Test protocol and open work

1. Effect and target quantity: what is reached, for how long, with what tolerance? Which failures are possible? Specify in advance which quantity f is to be predicted.
2. Candidates and context: states, trajectories, preparation, system boundary, environment and equivalences.
3. Conditions: what comes from independent measurements, existing theory and new assumptions? For a cost/action family, also give the physical origin of signs and ratios, conventions, renormalization conditions, valid scale and uncertainties. Keep track of all members that are still admissible; an artificially small class is no additional evidence.
4. Input and weight: cost vector, any conversion, measure, weighting rule, normalization and ε. No hidden choice of a desired winner.
5. Process: transition law, time scale, drive, barriers, memory and adaptation. Check stated stationary weights against the dynamics.
6. Look for simultaneous saturation: collect proven bounds and their hypotheses; check mutual dependence; write out the equality conditions; investigate whether they are jointly realizable; determine all equality cases. An infimum that is not attained does not count as a realized minimum. State separately whether saturation is derived from the model or imposed as an additional selection hypothesis.
7. Determine closure: remove only physically equivalent descriptions; report the dimension per component, counts where discrete, and the set of possible f-values. Dimension zero is not the same as a single outcome. Also check whether remaining free directions affect the target quantity.
8. Ablation: determine minimal closing subsets for that same f. Also test which conditions are needed for existence and for saturation. ‘Minimal’ depends on what counts as one condition. Full enumeration can require an exponential amount of work.
9. Negative controls: record what the package does not determine. New hypotheses may turn a negative control into a test case only after visible registration.
10. Independent test: do not use the same outcome for model choice, tuning and confirmation. A retrodiction that has already been used for the design is not an independent prediction.
11. Decision rule: expected effect size, uncertainty, comparison model, falsification criterion and discriminating power. ‘Undecided’ remains a valid outcome.
12. Self-application: also check whether EEML introduces extra assumptions, redefines the effect after the fact, or merely renames known theory. Closure within a class is not a derivation that nature realizes that class.

Origin and date of inclusion: review of 28 September 2026; included on 28 September 2026. According to the latest clarification, EEML-C1 is an operational proposal from that review, not an external preregistration. CPH-ν remains the additional candidate hypothesis dated earlier on 28-09-2026. The date attached to a proposal, the reproducibility of a calculation, the registration of a decision rule and the execution of a test are different matters.

EEML-C1: fixed proxy. The supplied linearization of the absolute stability bound is:

Solving for the critical top mass gives:

This is the proxy chosen by the author, in line with the explicit linearization in Buttazzo et al., Investigating the near-criticality of the Higgs boson, equation (64). Using it is not a new NNLO fit or a new analysis of experimental events.

The supplied inputs, fixed for this proposal, are:

These values are input from the review, not current world averages re-established here. The statistic is:

The mass interpretation remains explicit. The direct mass measurement is not simply a precise pole-mass measurement. Write \(M_t^{\rm pole}=M_t^{\rm direct}+b_{\rm map}\). For the central numbers below, \(b_{\rm map}=0\) is chosen as a reference and its uncertainty is varied separately. That is a scenario assumption, not a derivation that the conversion is exactly zero. A nonzero central conversion also shifts the numerator of the statistic.

Substituting gives:

Error budget: checked calculation convention. For the following sensitivity calculation, the contributions are treated as independent and combined in quadrature. Here, interpreting the theory and conversion uncertainties as standard deviations is an additional statistical assumption.

Hence:

Visible arithmetic correction: the supplied summary ‘about 1.5–2σ’ was a rough indication. The full band from this error budget is, rounded, 1.41–2.24. This is a band across two chosen uncertainty scenarios, not a confidence interval for a single estimated parameter. Nor do the numbers prove that 0.5–1.0 GeV is the correct conversion uncertainty; that scale is model information supplied as input here.

When correlations are available, the calculation must be replaced by the corresponding covariance propagation. With \(f=M_t^{\rm crit}\), \(v=(1,-\partial_{M_H}f,-\partial_{\alpha_s}f)\) and input covariance Σ, the linear form is \(v^{\mathsf T}\Sigma v\), plus only the genuinely separate extra contributions. Non-Gaussian systematic errors require a suitable model. Without such a statistical foundation, there is no conversion of this normalized distance into a p-value or discovery significance.

