Conditional benchmarks and quantum context

Status: conditional mathematical theorem. Let 𝒜 be the class of smooth, closed, connected, oriented spin four-manifolds that admit a Riemannian Einstein metric and for which σ≠0. Closed means compact without boundary; Riemannian means a positive-definite metric.

Within precisely this class, the following chain is central:

The two steps have different foundations:

1. Einstein geometry: Hitchin–Thorpe gives 2χ≥3|σ|.
2. Smooth spin topology: Rokhlin gives σ∈16ℤ. With σ≠0, it follows that |σ|≥16.

Together they yield χ≥(3/2)·16=24. The number 24 is not put into a functional to be fitted. It is the sharp, reachable lower bound that follows from these conditions. See Furuta, introduction, for Rokhlin and LeBrun, Lemma 1, for Hitchin–Thorpe and its rigidity.

#Why equality yields K3

For K3 with its complex orientation, σ=−16 and χ=24. A Ricci-flat K3 metric is Einstein and reaches both bounds. Conversely, within 𝒜, χ(M)=24 forces |σ(M)|=16 and 2χ(M)=3|σ(M)|.

The equality case of Hitchin–Thorpe is Ricci-flat and is finitely covered by a flat torus or a Ricci-flat K3. The torus possibility is ruled out by σ≠0. For the remaining covering K3→M of degree d, the following holds:

Hence:

possibly with reversed orientation. This identifies the underlying smooth four-manifold, not one unique complex structure, metric or size.

EEML meaning: within 𝒜, K3 is the realization for which both structural bounds are saturated simultaneously. Therefore minₘ∈𝒜χ(M)=24. The choice that nature uses this class and realizes this minimum, and the process by which it reaches it, remain separate physical research questions. The derivation combines existing mathematical theorems; it is not a new empirically confirmed law of nature.

#Conditional cost derivation without a pre-imposed Einstein condition

We vary both the smooth space M and its metric g. M is closed (compact without boundary), connected, oriented, four-dimensional and spin. The metric is Riemannian, hence positive-definite. We impose σ(M) ≠ 0.

We do not assume in advance an Einstein condition, Ricci-flatness, Kähler structure, simple connectedness, string theory, K3 topology or the value 24.

All norms are full tensor norms, for example |Rm|² = RᵢⱼₖₗRⁱʲᵏˡ. Norms of curvature as an operator on two-forms sometimes use different factors. W⁺ and W⁻ are the two orthogonal parts of the Weyl curvature; Ric₀ is the trace-free Ricci tensor and R the scalar curvature.

The basic identities are:

See Marigliano, chapters 3–4 for these conventions and identities. The optimization below is built explicitly from these identities here.

#The minimum theorem

Define the non-negative numbers

For arbitrary fixed a,b,c > 0 we choose

This describes a family of costs with positive, orientation-symmetric weights. The original full curvature norm is the case a=1, b=2, c=1/6. This is not the most general possible physical action.

From the signature formula it follows that U₊−U₋ = 3σ/2. Since x+y = |x−y|+2 min(x,y) for x,y ≥ 0:

For closed smooth spin four-manifolds, Rokhlin's theorem gives σ ∈ 16ℤ. Together with σ ≠ 0 this yields the exact identity

See Walpuski, Spin Geometry, Theorem 30.9 for Rokhlin. The four terms on the right show exactly where costs above the lower bound come from.

For equality, the following are necessary and, for an existing metric, sufficient:

1. |σ| = 16;
2. one of W⁺, W⁻ vanishes;
3. Ric₀ = 0;
4. R = 0.

It follows that Ric = 0. The Einstein condition and Ricci-flatness are conclusions here, not assumptions. From Gauss–Bonnet it then follows that χ = 24.

The classification of compact Ricci-flat half-conformally flat four-manifolds gives a finite covering by a flat torus or a Ricci-flat K3. The torus case has signature zero. In the other case, the covering degree d satisfies 16 = d|σ(M)|. For |σ(M)|=16, d=1. So M is K3, possibly with reversed orientation. The classification result and its explanation are in LeBrun, Lemma 1.

Ricci-flat K3 metrics exist and reach this lower bound. As a result we prove not just an infimum: the minimum is actually attained. The result selects a smooth topology and a class of metrics, not a unique shape or size.

Within this positive cost family, a, b and c need not be tuned to K3. However, this cost family, four internal dimensions, spin, closedness, Riemannian signature and a nonzero geometric signature remain model conditions. A full physical action can have extra contributions. Without the nonzero-signature requirement, the flat torus has cost zero. The static minimum theorem gives no formation process, moduli stabilization or prediction of a constant of nature.

#Physical connection: quadratic gravity, without a solved compactness problem

The invariants used also appear in local gravitational actions with terms that are quadratic in curvature. That is an existing physical connection for the choice of a model, not an independent derivation of the positive coefficients, the candidate class or the full action from EEML. See Stelle, Renormalization of higher-derivative quantum gravity.

With \(U=U_++U_-\) and the full tensor norms used here, Gauss–Bonnet gives:

On a fixed closed topology, the last term is constant. When comparing topologies, it is not. The three positive terms from the minimum problem may therefore not be replaced by a ‘standard action’ in which the Euler contribution has been silently omitted. Einstein–Hilbert, cosmological and matter terms are likewise not automatically included in this cost family.

#Completed comparison with entanglement equilibrium

Check of the proposed bridge, 28 September 2026. Alonso-Serrano and Liška, arXiv:2212.03168, equation 37 write the action that corresponds to their linearized equations as \[ S=\frac1{16\pi G}\int\sqrt{-g}\, \left[R+d\left(W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma} -\frac13R^2\right)\right]d^4x,\qquad d=\frac{5\mathcal C\,l_P^2}{168\pi}. \] The derivation uses, among other things, entanglement equilibrium, a logarithmic entropy correction and conformal matter in the linearized regime. It does not provide a full nonlinear derivation of our internal geometry costs.

Compare the quadratic coefficients in the same formal invariant basis, after separating out an Euler term: \[ \begin{array}{c|cc|c} & W^2 & R^2 & R^2/W^2\text{ coefficient}\\ \hline \text{EEML cost} & a+b/2 & c+b/12 & (c+b/12)/(a+b/2)>0\\ \text{action from the paper} & d & -d/3 & -1/3 \end{array} \]

Outcome: the signs do not match for \(a,b,c>0\) and \(d\ne0\). A common normalization or sign flip does not change the relative ratio. A free Euler term does not resolve this either. Write \[ d(U-Q/3)+\tau\chi=aU+bP+cQ. \] Then \(a=d+\tau,\ b=-2\tau,\ c=-d/3+\tau/6\). Positive b requires \(\tau<0\); positive a requires \(d>-\tau>0\); positive c requires \(d<\tau/2<0\). That is contradictory. The general exclusion follows from this algebra. In addition, 81 basic identities have been recomputed exactly as an implementation check.

This refutes the proposed direct identification, not the conditional K3 minimum theorem or the paper. A Lorentzian spacetime action is not automatically a positive Riemannian cost for internal geometry. A different reduction or physical cost derivation may yield a different comparison, but must be carried out explicitly. On a Ricci-flat K3, P and Q are zero; comparing only the value on that minimum family would hide the contradictory signs outside the minimum. The result is therefore recorded as connection via the same invariants; coefficient bridge fails in this form.

#Lesson for EEML: robust families and scale-dependent visibility

Refinement of 28 September 2026. The breadth of a family and its physical grounding are different properties. That the same K3 minima work for all positive a, b, c is a genuine robustness result: within that family no fine-tuning is needed. It does not determine the weights and does not prove that nature belongs to that family. “The larger the family, the less it says” is therefore not a general rule. A statement that is proven for a larger class can in fact be stronger; a prediction that leaves more possibilities open can be less sharp.

Also, the entanglement derivation mentioned has not yet empirically identified the realized member. It provides a conditional candidate action. The sign difference found shows that this candidate is not directly our positive cost. The absolute statement “Euclidean negative R²” moreover requires an explicit continuation and action convention; the opposite relative signs are the convention-independent core of our comparison under a common sign flip.

For a characteristic variation length L, the derivative counting in this candidate action gives, when the leading Einstein contribution does not vanish, schematically: \[ \frac{\text{quadratic correction}}{\text{Einstein contribution}} \sim \frac{|d|}{L^2} =\frac{5|\mathcal C|}{168\pi} \left(\frac{l_P}{L}\right)^2. \] This is an order-of-magnitude estimate within the regime of validity, not a ratio of action integrals that would also hold for a zero denominator. At large L, differences are suppressed; they do not vanish exactly. Large coefficients, additional light fields and other integrated-out mass scales can change the regime. Close to the cutoff scale, it is precisely the truncation to quadratic terms that is insufficient. The leading gravitational description can moreover contain Λ and matter. “Corrections only play a role close to the Planck scale” and “exactly one theory remains” are therefore too strong. See the power counting in Burgess, §4.2.2.

