μ
Proton–electron ratio
The signed-branch ansatz yields 1836; the physical baryon and electron gaps remain to be calculated.
Ansatz / open spectrumDimension: 1
SIMPLIFIED DERIVATION & CONCEPTUAL CONNECTION
Can one temporal theory predict the proton and electron on the same scale?
The mass ratio removes a common dimensional normalization, making it a sharp unification target. A spectral route gives μ = εₚ/εₑ. A separate structural ansatz gives the integer 1836, but its energy rule and fractional correction remain unproved.
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The SMM/DTT connection
The common-substrate principle demands that electron and baryon records come from compatible sectors of one dynamics. Their energies should not be normalized with independently fitted clocks. The mass ratio is therefore particularly useful: the shared units cancel and expose the genuinely dimensionless spectral problem.
Use a common mass dictionary
mₑ = aεₑ/(c²τ*), mₚ = aεₚ/(c²τ*).
The same a, c, τ* and vacuum reference must apply to both sectors. A proton sector additionally needs its interacting baryon dynamics, rather than an arbitrary mode labeled “proton.”
Cancel the conversion factors
μ ≡ mₚ/mₑ = εₚ/εₑ.
This is an exact algebraic consequence of the common dictionary. It avoids an arbitrary overall energy unit, but it does not calculate either gap. Entering measured or fitted gaps would still import the result.
The separate 1836 structural ansatz
Start with the valuation polynomial
N₃(b) = b² + 3b + 7.
N₃(b) = 137 ⇒ b² + 3b − 130 = 0 ⇒ b₊ = 10, b₋ = −13.
N₃(b) = 137 ⇒ b² + 3b − 130 = 0 ⇒ b₊ = 10, b₋ = −13.
These roots are exact. The negative root is not a positive counting dimension and has no automatic interpretation as antimatter.
Declare an inertia rule and evaluate it
μ₀ := |b₋|N₃(b₊) + b₊(b₊ + 1)/2
= 13 × 137 + 55 = 1836.
= 13 × 137 + 55 = 1836.
The multiplication by 13 and the triangular term 55 are an added baryonic inertia rule. They need a Hamiltonian or transfer-theory derivation. The closeness to the reference μ = 1836.152673426 is a motivation for investigation, not a proof of that rule. The missing residual is 0.152673426.
Test a proposed common correction
Suppose both α⁻¹ and μ depend only on a continuous effective valuation b:
X(b) = b² + 3b + 7,
M(b) = (b + 3)X(b) + b(b + 1)/2.
M(b) = (b + 3)X(b) + b(b + 1)/2.
An exact finite-dimensional rank cannot change continuously, so this deformation is an additional effective ansatz. Fitting X to 137.035999177 yields b ≈ 10.0015650751 and M ≈ 1836.698895446. It misses the mass-ratio reference by about 0.54622202.
What about a three-component color sector?
A complex three-dimensional color space with a fixed Hermitian form and volume form has SU(3) symmetry, and three fundamentals admit a determinant singlet. Those representation facts do not calculate confinement, proton binding energy, flavor or spin structure. An internal color triplet is also not the same object as the real rank-three spatial readout.
Try the workbench
Use the signed-branch route to recover 1836 at b = 10, then run the one-input α cross-check. Switch to the spectral route and scale both gaps by the same factor: their ratio stays fixed. This cancellation is the reason the ratio is a good dimensionless target, while the gap values still need a derivation.
Source basis: C · §11 and §§14–15; pp. 18–19, 22–25. These explanations summarize the stated models; they do not add a new physical selection theorem.
SHARED WITH THIS DERIVATION
Follow the common premises
Open a premise to inspect its role and jump to every constant that uses it.
NEXT MISSING RESULT
Derive a common baryon–electron spectrum
Replace the signed inertia ansatz by physical transfer sectors and confinement dynamics; compute the residual rather than naming it a correction.