Claim and decision rule: the canonical EEML core contains no generic mechanism for exact criticality. This is the previously delimited theoretical conclusion; C1 predicts no future value or band for the distance. Decision rule: none. A change in \(\delta_{\rm crit}\) therefore does not by itself refute EEML. A later hypothesis about proximity to the bound requires, in advance, its definition, treatment of uncertainties and exclusion limit. A proxy, mass conversion or error budget changed later must be recorded as a visible protocol change.

The comparison with the stronger MPP requirement \(\lambda(\mu_\star)=\beta_\lambda(\mu_\star)=0\) remains open. This requirement and the stability-line proxy are not the same executed test. The proposal starts no automatic monitoring and contains no new empirical dataset.

CPH-ν: Majorana as an additional hypothesis. The original idea is that, for the same observed effect, less remaining continuous charge freedom is preferred, so that a lepton-number-violating/Majorana description is expected rather than an exact Dirac scenario. That is extra model content. A Majorana term can remove one charge freedom and at the same time add other parameters; other interactions can likewise remove freedom. The freedom to be minimized and a necessary or quantitative prediction therefore still have to be specified.

A limited test can, however, be described precisely. Under three light Majorana neutrinos, dominance of their standard exchange and fixed remaining interaction and nuclear-physics assumptions:

With \(a_i=|U_{ei}|^2m_i\), the minimum over the free Majorana phases is \(\max(0,2\max_i a_i-\sum_i a_i)\). An established inverted ordering plus exclusion of the entire band fixed in advance, including experimental and theoretical uncertainties, would refute this limited model package. The sensitivity must be stated quantitatively; ‘ton-scale’ alone does not suffice.

This criterion does not automatically refute all Majorana theories, general CPH or EEML. With normal ordering, mutual cancellation and blind regions can exist. Alternative mechanisms or interference may not be added after the fact to the same tested model. A discovery of neutrinoless double-beta decay supports, under the usual conditions, lepton-number violation and a Majorana contribution, but does not automatically identify the dominant mass mechanism or CPH as the cause. See Duerr, Lindner and Merle.

Operational protocol still incomplete: there is as yet no fixed combination of isotopes, sensitivities, parameter intervals, nuclear matrix elements, statistical decision rule and quantitative CPH probability claim. Nor does the core of EEML predict a unique neutrino mass ordering. The independent test coverage must be reported; an unquantified ‘preference’ cannot be sharply refuted by a single outcome.

The wording ‘risk low as long as normal ordering holds’ remains recognizable in the registration history as a qualitative assessment of limited test coverage. It is not a calculated probability of discovery, exclusion or a false rejection. Normal ordering allows blind regions; inverted ordering can, for the limited standard-exchange model, give a band that can be fully covered. The stated assumptions and uncertainties remain necessary for that.

#Other components: registration and execution kept separate

‘Not yet registered’ does not mean that no substantive calculation has ever been done on it; nor does ‘already discussed or calculated’ mean that a test protocol existed in advance. This inventory preserves that distinction without claiming additional results.

Execution: 28 September 2026; constructive model checks, not an empirical test or a prediction registered in advance. The question is whether the current package of structural conditions fixes the intended quantity. For each control, two allowed inputs were constructed that give different outcomes. That refutes uniqueness within that package; it does not say that both models reproduce all measurements of nature.

For the particle controls we use three families with the fixed representations and Yukawa-invariant hypercharges, without CPH, imposed CP symmetry, axion mechanism or additional flavor law. In the left-chiral convention: \[ (q,u,d,l,e,h)=(1/3,-4/3,2/3,-1,2,1). \] The seven residuals \(q+u+h,\ q+d-h,\ l+e-h,\ 2q+u+d,\ 3q+l,\ 6q+3u+3d+2l+e\) and \(6q^3+3u^3+3d^3+2l^3+e^3\) are, in rational arithmetic, all exactly zero. Yukawa invariance fixes charges, not the coefficients of the allowed interactions. The three-family choice also has an even number of SU(2) doublets. This is not a full UV consistency test.

The mixing-matrix check gives a maximum unitarity error of \(2.22\cdot10^{-16}\); the singular values and determinant were checked numerically. The Λ residuals are exactly zero in rational arithmetic. For Λ, it is therefore the corresponding solution that was changed, not the constant within the same flat solution. The chosen units and numbers serve as a counterexample, not as a cosmological estimate.

