Statistics: measure, multiplicity and limits

The distinction between possibilities, joint consistency, costs and weight remains the core. The following elements can also stay:

• An effect and the model class under study are described in advance.
• Hard constraints are checked jointly.
• With several cost components, the Pareto frontier shows which candidates have avoidable overhead, without immediately choosing an exchange rate between components.
• A proven and attained lower bound is an exact minimum within the stated class.
• Algebraic or discrete closure can determine a ratio or choice without minimizing costs.

Two formulations need precision. Pareto is independent of a scalar weighting for a fixed cost vector, not independent of the choice of costs. And closure by constraints is not in itself proof of efficiency: a class can leave a single candidate without any cost criterion doing any work.

The K3 result remains the already verified conditional example: for closed, smooth, connected, oriented spin four-manifolds with a Riemannian Einstein metric and nonzero signature, one has χ ≥ 3|σ|/2 ≥ 24. K3 attains the bound. The physical choice of this class and of χ as selection criterion is still needed separately. The proof and the sources are in Part VI.

For separate, physically distinct possibilities with positive base measures μᵢ, finite costs Sᵢ and ε > 0:

S and ε have the same units; Z must be dimensionless relative to a fixed reference measure. For k states counted with equal weight and with the same cost S*, one has exactly:

for μᵢ = 1. Two such states against one yield a ratio of 2:1 for equal macro base measures. The outcome with two states gets probability 2/3, not certainty.

This lowers the effective cost of the combined outcome, not the physical cost S* of each individual process. No energy appears for free. If S is energy and ε = kBT, this correction has the familiar entropic meaning. In other applications the meaning of S and ε has to be established.

A second name for the same physical possibility must not increase the weight. If one possibility with measure μ is split into seven descriptions, their measures must still add up to μ. Giving each copy the full measure again artificially adds a factor of seven. The computational check confirms that measure-preserving relabeling preserves F exactly.

We therefore replace “equivalent realizations” with: distinguishable physical realizations of the same macro-effect, counted with a specified measure. Descriptive redundancy and gauge equivalence are not extra physical chances.

A precise statistical working version is:

Here ν is the base measure over macro-outcomes and μλ the measure within each outcome. The joint measure must be fixed in advance; if you normalize μλ separately, the removed total measure must reappear in ν. Otherwise you change the prediction.

The formula requires a finite, positive total normalization. Macro-classes must form a consistent partition. Overlapping classes must not silently double-count. For continuous macro-variables a suitable decomposition of the joint measure is needed; simply integrating over a geometric surface of measure zero is not enough.

Moreover, a model parameter that can be chosen freely is not automatically a random variable of nature. A partition function at a given λ is no proof of a physical ensemble in which λ varies. That requires an extra ensemble, dynamics or an explicit model prior. For continuous λ, what matters are probability densities relative to ν and probabilities of regions. Under a continuous distribution a single point has probability zero; the highest density is not defined without a reference measure either.

Numerical example: two equal minima against one give, for equal ν, a probability of 2/3 for the first class. With macro base measures 0.1 and 0.9 this becomes 2/11 ≈ 0.182. So the measure demonstrably does substantive work.

The physical Multiple Point Principle (MPP) concerns the hypothesis that couplings allow different vacuum phases with equal, approximately zero vacuum energy density. That is more than counting equally costly possibilities. See Froggatt et al., The Two Higgs Doublet Model and the Multiple Point Principle.

The conditional calculation is correct within a restricted class. For finitely many discrete states, a common minimal cost S*, positive measures and a positive gap Δλ to higher costs, one has:

Hence:

With equal individual measures, Mλ is proportional to the number of minima kλ. If only these minima are included, the counting rule is exact. If higher states are present, it is an asymptotic approximation, not a general formula at finite ε.

