推导——特别是精确解释为什么 $\alpha_\text{em}$ 应取Thomson极限而 $\alpha_s$ 应取 $M_Z$ 值——仍是开放问题。

§5 对称性Ansatz与色精确对称

§5.1 $\delta_1 = \delta_2 = \delta_3$

同侧三对4DD block的频率比（$\delta_1, \delta_2, \delta_3$）可能相同。这是一个对称性ansatz，不是定理。

动机来自big crunch对称性：宇宙坍缩到reset点时，所有DD层级已经坍缩，三对4DD block完全不可区分。如果频率比在轴标号出现之前就被固定，且之后不再发生轴依赖的renormalization，则 $\delta_1 = \delta_2 = \delta_3$。

此ansatz compatible with实验中色对称 $SU(3)_c$ 从未被观测到破缺的事实。

此ansatz的隐含假设是频率不对称在3DD展开之前就已确定。如果频率比可以在3DD展开之后通过自发对称破缺获得轴依赖性，则 $\delta_1 = \delta_2 = \delta_3$ 不是必然的。

§5.2 对 $S^3$ Packing的影响

在 $\delta_1 = \delta_2 = \delta_3$ 的ansatz下，$S^3$ 上六个block的closest legal packing问题具有三重对称。packing的对称branch给出唯一的leading-order几何常数 $c = \pi/(3\sqrt{2})$。

没有此ansatz，packing问题允许轴依赖的修正，$c$ 可能带有轴编号的微扰项。在对称ansatz下，$c$ 是精确的leading-order值。

§5.3 Gauge-coupling子包的唯一连续输入

在本篇讨论的gauge-coupling子包中：

$\sin^2\theta_W = 3/13$ 是纯DD结构定理，不含连续参数
$\alpha_G = \alpha_\text{em}^65/4$ 由 $\alpha_\text{em}$ 唯一确定
$\alpha_s = (65/4)\,\alpha_\text{em}(1 - c\,\alpha_\text{em})$ 由 $\alpha_\text{em}$ 唯一确定

因此，在此子包内，$\alpha_\text{em}$ 是唯一的未定连续输入。所有结构关系由DD组合数 $\{4, 13, 15, 65\}$ 和 $S^3$ 几何常数 $c$ 确定。

本篇不claim "整个物理宇宙只有一个自由参数"。Higgs真空期望值 $v$、个体费米子质量、CKM/PMNS矩阵参数等是否也能压到 $\alpha_\text{em}$ 上，是后续工作。

§6 讨论

§6.1 结果层次

定理级（已发表）：

关系	偏差	来源
$\sin^2\theta_W = 3/13$	$0.20\%$	篇III
$\ln\alpha_G/\ln\alpha_\text{em} = 65/4$	$0.044\%$	前置篇
$(N,d) = (12,4)$ 唯一性	—	前置篇

Structured conjecture（本篇）：

关系	偏差	支撑
$\alpha_s = (65/4)\alpha_\text{em}(1-c\alpha_\text{em})$	$0.05\%$	$Z(t)$ 模板, $c$ 双重出现

强对称性ansatz（本篇）：

ansatz	与实验一致性
$\delta_1 = \delta_2 = \delta_3$	SU(3)_c 精确对称从未被破缺

§6.2 向Anti-GUT读法的展望

本文的结果提供了对传统大统一叙事的一种方法论挑战。

传统GUT假设四种力在某一极高能标处汇聚为同一种力，由单一李代数描述。本文的 $Z(t)$ 结构提示一种不同的图像：规范力（$W_\text{OR}$, union-like）和引力（$W_\text{AND}$, coincidence-like）可能是同一个生成对象的两种非等价读出。李代数是线性空间，其运算是加法的；如果引力的耦合结构本质上是指数的（coincidence-like），则单一李代数可能不足以穷尽"力的生成规则"。

这不是本文已完成的反证。$W_\text{OR}$ 和 $W_\text{AND}$ 的物理归属——为什么单体属性力对应union读出，为什么双体关系力对应coincidence读出——仍是开放问题。如果未来的工作能从DD结构严格推出这一归属，则anti-GUT将从方法论挑战升级为结构性论证。

§6.3 可检验预测

本篇最直接的可检验预测是 $\alpha_s$ 的修正值：

$$\alpha_s^\text{pred} = \frac{65}{4} \times \frac{1}{137.036} \times \left(1 - \frac{\pi}{3\sqrt{2}} \times \frac{1}{137.036}\right) = 0.11794$$

当前PDG世界平均值 $\alpha_s(M_Z) = 0.1180 \pm 0.0009$，偏差在 $0.07\sigma$ 以内。需注意当前误差条（$\pm 0.76\%$）仍然较宽，$0.07\sigma$ 更多是encouraging的中心值重合，尚不构成强约束。未来lattice QCD或高精度 $\alpha_s$ 测量可以检验此预测。如果 $\alpha_s$ 的精确值偏离此预测超过数个标准差，conjecture被证伪。