This freedom is in line with the parameter structure of the Standard Model, PDG. The check concerns the packages explicitly named here, not two fully constructed string vacua that simultaneously satisfy every condition in this dossier. The layer-III measure, a process selection or a new additional law may later provide extra information, but is not available here.

Conclusion of the four controls: the current package closes hypercharges under conditions, but not Yukawa magnitudes, mixing angles, strong CP or Λ. This is a result of the protocol as executed. It is not a failed numerical fit, nor a physical refutation of an EEML prediction made in advance: for these quantities, no closed prediction had been derived in the first place.

Start with a process whose events, waiting times, support and effect are separately measurable. For a fully observed time-homogeneous Markov chain, transition counts \(N_{ij}\) and total residence times \(T_i\) can be used to estimate rates:

This requires that observation and model justify this estimate. With limited data, an unobserved transition is not proven impossible.

Check whether a single reconstructed law simultaneously describes occupation, currents and waiting times. Then predict a new blockage or load, fixed in advance, without retuning. Keep track of alternatives that explain the same existing data. A successful reconstruction first supports that process model; for a result that is EEML's own, it must be clear in advance which extra relation EEML predicts.

H1a, H1b and H2 from Part V remain separate hypotheses. Quasipotential agreement is something other than reversibility. Minimal dissipation over all generators is not a proven general law. Setting a cost equal to \(-\varepsilon\log\pi\) after the fact makes agreement tautological and provides no independent test.

Revised test proposal, 28 September 2026. D1 operationalizes a conditional elimination claim, not a universal prohibition and not an automatic dichotomy based on the outcome. Physarum is the chosen development example. Literature and outcomes already examined are development information. The registration is frozen only after the measurement and calibration fields have been filled in, and before any inspection of independent test data.

#Revised status: testing conditions, no outcome as proof of class

The strong H_D formulation remains above as history. D1 now investigates which process conditions, fixed in advance, justify elimination, and then tests their quantitative consequence. Survival alone does not identify the dynamics. If a prediction fails, the process model, the coupling to the measured quantities and the hypothesis are examined; the conditions are not adjusted afterwards to make the outcome fit after all.

The connection to evolutionary game theory is useful but not literally identical:

• Pareto dominance compares different cost components for the same effect.
• Strict dominance in a game compares a single payoff for every possible composition of opponents: \(f_Y(z)>f_X(z)\) for all relevant z.
• The conversion of a cost vector into that payoff is additional domain input. “Nowhere worse and somewhere better” across opponents is weak dominance, for which elimination does not follow in general.
• An alternative that initially has a positive share is more than an alternative that is merely reachable. Pure replicator dynamics does not discover a strategy with zero initial share.

For regular dynamics \(\dot x_i=x_i g_i(x)\), payoff monotonicity means that \(f_i>f_j\) also gives \(g_i>g_j\). Convex monotonicity requires the stronger comparison with mixed strategies: \(\sum_i p_i f_i>f_j\Rightarrow\sum_i p_i g_i>g_j\). Within that class, Hofbauer–Weibull (1996) proves elimination of iteratively strictly dominated pure strategies from an interior initial state. The different dominance/monotonicity classes are explicitly distinguished by Viossat. “Every below-average strategy shrinks” is not a complete substitute for these conditions; nor does imitative mean gradient-like.

The negative claim from the literature is also limited. Under continuity, positive correlation, Nash stationarity and innovation, Hofbauer–Sandholm (2011) gives an existence result for games with surviving strictly dominated strategies. This does not mean that they survive in every innovative process. Moreover, there are imitation dynamics with such survival: Mertikopoulos–Viossat (2022). The dichotomy “imitation eliminates, innovation circulates” is therefore withdrawn.

#An exact, finitely testable reference

For the replicator law \(\dot x_i=x_i(f_i-\bar f)\), positive initial support on X and Y and a uniform advantage \(f_Y-f_X\ge\delta>0\), in the chosen unit of time:

For a detection tolerance \(\tau>0\), this yields a guaranteed upper bound \(x_X\le\tau\) as soon as \[ t\ge \max\!\left(0,\frac1\delta \log\frac{x_X(0)}{\tau x_Y(0)}\right). \] If X is initially absent, its share in this replicator reference remains zero. This is a conditional model result, not a universal EEML clock. The payoff normalization sets the time scale. For a time-dependent advantage, the integral of the advantage is decisive. With constant payoffs \(f_Y=1,f_X=0\) and equal initial shares, one has exactly \(x_X(5)=1/(1+e^5)=0.006693\), below the bound \(e^{-5}=0.006738\).

