Light Speed, Dimensional Breakthrough, and the Coupling Constant Hierarchy
SAE Physics Series — Four Forces Prequel

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Light Speed, Dimensional Breakthrough, and the Coupling Constant Hierarchy

SAE Physics Series — Four Forces Prequel

Han Qin (秦汉)

Independent Researcher

ORCID: 0009-0009-9583-0018

2026

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Statement: This paper, based on the Self-as-an-End (SAE) framework, derives three results from SAE axioms. First, the DD Breakthrough theorem: the speed of light c is the causal limit defined at 1DD's birth; remainder development reaching c triggers breakthrough to the next DD; the characteristic quantity of (n+1)DD is E/c^n. Second, the DD Splitting theorem: 1DD (point) cannot split; 2DD (line) splits symmetrically into left-handed and right-handed 3DD; each 3DD splits along three spatial axes into dual 4DDs; yielding 12 4DDs total. Third, from SO(12) = 66 generators of the 12 4DDs' internal symmetry, modulo SO(2) (the splitting phase redundancy of 2DD), the coset space SO(12)/SO(2) has dimension 65. Normalized by 4 spacetime dimensions, the coupling constant relation α_G(M_Z) = α_em(M_Z)^{16.25} follows, matching observation to 0.044%. All forms of falsification are welcome.

Firewall: The structural predictions Λ = 2(ω₂² − ω₁²)/c² (Cosmo Paper I) and a₀ = (π/2)·c(ω₂ − ω₁) (Cosmo Paper III) do not depend on any result in this paper.

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1. Abstract

Standard physics accepts the speed of light as an empirical constant and offers no structural explanation for the 34-order-of-magnitude gap between the electromagnetic and gravitational coupling constants. This paper derives both from SAE axioms.

The argument proceeds in three stages.

First, the DD Breakthrough theorem (§2–3): chaos (0DD) must develop (SAE axiom), creating distinction (1DD) and simultaneously defining c. Within each DD, the remainder develops until it reaches c, then breaks through to the next DD. Each breakthrough divides the remainder's potency by c. The characteristic quantity of (n+1)DD is E/c^n: energy E (1DD), momentum p = E/c (2DD), mass m = E/c² (3DD), and the pair-production parameter m/c = E/c³ (4DD, intelligible but not directly measurable). c is universal because all DDs share one spacetime, and finite because 4DD is only half of a dual structure.

Second, the DD Splitting theorem (§4): a point (1DD) has no internal symmetry and cannot split. A line (2DD) embedded in higher-dimensional space has chirality — it can be wound left or right. Therefore 2DD splits symmetrically into left-handed and right-handed 3DD, which is the geometric origin of parity. Each 3DD has three spatial axes; each axis produces a dual 4DD (causality + reverse-causality) upon breakthrough. This yields 2 × 3 × 2 = 12 4DDs total. Our observable universe is one side of one dual pair within the left-handed 3DD.

Third, the Coupling Constant relation (§5): the 12 4DDs, arising from continuous breakthroughs with non-overlapping light cones, span an internal Euclidean space R^12 with symmetry group SO(12) (66 generators). The 2DD splitting imparts a common phase to all dual-4DD planes; this SO(2) redundancy is quotiented out, yielding the coset space SO(12)/SO(2) of dimension 65. Normalized by the 4 spacetime dimensions, the internal complexity density is κ = 65/4 = 16.25. The postulate that κ determines the electromagnetic-gravitational coupling hierarchy predicts:

α_G(M_Z) = α_em(M_Z)^{16.25}

Observation: |ln α_G(M_Z)| / |ln α_em(M_Z)| = 16.2572. Deviation: 0.044%, within experimental uncertainty.

Terminology: DD = Dimension Degree. Full definitions: SAE Methodological Overview (DOI: 10.5281/zenodo.18842449).

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2. The Birth of c

2.1 0DD: Chaos

0DD is undifferentiated chaos. No space, no time, no distinction, no concept of speed.