EEML reading: different microscopic theories can describe the same leading operation at a given scale, within a chosen accuracy. That is effective universality, not a proven selection of the cheapest fundamental theory. The original members and their parameters do not disappear from the candidate space because their differences become hard to measure.

The physical constraints must then be justified separately:

• Freedom from tachyons, absence of ghosts, Euclidean boundedness from below and a controlled effective expansion are different requirements. One of them does not prove the others; the earlier higher-derivative warning still applies.
• If positive a and c are taken to mean the conformal anomaly coefficients, they are called \(a_{\rm CFT},c_{\rm CFT}\) here. They are not our cost weights a and c. Positivity conditions can bound ratios under their own hypotheses, but do not by themselves identify the full gravitational action. See Hofman–Maldacena.
• Local counterterms and renormalization conditions remain relevant: the \(\Box R\) anomaly term is sensitive to the local R² counterterm. Positivity of anomaly coefficients does not by itself fix that freedom. See Asorey, Gorbar and Shapiro.

Working rule: keep all members that are compatible with the physical information fixed in advance. Narrow that class only with independently motivated conditions, not with the wish to prove a minimum. Record the scale, conventions, sign conditions, uncertainties and the translation from action to any costs. A remaining theorem is then physically better motivated, but remains conditional until class and prediction have been tested independently. For the candidate with ratio −1/3, that cost translation is needed first; simply minimizing its action does not yield a new K3 selection.

A lower bound is not coercivity. Coercivity would, in a specified space of metrics and after treatment of symmetries, have to prevent geometries with bounded costs from escaping uncontrollably. Here the scaling direction in fact provides an explicit counterexample. In four dimensions, for \(g_\ell=\ell^2g\):

Quadratic curvature densities scale as \(\ell^{-4}\), the volume element as \(\ell^4\). A fixed Ricci-flat K3 can therefore keep the minimum cost 24a for every \(\ell>0\) while its volume becomes arbitrarily large or small. As a result, the unfixed sublevel sets are not compact in the usual smooth geometric sense. This is not resolved by the lower bound. A volume normalization removes this specific scaling direction, but does not by itself prove compactness of all remaining moduli.

The degeneracy is concrete even at fixed topology: modulo diffeomorphisms, the space of smooth Ricci-flat K3 metrics locally has 58 real parameters, of which 57 at fixed volume and one scale parameter. The 57 come from positive three-planes in a space of signature (3,19): \(3\cdot19=57\). Singular loci and discrete identifications require separate treatment. Every smooth Ricci-flat K3 metric in this family reaches the same cost 24a. See Tripathy and Trivedi, Compactification with flux on K3 and tori, §2.3.

The result therefore reads: the cost family is bounded below, but fixes no volume, shape or unique metric. The explicit scaling direction prevents compact sublevel sets in the unfixed metric space. Incidentally, the existence of flat cost directions is not by itself a general proof of non-compactness; for that, their geometry is also relevant.

Within a separately defined ensemble that consists solely of these minima, the exponential cost factor is constant and the chosen measure determines their relative weights. That measure is not derived from EEML. However, an ensemble over all metrics cannot simply be restricted to the set of minima: this set can have measure zero in the original measure. Integration over nearby fluctuations can produce extra determinant and measure factors. Reachability also remains a process question. ‘The measure does the work’ thus holds for the specified minimum ensemble, not as a general replacement for the other layers.

The existence of a minimum is nevertheless known in this example: existing Ricci-flat K3 metrics reach the derived bound. That is a different argument from coercivity. Choice of scale, moduli stabilization, normalization of an integral over metrics and dynamical reachability remain separate questions.

Euclidean positivity does not prove a healthy Lorentzian quantum theory either. Generic local higher-derivative additions to Einstein gravity contain, in the usual linear analysis, an extra spin-2 mode with the wrong energy sign; under standard quantization, the well-known ghost/unitarity problem arises. Which degrees of freedom the precise chosen theory has must be checked separately. See Stelle, Classical gravity with higher derivatives.

Status: a robust application of existing geometric identities within a chosen cost family. Here the dossier shows no new general mathematical theorem, no solved quantum gravity model and no empirical topology selection.

K3 and the torus remain different candidates. Ricci-flatness alone does not make K3 more favorable than \(T^4\): both have \(\mathrm{Ric}=0\), while the torus is completely flat. A cost that uses only \(\int\|\mathrm{Ric}\|^2\) does not distinguish them. A tie remains a valid result.

The Kummer connection: the quotient of a complex two-torus under inversion, \(T^4/\mathbb Z_2\), has sixteen fixed points. The Euler characteristic of the quotient is \((0+16)/2=8\). Resolution of the sixteen singularities replaces each point by an exceptional two-sphere: net \(8+16=24\). The result is a Kummer K3. This is a construction relation, not a proof of cosmic evolution, and not every K3 is a Kummer surface. See Huybrechts, Lectures on K3 Surfaces.

String-theoretic dualities connect specifically chosen theories and compactifications. They do not say that K3 and \(T^4\) are always the same. Theory, dimensions, bundles, fluxes, scales and observables remain necessary. The class in the proof of the bound has not yet been derived independently from our universe. Without further index and bundle analysis, four-dimensional chirality does not imply the required internal signature. There is no derivation here that K3 is our four-dimensional spacetime, or a valid parameter-free derivation of α.

#Conditional physical reading: six-dimensional (2,0) supergravity

In ordinary 6D (2,0) supergravity with one gravity multiplet and, as extra massless matter, only abelian tensor multiplets, cancellation of the gravitational anomaly fixes their number at \(n_T=21\). This statement concerns that field content and consistency conditions; extra sectors or anomaly inflow must be treated separately. See Riccioni, Tensor Multiplets in Six-Dimensional (2,0) Supergravity.

Type IIB compactification on a smooth K3 realizes this low-energy theory with 21 tensor multiplets. The local scalar moduli space is \(SO(5,21)/(SO(5)\times SO(21))\). See Heydeman, Schwarz, Wen and Zhang.

This is a useful connection between geometry, spectrum and anomaly cancellation. It is not an observation of 6D (2,0) supergravity in our universe, not a unique inverse reconstruction of K3 and not a prediction of 21 dark matter channels.

#What exactly is closed?

Fixed: one generation, the usual non-abelian representations, one Higgs doublet, no right-handed neutrino and real hypercharges. The goal is to determine the labeled fermion charges relative to the Higgs charge uniquely. We take h≠0 and choose the computational unit h=1. This only fixes the unit of hypercharge; in this unit, the electromagnetic generator is \(T_3+Y/2\).

The seven conditions are:

“Minimal” here means minimal under omission of conditions. This must be distinguished from the smallest number of conditions in some arbitrary other formulation. The result also depends on what is counted as one condition.

#The 2-of-3 derivation is correct

With the three Yukawa conditions fixed, q,l,h remain. Then

For h≠0, each pair enforces the known ratios. No single condition does so. For h=1, for example, \((q,l)=(0,0)\), \((0,-1)\) and \((1/18,0)\) are counterexamples for only \(A_2\), only G and only C, respectively.

The motivation for h≠0 must remain precise: it concerns the usual electroweak-electromagnetic embedding. Merely saying that “a U(1) remains” after breaking does not exclude h=0; after all, an uncoupled spectator U(1) can remain. The nontrivial Higgs hypercharge remains a named input.

#The full ablation gives nine sets

There is also a second replaceability:

as a polynomial identity. Each pair from \(\{Y_u,Y_d,A_3\}\) therefore implies the third condition.

All minimal closing sets have exactly the form

That makes \(1\times3\times3=9\) sets, each with five conditions. They give

The check uses exact rational elimination of the linear conditions and an analysis of the real zeros of the remaining cubic. It examines all 128 subsets, not just a numerical sample. The known solution is used as a reference point to examine uniqueness; this is an ablation of a known derivation, not a blind prediction.

The previously stored exact check reports 128 examined subsets and 256 additional point checks. This compilation does not rerun that calculation.

#Why the route without Yukawas is different

The four anomalies without Yukawas give

There are thus several branches. The branch \(q=0,u=-d,l=e=0\) is vector-like for the U(1) charges, but not automatically vector-like for the full gauge group: the quark doublet, for example, has no partner in its conjugate full representation. General “chirality” therefore does not exclude that branch. Specifically requiring that the U(1) assignment be chiral is a different, additional condition.

