# Apeiron/AME-1: Terminal Numerical Classification of a Frozen Homogeneous Model and a Prospective Falsification Program

**Venue-neutral master manuscript — controlled technical draft**  
**Author(s):** [AUTHORSHIP TO BE PROVIDED]  
**Affiliation(s):** [AFFILIATION TO BE PROVIDED]  
**Corresponding author:** [TO BE PROVIDED]  
**Status:** technical core documented for preprint packaging; venue-specific and administrative completion outstanding  
**Numerical reference:** terminal v7.13 Stage-0 HARD-PASS; Float64 crossing schedule complete and frozen

## 1. Title and cautious abstract

### Abstract

We report the terminal numerical classification of a frozen homogeneous Apeiron/AME-1 model and separate this result from physical, empirical and ontological claims. The model is defined operationally on a spatially flat expanding FRW background by a noncanonical pressure function, a portal-coupled renormalized mode source and a discrete fixed-point map on 15361 time nodes. At the binary64 endpoint \(\tau=0.47878124999999994\), corresponding to the exact protocol decimal \(0.47878124999999994315658113919198513031005859375\), the inherited v7.13 state already satisfied the fixed-point criterion and required no Newton or GMRES update. The recorded fixed-point, Friedmann, continuity, Raychaudhuri, quantum-source and Wronskian diagnostics are \(4.6838977141305804\times10^{-11}\), \(3.818888298257304\times10^{-12}\), \(3.7214611922998273\times10^{-6}\), \(1.4051164250399966\times10^{-5}\), \(3.883856960984437\times10^{-8}\) and \(7.432742701266731\times10^{-9}\), respectively. All preregistered v7.13 gates passed and no finalization failure was recorded, yielding a HARD-PASS under the frozen protocol. The adjacent binary64 endpoint \(\tau=0.47878125\) is an immutable v6.74 NONPASS because its fixed-point and Friedmann gates failed. The classified endpoints are adjacent representable values; the exact protocol midpoint rounds to the NONPASS endpoint, terminating the frozen Float64 schedule. The release includes the operational equations, exact gate table, hash manifest, immutable artifacts and a read-only integrity verifier. This verifier does not rerun the solver or recompute physical diagnostics. The result therefore establishes compliance with the stated numerical criteria for this discretization, not continuum convergence, empirical confirmation or an Apeiron ontology. AP1–AP3 are presented only as a prospective falsification program; no observable curve, data fit or empirical preference is claimed.

**Keywords:** numerical cosmology; fixed-point residual; gate classification; binary64 boundary; reproducibility; falsifiability

## 2. Motivation and problem statement

Foundational model programs face distinct burdens of evidence. First, their internal equations and numerical decisions must be explicit enough to audit. Second, their computations must satisfy declared criteria without retrospective changes. Third, their physical interpretation must be converted into observables and compared with data. Passing one burden does not discharge the others.

The present work addresses the first two burdens for one frozen homogeneous Apeiron/AME-1 calculation. It asks:

1. What model and discrete map were actually evaluated?
2. What exact rules produced the terminal PASS/NONPASS classifications?
3. What does the terminal result establish, and what remains open?
4. How can later observable tests be preregistered without changing the completed numerical line?

The v7.13 result and its adjacent v6.74 counter-endpoint are treated as immutable. No solver rerun, gate change, parameter adjustment or new physical assumption is introduced in preparing this manuscript.

## 3. Definition and delimitation of the Apeiron model

### 3.1 Operational definition

The paper uses “Apeiron/AME-1” as the name of a frozen homogeneous numerical model. Its authoritative publication definition is operational: the variables, parameters, pressure function, portal coupling, quantum-source prescription, background equations, initial data, discretization and active source hashes are stated in APEIRON_CANONICAL_MODEL_SPECIFICATION_LATEST.md.

