This entire page also exists as a plain-language reading version, with a link back to the formal original per section. To Mathematics in Words →

In plain language, the formulas below say exactly this:

1. Nothing-cannot-be, so everything exists. Non-existence is the unstable option; every stable, self-consistent configuration is real. Formally: the existence set 𝔄 (Addendum 10.3).
2. Existing is not the same as being selected. Existence is free; selection is a matter of weight. The two are kept consistently apart on this page.
3. Least resistance is a preference, not a prohibition. P = e^−S (Theorem 3.1) gives paths of greater resistance not zero weight, but exponentially less. The efficient configuration is preferred; the others do not cease to exist.
4. The preference comes from the law itself. No external criterion applied to reality — least resistance is reality filtering itself. The law applies to itself: f(EE) = EE (§9). No judge beside the game.

This page formalises what the philosophy page describes conceptually. Go to philosophical foundation →

For independent configurations A, B (i.e. A ⊥ B in 𝒞):

P(A ⊕ B) = P(A) · P(B)

Independence: there is no information about A that says anything about B. Composition ⊕: combination as disjoint configuration.

P: (𝒞, τ) → ([0,1], standard topology) is continuous. Equivalently: for every convergent sequence C_n → C, P(C_n) → P(C).

There exists a real-valued functional S: 𝒞 → ℝ ∪ {+∞}, the resistance-action, such that for independent A, B:

S(A ⊕ B) = S(A) + S(B)

Under axioms A2, A3, A4, there exists a unique continuous function φ: ℝ → [0, 1] such that P(C) = φ(S(C)), and this necessarily has the form:

φ(s) = e−s   (up to scale normalisation)

Machine-verified (13 August 2026): the core theorem has been formalised and accepted in Lean 4 against mathlib, including the point that monotonicity need not be assumed but follows from the codomain (P(s+t) ≤ P(s) because P(t) ≤ 1). The proof rests on exactly Lean's standard axioms (propext, choice, quotients): the same trio as in the price list. Script archived (Kern.lean).

f(x + y) = f(x) · f(y)   for all x, y ∈ ℝ

Since x ↦ e^−x is strictly monotonically decreasing: arg max_{C ∈ 𝒞} P(C) = arg min_{C ∈ 𝒞} S(C). The configuration of least resistance carries the highest preference. Infinitesimal form:

δS = 0

This is the principle of least action — not assumed but derived from the three axioms. Note the reading key: "highest preference" is not "only possible". Configurations of greater resistance retain weight (exponentially less), they exist; δS = 0 marks the preference, not a prohibition on the rest. The distribution over all configurations returns in Addendum 10.3.

The main theorem does not need the continuity assumption. Since P lies in [0,1] by definition (Definition 1.2), h = ln φ is an additive function bounded above everywhere, and an additive function bounded above on a set of positive measure is linear (Ostrowski 1929; the pathological Hamel solutions are unbounded on every such set). With S ≥ 0 and non-triviality, P = e⁻ˢ follows directly from A2 + A4 + the codomain. The system thereby compresses from three axioms to two, and continuity degrades from assumption to consequence: f(EE) = EE in practice, the law itself became simpler. Caveat: the range of S must contain an interval; for purely discrete range, e⁻ˢ is the canonical but not unique interpolation. Full proof: working document kernwerk-12aug.

For a sufficiently smooth configuration C ∈ 𝒞 there exists a decomposition

Seff(C) = Sdyn(C) + Kdesc(C) + Istruct(C)

corresponding to a splitting of the tangent space T_C𝒞 into three orthogonal subspaces:

• T_t: tangent to time-evolution trajectories (diffeomorphism orbit)
• T_h: normal to the critical-point surface (Hess S|_C)
• T_r: parametrisation freedom (gauge orbit)

under the L²-inner-product structure on T_C𝒞.

K(C) = Ktopo(M) + Kmoduli(M) + Kfields(φ | M)

Ktopo(K3) = 0    ← K3 unique in class
Kmoduli(K3) = 58 · log2(1015) ≈ 2890 bits   ← dim 𝓜Einstein = 57 + volume
Kfields(SM) ≈ 950 bits
Ktotal ≈ 3840 bits

Let V: 𝒞 → ℝ be the potential and H_C := D²V|_C the Hessian with eigenvalues {λ_i}. Define:

I(C) = Itach(C) + Ift(C)

Itach(C) = Σi: λᵢ < 0 τᵢ · |λᵢ|
Ift(C)   = Σi: 0 < |λᵢ| < Λ²nat log(Λ²nat / |λᵢ|)

E[C] = αS · Sdyn(C) + αK · Kdesc(C) + αI · Istruct(C)

Convention: αS = 1. Naturalness: αK, αI > 0.