Even if q≠0 is imposed, the ratio to h remains free: h does not appear in any of these anomalies. Take

All four anomalies vanish and the fermion ratios are standard, but q/h=1 instead of 1/3. The Higgs interactions do not close. Moreover, the interchange of the two quark singlets must be dealt with if one wants to determine their labels.

Conclusion: without Yukawas, fermion ratios can be constrained under additional conditions. That is not the same closure of the full Higgs-plus-fermion question. This distinction belongs explicitly in the ablation protocol.

Under suitable regularity, the multiplicative Cauchy relation gives an exponential form. In addition, positive decreasing weights and complex phase factors require different conditions. The constant ℏ, the measure, the Born rule and the choice of the physical action do not follow from this functional equation.

For a suitable Hamiltonian, analytic continuation shows the well-known operator connection:

In field theory, Euclidean correlation functions that satisfy the full applicable Osterwalder–Schrader conditions yield a reconstruction of a Lorentzian quantum field theory. Reflection positivity is essential, but not the only condition: regularity/growth bounds, covariance and symmetry, among others, also play a role. This concerns correlation functions and their reconstruction, not merely replacing i in an arbitrary exponent. Osterwalder and Schrader, 1975.

Moreover, reflection positivity is not the same as a pointwise positive classical weight function. Nor is every Gibbs distribution automatically the analytic continuation of a physical quantum field model. A thermal quantum state also comes with the imaginary-time period \(\beta\hbar\) and the associated boundary condition.

For the naive Euclidean Einstein–Hilbert integral, the conformal direction makes the action unbounded below. That is a real problem for that definition of the integral. It is not a proven impossibility of every Euclidean-Lorentzian connection in quantum gravity. Research on measures and contours addresses precisely this issue. Dasgupta and Loll, A proper-time cure for the conformal sickness in quantum gravity.

Formulation to retain: positive Euclidean and coherent Lorentzian descriptions can be connected where a suitable analytic reconstruction exists. The conditions and the chosen definition of the integral must be checked per model. This does not replace a Born postulate with an EEML derivation.

Status (addition of 28 September 2026): known physics, with a possible EEML interpretation. The statement “gravity and interactions are based on EEML” is not demonstrated by this.

Stationarity is not a general efficiency minimization. The usual classical descriptions of gravity and the gauge fields have action principles. There, \(\delta S=0\) is the stationarity condition; a stationary point can also be a saddle point. The Lorentzian action is not a generally non-negative measure of effort or dissipation. It therefore does not follow from its stationarity that every process has the lowest costs. The specific action and couplings are input as long as EEML does not derive them separately.

Free fall makes one meaning of effortlessness concrete. An ideal freely falling test particle without non-gravitational forces has zero proper acceleration:

That does not rule out coordinate acceleration or tidal effects on extended bodies. A timelike geodesic locally maximizes proper time in a suitable convex normal neighborhood; this is not an unrestricted global statement. Here, the absence of a required propulsive force is the useful connection to “effortless effect”. See Tong, geodesic motion and Reall, General Relativity, §2.

The coherent branch explains stationarity conditionally. In a suitable path-integral description we write schematically:

In a valid semiclassical approximation, rapidly varying phases largely cancel each other out. Neighborhoods of stationary paths can supply the leading contributions, even when there are several. This uses a given action, measure and quantum theory; it does not derive those ingredients from EEML. At finite \(\hbar\), non-stationary contributions are not generally exactly zero. See Feynman, the action principle and quantum amplitudes.

Jacobson is a concrete conditional reconstruction. His derivation assumes, among other things, horizon entropy \(S_{\rm hor}=\eta A\), the Unruh temperature, local Rindler horizons in local equilibrium and \(\delta Q=T\,dS_{\rm hor}\). Using the Raychaudhuri equation and the local conservation law, one obtains, in units \(c=k_B=1\):

The unknown entropy density \(\eta\) thus determines G; the derivation does not predict its value. Likewise, \(\Lambda\) remains an integration constant. This is more than multiplicity alone: geometry, causality, temperature and energy flow are involved. See Jacobson, 1995, Thermodynamics of Spacetime.

What EEML can do with it. This example fits the protocol of having independent descriptions close jointly. A standalone extension would have to derive a required relation or a measurable deviation that was not yet input. The assertions “this is the only lead” and “nothing comparable exists for gauge interactions” are not adopted: this check does not substantiate such exclusionary claims about the literature.

Connection to EEML's own claim about nature: neither stationary phase, nor free fall, nor Jacobson's reconstruction proves the prohibition of persistently dominated realizations. That prohibition compares reachable alternatives with the same effect and a predetermined cost vector. That comparison structure is not automatically present in \(\delta S=0\) or \(\delta Q=T\,dS_{\rm hor}\). For that, EEML-D1 remains a separate test proposal.

Status (proposal of 28 September 2026): an additional hypothesis, designated H_R. It is separate from the historical H_D and the conditional D1 test. The connection to unitary quantum theory is known; EEML has not derived unitarity independently.

#What the pattern does and does not say

Under closed unitary evolution, \(\rho(t)=U(t)\rho(0)U(t)^\dagger\), the eigenvalues of \(\rho\) are conserved. Hence the fine-grained von Neumann entropy is conserved as well:

This is a consequence of the assumed dynamics. It does not show a competition in which nature chooses the most efficient member of every family of theories. The entropy of a subsystem can increase at the same time, and fine-grained entropy is not simply the thermodynamic entropy or local entropy production. In a complete Hamiltonian description, a related conservation of the fine-grained Gibbs entropy holds, given a suitable measure and boundary conditions.

Three concepts remain separate:

T violation has been measured directly in B-meson transitions. That is not proof of intrinsic loss of the complete quantum state. CPT is a theorem under the applicable quantum-field-theoretic conditions and is tested experimentally; “exact in every possible model of nature” is stronger. See BABAR, T violation and PDG, CPT tests and hypotheses of the theorem.

The claim that reversible microdynamics “has so far been reconstructed for everything” is not adopted. The models discussed do not provide a complete, proven microtheory of measurements, black holes and the universe. Classical deterministic equations, too, give a backward calculation only on their valid, well-posed domain of evolution; singularities require separate treatment.

#Two known mechanisms, without mixing up the layers

Statistical reduction. Multiplicity helps explain why some macrostates are typical. For the course of events, dynamics, the chosen macrovariables and preparation are needed as well. A static count produces no transition rate or arrow of time. In the usual statistical explanation, a special low-entropy preparation plays an essential role; a universal cosmological explanation of it is not thereby provided. See Lebowitz, microscopic dynamics and the macroscopic arrow of time.

The exact Mori–Zwanzig redescription brings hidden dynamics back as, among other things, memory and fluctuation terms. A practically irreversible description, for example a Markovian one, requires further conditions or approximations. Merely making a description coarser is not physical heat production. Dissipation is therefore not exclusively “layer III that counts”: layer IV, preparation and the chosen system boundary also play a part. See Mori, 1965.

Effective field theory. Within a controlled expansion, contributions from higher-dimension or higher-derivative operators can be small at energies well below the relevant heavy scale. Symmetries, light degrees of freedom, coefficients and the valid power counting remain input. The leading law is then an approximation; the other allowed terms are not necessarily absent. Einstein–Hilbert is the leading two-derivative term in the corresponding metric expansion, but the zero-derivative cosmological term may not in general be omitted. This does not explain a universal preference for zero dissipation and does not select a unique fundamental theory. See Burgess, effective field theory.

#H_R: conservation of fine-grained information

Additional hypothesis: the complete physical dynamics does not intrinsically and irretrievably erase state information. In the quantum formulation, a truly closed whole evolves unitarily. It must be possible to account for effective information loss through an explicit dynamics of omitted degrees of freedom, correlations and the preparation.

“Fundamental” is not defined here as “reversible”. H_R is precisely a claim about the complete dynamics. In this specific hypothesis, the name “effortless” refers to conservation of information; the hypothesis does not prove 100% thermodynamic efficiency, minimal costs, automatic motion or the dominance prohibition. The universal extension is open, even when a concrete closed quantum experiment agrees with unitarity.

#Negative control: information loss without heating

A simple dephasing model with fixed \(H=(\hbar\omega/2)\sigma_z\) and \(\gamma>0\) reads:

Because H commutes with \(\sigma_z\), \(\operatorname{Tr}(H\rho)\) remains constant. Yet the coherence decreases. Starting from \(|+\rangle=(|0\rangle+|1\rangle)/\sqrt2\), the eigenvalues are:

The entropy rises from zero to \(k_B\ln2\), without any change in the mean energy. For \(t>0\), the channel has no physical inverse on all states: formally calculating backward would yield negative eigenvalues for some valid input states. This distinguishes physical reversibility from mere algebraic invertibility.