The background is spatially flat FRW with scale factor \(a\), e-fold variable \(N=\ln a\), Hubble rate \(H>0\), natural units \(c=\hbar=1\) and reduced Planck mass \(M_{\rm pl}=1\). The numerical time coordinate is

\[
\tau=H_\star t,\qquad \frac{d}{d\tau}=\frac{1}{H_\star}\frac{d}{dt}.
\]

### 3.2 Scope

The registered calculation is homogeneous and background-level. It does not contain:

- a unique fundamental covariant action;
- a closed scalar, vector or tensor perturbation theory;
- a continuum-limit demonstration;
- a global or nonlinear stability theorem;
- an observational likelihood analysis;
- an ontological test of an Apeiron medium.

The lack of a unique action-level completion is not concealed. It limits the article to the exact operational discrete model that was computed.

### 3.3 Meaning of the control parameter

\(\tau\) is the registered numerical time/control endpoint of the frozen calculation. A unique observational or cosmological mapping beyond \(\tau=H_\star t\), including any direct mapping to redshift, remains an AP1 task. The terminal Float64 bracket must therefore not be described as a measured physical constant or exact physical phase boundary.

## 4. Frozen assumptions, degrees of freedom and equations

### 4.1 Stored state and degrees of freedom

The full trajectory container is

\[
\mathbf y=(\sigma,\dot\sigma,\theta,\dot\theta,\chi,\dot\chi,H,N).
\]

The iterated state uses six columns,

\[
\mathbf u=(\sigma,\dot\sigma,\theta,\dot\theta,H,N),
\]

while the homogeneous \(\chi\) bookkeeping coordinate is fixed to zero. The portal sector enters through renormalized mode expectation values \(\rho_q\), \(p_q\) and \(\langle\chi^2\rangle_q\). This six-column numerical state is not claimed to be a count of fundamental degrees of freedom in an unknown covariant completion.

The fixed-point vector is the flattened scaled state

\[
\mathbf x=\operatorname{vec}\!\left(
\frac{\sigma}{10^{-2}},
\frac{\dot\sigma}{10^{-8}},
\theta,
\frac{\dot\theta}{10^{-8}},
\frac{H}{10^{-6}},
\frac{N}{10^{-1}}
\right),
\]

with 15361 nodes, 92166 Float64 components and a full stored solution array of shape \(15361\times 8\).

### 4.2 Homogeneous pressure

Define

\[
X=\frac12\dot\theta^2,\quad z=\frac{\sigma}{f_\sigma},\quad
b=b_\star+b_{\rm slope}\frac{\theta}{M_{\rm pl}}-\frac{5X}{\Lambda^4},
\]

\[
F(z,b)=\frac14z^4-\frac12z^2+bz .
\]

The registered pressure is

\[
P(X,\sigma,\theta)
=X+\frac{5}{3\Lambda^4}
\left(z-\frac{2}{\sqrt3}\right)^2X^2
-E_cF(z,b)-V_s(\theta),
\]

where \(V_s(\theta)=\sum_{i=0}^{4}c_i\theta^i\) and the five frozen coefficients are given in the canonical specification. The executable derivatives \(P_X\), \(P_{XX}\), \(P_\sigma\), \(P_\theta\) and \(P_{X\sigma}\) are stated there in full.

The AME stress is

\[
\rho_{\rm AME}=\frac12\dot\sigma^2+2XP_X-P,\qquad
p_{\rm AME}=\frac12\dot\sigma^2+P.
\]

### 4.3 Portal and quantum source

The field-dependent portal mass is

\[
m_\chi^2=m_{\chi,0}^2+
2\lambda(v+\sigma)^2\cos\!\left(\frac{2\theta}{f_{\rm phase}}\right).
\]

The quantum routine uses four Pauli–Villars sectors with coefficients \((1,-3,3,-1)\), regulator indices \((0,1,2,3)\), 128 resolved momentum nodes on the initial physical support \(0\le k\le0.6\Lambda\), an analytic order-0 tail and auxiliary adiabatic curvature orders 2 and 4. Mode functions are advanced by a three-substep Yoshida composition of an implicit-midpoint update.