Two counter-checks carried out:

1. For the replicator with payoff matrix \[ A=\begin{pmatrix}0&0&1\\0&0&0\\-1&-1&-1\end{pmatrix} \] strategy 1 dominates strategy 2 only weakly. Starting from \(x=(1,1,1)/3\), one has \(f_2=0\) and \(\bar f=-x_3(1-x_1)\le0\), so \(x_2\) does not decrease. Numerically, \(x_2(30)=0.408248\). Halving the integration step changes the final distribution by less than \(2.4\cdot10^{-12}\). Applying a strict-dominance theorem here would be wrong.
2. With a constant advantage for Y, add symmetric inflow: \[ \dot x=x(1-x)+\mu(1-2x),\qquad x=x_Y. \] At \(\mu=0.1\), the stable equilibrium is \(x=0.909902\). X therefore retains \(9.01\%\), without a cycle: this is autonomous one-dimensional dynamics. This example accounts for inflow, but does not claim to satisfy all conditions of the Hofbauer–Sandholm theorem.

#What is actually measured?

The classification therefore has several axes. Selection monotonicity orders growth by payoff. H1b requires detailed balance of a specified stochastic process law. Replicator dynamics can circulate in non-potential games; conversely, a reversible noise process can keep visiting worse states. Neither axis is read off from a stationary distribution alone. Limited measurement data can exclude a class or be consistent with it, but do not prove all global conditions of a theorem.

#Kramers: a mean waiting time, not a hard end time

For the separate reference \[ dX=-V'(X)\,dt+\sqrt{2\varepsilon}\,dW \] with small noise, a suitable smooth potential, a nondegenerate local minimum \(x_m\) and a separating maximum \(x_s\), the one-dimensional Eyring–Kramers approximation with unit mobility gives: \[ \mathbb E\tau_{\rm escape}\sim \frac{2\pi}{\sqrt{V''(x_m)|V''(x_s)|}}\, e^{[V(x_s)-V(x_m)]/\varepsilon}. \] See Berglund, Kramers' law.

This is an asymptotic mean time for the chosen transition; it is not an upper bound for every realization. Barrier, noise and prefactor must be determined independently. A lower destination does not prove a low barrier. Return remains possible; with finite noise, a higher-cost state can have positive stationary occupation, even under detailed balance. The limits \(\varepsilon\to0\) and \(t\to\infty\) cannot be interchanged without conditions: removing the noise first can permanently lock in a local minimum. D1 therefore reports residence times as well as return and occupation.

#The general comparison form

This form determines effect, costs, dominance and reachability. A subsequent elimination claim additionally requires the independently tested process conditions described above. “Violation” means violation of that claim as named in advance, not automatically that every surviving dominated candidate breaks a law of nature.

Starting condition: choosing a process alone is not sufficient to fill in these fields numerically. The measurement setup and independent calibration determine resolution and error margins. These must not be chosen on the basis of the desired D1 outcome.

#Filled-in Physarum design D1-P (before a new measurement)

Status: concrete design, not a frozen measurement registration. The choices below are protocol input. The literature data were first assessed for usability; no suitable complete dataset has been verified yet. Therefore no measurements, physiological constants or numbers of organisms are invented.

A finite library of reachable Y's can yield a dominance witness, but cannot prove that X is globally undominated. For the replicator reference, a conservative advantage bound from calibration is used to fix the time prediction. A Kramers prediction requires a separately validated noise/barrier model; the exponent is not back-fitted from the same recovery time and then called “predicted”.

Data gate before execution: the complete time series of geometry, flow or tracer, drive, preparation and permitted interventions must be available, with independent replicates and error estimates. The checked Dryad descriptions do not provide all of these data. The mediaTUM repository remained inaccessible on rechecking. The literature survey has thus been carried out, but the analysis of literature data has not yet proved possible. Numbers of organisms, numerical material limit, measurement tolerances and statistical discriminating power are fixed only from independent calibration. Version freezing follows after that; until then, this remains a design.