For qualitative orderings between topological classes or discrete parameters, the minimiser of E is independent of the exact values of α_K, α_I as long as α_K, α_I > 0 and competitive terms point in the same direction.

The observer-conditioned measure factorises per member into structure F (weight 2⁻ᴷ) and address coordinates (conditional density). For observed data the total surprise decomposes additively: −log₂ P = K(F) + typicality bits. Model comparison at fixed data must therefore use the sum: K is the prior, typicality the likelihood, and the "combined ledger" of the testing page is nothing but Bayesian model comparison with the universal prior. Bare K-minimisation is the special case of equal likelihoods. The methodological rule of 12 August (and the axion reasoning built on it) is thereby a theorem rather than a convention.

Each building block of E is itself a minimal choice within its own category:

• S_dyn: Einstein-Hilbert unique under local/covariant/≤2nd-deriv (Lovelock 1971).
• K_desc: MDL unique invariant complexity measure (Rissanen 1978, 1983).
• I_struct: Hessian spectrum unique linear-response stability measure (Morse theory).

f(EE) = EE is not merely philosophical but operational.

K(Cinit, Cfinal) = ∫𝒫 𝒟x e−Seff[x]/ℏ   (Euclidean)

wk ∝ det(D²Seff|Ck*)−1/2

Intuitive: "what exists", "with what weight" and "what is actual" are three different questions. All stable configurations exist; the measure distributes the weights; and "which one is actual?" turns out to be answerable only from the inside. This replaces the unprovable requirement that only one universe could be possible — the necessity lies not in the one universe but in the distribution.

𝔄 := { C ∈ 𝒞 : Seff(C) < ∞, ∇Seff(C) = 0, D²Seff|C ⪰ 0 }   existence — many
μ(Ck) ∝ e−Seff(Ck)/ℏ · det′(D²Seff|Ck)−1/2          measure — weights (Theorem 10.2)
x* ∈ 𝔄                                                  actuality — indexical

The chain H ⊇ X ⊇ 𝔄 ∋ x* is precisely the "Three Modes of Existence" of the philosophy page: 𝔄 is mode A (stable subset), now with an exact definition. No convexity of S_eff is required; 𝔄 may be large or infinite.

actuality is indexical; the objective, necessary structure is the pair (𝔄, μ)

Intuitive: the stability term turns out not to be independent. Tachyons are not a penalty but a disqualification (what decays does not exist — does not belong to 𝔄), and fine-tuning is description cost: tuning a parameter to precision ε costs log(1/ε) bits — which K already measures (Def. 6.1, K_fields). This eliminates α_I, and α_K is fixed at ln 2: one extra bit of description = a factor 2 less preference.

E[C] = Sdyn(C) + ln 2 · Kdesc(C)   on 𝔄 := {Seff < ∞, δSdyn = 0, D²V ⪰ 0}

Intuitive: may one sum over all existing universes, or does the sum explode? Answer: the complexity term itself enforces convergence. Simplicity is exponentially preferred, and that is exactly enough to render an infinite landscape harmless.

ΣC ∈ 𝔄disc e−Sdyn/ℏ · 2−K(C) · det′−1/2 ≤ D · e−B/ℏ · Σ 2−K ≤ D · e−B/ℏ < ∞

Intuitive: what does the distribution μ mean for us? "Actual" is indexical (Corollary 10.5), and the question "which universe is actual?" can only be asked in a universe that sufficiently represents itself. Predictions therefore follow from μ conditioned on self-representing universes, plus typicality within them.

The referee question: how does the asymmetric, complex physical reality emanate from a mathematically perfectly optimised law? The three-layer gives a clean answer to both halves.

The question. Why does a quantum branch weigh as |ψ|², and not |ψ| or |ψ|⁴? This is a famous open foundation (Everett, Gleason, Zurek, Deutsch-Wallace, Sebens-Carroll each have a route). EEML's three-layer is already structurally many-branched-with-a-measure; the question is whether its own axioms fix the measure over branches.