The normalized quantum-source diagnostic represents

\[
\dot\rho_q+3H(\rho_q+p_q)
-\frac12\dot m_\chi^2\langle\chi^2\rangle_q=0.
\]

### 4.4 Registered background map

With \(\rho_{\rm tot}=\rho_{\rm AME}+\rho_q\) and \(p_{\rm tot}=p_{\rm AME}+p_q\), the image map integrates

\[
\ddot\sigma=-3H\dot\sigma+P_\sigma
-\frac12\partial_\sigma m_\chi^2\langle\chi^2\rangle_q,
\]

\[
\ddot\theta=
\frac{
P_\theta-\frac12\partial_\theta m_\chi^2\langle\chi^2\rangle_q
-3HP_X\dot\theta-P_{X\sigma}\dot\sigma\dot\theta
}{
P_X+2XP_{XX}
},
\]

\[
H=+\sqrt{\frac{\rho_{\rm tot}}{3M_{\rm pl}^2}},
\qquad \dot N=H.
\]

The Raychaudhuri relation

\[
\dot H=-\frac{\rho_{\rm tot}+p_{\rm tot}}{2M_{\rm pl}^2}
\]

is retained as an independent diagnostic. Hyperbolicity gates require \(P_X>0\) and \(P_X+2XP_{XX}>0\).

### 4.5 Fixed-point definition

For the hash-identified image map \(\mathbf G\),

\[
\mathbf R(\mathbf x;\tau)=\mathbf x-\mathbf G(\mathbf x;\tau),
\qquad
\delta_{\rm FP}=\|\mathbf R\|_\infty .
\]

All parameters, source files, gates, tolerances, discretization and solver budgets were frozen before the terminal classification.

## 5. Numerical method and reproducibility

### 5.1 Discretization and nonlinear budget

The grid spacing is \(\Delta\tau=\tau_{\rm end}/15360\). The registered finite-difference step for Jacobian-vector products is \(10^{-5}\). The allowed nonlinear budget was:

- at most 6 Newton steps;
- GMRES restart 10;
- at most 16 GMRES cycles per Newton step;
- GMRES relative tolerance 0.05 and absolute tolerance 0;
- line-search factors \(2^{-j}\), \(j=0,\ldots,12\).

These values describe the frozen allowance. The terminal v7.13 evaluation consumed no Newton update because the inherited state already passed the fixed-point gate.

### 5.2 Persistence and immutability

The v7.13 result JSON, solution NPZ, gate register, protocol, crossing decision, terminal checkpoint state and active runtime sources are individually hashed in APEIRON_REPRODUCIBILITY_MANIFEST_LATEST.json. The adjacent v6.74 result, solution and gate register are also included. The v6.74 gate table is preserved for its own classification and is not substituted for the strict v7.13 table.

### 5.3 Read-only verification

The accompanying verifier:

1. checks every registered SHA-256 digest;
2. checks cross-document identities and terminal metadata;
3. checks NPZ keys, shapes and Float64 dtypes;
4. compares the stored diagnostics with the immutable gate table;
5. checks binary64 endpoint adjacency and exact-protocol midpoint rounding.

It imports no solver module, evaluates no physical map and modifies no file. Its passing terminal message is APEIRON_PUBLICATION_PACKAGE_VERIFY_PASS. This is package-integrity verification, not independent numerical reproduction.

### 5.4 Reproducibility levels

| Level | Meaning | Status |
|---|---|---|
| V0 | narrative and filenames | passed |
| V1 | narrative and structured handoff agree | passed |
| V2 | immutable artifacts resolved and hashes verified | passed |
| V2.5 | stored gates, array forms and crossing identities checked read-only | passed |
| V3 | physical diagnostics independently recomputed from the stored state | not performed |
| V4 | independent implementation reproduces the classification | open |
| V5 | observable templates preregistered and tested | not begun |