2.2 0DD → 1DD: Distinction is born

The SAE axiom "remainder must develop" forces chaos to differentiate. The first differentiation — separation of subject and object — is 1DD. This simultaneously creates spatial extension, temporal sequence, and the possibility of propagation. The maximum rate of propagation is c.

c is not imported from outside. It is defined at the moment 1DD is born.

2.3 c is universal and finite

Universal: all DDs from 1DD to 4DD are built within the same spacetime. One spacetime, one causal limit. c is not a property of electromagnetism; it is a property of the shared spacetime born at 0DD→1DD.

Finite: 4DD splits into dual 4DDs with opposite time arrows (Cosmo Paper V, DOI: 10.5281/zenodo.19329771). We observe only one side. Complete dual-4DD = 0DD = no time = speed concept not applicable. Half of 4DD = time exists but is one-directional = c is finite. c and t are born as a pair from the 4DD split.

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3. The DD Breakthrough Theorem

3.1 Mechanism

"Remainder must develop" does not stop at 1DD. Within each DD, the remainder continues to develop. When its development speed reaches c — the limit of the shared spacetime — the remainder breaks through into the next DD.

3.2 Why each breakthrough divides by c (dimensional postulate)

At breakthrough, the remainder's entire developmental velocity is at c. The SAE scaling postulate is: each breakthrough divides the remainder's potency by c, the causal limit. This is dimensionally consistent — [E/c] = momentum, [E/c²] = mass — and empirically anchored by the two known relativistic identities E = pc and E = mc². The postulate extends this pattern to the full DD sequence. It is not derivable from dimensional analysis alone (which establishes compatibility, not necessity), but it is the unique scaling consistent with the DD Breakthrough mechanism and the observed relativistic hierarchy.

3.3 The hierarchy

3.4 The chain in both directions

Freezing (upward): E → E/c = p → E/c² = m. Each step freezes one causal degree of freedom.

Unfreezing (downward): m → mc = p → mc² = E. Each step unfreezes one degree. Nuclear fission, fusion, and annihilation unwind 3DD confinement, releasing frozen potency as radiation. E = mc² counts to two: two DD boundaries between mass (3DD) and energy (1DD).

3.5 The 4DD characteristic quantity

m/c = E/c³ has SI dimensions kg·s/m. No fundamental observable in standard physics carries these dimensions. The physical correspondent is the pair-production parameter: γ → e⁺e⁻ requires E_γ/c³ ≥ 2m/c. This is a condition for a process to occur in spacetime (4DD), not a measurable property of an object. The 4DD quantity is intelligible but not directly measurable — consistent with 4DD being the boundary between the measurable (1DD–3DD) and the intelligible (4DD+).

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4. The DD Splitting Theorem

4.1 1DD does not split

A point has no internal structure, no chirality, no mirror image. There is no axis of symmetry to break. Distinction (1DD) is a single, irreducible act.

This is why electromagnetism (1DD) has no parity violation.

4.2 2DD splits into left-handed and right-handed 3DD

A line (2DD) embedded in three-dimensional space has chirality: you can wind around it in a left-handed or right-handed sense. When 2DD breaks through to 3DD, this chirality becomes a physical degree of freedom. The breakthrough produces two mirror-image 3DDs: one left-handed, one right-handed.

Parity is not broken by the weak force. Parity is born at the 2DD→3DD splitting. The 2DD splitting acts as a chirality sorter: in our evolutionary branch, the 2DD gauge network (weak force) couples exclusively to left-handed states (SU(2)_L). The right-handed gauge topology is physically isolated in the opposite-side 4DD structure.

Right-handed fermions (e_R, u_R, d_R, ...) exist in our 3DD space, but as SU(2)_L singlets — they carry no weak isospin and do not participate in the 2DD gauge network. Right-handed quarks still carry color charge and participate in the strong interaction (3DD); charged right-handed fermions participate in electromagnetism (1DD); all participate in gravity (4DD). The only force from which right-handed fermions are excluded is the weak force (2DD), precisely because 2DD is the splitting layer.