Step 1 [derived]: composition forces a power law. For independent subsystems amplitudes multiply (c_total = c₁·c₂), and axiom A2 demands that weights of independent events multiply. So f(|c₁||c₂|) = f(|c₁|)·f(|c₂|) on (0,1]: the multiplicative Cauchy equation. With the codomain [0,1] and a logarithmic substitution this is exactly the situation of the A3 sharpening: an additive function bounded above is linear (Ostrowski), hence f(x) = x^α: a branch's weight is necessarily a power of its amplitude. No continuity needed; same theorem, third application (after e⁻ˢ and the complex numbers).

Step 2 [derived, conditional]: branching consistency forces α = 2. If a branch splits symmetrically into two equal sub-branches (c → c/√2, c/√2), weight conservation demands 2·(|c|/√2)^α = |c|^α, which holds only for α = 2. More generally: Σ|c_i|^α is invariant under the unitary dynamics only for α = 2: any other power lets total probability mass drift in time or depend on how finely branches are sliced. Weight conservation under refinement plus A2 thus gives Born: P = |ψ|².

Honest conditions, named: (i) that the weight depends only on |amplitude| rests on decoherence/envariance (phases of decohered branches are inaccessible in principle; Zurek); (ii) the preferred-basis question is answered by decoherence, not by us; (iii) weight conservation under branching seemed a genuine assumption, but was dissolved the same evening: see below. Within those conditions the Born rule is not a postulate but the third output of the same two axioms: one weighing engine, three laws: a-local over the members, local per branch and per step. Relation to the literature: the ingredients are known (Everett's additivity, Gleason's uniqueness, Zurek's envariance); the contribution here is the unification: that exactly the EEML axioms which fixed the existence measure also fix the branch measure.

Addendum (same evening): the no-drift assumption dissolved. What still stood as an assumption turns out to decompose into three things the framework already carried. First, the timeless ontology: weights live on complete histories, not on moments; every branch cross-section at every time partitions the same set of histories, so there is no level at which total probability mass could drift: "drift" is time-talk applied to the timeless layer, the same category error as at the zero point (11.2). Second, the additivity of the measure over disjoint alternatives: already at axiom level. Third, decoherence, which was already condition (i): decohered branches are orthogonal and their amplitudes then add Pythagoreanly, |c|² = Σ|c_i|², without interference terms. And this makes step 2 more elegant than via dynamics: additivity demands P(branch) = Σ P(sub-branches), the power law gives P = |c|^α, and Pythagoras tolerates that only for α = 2. The power two is thus not a dynamical fact but the unique point where the additivity of the measure and the geometry of orthogonal branches coincide. What honestly remains: decoherence is approximate (interference is never exactly zero), so Born is exact in the decohered limit; and the Pythagorean structure comes from the inner product of quantum kinematics itself, which we use but do not derive: no circularity with the probability rule, but a dependence on the kinematics, named as such.

Measure. Over the Wedderburn-classified finite spectral triples the description cost is frozen as K_F = K_A + K_diag with prefix-free (Elias-gamma-style) coding; it adds under composition and thus qualifies under axiom A4, weighted at ln 2 per bit per Reduction 10.6.

Theorem S′ (first-order obstruction). In every Krajewski diagram where the leptons occur exclusively as a colour component of one fundamental irrep of a simple factor (SU(4) or SU(5) embedding), every admissible Dirac component acts as the identity on the shared index, and quark-lepton Yukawa equality at the breaking scale is forced. Basis of proof: the exact condition [[D,a],b°] = 0 admits only module maps per component; verified numerically-exactly by rank computation.

Result. Under observer conditions (an electron must be able to exist) and the measurements (neutrino mass; bottom ≠ tau, which via S′ eliminates all embedding classes), the standard model with right-handed neutrinos is the unique K_F-minimal survivor within the declared universe, robust up to matrix size n = 5 and against four building blocks. The hypercharges follow from the anomaly conditions themselves. Full chain, criteria and scripts: testing page, section 2.