## 6. Gates, tolerances and classification rules

### 6.1 Exact v7.13 gate table

| Gate | Operator | Threshold | Recorded value | Result |
|---|---:|---:|---:|---|
| fixed-point infinity norm | < | \(5.0\times10^{-11}\) | \(4.6838977141305804\times10^{-11}\) | pass |
| Friedmann constraint | < | \(1.0\times10^{-10}\) | \(3.818888298257304\times10^{-12}\) | pass |
| continuity residual | < | \(1.0\times10^{-5}\) | \(3.7214611922998273\times10^{-6}\) | pass |
| Raychaudhuri residual | < | \(2.0\times10^{-5}\) | \(1.4051164250399966\times10^{-5}\) | pass |
| quantum-source identity | < | \(1.0\times10^{-7}\) | \(3.883856960984437\times10^{-8}\) | pass |
| Wronskian diagnostic | < | \(2.0\times10^{-8}\) | \(7.432742701266731\times10^{-9}\) | pass |
| minimum total density | > | 0 | \(2.0729288609194362\times10^{-14}\) | pass |
| minimum H | > | 0 | \(8.312498343027667\times10^{-8}\) | pass |
| minimum \(P_X\) | > | 0 | \(1.0523451240627972\) | pass |
| minimum \(P_X+2XP_{XX}\) | > | 0 | \(1.052350229025336\) | pass |
| finite | is | true | true | pass |
| finalization failure | is | null | null | pass |

### 6.2 Classification logic

HARD_PASS requires every listed numerical gate to pass and the finalization-failure field to be null. PASS/NONPASS is a deterministic protocol classification. It is not a statistical \(p\)-value, likelihood statement, model-selection score or empirical verdict.

## 7. Results of the terminal v7.13 calculation

### 7.1 v7.13 endpoint

The terminal exact protocol value is

\[
\tau_{\rm HP}^{\rm exact}
=0.47878124999999994315658113919198513031005859375,
\]

stored as binary64

\[
\tau_{\rm HP}^{64}=0.47878124999999994.
\]

The terminal record reports one physical-map evaluation, zero Newton steps, no iteration log, no finalization failure and HARD_PASS. The exact diagnostic values and thresholds are listed in Section 6.

### 7.2 Immutable v6.74 counter-endpoint

The adjacent endpoint

\[
\tau_{\rm NP}^{64}=0.47878125
\]

is classified NONPASS in the immutable v6.74 record. Its fixed-point diagnostic \(1.8866305739884126\times10^{-10}\) and Friedmann diagnostic \(1.7440413453847122\times10^{-10}\) failed the gates registered for that result; its remaining documented gates passed. The counter-endpoint is used only for the frozen crossing classification.

### 7.3 Float64 crossing resolution

| Quantity | Value |
|---|---|
| exact protocol bracket width | \(5.684341886080801486968994140625\times10^{-17}\) |
| binary64 bracket width | \(5.551115123125783\times10^{-17}\) |
| next exact protocol midpoint | \(0.478781249999999971578290569595992565155029296875\) |
| midpoint converted to binary64 | \(0.47878125\), the NONPASS endpoint |
| endpoint relation | adjacent representable binary64 values |
| frozen decision | schedule complete; further Float64 bisection forbidden |

The bracket is a representation- and protocol-dependent classification result. It is not an exact boundary in the real numbers or a measured physical uncertainty.

## 8. Physical interpretation

The result establishes a narrow technical fact: the stored v7.13 state meets the frozen numerical acceptance criteria, while the adjacent stored v6.74 endpoint is NONPASS. This closes the registered Float64 continuation schedule without reopening the solver.

The result does not demonstrate that the operational equations are unique, fundamental or empirically adequate. Passing positivity and hyperbolicity gates at the terminal trajectory does not prove global stability. Small residuals do not provide observational evidence. The ontology of an Apeiron medium remains a hypothesis outside the reach of the present calculation.

Physical plausibility requires separate work: action-level interpretation or another principled completion, perturbation dynamics, regulator and scheme analyses beyond the registered discrete scope, continuum/refinement evidence, comparison with established constraints and ultimately preregistered data tests.

## 9. Limits and unanswered questions

The principal open questions are:

1. Does the operational homogeneous model admit a defensible and sufficiently unique covariant completion?
2. Do the results persist under controlled grid and method refinement?
3. What are the linear scalar, vector and tensor perturbation equations?
4. Are the relevant sectors stable beyond the terminal background trajectory?
5. How is \(\tau\) mapped uniquely to observable cosmological variables?
6. Does one common parameter set satisfy expansion, growth, lensing, local and gravitational-wave constraints?
7. Can an independent implementation reproduce the terminal classification?