Nor does observed dephasing prove fundamental loss. A unitary controlled-Z on two qubits initially prepared in \(|+\rangle\) makes each one individually maximally mixed, while the whole remains pure. This coupling commutes with the system energy \(H\otimes I\). The same complete dephasing of the system is thus compatible with information in correlations. Both examples have been checked analytically and numerically; they are not measurements of nature.

More generally, a finite-dimensional quantum channel has a unitary extension with a suitable environment. That mathematical representation does not prove that the environment in question physically exists, nor that it yields a local microtheory over all times without further conditions. It does show why a measured reduced channel alone does not determine the fundamental interpretation. See Kretschmann, Schlingemann and Werner, Stinespring representation.

#Collapse as a test direction, with a delimited interpretation

GRW/CSL-type theories add objective stochastic collapses. Common CSL variants predict, among other things, suppression of coherence, momentum diffusion and spontaneous heating or radiation. Not every variant has the same heating signal: dissipative extensions change the energy dynamics. Heating and fundamental non-unitarity are therefore not synonyms. See Carlesso et al., experimental collapse tests and Smirne and Bassi, dissipative CSL.

Checked existing exclusion: in March 2026, XENONnT published stronger bounds on the investigated Markovian CSL and Diósi–Penrose models via an X-ray search. They cover values from originally proposed CSL parameter regions. It is not an exclusion of all collapse models and not an EEML prediction made in advance. See XENON Collaboration, PRL 136, 120201.

A test must fix in advance the collapse model, parameter region, noise spectrum, environment model, observables, uncertainties and decision rule. A single unexpected heat signal is insufficient. Mutually consistent mass, distance and time dependences must distinguish between the alternatives laid down. A convincingly demonstrated intrinsically non-unitary process would refute H_R; excluding a limited environment model does not yet demonstrate that universal character.

Ever smaller upper bounds leave H_R standing for now within the tested range, but do not prove an exact zero and do not distinguish EEML from ordinary unitary quantum theory. An unbounded appeal to an unknown environment must not make the hypothesis irrefutable after the fact. The universal H_R is therefore not yet a fully operational test; concrete model comparisons can be.

#Registration of H_R: “effortless member”

Dossier registration: 28-09-2026; hypothesis with a test direction, no external preregistration. “Fundamental” here means an assumed complete process law of the physical degrees of freedom within an explicitly specified domain of validity, distinguished from a reduced description by separate information about couplings and environment. Reversibility is not part of that definition. That a laboratory system is truly completely isolated is not a directly measurable fact; the environment boundary remains part of the model package to be tested.

The test coverage is limited: as \(\lambda\to0\), CSL approaches the reference model, so finite precision does not exclude every positive collapse rate. The link to an existing experimental program makes the hypothesis open to investigation, but gives EEML no exclusive prediction.

1. 331: checked closure within a specific class. Take the gauge group \(SU(3)_C\times SU(3)_L\times U(1)_X\), \(N_f>0\) lepton families as antitriplets, n quark families as triplets and the rest as antitriplets. The cubic \(SU(3)_L\) anomaly gives:

With three colored Dirac flavors per family, without extra colored matter or scalars, the additional requirement of QCD asymptotic freedom above the relevant thresholds gives \(b_0=11-2N_f>0\), so \(N_f\le5\). Combined, exactly \(N_f=3\) remains. The representation assignment and asymptotic freedom of this extension are input. Other lepton/quark assignments change the count. See Pisano and Pleitez and Calabrese et al.. This is discrete closure, not a new efficiency minimization.

2. Supersymmetric 6D route via Bershadsky–Vafa: proposed, not yet independently reconstructed. The historical work imposes congruence conditions on specific six-dimensional gauge and matter sectors. On this basis, a supplied in-house application claims \(N_f\equiv0\pmod3\), but its full spectrum, anomaly polynomial, tensor charges and Green–Schwarz quantization are missing here. It is therefore not counted as a second successful derivation. See Bershadsky and Vafa. The modern treatment shows why the global definition of the Green–Schwarz mechanism must also be checked: Lee and Tachikawa.

3. Dobrescu–Poppitz: historical route, modern reassessment needed. The original non-supersymmetric 6D model links the chosen chiral content and a historical SU(2) anomaly condition to multiples of three: Dobrescu and Poppitz. According to the modern bordism analysis, the argument ‘non-trivial \(\pi_6\), hence a separate global anomaly’ does not suffice. For the spin theories treated with SU(2), SU(3) or \(G_2\), the seven-dimensional bordism groups in question vanish; relevant older congruences must be assessed via local anomaly cancellation and the full compensation structure. See Davighi and Lohitsiri and Lee and Tachikawa.

This correction does not prove that every multiple-of-three condition disappears. It does prevent the old homotopy step from being presented, without recalculation, as an independent conclusive proof. Even after six-dimensional consistency, a compactification/projection to four dimensions must yield the desired family spectrum.

In addition, the historical route from older EEML texts: the standard embedding. In heterotic compactifications with the standard embedding, the net number of families is |χ|/2 (Candelas, Horowitz, Strominger and Witten 1985). Three families therefore require |χ| = 6; the classic example is the Tian–Yau space with χ = −6, which is not simply connected. This is a conditional count: nothing in this route selects |χ| = 6, and spaces with χ = 0 exist too. The claim in those texts that 6 is the smallest possible value is wrong (see E11).

Combined status: one explicitly checked 331 count, one incompletely supplied supersymmetric calculation and one historical 6D route under reassessment. These are not three independent confirmations that EEML predicts three generations.

Question: can the current EEML derive the fine-structure constant? Here α refers to electromagnetism, ultimately to the low-energy value α(0). The curvature coefficient a from VI.A is a different quantity. This attempt does not use 137 to choose parameters or integers.

#E1. Which part must the derivation fix?

In natural, rationalized units, with the electron charge as the unit in the covariant derivative, we write the Euclidean kinetic term:

After canonical normalization of A, e appears in the matter coupling. Hence:

These normalization conventions are explained, for example, in Tong, Supersymmetry, §4.1. The electromagnetic sector used here need not be supersymmetric. The coefficient k is not an electrical resistance in ohms, nor is it a measure of dissipation.

The existing K3 theorem does not determine k. A minimal extension with an independent, positive Maxwell term leaves the same geometric minimum cost 24a in place on a configuration F=0, for every k>0. In this extension, a vectorlike charged Dirac sector can retain the same charges and anomaly cancellation. As soon as charged matter is used as a probe, different k do have different observable interaction strengths. So this is not a mere change of units.

Scope: this is an explicit freedom in a factorizable classical/low-energy model. It is not a proof that every possible complete theory of quantum gravity must leave k free. It does show that the already proven geometric bound and the stated charge conditions do not supply that missing relation. Quantum corrections or a non-factorizable action must be computed separately.

Nor is a itself determined: for t>0 we have \(\mathcal E_{ta,tb,tc}=t\mathcal E_{a,b,c}\), with the same minimizers. The value of an action relative to ℏ or ε can be physically relevant; that normalization does not follow from the minimizer.

#E2. A more concrete geometric bridge

For harmonic two-forms on K3 we must distinguish two matrices:

Q is the topological intersection form in a fixed integral basis. H is positive definite and contains metric information via the Hodge star. A fixed Q does not determine H.

This distinction has a physical application: under reduction of the M-theory field \(C_3=A^I\wedge\omega_I+\cdots\) on K3, H appears in the kinetic matrix of the resulting gauge fields. In the seven-dimensional Einstein frame there is moreover a volume factor \(V_{K3}^{-3/5}\). This is a 7D result; further compactification and identification of the electromagnetic field are needed for α in four dimensions. See Castellano, Herráez and Ibáñez, §5.2, equations 5.11–5.13.

A uniform rescaling of the internal four-dimensional metric does not change the Hodge star on two-forms. Changing the shape, however, can change H. The volume factor and any further reduction must therefore not be confused with the shape dependence.

For an exactly computable slice we use the hyperbolic lattice block U, which occurs in the K3 lattice. Define for x>0:

Then the following hold exactly:

This model therefore has one unchanged intersection form and yet a continuous family of positive Hodge norms. The relation to the lattice and the description in terms of positive three-planes are given in Tripathy and Trivedi, §2.3.

Limitation: the 2×2 matrices form an algebraic toy model of their own here. A fully smooth K3 realization requires an admissible positive three-plane, treatment of roots with square −2 and exclusion of singular loci. We have not constructed a smooth compact physical solution for every x.