#Costs: what the √k term really means

For a round tube with length L, radius r and viscosity μ:

D is the conductivity factor before division by length; k is the full hydraulic conductance of the tube. The material/fluid volume of this cylinder is:

“√k costs” therefore require a specified k convention and the correct length factor. This gives a volume proxy. The assumption that metabolic maintenance is proportional to it requires a separate justification. A model-based cost comparison is not a measurement of maintenance energy. Watanabe and Takamatsu (2014) use loss, construction/maintenance costs and vulnerability as different performance measures in a Physarum-inspired algorithm.

In the stationary or justified quasi-static Poiseuille approximation, the first component can be chosen as \[ R_{\rm diss}=\frac1T\int_0^T\sum_e\frac{Q_e(t)^2}{k_e(t)}\,dt. \] With changing tube volumes and active contractions, the full drive, storage and dissipation accounting must be checked. Flow derived from a tube model may not be presented as independently measured flow. For recovery, “not returned within the measurement window” is a censored outcome; it is not silently entered as recovery exactly at the end time.

#Reachability without circular reasoning

A purely optimized network Y is not yet a physically reachable Y. A witness can be an independently validated trajectory or a replicated realization within the same initial and environment class described in advance. Preparation differences, measurement uncertainty and required control are recorded explicitly.

A deterministic process law with one complete initial condition and one fixed input usually yields one trajectory. A set of alternatives arises only through variations, noise or control permitted in advance. Those variations must be included in C and z₀. Nor may the reachability law already assume that all dominated final states disappear: the test would then presuppose its own outcome.

If both Y’s reachability and its low costs are computed exclusively by the process model, the conclusion is conditional on that model. For a physical refutation, this combined measurement and model package must be sufficiently substantiated independently.

#Assessing dominance and persistence separately

Write \(\Delta_a=R_a(Y)-R_a(X)\). Let \([L_a,U_a]\) be joint uncertainty bounds, with the treatment, fixed in advance, of shared measurement errors and the number of candidates examined. A conservative witness of dominance satisfies:

\[ U_a\le0\quad\text{for every component }a,\qquad U_b<-\delta_b\quad\text{for at least one component }b, \] with \(\delta_b>0\) fixed in advance. Effect, constraints and reachability must be sufficiently certain at the same time. Not being significantly more expensive is not proof of not being more expensive. Introducing a small permitted deterioration means an explicit, relaxed variant of dominance and must not be done afterwards.

A found, confirmed Y is sufficient to demonstrate dominance. Not finding a Y with an incomplete search procedure does not prove that no Y exists. Therefore also record the coverage of the search space; without sufficient coverage, “no violation found” remains a limited outcome.

“X persists” gets its own graph/capacity tolerance, observation duration and decision rule. Under its conditions, the replicator reference yields the time bound derived above; the Kramers reference yields an asymptotic mean time. Where such a derivation is lacking, a finite period is additional hypothesis input. A tested \(H_D(T)\) is not equated afterwards with the unbounded “X eventually disappears”. For statistical claims, the permitted survival probability is also fixed in advance.

#Both a negative and a positive control

The negative control contains a cheap Y that demonstrably lies outside the reachable set fixed in advance. That Y must not count as a witness of a violation. The positive control contains a demonstrably reachable Y with the same effect and lower costs; the method must recognize this Y.

Correction to the supplied formulation: the fact that the method finds no violation anywhere does not by itself prove that reachability is too narrow. It can also be a correct outcome, insufficient search coverage or insufficient measurement precision. The positive control tests whether the method can find a known dominance. Moreover, a wrong control answer can be due to the implementation or the measurement model, not exclusively to the reachability definition.

Six synthetic controls have been carried out: unreachable Y, reachable dominating Y, a cost trade-off, uncertainty about not being more expensive, a Y outside the time limit and a Y with the wrong effect. They give, respectively, inadmissible, dominance demonstrated, non-dominating, undecided, inadmissible and inadmissible. The volume identities have also been checked on 36 combinations of radius, length and viscosity. This validates only those computational steps and the decision logic.

#Which literature data can help?