Items 1–7 constrain interpretation and later claims. They do not alter the already completed v7.13 classification. AP1–AP3 are not prerequisites for reporting this narrowly scoped numerical result.

## 10. Falsifiable predictions and future work

Appendix A defines a prospective program, not completed evidence.

### 10.1 AP1 — joint expansion, growth and lensing

AP1 is intended to derive \(H(z)\), \(w_{\rm ap}(z)\), \(f\sigma_8(z)\), \(\mu(k,z)\), \(\eta(k,z)\) and \(\Sigma(k,z)\) from one frozen parameter core. Before data contact it requires a unique physical \(\tau\) mapping, closed perturbation equations, stability gates, fixed grids, priors, masks, likelihoods and versioned output templates. No such curves are reported here.

### 10.2 AP2 — residual filament gravity

AP2 prospectively tests a residual lensing profile after reconstruction of galaxies, gas and the declared comparison-model matter distribution:

\[
\Delta\kappa_{\rm ap}
=F(r_\perp,z,d_{\rm node},{\rm curvature}\mid\theta_{\rm ap}).
\]

It must reuse the AP1 parameter core and pass prespecified void and rotated-axis null tests. No filament signal is claimed in this paper.

### 10.3 AP3 — gravitational-wave propagation

AP3 treats \(c_{\rm GW}=c\) as a survival condition and prospectively derives amplitude, polarization or line-of-sight observables from a closed tensor sector. No modified-propagation detection is claimed.

### 10.4 Cross-channel decision rule

AP1–AP3 must use one preregistered parameter core. Channel-by-channel retuning is forbidden. A conflict with any hard gate produces NONPASS for the affected hypothesis rather than retrospective adjustment of v7.13.

## 11. Conclusion

The frozen Apeiron/AME-1 v7.13 Stage-0 state at binary64 \(\tau=0.47878124999999994\) satisfies every exact registered gate and is classified HARD_PASS. The adjacent binary64 endpoint \(\tau=0.47878125\) is an immutable v6.74 NONPASS because its fixed-point and Friedmann gates fail. The exact protocol midpoint converts to the NONPASS binary64 endpoint, so the registered Float64 crossing schedule is complete.

The publication package now states the operational homogeneous model, exact v7.13 thresholds, artifact hashes and a read-only verification path. These additions close the former documentary blockers without recomputing the physics. They do not establish continuum convergence, empirical confirmation or an Apeiron ontology. The technical core is suitable for controlled preprint packaging after editorial and administrative completion; venue-specific journal submission preparation remains outstanding. AP1–AP3 remain separate follow-up work.

## 12. Methods and reproducibility supplement

### 12.1 Controlled file manifest

| File | Role |
|---|---|
| APEIRON_CANONICAL_MODEL_SPECIFICATION_LATEST.md | operational model, equations, units, state and source hashes |
| APEIRON_V7_13_GATE_SPECIFICATION_LATEST.json | exact thresholds, operators, values and classification |
| APEIRON_REPRODUCIBILITY_MANIFEST_LATEST.json | SHA-256 manifest |
| README_REPRODUCIBILITY.md | verification scope and instructions |
| verify_apeiron_publication_package.py | read-only verifier |
| TECHNICAL_SOURCES/STAGE00_HARD_PASS_RESULT_v7_13.json | immutable terminal result |
| TECHNICAL_SOURCES/STAGE00_HARD_PASS_SOLUTION_v7_13.npz | immutable terminal arrays |
| TECHNICAL_SOURCES/RECOVERY_GATE_REGISTER_v7_13_STAGE00.json | immutable terminal gate register |
| TECHNICAL_SOURCES/RECOVERY_PROTOCOL_v7_13.json | immutable protocol |
| TECHNICAL_SOURCES/CROSSING_RESOLUTION_RESULT_v7_13.json | immutable crossing decision |
| TECHNICAL_SOURCES/STAGE00_NONPASS_RESULT_v6_74.json | immutable adjacent NONPASS result |
| TECHNICAL_SOURCES/STAGE00_NONPASS_SOLUTION_v6_74.npz | immutable adjacent NONPASS arrays |
| TECHNICAL_SOURCES/RECOVERY_GATE_REGISTER_v6_74_STAGE00.json | immutable adjacent NONPASS register |
| TECHNICAL_SOURCES/STATE_SNAPSHOT | terminal read-only checkpoint snapshot |
| TECHNICAL_SOURCES/ACTIVE_RUNTIME | hash-frozen active source path |

