From Effortless Effect
to α

1. Which number are we seeking?

The target is the electromagnetic fine-structure constant in the low-energy limit α(0), whose inverse is about 137.036. A coupling at another energy scale is not the same target without running and threshold corrections. See NIST, the fine-structure constant.

The measurement is the final check. Selecting a normalisation or modulus because it yields 137.036 is fitting. A derivation must fix those choices independently.

2. What the verification restores

The exponential theorem was formally reverified. The K3 chain c₂→χ=24→a₄=4/15→ζΔ(0)=−11/15 holds in the stated scalar Ricci-flat setting. The modular calculation below is reproducible as well.

These are separate steps. ζΔ(0) is not the spectral determinant, and neither is already α without an action containing correctly normalised electromagnetic fields.

3. A correct modular calculation

Write q=e^2πiτ for Im τ>0 and define Dedekind eta:

η(τ) = exp(πiτ/12) ∏ₙ≥₁ (1−qⁿ)
V(τ) = Im(τ) |η(τ)|⁴

The transformations η(τ+1)=e^πi/12η(τ) and |η(−1/τ)|=|τ|^1/2|η(τ)| give V(τ+1)=V(τ) and V(−1/τ)=V(τ). The factor from Im(−1/τ)=Im(τ)/|τ|² cancels exactly. Thus V is modular invariant. Source: NIST DLMF, modular transformations.

Recalculated with 60-digit working precision and 100 product terms, for ρ=½+i√3/2:

At these arguments the product tail is far smaller than the displayed rounding. This verifies the calculation. It does not supply a complete physical threshold correction: group factors, possible additive constants, operator, scale and matching convention must come from the model. Comparing two points also does not itself prove a global extremum or physical selection of ρ.

4. Two concrete problems in the old α route

The scalar energy matrix. The old numerical K3 matrix assigns nonzero energy to the constant function. Its precise spectra and the ζ′(0) estimates based on them therefore remain unverified inputs.

A factor two in normalisation. The old alpha_u_toets.py states, for n triplets:

[f₀ n / (96π²)] g² = 1/4
⇒ g² = 24π²/(f₀ n)

Its function instead returns 12π²/(f₀n), giving 1/8 rather than 1/4 in its own equation. This is an internal algebraic inconsistency. The correct physical trace convention also requires independent justification. The old band cannot return as a standalone precision result.

5. What coupling running reproduces

The original one-loop and two-loop gauge equations were also recalculated with an independent adaptive ODE solver. In the two-loop model without Yukawa terms, the three pairs cross at different scales:

Forcing g₁=g₂=g₃ at the last scale and running back to M_Z gives α_em⁻¹(M_Z)≈143.122, against the input value 127.951, about 11.86% higher. This confirms the mismatch for this specific unchanged model choice.

The calculation starts from measured electroweak and strong couplings; it is a consistency test, not a derivation without measurement inputs. The old script also adds the measured difference 137.035999−127.951 to report a low-energy number. That is not independently calculated running to α(0). Yukawa terms and model-specific thresholds are absent here.

6. The chain that still needs to close

1. Selection criterion. Define resistance and self-application without hidden α input.
2. Geometry. Justify K3's relevance and compare admissible alternatives.
3. Physical model. Specify fields, gauge group, action and operator. Derive coupling normalisation from that action.
4. Fixed data. Determine moduli and scales through an independently motivated mechanism.
5. Reliable calculation. Repair the energy matrix, verify null modes and convergence, then calculate the required determinants and thresholds.
6. Prediction. Run to α(0), report uncertainties and then compare with measurement.

The failure of one normalisation, spectrum approximation or unification condition does not rule out this entire research aim. It identifies where that particular route does not yet work.

7. An exact pre-test: fixing size and coupling together

Worked out on 16 September 2026, after a proposal by Robin Genis. This part bounds concrete cost functions. It is not an impossibility proof for every spectral theory or for EEML, and not a derivation of the measured constant.