§6.4 开放问题

OR→AND的严格DD推导。 从DD结构推出"属性力对应union读出，关系力对应coincidence读出"。
$Z(t)$ 多变量扩展。 构造 $Z(t_W, t_Y)$ 以从通道分割中读出 $\sin^2\theta_W = 3/13$。
$c$ 在 $\alpha_s$ 中出现的独立几何推导。 证明 $\alpha_s$ 的一阶修正系数必须等于 $R_1$ 的packing常数 $c$。
$0.05\%$ 残差的来源。 $R_1$ 和 $\alpha_s$ 的残差都在 $0.05\%$ 附近。可能的来源：(a)同pair内部的non-exact antipodal配置（与parity violation相关），(b)两个3DD之间的左右非对称（$T_1 \neq T_2$）。
$v = 246$ GeV。 如果 $\alpha_s$ conjecture确认，$v/\Lambda_\text{QCD}$ 的推导链条可以继续。

§7 结论

本篇将SAE四力系列的两条已发表定理（$\sin^2\theta_W = 3/13$, $\alpha_G = \alpha_\text{em}^65/4$）和一条新的structured conjecture（$\alpha_s = (65/4)\alpha_\text{em}(1 - c\alpha_\text{em})$）收入同一个计数框架。三条关系由DD组合数 $\{4, 13, 15, 65\}$ 与 $S^3$ 几何常数 $c = \pi/(3\sqrt{2})$ 共同组织。

$65/4$ 的两次出现——在 $\alpha_s/\alpha_\text{em}$ 中作为线性系数，在 $\alpha_G$ 中作为指数——由formal analytic generator $Z(t)$ 的两种非等价读出统一。纯 $S^3$ 几何常数 $c = \pi/(3\sqrt{2})$ 同时控制质量层级比和耦合常数比的一阶修正，未引入新的拟合参数。

$Z(t)$ 的完整物理字典——从DD结构推出union和coincidence读出的精确归属——以及 $3/13$ 的生成函数级读出，是明确的下一步工作。

附录A 数值验证表

取 $\alpha_\text{em}(0)^-1 = 137.035999177$（CODATA 2022），$\alpha_\text{em}(M_Z)^-1 = 127.930$（$\overline{\text{MS}}$, PDG），$\alpha_s(M_Z) = 0.1180 \pm 0.0009$（PDG 2025），$c = \pi/(3\sqrt{2}) = 0.740480$。

$\alpha_s$ 预测值

公式	预测值	偏差	偏差($\sigma$)
$(65/4)\,\alpha_\text{em}(0)$	$0.11858$	$+0.49\%$	$+0.65$
$(65/4)\,\alpha_\text{em}(0)(1-c\,\alpha_\text{em}(0))$	$0.11794$	$-0.05\%$	$-0.07$
$(65/4)\,\alpha_\text{em}(M_Z)$	$0.12702$	$+7.65\%$	$+10.0$
$(65/4)\,\alpha_\text{em}(M_Z)(1-c\,\alpha_\text{em}(M_Z))$	$0.12633$	$+7.06\%$	$+9.3$

最佳吻合：$\alpha_\text{em}(0)$ 加 $c$ 修正，偏差 $0.07\sigma$。

$\alpha_G$ 对数比值

量	值
$\alpha_G(M_Z) = (M_Z/M_P)^2$	$5.579 \times 10^-35$
$\ln\alpha_G(M_Z)/\ln\alpha_\text{em}(M_Z)$	$16.2572$
$65/4$	$16.2500$
偏差	$0.044\%$

$\sin^2\theta_W$

量	值
$3/13$	$0.23077$
实验值（$\overline{\text{MS}}$, $M_Z$）	$0.23122 \pm 0.00004$
偏差	$0.20\%$

$c$ 的双重出现

量	Leading order偏差	修正后偏差	修正符号
$R_1 = m_\mu/m_e$	$-0.59\%$	$-0.05\%$	$+c$
$\alpha_s/\alpha_\text{em}$	$+0.49\%$	$-0.05\%$	$-c$

附录B $S^3$ Closest Legal Packing

$S^3$ 上六个4DD block构成三对，每对沿一个轴方向对径排列。在 $\delta_1 = \delta_2 = \delta_3$ 的对称ansatz下，closest legal packing问题具有三重对称。每个block的时间箭头与其他block的几何约束给出最近合法配置，其leading-order偏离量为

$$c = \frac{\pi}{3\sqrt{2}} = 0.740480\ldots$$

作为heuristic reading，$c$ 的因式结构可能反映 $S^3$ 曲率（$\pi$）、三重对称（$3$）和正交方向对角因子（$\sqrt{2}$）的几何含义，但此分解目前是解释性的，不是严格推导。