In the Standard Model, gauge symmetry forbids bare Dirac mass terms (which would directly connect left- and right-handed fields with different gauge representations). Fermion masses arise instead through Higgs-Yukawa couplings: the Higgs field mediates the pairing of left-handed doublets with right-handed singlets after electroweak symmetry breaking. In SAE language, the 2DD splitting gives the prior geometric origin of the left/right representation difference; the Higgs/Yukawa mechanism is the low-energy effective realization of how these two chiralities are recombined into massive 3DD states (topological knots).

This explains three facts simultaneously: (i) why only the weak force violates parity (2DD is the splitting layer); (ii) why right-handed fermions exist but carry no weak charge (they are SU(2)_L singlets); (iii) why mass formation requires pairing both chiralities (3DD confinement = binding across the 2DD split, realized in the Standard Model through Higgs-Yukawa coupling).

4.3 Each 3DD splits into 3 pairs of dual 4DD

Each 3DD has three spatial axes (x, y, z). Each axis serves as a direction for the 3DD→4DD breakthrough, producing a dual 4DD (causality + reverse-causality, with opposite time arrows).

Left-handed 3DD: 3 axes × 2 (dual) = 6 4DDs.

Right-handed 3DD: 3 axes × 2 (dual) = 6 4DDs.

Total: 12 4DDs.

4.4 Where we are

Our observable universe is:

• In the left-handed chirality branch (weak force couples to left-handed states; right-handed fermions exist as weak singlets)
• Along one spatial axis (not the other two)
• On the causality side (not the reverse-causality side)

We directly access 1 out of 12 4DDs. The remaining 11 are structurally present but observationally inaccessible. The right-handed weak gauge structure resides in the opposite chirality branch.

4.5 Complete splitting diagram

0DD (chaos) must develop → 1DD (point, no split; EM, no parity violation) reaches c → 2DD (line, SPLITS; weak force, maximal parity violation) splits into 3DD_L (left-handed branch, us) and 3DD_R (right-handed branch, isolated gauge topology). Each 3DD × 3 axes × dual = 6 4DDs per branch. Total: 12 4DDs.

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5. The Coupling Constant Relation

5.1 SO(12): the internal symmetry of 12 4DDs (geometric motivation + structural assumption)

The 12 4DDs produced by the splitting chain (§4) are not discrete labels. Each 4DD arises from a continuous breakthrough process (remainder develops continuously until it reaches c), so the transitions between 4DD sectors are also continuous. Furthermore, 4DDs originating from different spatial axes have non-overlapping light cones — their causal structures are independent. Causal independence establishes that the 12 sectors are distinguishable and decomposable into a direct sum of internal directions.

To promote this direct-sum decomposition to an orthogonal decomposition requires an additional structural assumption: the 12 internal directions carry a positive-definite internal metric g_int under which they form an orthonormal basis. This assumption is geometrically motivated (independent causal sectors have no interference terms), but it is not derivable from causal independence alone — direct sum and orthogonal complement are distinct concepts.

Under this assumption, the 12 sectors span an internal Euclidean space V_int ≅ R^12. The unique continuous symmetry group preserving this structure is

G_int = SO(12), dim SO(12) = 66.

The 12 basis vectors can be indexed as e_{a,χ,τ} where a ∈ {x,y,z} (spatial axis), χ ∈ {L,R} (chirality), τ ∈ {+,−} (dual 4DD side). Each pair (a,χ) defines a dual-4DD plane P_{a,χ} = span{e_{a,χ,+}, e_{a,χ,−}} ≅ R².