The spectral costs on their own select no finite size. Take a fixed Ricci-flat K3 and vary only the overall size, g_L=L²g₁, at fixed cutoff and reference scale. The heat trace Σ exp[−λₙ/(Λ²L²)] increases strictly with L and approaches the number of zero modes as L↓0. The determinant of the scalar Laplacian with the zero mode removed scales, with the previously verified value ζ(0)=−11/15, as

log det′(Δ_L/μ²) = C + (22/15) log L

and runs to −∞ as L↓0 once it enters the cost with a positive coefficient. A finite stable size requires, in these models, an additional contribution or a justified restriction of the domain. The general claim that every spectral cost function is monotone is too strong: (λ₁/(Λ²L²)−1)² has a minimum at L=√λ₁/Λ. The lemma holds per named variant, under its assumptions.

A stable size is not yet α. The model R(L,e)=L⁴+n²/(e²L⁴), with a fixed integer n>0, has opposite scaling powers, contains the coupling and uses a quantised coefficient. For every e a stable size L⁴=n/e appears, but after substitution R=2n/e remains, with no minimum at finite e. In log coordinates (−1/4, 1) is a joint rescaling of degree −1: a genuine escape direction. The hydrogen atom has the same limitation: it fixes the size for a given coupling, not the coupling itself.

The exact criterion. For a finite sum of power terms with fixed positive coefficients, R=Σ cᵢ Π xⱼ^aᵢⱼ, the cost is convex in log coordinates. With P the convex hull of the exponent vectors aᵢ and A the matrix with those vectors as rows:

A joint rescaling y→y+tv with aᵢ·v=k for all i gives R(y+tv)=e^ktR(y): for k≠0 an escape direction, for k=0 a flat direction along which minimisers can exist. The absence of such a rescaling is not sufficient: L+e+Le decreases to zero as both shrink. A unique minimum in m variables needs at least m+1 terms that enclose the origin. For α alone that is too strict: e²+e⁻² fixes e=1 while L stays free. For α=e²/(4π) the test is: 0 in the relative interior, and w=(0, 2) orthogonal to ker A.

The three-term family. With a third term the example becomes

R(L,e) = L⁴ + n²/(e²L⁴) + c·e²
e*³ = n/c,   L*⁴ = n^(2/3) c^(1/3)

The exponent vectors (4,0), (−4,−2) and (0,2) enclose the origin, so there is one global minimum. All three contributions there equal t=n^2/3c^1/3; the Hessian in log coordinates is t·[[32, 8],[8, 8]] with determinant 192t²>0. If e is the correctly normalised electromagnetic coupling, it follows conditionally that

α* = (1/4π) · (n/c)^(2/3)

With general prefactors R=A·L⁴+B/(e²L⁴)+C·e² one gets e*³=√(AB)/C, unchanged under rescaling of the reference length or of the whole cost. Replace c·e² by c·L^ae^b; the convex weights for the origin are then (2b−a, 2b, 4)/(4+4b−a), and a unique finite minimum exists exactly when

b > 0   and   a < 2b

So c·e² passes, while c·L⁴e² sits on the boundary and yields no finite minimum. This is a concrete requirement on the field energy still to be derived, including its scale dependence.

Four test criteria for candidate models.

1. The derived total cost yields a finite, stable size.
2. The construction demonstrably determines the electromagnetic gauge term.
3. All coefficients that influence α follow independently. Quantisation can help but is no guarantee; n and its selection must also follow from the model.
4. All selected solutions give the same α, also when scale and coupling may vary jointly. Flat directions may remain as long as they do not change α.

State of affairs: within the class of finite positive posynomials an exact pre-test is now available for the existence of minima and for fixing a given quantity. The three-term example passes mathematically. The origin of its coefficients, the connection to electromagnetism and the transition to α(0) remain open. The missing physical step is to derive the third contribution and its coefficient from the action, so that the minimum becomes more than an adjustable formula.

The checks are exact and do not use the measured α: 16 symbolic check groups and 12 test cases for the criterion. Download the α route bundle · Reproduction and scope

8. Sources and earlier attempts

Threshold corrections in heterotic flux compactifications is a physical example investigating K3 geometry and coupling corrections together. It does not derive α from EEML. The historical article and historical preprint remain available with their original claims.

Download the verification bundle · Reproduction and scope

v0.24 · 16 September 2026