This literature survey uses paper content and data descriptions. No raw datasets have been downloaded or reanalyzed. Published results are already known and count as development/retrodiction for this dossier. An independent test requires unused data separated in advance, or a new measurement. The statement that the complete form can be filled in directly with existing data has not yet been demonstrated.

Evidential weight of an outcome: a valid counterexample refutes the universal prohibition formulation insofar as that formulation actually excludes this case. It does not automatically make all other EEML models and mathematical results incorrect. A domain claim restricted afterwards is a visible revision, not a rescue of the same universal claim. A successful domain test does not prove universality; it only supports the formulation specified in advance and tested, within its scope.

Current D1 status: the scientific status has been revised, the class features and reference predictions have been worked out, and the Physarum design and the data gate have been filled in. The synthetic controls have been carried out. Measurement calibration, numbers of independent organisms and sufficiently complete data are still missing for a frozen test against nature. A miss is not automatically relabeled as “another class”: first, the independently tested premises and measurement errors are examined.

Origin: submitted proposal of 28 September 2026, checked and sharpened in this master version. Status: proposal plus algebraic and synthetic checks; not yet a full preregistration, new dataset or physical confirmation. The existing T 0.3 theorem is retained. The experimental identification of its variables and the choice of an organism or experimental setup are additional hypotheses.

The central question is concrete: can a single pre-calibrated process law predict the course after a new blockage, including route use and structural memory, without retuning? Physarum is a candidate. The model below initially applies to three parallel routes with positive, unambiguously defined transport. A real Physarum network must be tested against that reduction.

#P1.1. Do not confuse measured quantities with effect

Measure \(Q_i(t)\), \(Q_{\rm tot}(t)=\sum_iQ_i(t)>0\), route capacities \(D_i(t)>0\), and a transport performance over a time window, fixed in advance. Define, where the measured flows are non-negative:

What D measures (for example a hydraulic conductivity factor) must be fixed in advance. Do not use a D back-calculated from x to then test the relation between x and h. D, route geometry, pressure/drive and flow must be measured separately where possible; shared measurement errors remain in the uncertainty budget.

Shares do not predict total performance. The same x and h fit both a total throughput of 1 and one of 10. The T 0.3 model contains no equation for \(Q_{\rm tot}\), failure probabilities or the absolute magnitude of capacity. P1 initially tests the distribution and memory development, conditional on the specified drive. Total throughput and effect are checked separately. A prediction of effect success probability first requires an extension with absolute capacity, drive, noise and possible failures. Failed trials must not be silently filtered out.

Physarum moreover exhibits back-and-forth, contraction-driven flow. A raw instantaneous flow is therefore not automatically a probability share. Direction, measurement cross-section, time averaging and, where applicable, tracer transport must be chosen in advance. Summing \(|Q_i|\) measures activity and is not by itself net transport from source to destination. See Alim et al., Random network peristalsis in Physarum polycephalum organizes fluid flows across an individual.

#P1.2. Fully specify the model and the blockage

For this first candidate, choose fixed, independently determined costs \(c_i\) and \(\Phi(x)=\sum_i c_ix_i\). Costs and ε have the same units. The costs must not be chosen after the fact to reproduce measured shares. Fixed length can be a route characteristic; actual hydraulic resistance can change during structural adaptation and is therefore not simply a constant c.

Let A be the set of open routes within one phase. Define:

x lies on the simplex of A; h stays on the simplex of all three routes. On the closed route x=0, and the memory decays as \(h_i(t)=h_i(t_b)e^{-\eta(t-t_b)}\). That memory is not set to zero.

As an explicit ideal intervention law upon closing, we choose \(x_i^+=0\) outside A and \(x_i^+=x_i^-/\sum_{j\in A}x_j^-\) inside A, provided the denominator is positive; \(h^+=h^-\). This is an imposed instantaneous redistribution, not a derived law of fluid flow. A real setup with storage, a different flow response or a changing throughput requires a different intervention law, described in advance. Upon reopening, only A changes, and x and h are not reset.

With the convention \(0\log(0/h)=0\), we keep \(\Psi=c\cdot x+\varepsilon D_{\rm KL}(x\|h)\). Within a fixed phase with positive shares on A:

The KL terms between x and p are taken over their common support A; the last sum covers all three routes. The same derivation works because the constant normalization term drops out against \(\sum_{i\in A}\dot x_i=0\). The jump upon closing is allowed. Upon reopening, Ψ is continuous for this intervention law, while a reopened route with x=0 and p>0 gives a one-sided infinite entropy slope. Use finite time differences there; do not add an artificial small flow.