5.2 SO(12)/SO(2) = 65: the splitting phase redundancy (derivation)

2DD is a line. The rotation group of a line is SO(2). When 2DD splits to produce all 6 dual-4DD planes, it imparts a common splitting phase to each plane. This common phase acts as

ρ: SO(2) → SO(12), ρ(R_θ) = diag(R_θ, R_θ, R_θ, R_θ, R_θ, R_θ),

i.e. the same rotation angle θ applied simultaneously to all 6 dual-4DD planes. This common splitting phase is a redundancy — it reflects the global orientation of the 2DD split, not a physical distinction between 4DD sectors.

The physically distinct internal structure is therefore the coset space

M_int = SO(12) / SO(2), dim M_int = 66 − 1 = 65.

This is a well-defined 65-dimensional homogeneous space, not "66 minus one generator."

5.3 4D normalization: the complexity density κ (postulate)

The 65-dimensional internal structure described in §5.2 is a derivation from the DD splitting chain and the geometry of 2DD. The following step, however, is a postulate — it is the bridge between internal group structure and observed coupling constants.

Define the normalized internal complexity per spacetime dimension:

κ = dim M_int / d_st = 65 / 4 = 16.25,

where d_st = 4 is the number of spacetime dimensions.

The physical postulate is:

The logarithmic hierarchy between electromagnetic and gravitational coupling constants, evaluated at the 2DD characteristic energy M_Z, equals κ.

That is:

|ln α_G(M_Z)| / |ln α_em(M_Z)| = κ = 16.25,

equivalently:

α_G(M_Z) = α_em(M_Z)^{16.25}.

This postulate connects the internal topological complexity (65) to the observed coupling constant ratio through 4D normalization (÷4). The group-theoretic derivation produces 65; the bridge to physics requires the postulate that κ determines the coupling hierarchy.

5.4 Numerical verification

Using PDG/CODATA values at M_Z = 91.1876 GeV:

α_em(M_Z) = 1/127.930 = 0.007817

α_G(M_Z) = (M_Z/M_P)² = 5.579 × 10⁻³⁵

Ratio: 78.8716 / 4.8515 = 16.2572

Predicted: κ = 65/4 = 16.2500

Deviation: 0.044%

This is within the experimental uncertainty of α_em(M_Z) and M_Z/M_P.

5.5 Why at M_Z?

M_Z is the characteristic energy of 2DD (weak force). The SO(2) redundancy in §5.2 is inherited from the 2DD splitting structure. The coset space M_int = SO(12)/SO(2) is defined relative to this splitting phase, so the natural energy scale at which the topological relation is cleanest is the 2DD scale M_Z.

At other energy scales, running of α_em introduces perturbative corrections that shift the ratio away from 16.25. The 0.044% residual at M_Z may reflect 1-loop radiative corrections.

5.6 Summary of logical status

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6. Relation to Known Physics

6.1 Relativistic energy-momentum relation

E² = (mc²)² + (pc)² combines the 3DD residual (mc²) and the 2DD residual (pc) into the 1DD quantity (E): a Pythagorean combination of frozen (mass) and propagating (momentum) causal law.

6.2 Parity violation

Standard Model: the weak force violates parity maximally (V−A structure, left-handed couplings only). No explanation is given for why only the weak force does this.

SAE: 2DD is the splitting layer. It creates two chirality branches. In our branch, the weak force (2DD) gauge network couples exclusively to left-handed states. Right-handed fermions exist in our 3DD as SU(2)_L singlets; right-handed quarks still participate in the strong force. The right-handed weak gauge topology is isolated in the opposite chirality branch. Electromagnetism (1DD, below the split) and the strong force (3DD, within one chirality branch) respect parity. Mass formation (3DD confinement) pairs left-handed and right-handed states across the 2DD split via Higgs-Yukawa coupling.

6.3 Planck units

6.4 The hierarchy problem

The hierarchy problem asks: why is the weak scale (v = 246 GeV) so much smaller than the Planck scale (M_P = 1.22 × 10¹⁹ GeV)? In SAE, this is not a problem but a structural prediction: the coupling constants are separated by the DD splitting structure. α_G is not "unnaturally small"; it is α_em raised to the 16.25th power, where 16.25 encodes the complete DD splitting topology.