The usable conclusion is: with equal minimal costs, equal measures per state, equal macro base measures and negligible contributions above the minimum, the class with more minima weighs more. That is not yet a mechanism that first makes the vacua equal in energy, fixes their energy at zero, or lets the couplings evolve toward such a point.

#Check A: one broad minimum beats two narrow ones

Take dimensionless x and measure dx. Model A consists of two disjoint copies of the real line, each with S(x) = 100x²/2. Model B consists of one copy with S(x) = x²/2. All minima have zero cost. The Gaussian integrals give exactly:

With equal macro base measures, one broad minimum weighs five times as much as two narrow minima. This does not refute the discrete counting theorem: it shows that “equal measure per minimum” has additional content for continuous landscapes. The neighborhood of each minimum also counts.

More generally, for small ε, isolated, nondegenerate minima in the same dimension d yield the following Laplace approximation, provided the measure is sufficiently regular there and the integral is dominated by the neighborhoods of these minima:

H is the positive Hessian of S and ρ the density of the measure. Flat directions or continuous sets of minima require a different treatment. Counting k alone is then insufficient.

#Check B: the point with equal minima can in fact be unfavorable

Take two states with positive costs:

At λ = 0 both minima are equal. Yet:

F has its maximum at λ = 0. With a flat macro-measure, the probability density is smallest there. For ε = 0.5 the density at the endpoints is 3.762 times as large.

This example deliberately violates the condition that Smin is independent of λ. It therefore does not refute the conditional formula above. It shows why that condition must be derived physically before the formula can serve as an MPP mechanism.

For a fixed finite set with fixed positive measures:

That minimum can be 3 or any other number. ε → 0 does not mean zero effort, zero resistance or the philosophical zero point. Likewise, “minimally sufficient structure” follows only if S actually measures that structure. Nor does a non-sharp lower bound need to be saturated.

With equal minima, the multiplicity can remain visible in the probabilities while it disappears from the limiting value of F. For two minima of cost 3 against one:

Looking only at the limiting value of F therefore loses information. For infinite systems, moreover, the measure, the growth of state counts and the order of limits matter, for example when the number of states grows exponentially with system size N. Under suitable hypotheses the free-energy density can then take the form \(f(\varepsilon)=\inf_e[e-\varepsilon s(e)]\), with s the entropy density. The limits \(\varepsilon\to0\) and \(N\to\infty\) must not be silently interchanged.

A positive, nontrivial weight function that multiplies under independent composition and satisfies suitable regularity has an exponential form. The decreasing choice e^(−S/ε) additionally requires that higher S receives less weight. The additive costs and the independence are hypotheses that must be justified for each application.

For independent systems with additive costs, a product measure and the same ε, it follows exactly that:

There is also a precise form of consistent redescription. For a disjoint partition into blocks Bⱼ, define:

Then, with the measure already included in Fⱼ:

Bundling first and weighting afterwards gives the same total as weighting directly. This follows from the associativity of addition; it is a checkable consistency requirement, not a new law of nature.

“EEML(EEML) = EEML” remains a program for now, not a defined mathematical fixed-point equation. It lacks an operator with a domain and a meaning of equality. The usable self-application rule reads:

Apply consistency checking, ablation and the removal of demonstrably superfluous assumptions to EEML itself as well. In doing so, do not silently change the effect that is to be explained.

Nor is the positive counting rule automatically a quantum rule. Coherent alternatives require the addition of amplitudes and can cancel each other out. Born and the physical choice of a coherent or positive weighting remain additional structure.

A weight does not say how quickly a state is reached, which barriers lie along the way, or whether there is a net current. Even the same stationary distribution does not determine that current.

Check on a ring with three states:

Events occur in both models; zero net current is not an absence of events. The statistical landscape is the same; the process differs. These Markov models are mathematical counterexamples, not a thermodynamic explanation of the drive.

That is why the process idea of EEML is explicitly retained: reachability, transition rates, memory, adaptation and drive must be added or derived. A static sum over possibilities does not replace that layer.