Renormalizing memory changes the equation. For \(H_A=\sum_{i\in A}h_i\) and \(z_i=h_i/H_A\) on A, the following holds:

Replacing h by z with unchanged η is therefore a different model. With time-dependent costs, \(\dot c\cdot x\) is added to \(\dot\Psi\); with costs that depend on structure, the corresponding chain-rule term is added. “Within a phase” therefore also means that these hypotheses actually hold.

#P1.3. When is capacity really the T 0.3 memory?

Normalizing a capacity does not automatically make its evolution law \(\dot h=\eta(x-h)\). For the explicitly chosen reinforcement law \(\dot D_i=aQ_i^\mu-\gamma D_i\), with non-negative \(Q_i=Q_{\rm tot}x_i\), it follows that:

For μ=1 this becomes the T 0.3 form, but with \(\eta_{\rm eff}=aQ_{\rm tot}/D_{\rm tot}\). A constant η requires a constant ratio or a justified approximation. With fixed Q and preparation \(D_{\rm tot}=aQ/\gamma\), that ratio is indeed constant and η=γ. This is a conditional connection to known adaptive flow models, not a proof that every measured capacity memory follows this law.

There is moreover an exact overlap with ordinary flow physics. For three passive parallel channels with the same pressure drop and with conductivity factor D, the following holds: \[ Q_i=\frac{D_i}{L_i}\Delta P,\qquad x_i=\frac{h_i/L_i}{\sum_jh_j/L_j}. \] The algebraic logit target distribution has the same form if one chooses \(c_i=\varepsilon\log(L_i/L_0)\). That is a demonstration of possible indistinguishability, not permission to introduce this cost after inspecting the data. The inertia, capacity law and independently fixed cost remain testable. If D is already the full channel conductance including length, this simple hydraulics gives precisely x=h.

A hydraulic adaptation model is therefore a necessary reference as soon as P1 is applied to real tubes. Ito, Johansson, Nakagaki and Tero describe an existing combination of Kirchhoff flow and flow-reinforced conductivity. Recognizing a related memory law is not in itself a new result about nature.

#P1.4. Calibration requires change, not only a quiet baseline

In an equilibrium of T 0.3, x=h=p. For every route in use, the constants c_i are then equal. With a strictly cheapest route, the limit can lie on the boundary. With equal costs and x=h, all positive κ, η and ε are compatible with exactly the same resting measurement. A quiet baseline therefore does not identify these parameters; interior equilibria with different constant costs do not exist here.

Therefore use a calibration phase with sufficient transients, separated in advance: known preparations or small, predetermined perturbations. Keep the later blockage/reopening trials as test data. Before that test, check whether the time resolution can distinguish κ and η, and whether the cost contrasts can determine ε. Report the joint parameter uncertainty; freeze the estimation method and the uncertainty distribution, not just three rounded numbers.

Two additional relations make the content concrete:

\[ h_i(t_2)-h_i(t_1)=\eta\int_{t_1}^{t_2}(x_i-h_i)\,dt, \] \[ \log\frac{(x_i+\dot x_i/\kappa)h_j} {(x_j+\dot x_j/\kappa)h_i} =-\frac{c_i-c_j}{\varepsilon}\qquad(i,j\in A). \]

The second relation holds where the reconstructed p shares are positive; numerically differentiating noise requires a measurement model fixed in advance. The first can be tested directly with integrals. Both require the same parameters for all routes and test phases. Earlier, a nonlinear Φ was used in T 0.3; the interior equilibria found there are no justification for a flat baseline in this new linear variant.

#P1.5. References must be as concrete as EEML

“A freer memory model” does not yet specify a testable rival. A delimited candidate family is: \[ p_i^{A,\rho}\propto\mathbf1_{i\in A}h_i^\rho e^{-c_i/\varepsilon}, \qquad \dot x=\kappa(p^{A,\rho}-x),\qquad \dot h=\eta(x-h). \]

Here M0 is given a passive structural variable so that the same measurements can be assessed; h does not steer its route choice. A purely static formula without a time law is not a full-fledged rival for the recovery rate.