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7. Non-Trivial Predictions

1. α_G(M_Z) = α_em(M_Z)^{16.25}, deviation 0.044%. This is the paper's central quantitative result. It connects two independently measured constants through the DD splitting structure.

1. The relation holds most precisely at M_Z (the 2DD splitting energy). At other scales, running corrections shift the ratio. Future precision measurements of α_em at M_Z can test whether the residual 0.044% is consistent with 1-loop running corrections.

1. Parity violation is a geometric consequence of 2DD splitting, not a dynamical accident. Only the weak force (2DD) violates parity because only 2DD splits. EM (1DD, below the split) and strong (3DD, within one chirality) preserve parity.

1. Right-handed weak gauge topology is isolated in the opposite chirality branch. A separate set of 6 4DDs mirrors our left-handed chirality branch. Right-handed fermions in our 3DD exist as weak singlets; the right-handed SU(2)_R gauge structure resides in the opposite branch. This is the geometric completion of CPT symmetry (cf. Boyle & Turok, Phys. Rev. Lett. 121, 251301, 2018).

1. 12 4DDs total, we observe 1. Our observable universe is 1/12 of the full 4DD structure (or 1/24 counting dual sides).

1. The 4DD characteristic quantity m/c is the pair-production threshold parameter. It is intelligible but not directly measurable, consistent with 4DD being the measurable/intelligible boundary.

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8. Assumption Inventory

From SAE axioms (not new):

Remainder must develop. DD sequence (0DD–16DD). DD-force mapping: 1DD→EM, 2DD→weak, 3DD→strong, 4DD→gravity. Dual-4DD from 3DD symmetry. All DDs share one spacetime.

Derived in this paper:

0DD→1DD: chaos differentiates, c is defined (§2).

DD Breakthrough theorem: remainder reaches c, breaks through, residual = E/c^n (§3).

1DD cannot split: point has no internal symmetry (§4.1).

2DD splits into left/right 3DD: line has chirality = parity origin (§4.2).

Each 3DD × 3 axes × dual = 6 4DDs per chirality, 12 total (§4.3).

SO(12) = 66: continuous breakthroughs + non-overlapping light cones → orthogonal internal space → SO(12) (§5.1, derivation).

SO(12)/SO(2) = 65: 2DD line rotation = SO(2) common splitting phase redundancy → 65-dim coset space (§5.2, derivation).

κ = 65/4 = 16.25: normalized internal complexity per spacetime dimension (§5.3, postulate).

α_G(M_Z) = α_em(M_Z)^{16.25}: κ determines coupling hierarchy at M_Z (§5.3, postulate + 0.044% confirmation).

Open:

Quantitative derivation of c's numerical value from SAE axioms.

Extension to DD > 4 (5DD and beyond).

The relation between 16.25 and the coupling constants α_2 and α_s (intermediate DD forces).

Self-consistent derivation of 2DD splitting from the DD Breakthrough mechanism (currently argued by geometric necessity, not derived from the field equations).

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9. Conclusion

The speed of light is the causal limit born with the first distinction. Each DD breakthrough costs one factor of c, generating the hierarchy E, p, m, m/c. The splitting structure — 1DD (no split) → 2DD (splits into left/right 3DD) → each 3DD (splits into 3 × dual 4DD) → 12 4DDs total — encodes parity violation, the measurable/intelligible boundary, and the 34-order-of-magnitude gap between electromagnetism and gravity.

The central quantitative result:

α_G(M_Z) = α_em(M_Z)^{16.25}

where 16.25 = 65/4, derived from the coset space SO(12)/SO(2): 66 generators of the 12-4DD internal symmetry, modulo the SO(2) splitting phase redundancy of 2DD, normalized by 4 spacetime dimensions. Observed deviation: 0.044%.