The split carries over. Mathematics too has multiple consistent rulebooks (the set-theoretic multiverse), and the same law/address division works there: whatever every arithmetic-carrying system enforces is law; the famous independent statements (the continuum hypothesis, the value of 2^ℵ₀, V=L) are address: the α's of mathematics. A century of "but is CH true?" receives the same answer as "why is α 1/137": a category error of the address type, now with a framework explaining why those questions had to prove undecidable.

The history of the axiom of choice as a matching retrodiction. Choice (AC) and determinacy (AD) cost exactly the same under the measure (each: the base rules plus one axiom), so the practice likelihood decided: AC won globally (it compresses nearly all of mathematics), AD won locally in the one subfield it cleans up, and the field eventually bought both compressions at once via the large-cardinal tower (which proves AD holds in its own wing L(ℝ)). That is exactly the Bayesian pattern of the scale, and it is how history actually went.

And the likelihood has now been measured for the first time. Over a formalised corpus of 439,764 theorems (Lean/mathlib; sample 2000, seed 137, fully reproducible): 90.5% depends on propositional extensionality, 83.7% on quotients, 70.2% on the axiom of choice [68.2-72.3]; the purely constructive core is 7.4%. The refinement resolving Diaconescu's conflation (in Lean, choice also supplies classical logic): genuine witness extraction is at most 18.7% [14.3-23.1]; at least 53.0% needs only excluded middle. Giving up classical logic thus costs ~72% of the library; dropping genuine choice alone at most ~19%. Three historical outcomes (intuitionism lost outright, AC scepticism faded, constructive mathematics remained a real niche), three matching numbers: the first empirical price list of an axiom system.

And registered forward. The same frozen rubric, calibrated on the V=L case (it rejects it, exactly as the field did), weighs the two live candidate extensions: today Martin's Maximum wins (fully constructed, forty years of yield) over Ultimate-L (superior decisiveness, but its existence hangs on unproven conjectures); the flip trigger is fixed (a proof of the iterability conjectures tips the verdict) and the actual expectation is synthesis: the field will eventually buy both again, as with AC and AD. Testable against the coming decades of set theory.

— Theorem 3.1: P = e^−S from A2+A3+A4 (Cauchy, rigorous)
— Corollary 3.2: δS = 0
— Theorem 8.2: qualitative robustness
— Theorem 9.1: self-consistency
— Extension 10.1: path integral
— Theorem 10.2 (negative part): deterministic descent ≠ arg min; Laplace weights det^−1/2 (finite-dim.)
— Theorem 10.4: causal dichotomy for actuality (logical)
— Theorem 10.7: normalisability of the discrete part via Kraft (given prefix-free MDL + sector lower bound)
— Reading rule 10.8, half theorem: self-representation ⟹ compressibility (AIT)
— 11.1, half: gauge ⊥ physical via Ebin's slice theorem
— Theorem S′ (§12): first-order condition ⟹ quark-lepton Yukawa equality for embedded leptons (Aug 2026)
— Selection result §12: SM + ν_R uniquely observer- and data-compatible within the declared universe (Aug 2026)

— Lemma 4.1: decomposition orthogonality (remainder: T_t vs T_h within the slice)
— Definition 6.1: MDL as specific choice for K
— Definition 7.1: Hessian spectrum (reduced via Reduction 10.6: I_tach → membership, I_ft → bits)
— Caveat 5.1: repair of the conformal-factor problem (rotation conditional; K3 sector restriction hard within sector)
— Theorem 10.2 (field theory): det as ζ-regularised one-loop determinant
— Corollary 10.5: indexical actuality (conditional on the HERE axioms)
— Reduction 10.6: α_I eliminated, α_K = ln 2 (given MDL)
— Theorem 10.7, continuous part: finite moduli volumes (Douglas–Lu)
— Remark 10.9: symmetry breaking (law symmetric, member broken) and complexity (simple law, rich behaviour)

— 11.1 remainder: T_t ⊥ T_h within the physical slice
— 11.2 zero-point dynamics (candidate: field-theoretic Langevin/annealing)
— 11.3 remainder: factorisation e^−S_dyn/ℏ · 2^−K
— 11.4 heat-kernel a₆ on K3 — solved July 2026: a₆ = −0.71 ± 0.02 (see 11.4 card)
— 11.5 remainder (downgraded): stable saddles; boundary λ = 0
— 11.6 remainder: boundary λ → 0 in det′
— 11.7 convergence conjecture: low K ⟹ self-representable