#E3. The attempt with EEML: let two contributions bound each other

Fix a nonzero integral two-form class \((n,m)\), with n and m both different from zero. As an additional model cost, take its quadratic norm, with a fixed positive prefactor omitted:

This is the structure of a positive field-energy term at fixed flux. The chosen class is input: the effect or the flux sector must be fixed independently. The cost family of VI.A did not supply this flux.

Differentiating gives:

Hence:

The global lower bound also follows directly:

The conditional result: as soon as the two specified contributions are genuinely present, minimal total input fixes one parameter that the earlier geometric costs left free. No choice of the desired final number is needed to prove this minimum theorem.

The ablation is decisive. For m=0 and n≠0 the infimum shifts to x→0; for n=0 and m≠0, to x→∞. If both may be zero, the empty sector costs zero for every x. The interior minimum thus rests on keeping both contributions jointly, in accordance with the working rule ‘compare at equal effect’.

Moreover, an equal minimum cost does not automatically fix the same parameter in different sectors. In this marked basis, (n,m)=(1,6) and (2,3) both have minimum cost 12, but x=6 and x=3/2 respectively. This is a check within the toy model, not a classification of physically inequivalent K3 vacua; for that, basis identifications and the embedding of the measured gauge field must also be taken into account.

#E4. Static selection is not yet a law of motion

We can choose a process separately. Set z=log x and take:

Then:

For two nonzero contributions, \(\mathcal R(z)=n^2e^z+m^2e^{-z}\) goes to infinity at both ends and has a single minimum. This chosen relaxation converges to it. For n=1, m=2 and Γ=1, five initial states were computed; all reach x=2 with cost 4, with decreasing cost up to numerical rounding.

Status: this illustrates a possible process that realizes the minimum condition. The choice of the gradient flow, the coordinate mobility and any outflow is extra model input. A closed conservative system need not relax in this way. This process law is not derived from zero resistance or from EEML alone.

#E5. What does this yield for α?

Only if a consistent four-dimensional reduction with fixed charges actually makes the remaining electromagnetic coefficient \(k=\kappa x\) does it follow, at the chosen matching level, that:

After matching, mixing where needed and scale evolution, the schematic outcome would be:

This is a conditional formula with unknown input, not a computed constant of nature. κ must follow from the full action, scale factors and charge normalization. The flux sector (n,m), all other relevant moduli and the quantum corrections are not derived here.

Moreover, a two-form probe flux on K3 is not automatically an admissible background flux of the M-theory field C₃ mentioned above, whose field strength is a four-form. We have used the norm structure, but have not built a full compactification that fulfills both roles at once. A real background must respect its field equations, source/charge conditions and backreaction on the geometry. It can change the Ricci-flat background. The toy calculation is therefore not a proof of moduli stabilization in our universe.

#E6. Can spectral action or scale evolution fill in the missing number?

Spectral action. It can give relations between gauge couplings, but requires a spectral geometry and an action prescription. In the conventions of Chamseddine–Connes (1996), equation 3.18:

The spectral normalization f₄ affects the absolute strength. A fixed choice of the cutoff function can pin down that normalization within that model; EEML has not yet derived this choice or the required spectral data. Moreover, this historical calculation concerns a different geometric construction and is not a K3 calculation of α(0). See Chamseddine and Connes, The Spectral Action Principle.

Scale evolution. For the simple QED model with one active Dirac fermion of unit charge, well above its mass and without thresholds in the interval studied:

The one-loop coefficient is checked, for example, in Igarashi and Itoh, QED in the exact renormalization group. The equation transports an initial value. It does not fix it: at a scale ratio of 10, initial values 100 and 150 for α⁻¹ stay exactly 50 apart. This illustration is not a calculation through the electroweak or hadronic thresholds to α(0).

Outcome of the attempt: the current conditions do not yield a unique α. What has been found is an exact intermediate step: a specified double load can fix a shape parameter via a positive quadratic norm. To turn this into a derivation of α, a single physical model must supply that load as well as the normalization and the electromagnetic identification, and its other predictions must hold up independently.

#E7. Full shape space: a minimum is not yet closure for the coupling

The earlier x-model held the other shape directions fixed. Now we let them take part. At fixed volume, a positive three-plane Σ in a real space of signature (3,19) determines the Ricci-flat K3 metric, with the usual exclusions for singularities and discrete identifications. The local shape space has dimension 57.

Write a fixed flux vector v as v₊+v₋ according to the Hodge splitting. Then:

For Q(v,v)>0, equality means v∈Σ. The remaining two positive directions can still move in a space of signature (2,19): 38 shape parameters remain free. For Q(v,v)<0, equality means Σ⊂v⊥, and 54 shape parameters remain. These are local counts on admissible smooth branches, not a guarantee that every integral flux admits a smooth minimum. A vector of square −2 in the negative plane can in fact force a singular locus.

More generally, r independent positive and s independent negative directions, under compatible, mutually orthogonal saturation, locally leave \((3-r)(19-s)\) shape parameters free. Volume is not counted. The geometric parametrization is given in Tripathy and Trivedi, §2.3; the dimension count here follows by removing the fixed directions from the positive and negative planes.

A small exact counterexample makes the consequence for couplings visible. Take a real, not integrally normalized basis:

We have H_z>0 and H_zQ⁻¹H_z=Q. For the fixed flux v=(1,0,0), the minimum cost remains H_z(v,v)=1. Another fixed probe direction w=(0,1,0) has norm H_z(w,w)=cosh(2z), which does change. This is an algebraic demonstration of a free coupling direction; not a full K3 compactification.

Correction to a possible reading of E3: one fixed shape parameter does not mean full moduli stabilization. At the same time, not all moduli need to be fixed to predict one observable. The right test is whether the electromagnetic coefficient changes along the remaining minimum set.

#E8. A physical exclusion: does the intended photon stay massless?

A two-form flux has a concrete realization in heterotic line bundles on K3. There, the anti-self-duality condition stabilizes part of the shape space. But the same flux can give the corresponding U(1) gauge field a mass via coupled axion fields. The effective action moreover retains dilaton dependence in the gauge couplings. So this route does not automatically yield electromagnetism. See Louis, Schasny and Valandro, especially §4.4.

The relevant linear check is simple. With positive axion metric G and charge matrix N, the action contains \((db-NA)^{\mathsf T}G(db-NA)\). The quadratic mass block is NᵀGN. A combination A=vA_light escapes this mass exactly when Nv=0. Other mass mechanisms still have to be excluded separately.

For N=(1,−1), v=(1,1) is massless under this term. Its kinetic coefficient, however, remains vᵀKv and is not determined by Nv=0; for K=diag(2,5) that coefficient is 7. Charge normalization remains necessary for a physical coupling. Masslessness thus closes a condition, not a value of α.

#E9. From toy model to an explicit IIB flux family

We now choose one existing physical framework: type IIB on K3×T²/ℤ₂ with RR and NS three-form flux. In the leading-order supergravity description, the following apply: flux quantization, charge cancellation and, for the supersymmetric solutions studied here, the requirement that G₃=F₃−φH₃ is primitive of type (2,1). φ=C₀+i/g_s is the axio-dilaton; τ is the complex shape of the torus. This framework and the two-dimensional flux sector are given in Tripathy and Trivedi, §2 and §4.1.

Our own worked-out subfamily. Choose orthogonal integral K3 classes p and q with p²=q²=2, for example e₁+f₁ and e₂+f₂ in two hyperbolic lattice blocks. Let:

where F₃ and H₃, apart from the common unit factor, are a_x∧dx+a_y∧dy and b_x∧dx+b_y∧dy respectively. The even coefficients respect the flux quantization of this orientifold convention.

Set Ω=p−iq and choose the Kähler class orthogonal to p and q. The equations \((a_x-\phi b_x)^2=0\) and \(a_y-\phi b_y=\bar\tau(a_x-\phi b_x)\) give:

Then G₃ is proportional to Ω∧d\bar z: the required type (2,1). The stated orthogonality gives primitivity. The choice of an admissible smooth Kähler class and the remaining sources is part of the model; no new complete ten-dimensional solution has been constructed numerically here.

The positive norm gives the same balance. In the rectangular slice with C₀=Re τ=0, s=Im φ and t=Im τ, the normalized flux norm yields:

For BC=AD, s₀=A/B and t₀=C/A, the following holds exactly:

Both balances saturate simultaneously at (s₀,t₀). This is an explicit application of the sharpened protocol: the conditions have a joint realization, and the two parameters studied are then fixed within the chosen flux sector.

The sources bound the sectors. In the same convention:

Without anti-D3-branes and without the additional source contributions omitted here, this means AD≤3. The positive integer solutions of BC=AD≤3 have been fully enumerated: nine flux assignments before duality identifications. These are not necessarily nine physically distinct vacua. Five assignments have N_D3>0.