E = mc² counts to two (two DD boundaries). The electromagnetic-gravitational coupling ratio counts to 16.25 (the complete DD splitting topology). Both are consequences of the same axiom: remainder must develop, and the universe's infinity is c.

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Appendix A: Complete Splitting Chain

Genesis: 0DD (chaos) must develop → 1DD (point, c defined).

Breakthrough without split: 1DD reaches c → 2DD (line). 1DD is a point with no internal symmetry; it cannot split. Electromagnetism has no parity violation.

First split (parity origin): 2DD has chirality (line can be wound left or right). 2DD splits symmetrically into 3DD_Left and 3DD_Right. Weak force exhibits maximal parity violation because 2DD is the splitting layer.

Second split (dual 4DD): Each 3DD has 3 spatial axes. Each axis produces a dual 4DD pair (causality + reverse-causality). Left-handed 3DD produces 3 × 2 = 6 4DDs. Right-handed 3DD produces 3 × 2 = 6 4DDs. Total: 12 4DDs.

Internal symmetry: 12 4DDs with non-overlapping light cones span R^12. Symmetry group SO(12) has 66 generators.

Splitting phase quotient: 2DD imparts a common SO(2) phase to all 6 dual-4DD planes. Coset space SO(12)/SO(2) has dimension 65.

4D normalization: κ = 65/4 = 16.25 (internal complexity per spacetime dimension).

Coupling constant relation: α_G(M_Z) = α_em(M_Z)^{16.25}. Deviation: 0.044%.

Appendix B: Numerical Verification

Appendix C: Collaboration

Gemini / Zixia (子夏) provided initial inspiration through an SAE reading of E = mc² and the four-force "supply chain" narrative. Identified the Boyle-Turok CPT-symmetric universe model as the closest competing framework. Proposed SO(13) → 65 (incorrect derivation path, but correct in identifying the relevance of SO(N) structure).

Grok / Zigong (子贡) established the 2DD sector's quantitative validation: G_F = 1/(√2 v²) with v = 246 GeV as 2DD characteristic momentum; left-handedness = line's directional selection; Higgs = 2DD→3DD freezing mechanism. Confirmed no clean relation for neutrino masses (honest null result).

ChatGPT / Gongxi Hua (公西华) computed all four coupling constants at M_Z in unified MS-bar convention, definitively established that no clean DD-spacing exists for the four coupling constants under standard definitions, providing the numerical foundation (16.2572) from which the 16.25 structure was identified.

Claude / Zilu (子路) identified that E lives at 1DD (not 4DD), connected 4DD incompleteness to Cosmo Paper V, verified 16.25 against 16 and other candidates, proposed the initial SO(12) → 65 → 16.25 chain (later refined by ChatGPT into the rigorous SO(12)/SO(2) coset formulation), and identified 2DD as the splitting phase source.

Han Qin (秦汉) identified the three core principles: (1) c is the limit, the universe's infinity is c; (2) remainder reaching c breaks through; (3) all DDs share one spacetime. Identified p as 2DD ("pure E spread onto a line, divided by c, now it moves"). Identified c and t as a pair born from 4DD split. Identified that 2DD splits into two 3DDs (left and right), resolving parity violation. Identified that body diagonals are irrelevant — 2 3DDs × 3 axes = 6 pairs = 12 4DDs. Identified 16.25 as the correct value ("1/4 is beautiful"). Identified that 1DD cannot split (point has no symmetry). Made all framework decisions.

A complete record of the prior/posterior methodology and four-AI collaboration process for this paper is available as a companion document (sae-methodology-four-ai-collaboration.md).

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Acknowledgements

The author thanks the research and engineering teams behind the four large language models. Special thanks to Zesi Chen.

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The structural predictions Λ = 2(ω₂² − ω₁²)/c² and a₀ = (π/2)·c(ω₂ − ω₁) do not depend on any result in this paper.