在 $R_1 = m_\mu/m_e$ 中，$c$ 以正号出现（packing撑开质量间距）：

$$R_1 = \frac{3}{2\alpha_\text{em}}(1 + c\,\alpha_\text{em}) = 206.665 \quad\text{vs}\quad 206.768\text{（实验）}$$

在 $\alpha_s/\alpha_\text{em}$ 中，$c$ 以负号出现（packing收紧有效通道）：

$$\alpha_s = \frac{65}{4}\,\alpha_\text{em}(1 - c\,\alpha_\text{em}) = 0.11794 \quad\text{vs}\quad 0.1180\text{（实验）}$$

符号翻转：$S^3$ packing的同一个调整同时增大separation类量（$+c$）和减小overlap类量（$-c$）。

SAE四力系列第五篇。

引用本系列前述论文：前置篇 DOI: 10.5281/zenodo.19341042; 篇I DOI: 10.5281/zenodo.19342106; 篇II DOI: 10.5281/zenodo.19360101; 篇III DOI: 10.5281/zenodo.19379412; 篇IV DOI: 10.5281/zenodo.19411672; 世代篇 DOI: 10.5281/zenodo.19394500。

Four Forces Paper V: The Counting Origin of Coupling Constants and the Generating-Function Unification of 65/4

§1 Introduction

§2 DD Counting Foundations

§2.1 The origin of $(N,d) = (12,4)$

§2.2 Identities of the four counting constants

§2.3 Decomposition of the full relation algebra

§3 Three Coupling Relations

§3.1 $\sin^2\theta_W = 3/13$ (review)

§3.2 $\ln\alpha_G / \ln\alpha\text{em} = 65/4$ (review)

§3.3 $\alpha_s = (65/4)\,\alpha\text{em}(0)(1 - c\,\alpha\text{em}(0))$ (new result)

§3.3.1 Leading order

§3.3.2 Internal factorization of the leading-order coefficient

§3.3.3 $S^3$ closest-legal-packing correction

§4 Formal Analytic Generator $Z(t)$

§4.1 Definition

§4.2 Two non-equivalent readouts

§4.3 Physical grounds and precedent

§4.3.1 Meaning of the leading-order factorization

§4.4 Scale structure (working interpretation)

§5 Symmetry Ansatz and Exact Color Symmetry

§5.1 $\delta_1 = \delta_2 = \delta_3$

§5.2 Implications for $S^3$ packing

§5.3 Sole continuous input within the gauge-coupling subpackage

§6 Discussion

§6.1 Hierarchy of results

§6.2 Toward an anti-GUT reading

§6.3 Testable prediction

§6.4 Open problems

§7 Conclusion

Appendix A: Numerical Verification Table

$\alpha_s$ predictions

$\alpha_G$ logarithmic ratio

$\sin^2\theta_W$

Dual appearance of $c$

Appendix B: $S^3$ Closest Legal Packing

§1 引言

§2 DD计数基础

§2.1 $(N,d) = (12,4)$ 的来源

§2.2 四个计数常数的身份

§2.3 全关系代数的分解

§3 三条耦合关系

§3.1 $\sin^2\theta_W = 3/13$（回顾）

§3.2 $\ln\alpha_G / \ln\alpha\text{em} = 65/4$（回顾）

§3.3 $\alpha_s = (65/4)\,\alpha\text{em}(0)(1 - c\,\alpha\text{em}(0))$（新结果）

§3.3.1 Leading order

§3.3.2 Leading-order系数的内部因式分解

§3.3.3 $S^3$ Closest Legal Packing修正

§4 Formal Analytic Generator $Z(t)$

§4.1 定义

§4.2 两种非等价读出

§4.3 物理根据与Precedent

§4.3.1 Leading-order因式分解的含义

§4.4 Scale结构（working interpretation）

§5 对称性Ansatz与色精确对称

§5.1 $\delta_1 = \delta_2 = \delta_3$

§5.2 对 $S^3$ Packing的影响

§5.3 Gauge-coupling子包的唯一连续输入

§6 讨论

§6.1 结果层次

§6.2 向Anti-GUT读法的展望

§6.3 可检验预测

§6.4 开放问题

§7 结论

附录A 数值验证表

$\alpha_s$ 预测值

$\alpha_G$ 对数比值

$\sin^2\theta_W$

$c$ 的双重出现

附录B $S^3$ Closest Legal Packing

§3.2 $\ln\alpha_G / \ln\alpha_\text{em} = 65/4$ (review)

§3.3 $\alpha_s = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$ (new result)

§3.2 $\ln\alpha_G / \ln\alpha_\text{em} = 65/4$（回顾）

§3.3 $\alpha_s = (65/4)\,\alpha_\text{em}(0)(1 - c\,\alpha_\text{em}(0))$（新结果）