M1 contains M2. Therefore “M2 must always predict better than M1” is not a necessary consequence of M2. Equivalent prediction may in fact mean that the extra freedom is not needed, or that the experiment does not distinguish them sufficiently. Compare prediction errors and uncertainty on unused data, with a predetermined equivalence margin. A required statistical win over every broader model is not a valid automatic falsification rule.

The ablations are likewise precise: η=0 switches off learning but keeps a fixed route preference; ρ=0 switches off the feedback of h on choice; equal costs remove the cost contrast. “h constant” and “η=0” are the same ablation within this law, not two independent pieces of evidence.

#P1.6. Decision rules: the decrease is necessary, but not sufficient

A different positive time scale can make the same Ψ decrease and still give the wrong recovery rate. The check below doubles both rates only after reopening: even then Ψ decreases, but the route shares deviate from the prediction frozen in advance. Therefore assess the full course of both x and h, with Ψ as an additional model check. The final occupancy alone does not count as a process test.

The proposed test statistic \[ V_+=\sum_k\max\{0,\Psi(t_{k+1})-\Psi(t_k)-\delta_k\} \] can be useful, but “approximately zero” is not a decision threshold. Measurement times, windows, δ_k, shared measurement errors, parameter uncertainty and a threshold for V_+ must be fixed ahead of the test data. Pointwise error margins do not automatically protect against a false alarm over many time steps. Calibrate the overall decision rule with the measurement model chosen in advance and report its discriminating power.

The decrease theorem applies to the deterministic model state. Measurement noise or physical fluctuations need not give a monotonic measured trajectory. A stochastic extension requires its own derivation; noise must not be invoked without limit after the fact to explain every deviation. Compute Ψ from independent observations with their measurement model, not only from the curve predicted by M2.

Still missing for the experimental registration: concrete setup, operational D and c, direction/averaging rule, intervention response, number of independent networks, calibration and test windows, blockage durations, measurement frequency, detection limits, error model, parameter ranges, prediction score and numerical decision thresholds. Several blockage durations are useful for distinguishing mechanisms, but their choice must be frozen before the test. A recovery criterion must also be able to report “not recovered within the measurement window”.

Provisional test claim: a single fixed M2 parameter distribution predicts, after new interventions, the route shares and the measurably defined memory, within predetermined error bounds and alongside explicit rivals. A verified violation rejects this M2-plus-measurement model in the named domain. A successful test supports that process model; a tie with known hydraulics does not yet yield a distinctive EEML law of nature.

#P1.7. What has now actually been checked

Without new measurement data, the following were carried out: 1,000 states for the Lyapunov identity, spread across open and blocked phases; the conversion to normalized active memory; 27 parameter combinations that give the same quiet baseline; 300 normalizations of capacity laws; 300 examples of hydraulic-logit overlap; and a three-part synthetic trajectory with blockage and reopening, including halving of the integration step.

The largest numerical deviation in the Lyapunov identity is \(1.78\times10^{-15}\). Halving the time step changes the computed trajectory by at most \(1.74\times10^{-12}\). The two processes with different rates after reopening both have decreasing Ψ, but differ by at most 0.239 in total variation distance of the route shares. These are checks of chosen, dimensionless examples; not measurement precision or an effect size from an organism.

Furthermore, counterexamples were checked for “decrease determines the recovery rate”, “the same decrease identity holds for a freer memory coupling”, “changing costs automatically preserve decrease” and “shares determine total throughput”. The results test the proposal for internal consistency. The experiment itself has not been carried out.

• An independent physical derivation of relevant costs, measure, noise scale and process law per domain.
• An extra relation between the four layers that differs empirically from existing domain models.
• Independent measurement calibration and execution of D1-P, with class conditions measured in advance and the corresponding quantitative prediction; in addition, a concrete experimental model package for H_R.
• The regime of validity of an EEML selection or adaptation law; a universal minimum has not been proven.
• A new independent dataset with sufficient discriminating power.
• Topology selection and cosmology: the K3 bound does not replace the missing physical selection process.

For anyone building on this, researcher or AI, the priority is: treat EEML as a proposed theory of how processes work; keep the corrections, choose one delimited process and formulate one new discriminating test. Material laws and constants may be explicit input. K3, 24, 137, a unique cosmic zero point or a mandatory winner must not be fixed in advance as the outcome.