Three examples from the checked list:

In the subset with g_s≤1, the values 1, 1/2 and 1/3 thus occur. This is not a general bound on string couplings: only this small family has been computed. g_s=1 offers no small string-loop parameter; nor do the other examples justify a precision measurement without an error analysis.

Why ‘least flux’ does not yield a justified winner. The flux norm is not the full four-dimensional vacuum potential. After the consistent source bookkeeping, the leading-order no-scale description has a non-negative contribution that, in this slice, is proportional to \(\mathcal R-N_{\rm flux}\). For all the supersymmetric solutions above, this contribution reaches zero. Less flux charge here means more D3-branes; comparing only the flux norm leaves out a changing part of the whole. The no-scale structure and the role of sources are treated by Giddings, Kachru and Polchinski.

So there is closure for φ within a fixed sector, but not yet closure for φ across the sectors studied. The weights, reachability or extra corrections that could single out a sector are not derived here.

How close does this come to α? For a D3 gauge sector, the tree-level kinetic function depends on the axio-dilaton, whereas a D7 sector also uses the wrapped four-cycle. In the conventions of Conlon, Quevedo and Suruliz, equations 12–13, \(f_{\rm D3}=S/(2\pi)\), with Re S=1/g_s and g_YM⁻²=Re f. For that named, normalized gauge sector, this would then give \(\alpha_{\rm D3}^{-1}=2/g_s\) at tree level. For the first two table rows, that gives 2 and 4. The third row contains no mobile D3 sector; we do not add a probe there without changing the charge bookkeeping.

This is not α_em(0): the charged spectrum, a massless electromagnetic field, its generator normalization and the scale evolution have not been constructed. A free volume modulus does not directly block this D3 tree-level coupling, but it can affect scales and corrections. No complete prediction or exclusion of a realistic model has been achieved.

Progress: the norm balance from the toy model has now been applied in a named flux sector and yields a fixed string coupling there. The open problem has shifted to selection between sectors and the electromagnetic identification. Neither K3 nor 24 alone resolves those two questions.

#E10. What fixes the volume? A sign rule from topology

In E9 the volume stays free: the leading no-scale description does not fix it. The best-known mechanism that does is the large volume scenario (LVS) of Balasubramanian, Berglund, Conlon and Quevedo. It combines two known contributions: an α′³ correction to the Kähler potential that is proportional to the Euler characteristic χ of the Calabi–Yau threefold (Becker, Becker, Haack and Louis, eqs. 3.18 and 3.24), and a non-perturbative term on a small four-cycle:

Here \(\mathcal V\) is the Einstein-frame volume in units \(\ell_s=2\pi\sqrt{\alpha'}\); in the units of Becker et al. themselves, \(\xi=-\chi\zeta(3)/2\). For a volume with one small cycle, \(\mathcal V=\alpha(\tau_b^{3/2}-\gamma\tau_s^{3/2})\), the potential at large volume becomes:

Minimizing gives, for \(a_s\tau_s\gg1\):

The sign rule. A solution requires \(\hat\xi>0\), so χ < 0, that is \(h^{2,1}>h^{1,1}\). For χ ≥ 0 the potential, after minimizing over \(\tau_s\), is negative everywhere at large volume and increasing in \(\mathcal V\). There is then no minimum: the model slides to small volume, where this description no longer holds. Cicoli, Conlon and Quevedo, §2.2 give the general conditions: \(h^{2,1}>h^{1,1}>1\) and at least one blow-up cycle with a non-perturbative term. The minimum has negative energy (anti-de Sitter) and breaks supersymmetry.

Rechecked (1 October 2026). Own check in the example \(\mathbb P^4_{[1,1,1,6,9]}[18]\) (\(h^{1,1}=2\), \(h^{2,1}=272\), χ = −540, α = 1/(9√2), γ = 1); the code is available on request via the contact address.

• With the dilaton fixed and only the Kähler moduli, two exact identities hold for every volume that is homogeneous of degree 3/2: \(K^{i\bar\jmath}K_iK_{\bar\jmath}-3=3\hat\xi/(4\mathcal V-\hat\xi)\) and \(K^{i\bar\jmath}K_{\bar\jmath}=4\tau_iY/(Y-3\mathcal V)\), with \(Y=\mathcal V+\hat\xi/2\). Derived with Sherman–Morrison and the Euler relation, checked numerically to 10⁻¹⁵. The form of Becker et al. with the dilaton included differs only at relative order \(\hat\xi/\mathcal V\).
• Full N=1 potential with \(g_s\) = 0.1, \(W_0\) = 1, \(A_s\) = 1 and \(a_s\) = 2π/10: a minimum at \(\tau_s\) = 41.63 and \(\mathcal V\) = 1.34·10¹¹, with \(V_{\min}\) = −2.45·10⁻³⁴ in the units of the code. All eigenvalues of the Hessian are positive. The boxed formulas give \(\tau_s\) = 41.08 and \(\mathcal V\) = 0.98·10¹¹. Minimizing the three terms exactly gives 41.627 and 1.343·10¹¹: equal to five digits. The first correction in \(1/(a_s\tau_s)\) is in Conlon and Pedro, eqs. 2.16–2.18. The sensitivity is exponential: 1.3% in \(\tau_s\) becomes 30% in \(\mathcal V\).
• Scan over 216 combinations: \(g_s\) ∈ {0.02, 0.1, 0.2}, \(W_0\) ∈ {0.01, 1, 10}, \(a_s\) ∈ {2π/5, 2π/10, 2π/20} and χ ∈ {−540, −100, −20, −6, 0, 6, 100, 540}, within the window \(\max(10,\,10|\hat\xi|)<\mathcal V<10^{40}\) and \(\tau_s\) ≥ 1. For χ ≥ 0 there is no minimum in any case. For χ < 0 the minimum lies exactly where the formula places it. When that place is outside the window (small \(|\chi|/g_s^{3/2}\), or \(\mathcal V\) > 10⁴⁰), there is none in the window.

What topology does here, and what it does not. It does: the sign of χ decides whether a minimum at large volume can exist, before there is anything to compare. And |χ| sets the volume exponentially through \(\tau_s\propto(-\chi)^{2/3}/g_s\). It does not: it picks no Calabi–Yau, and \(W_0\), \(g_s\) and \(a_s\) remain input. The minimum has negative energy; a universe like ours needs a separate, disputed uplift. And it is a minimum of a chosen effective potential: that nature minimizes that cost does not follow.

Limits of the rule. “Required” holds within this truncated description. Branes and orientifold planes shift the relevant χ to an effective value (Minasian, Pugh and Savelli). With extra corrections (string loops, F⁴ terms, α′² logarithms), proposals have been made that also give a minimum for χ ≥ 0; their coefficients or signs are unknown or chosen, see for example Ciupke, Louis and Westphal. The original LVS paper left the case ξ < 0 open; the scan above finds no minimum in this model.

Back to E9. The E9 family uses K3×T², with χ = 24·0 = 0. There ξ = 0 and this mechanism gives no minimum: the volume stays free at this level. Nothing changes for α: the D3 coupling of E9 depends only on \(g_s\).

Lesson for EEML. This is a clean example of layer I before layer II: a chosen cost has a minimum only if a topological condition holds. It is existing physics and does not distinguish EEML from the existing model. Historical note: older EEML texts (April 2026) used a term −χ ln \(\mathcal V\) without a derivation and required χ = +6. In the real mechanism, that sign is exactly the one that does not work.

#E11. Looking back: the old α derivation was circular

In April 2026, EEML texts circulated with a “derivation” of \(\alpha^{-1}=137.036\) from K3 and a Calabi–Yau space with χ = 6. Part II already books that route as historical. Here is exactly why, rechecked with the PDG values at \(M_Z\) (MS-bar: \(\alpha^{-1}=127.951\), \(\sin^2\theta_W=0.23122\)); the code is available on request via the contact address.

So every “derived” number is a measured value, or the gap to a measured value: input is output. The oldest versions also wrote \(\alpha^{-1}(M_Z)=137.036\); at \(M_Z\) it is about 128.

Where 24 does appear in the couplings. Run the three measured couplings at one loop with the supersymmetric extension (MSSM) and they meet at \(1/\alpha\approx24.3\), near \(2\cdot10^{16}\) GeV. That holds if the superpartners sit near \(M_Z\). At 1 TeV it becomes 26.0, at 3 TeV 26.8; two loops and thresholds shift it further. This unification is classic (Amaldi, de Boer and Fürstenau 1991). That the value lies close to χ(K3) = 24 depends on an unknown scale: it is a coincidence, not a derivation. With the Standard Model alone, the couplings do not meet.

Three more errors from the same texts. Calabi–Yau spaces with χ = 0 exist, so 6 is not “the smallest”. A Calabi–Yau threefold has no elliptic fibration over K3. And K3 carries no SU(2) instanton of charge 1: the expected dimension of that solution space is \(8k-12\), so −4 at \(k=1\); the smallest charge is 2.

New proposal, 28 September 2026. This section is a constructed research model, not a derived law of nature. It does not replace the earlier benchmarks. The hypothesis reads:

The links along which operation is possible are themselves physical degrees of freedom. Matter and links change jointly. An observable topology can emerge as a long-lasting pattern in that mutual dynamics.

So here a process that can change topology is chosen before a topological winner is named. The hypothesis that this approach can also describe space itself is an additional one. The closest existing relatives are Quantum Graphity and the Bose–Hubbard model with a dynamical graph by Hamma et al.. There, matter and changeable interactions are already coupled. This dossier claims no original discovery of that general idea; it provides an explicit, checkable EEML trial variant.

#F1. What is physical, and what is assumed?

Take N possible carriers of an excitation and, for every pair, a switchable link. A link is a two-state system with occupation \(\hat n_{ij}=0,1\). At most one bosonic excitation can be present on each carrier. A fixed, measured excitation energy may be added; at a fixed total number it is a constant.

Three parameters are model input, to be determined independently:

• \(\Delta_b>0\): energy difference between a switched-on and a switched-off link when no excitations are present;
• \(J>0\): energy of coherent exchange via an existing link;
• \(g\ge0\): quantum amplitude for changing a link.

All three have units of energy. These are not arbitrary exchange rates between material, reliability and attention. In a possible quantum simulator they must be calibrated spectroscopically. That such a model describes the universe is not thereby established.

As a concrete transition rule, a link may change only when its endpoints already have a common neighbor. Call the corresponding projector \(\hat F_{ij}\). Explicitly: \[ \hat F_{ij}=1-\prod_{k\ne i,j}(1-\hat n_{ik}\hat n_{kj}). \] With this we choose the Hamiltonian \[ \boxed{ \hat H_{\rm PT}= \Delta_b\sum_{i<j}\hat n_{ij} -J\sum_{i<j}\hat n_{ij} (\hat a_i^\dagger\hat a_j+\hat a_j^\dagger\hat a_i) -g\sum_{i<j}\hat F_{ij}\hat\sigma^x_{ij}. } \]

The common-neighbor rule is an explicit new model choice, not derived from EEML. It restricts changes to proximity as determined by the current network. \(\hat F_{ij}\) does not contain \(\hat n_{ij}\) and therefore commutes with the corresponding flip. The Hamiltonian is Hermitian. The model already has an evolution parameter: \[ i\hbar\,\partial_t|\Psi\rangle=\hat H_{\rm PT}|\Psi\rangle. \] Time itself and Lorentz symmetry are not thereby derived.

#F2. Bookkeeping and connectedness

For time-independent parameters, the closed unitary evolution conserves the total energy expectation value and the number of excitations \(\hat N_{\rm ex}=\sum_i\hat a_i^\dagger\hat a_i\). Matter can exchange energy with the link sector; those sectors need not have constant energy separately.

The chosen flips preserve the number of connected components. After all, a removed link has an alternative route via the common neighbor; an added link connects points that were already connected. A connected initial configuration therefore stays connected. One connected whole is here an initial/sector condition, not a derivation that only one universe is possible. The model needs a seed; under this flip rule an empty graph develops no links.

Unitarity alone does not cause cooling to the ground state. For thermal selection, a named environment must be able to absorb energy. In a cosmological reading, such degrees of freedom must be accounted for within the entire whole. The heat-bath approximation below is a separate, effective assumption; a lower energy guarantees neither a transition nor a prescribed recovery time.

#F3. A genuine, computed topology switch

Take three physically identifiable carriers, one excitation and a connected network. There are three distinct labeled lines with two links and one closed triangle with three links. For now there are no filled faces: the triangle is a cycle, not a disk.

In the limit of small g, the lowest energies of the fixed graphs are: \[ E_{\rm line}=2\Delta_b-\sqrt2J,\qquad E_{\rm cycle}=3\Delta_b-2J. \] The exchange spectra are \((-\sqrt2J,0,\sqrt2J)\) and \((-2J,J,J)\), respectively. It follows that \[ E_{\rm cycle}-E_{\rm line} =\Delta_b-(2-\sqrt2)J. \]

The cycle becomes energetically favorable when \[ \boxed{\frac{J}{\Delta_b}>\frac1{2-\sqrt2} =1+\frac1{\sqrt2}\simeq1.70710678.} \]

The extra link raises the link energy but lowers the energy of the joint excitation state more strongly. So no rule has been introduced that directly rewards cycles or prescribes a desired Euler characteristic. The number follows from this chosen three-carrier spectrum; it is not a new universal constant of nature.

A cycle is not yet a cycle current. This gain concerns the coherent binding/exchange energy. The real ground states of the fixed graphs have no net particle current. Here the model explains the preference for a link structure, not compulsory ongoing motion. Drive or a differently prepared state requires a separate process calculation.

Topologically, under a graph measurement the number of independent cycles \(b_1\) changes from 0 to 1; \(b_0=1\) stays the same. The graph Euler characteristic changes from \(N-|E|=1\) to 0. At finite g the state can be a coherent superposition of graphs. Then, without a measurement/decoherence description, there is no single sharply classical topology.

#F4. One quantitative prediction, with temperature and counting

Assume a weakly coupled environment that actually thermalizes the connected one-excitation sector, with independently measured temperature T. Next, take g small compared with the relevant thermal and spectral scales. Set \(\beta=1/(k_BT)\). For one labeled line and the cycle: \[ Z_{\rm line}=e^{-2\beta\Delta_b} [1+2\cosh(\sqrt2\beta J)],\qquad Z_{\rm cycle}=e^{-3\beta\Delta_b} [e^{2\beta J}+2e^{-\beta J}], \] \[ \boxed{P_{\rm cycle} =\frac{Z_{\rm cycle}}{3Z_{\rm line}+Z_{\rm cycle}}.} \]

The factor 3 counts three distinct switching configurations of identifiable carriers. It must not be adopted blindly when node labels are purely gauge labels. In that case the physical state space and measure must be constructed anew.

At low temperature this gives approximately \[ \frac{P_{\rm cycle}}{1-P_{\rm cycle}} \simeq\frac13 \exp\!\left[\frac{(2-\sqrt2)J-\Delta_b}{k_BT}\right]. \] Equal ground energy therefore does not automatically mean equal total probability for a line and a cycle. The statistical layer here follows from the specified Hamiltonian, thermal assumption and counting; efficiency and weight are not equated.

Calculation performed, \(\Delta_b=1,\ k_BT=0.05\) in the same energy units:

The finite-g column was computed by diagonalizing the full Hamiltonian, not by refitting the small-g formula. For this finite system it is an energy crossing at g=0 and a rounded quantum/thermal transition, not a proven thermodynamic phase transition.

The equilibrium formula at \(g\to0\) first requires exchange and equilibration at \(g>0\). At exactly \(g=0\) the graph stays unchanged under the closed evolution. The checked counter-test starts in a line at \(J=2\Delta_b\): the cycle probability stays zero, although the cycle has a lower ground energy. The limit of an equilibrium ensemble and a dynamically blocked preparation are different questions.

#F5. What this tests, and what is still missing

A first laboratory route is a programmable system with three identifiable excitation modes and separate switching dynamics. That also requires implementing the common-neighbor condition; no already available setup is claimed here. Calibrate \(\Delta_b,J,g,T\) separately, fix the measurement basis and the conservation of one excitation, check equilibration, and then predict the cycle occupation at unused J values without readjustment.

A reproducible deviation outside the joint uncertainty model refutes this implementation/Hamiltonian package. Agreement shows that the mechanism works; a simulator that mimics our Hamiltonian does not prove that space is fundamentally built this way. Actual cosmological support requires independent, discriminating consequences for systems not designed by us.

The step to the geometry of the universe remains large and concrete:

1. More than a graph: these one-dimensional complexes have no higher cells and cannot represent K3. Faces, volumes and their attachment rules must come from additional physical interactions; automatically filling in a triangle in fact changes the topology.
2. A large-scale limit: show that the dynamics yields stable, local geometries, with a derivable dimension and measure. It can also yield dense cliques, irregular networks or persistent superpositions. Those outcomes must not be filtered out afterwards.
3. Spacetime and physics: causality, Lorentz behavior, gravity, the matter spectrum and their measurable couplings are missing. A changing discrete graph does not automatically evade the constraints on smooth Lorentzian topology change; the conditions for a possible continuum limit must be tested separately. See Borde, Topology Change in Classical General Relativity.

Verification log: four connected graphs, a full 32-dimensional Hilbert space within this connected sector for links plus hard-core bosons, and a 12-dimensional one-excitation sector have been computed in full. Hermiticity holds exactly and the commutator with the excitation number is exactly zero in the matrices used. Numerical errors in norm and energy conservation are smaller than \(1.4\cdot10^{-15}\); the spectral formulas agree to within \(4.5\cdot10^{-16}\). The small-g partition function and the blocked-graph check have also been verified. No data from nature have been used.

Significance for EEML: in this proposal, topology is a changeable carrier structure of operation. The calculation shows how more links can require less joint energy, while reachability and energy removal remain separate conditions. The new physical content lies in the chosen carriers, couplings and flip rule. EEML has not yet derived these uniquely; PT 0.1 is therefore a concrete candidate to test and extend, not a new proof claim about K3, χ=24 or the shape of the universe.

Status, 28-09-2026: literature check and limited calculation. The submitted report uses T1–T5; their full rules had not been supplied at the time of this inclusion, nor were they found in this text. Therefore those labels are not introduced here as new axioms. This chapter checks the concrete claims from the report.

#G1. A 3D construction is not an experimental confirmation

Balasubramanian, Davydova and Lin, arXiv:2605.10943v1, dated 11 May 2026, present a local 3D stabilizer Hamiltonian with a long-lived logical qubit under specified thermal dynamics. The theorem concerns sufficiently low, nonzero temperature and a suitable encoding. The construction does not use a translation-invariant lattice; the authors discuss limitations of TQFT fixed points. A general prohibition for translation-invariant 3D codes is not thereby proven. The theorem and proof outline consulted here are not an independent verification of the full proof.

Roberts et al., Cored product codes report numerical memory times for up to 60,000 qubits. The work first appeared in October 2025, with a revision in January 2026. Within the models studied, it supports the importance of entropy suppression; it is not a measurement on a built memory.

The wording “T5 has been confirmed twice” is therefore not adopted. These model results do, however, support the familiar lesson that an energy barrier without control over the number of error paths gives insufficient information.

An elementary illustration: if a correctly derived error probability is bounded by contributions whose number is at most \(B^m\) and whose energy is at least \(m\Delta\), then for \(\beta\Delta>\ln B\):

Here energy is set against multiplicity. The required probability bound, local dynamics and conversion to a memory time must be proven separately. This geometric series does not replace a self-correction theorem.

#G2. What the lattice coefficient does and does not measure

Lucini, Rago and Rinaldi (2012) find, with Monte Carlo lattice calculations for pure SU(\(N_c\)) Yang–Mills:

These are numerical field theory data, not an observation of a Hawking–Page geometry. Their extrapolation first takes the thermodynamic and continuum limits and then large \(N_c\). A finite \(N_c\) does not rule out a sharp transition at infinite spatial volume. Moreover, this theory is not the same as \(\mathcal N=4\) SYM or minimal AdS₄ supergravity.

An additional algebraic check shows why \(0.458\) does not directly measure the shift of \(T_c\). Write \(T_c=T_0+t_1/N_c^2+\cdots\) and \(\sigma=\sigma_0+s_1/N_c^2+\cdots\). Then:

The measured coefficient mixes both corrections. So even \(t_1=0\) can give a nonzero correction to the ratio. The same power \(N_c^{-2}\) does not identify an Euler term, a log term or an EEML mechanism.

#G3. The shift that does follow

Take two saddle contributions in the same canonical theory, with the same boundary data and renormalization conditions. Use \(k_B=\hbar=1\) and define \(\Delta F=F_{\rm BH}-F_{\rm AdS}\). Assume a controlled expansion

Here \(N_c^2\) stands for the chosen large parameter; that scaling does not automatically hold for every AdS duality. First-order expansion of \(\Delta F(T_*)=0\) gives:

\(\Delta S_0=-N_c^2\Delta f_0'(T_0)\) is the leading entropy difference. With leading latent heat \(\mathcal L_0=T_0\Delta S_0\) and \(\Delta I_1=\Delta F_1/T_0\):

A positive extra black-hole free energy shifts the point of equality upward. If \(\Delta F_1=O(1)\) and \(\Delta S_0=O(N_c^2)\), then \(N_c^{-2}\) follows. With a surviving \(\log N_c\) term, the shift can contain \((A\log N_c+B)/N_c^2\). One-loop order and literally \(O(1)\) are therefore not always the same.

For a compact finite-volume problem, within a positive two-saddle approximation with prefactors included in \(I\), the following holds:

The width between 10% and 90% is then \(2\ln(9)T_0/\Delta S_0\). This is rounding around a transition of the large-\(N_c\) limit, not a finite-\(N_c\) jump in latent heat. Here too, equilibrium probabilities do not determine a switching time or reachability.

The topologies are those of competing Euclidean saddle points. Their competition does not prove a smooth Lorentzian topology change of one universe. The established AdS/CFT context is set out in Witten, Thermal Phase Transition and Confinement.

#G4. A checked AdS₄ partial result

David et al., JHEP 08 (2022) 043, table 1 and equations (3.11–3.15), use \((4\pi)^2a_4=-a_EE_4+cW^2+b_1R^2+b_2RF^2\). Their renormalized integrals give \(C_{\rm local}=0\) on thermal AdS₄. The local curvature invariants themselves are not all zero. For Schwarzschild–AdS₄, \(z=r_+^2/L^2\):

Our own substitution at the classical Hawking–Page point \(r_+=L\) gives:

for the minimal \(\mathcal N=2\) gravity multiplet, with \(a_E=-11/24,\ c=0\). The \(b_1\) contribution drops out there. This is a local log coefficient in the scaling convention of the paper. Zero modes, the choice of ensemble and finite determinant parts remain necessary for the full \(\Delta F_1\); extra fields change the result.

This is not a calculation of \(\delta T\). A log coefficient does not determine the whole function. In particular, \(11/6\) must not be substituted as \(\Delta F_1\), as \(0.458\) or as a universal EEML constant.

The thermodynamic conversion can, however, be done explicitly. For a neutral spherical black hole in AdS₄:

It follows that \(\delta T/T_0=G_4\Delta I_1/(\pi L^2)\). The suppression here is \(G_4/L^2\); the translation to \(N_c\) requires a duality dictionary.

For AdS₅, one has instead \[ T_0=\frac3{2\pi L},\quad \Delta S_0=\frac{\pi^2L^3}{2G_5}. \] With the leading AdS₅×S⁵ dictionary \(G_5=\pi L^3/(2N_c^2)\), it follows that \(\delta T/T_0=\Delta I_1/(\pi N_c^2)\). The AdS₄ coefficient from the previous paragraph cannot be inserted here.

#G5. Scope of the next genuine calculation

The smallest next step is a single AdS₄ benchmark: neutral spherical Schwarzschild–AdS₄ versus thermal AdS₄, fixed \(L,G_4\), the same field content, the same thermal boundary conditions, zero modes and one fixed renormalization prescription. Determine the complete relative one-loop free energy at \(T_0\), with error control, and insert it into G3. Any finite higher-curvature couplings must be fixed independently; otherwise this temperature correction, too, remains model-dependent.

A comparison with \(\mathcal N=4\) SYM on \(S^3\) requires a separate AdS₅×S⁵ calculation and matching at the same coupling. The thermal partition function is not automatically protected when going from weak to strong coupling. Kristensson and Wilhelm, From Hagedorn to Lee–Yang treat finite-\(N_c\) partition functions in the free theory; without further steps, that does not give the strong-coupling correction. The specific weak-coupling shift meant in the report has not been identified here as a fully suitable check.

Execution: 27 exact checks of the coefficient reduction; 16 numerical root checks of the temperature shift in AdS₄/AdS₅ with both signs of a synthetic constant \(\Delta F_1\); a check of \(dF_0/dT=-S_0\), the two-saddle width and the normalization counter-check. The largest linearization error is 0.236% at the smallest large parameter used and decreases by roughly a factor of four upon doubling. This checks the expansion, not a physical determinant. No new lattice data, memory experiment or full one-loop calculation has been carried out.

Consequence for EEML: a small sector contribution can determine the dominant contribution when the leading free energies are nearly equal. That is a conditional thermodynamic mechanism. To turn it into an EEML prediction of its own, EEML must supply an independently determined contribution or relation that the existing theory does not yet fix. The supplied T2/T4/T5 statuses are therefore not adopted as empirical confirmation.