All exact hashes are stored in the machine-readable manifest.

### 12.2 Verification command

From the package root:

    python3 verify_apeiron_publication_package.py

Expected terminal line:

    APEIRON_PUBLICATION_PACKAGE_VERIFY_PASS

### 12.3 Data and code availability — controlled draft

The accompanying publication distribution contains the canonical operational specification, exact gate table, immutable result/state/protocol artifacts, active source path and read-only verifier. A permanent repository identifier, license and venue-specific availability wording remain to be assigned. No observational dataset was analyzed.

### 12.4 Statistical statement

No statistical fit, parameter estimation, hypothesis-test significance, Bayes factor, information criterion or observational model comparison is reported. PASS/NONPASS denotes only deterministic numerical gate classification.

### 12.5 Declarations

- **Author contributions:** [TO BE PROVIDED]
- **Funding:** [TO BE PROVIDED]
- **Competing interests:** [TO BE PROVIDED]
- **Acknowledgments:** [TO BE PROVIDED]
- **Data/code license:** [TO BE PROVIDED]
- **AI-assisted drafting disclosure:** [TO BE DECIDED ACCORDING TO VENUE POLICY]

## 13. Figure, table and data inventory

### 13.1 Figures

| Figure | Content | Status |
|---|---|---|
| Fig. 1 | separation of numerical classification, physical interpretation and future empirical test | specified; artwork pending |
| Fig. 2 | adjacent v7.13 HARD_PASS and v6.74 NONPASS binary64 endpoints | data fully specified; artwork pending |
| Fig. 3 | AP1 → AP2 → AP3 dependency and common parameter freeze | concept specified in Appendix A; artwork pending |

### 13.2 Tables

| Table | Content | Status |
|---|---|---|
| Table 1 | exact v7.13 gate definitions and outcomes | complete in Section 6 |
| Table 2 | Float64 crossing resolution | complete in Section 7 |
| Table 3 | verification levels | complete in Section 5 |
| Table 4 | controlled file manifest | complete in Section 12 |

### 13.3 Machine-readable products

| Product | Status |
|---|---|
| terminal result JSON and solution NPZ | included and hashverified |
| exact v7.13 gate and protocol JSON | included and hashverified |
| crossing-resolution JSON | included and hashverified |
| adjacent v6.74 NONPASS artifacts | included and hashverified |
| terminal checkpoint snapshot | included and hashverified |
| active runtime source path | included and hashverified |
| read-only verifier and manifest | included; PASS |
| AP1/AP2/AP3 templates | future work; no values exist |

## References

### Internal controlled sources

1. Apeiron — Start Here, latest persistent checkpoint.
2. Apeiron current project state, latest structured checkpoint.
3. Apeiron/AME-1 — Canonical Operational Model Specification, version 1.0.
4. Apeiron v7.13 — Exact Gate Specification, version 1.0.
5. Apeiron — Reproducibility Manifest and Read-only Verifier, version 1.0.
6. Apeiron/AME-1 Appendix A: Falsifiable Predictions — Guide for the Next Three Work Packages, version 1.0, 30 August 2026.

### External primary and official context

- IEEE, IEEE Standard for Floating-Point Arithmetic, IEEE Std 754-2019 (2019).
- Y. Saad and M. H. Schultz, “GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems,” SIAM Journal on Scientific and Statistical Computing 7, 856–869 (1986), doi:10.1137/0907058.
- Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6 (2020), arXiv:1807.06209.
- eBOSS Collaboration, S. Alam et al., “The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory,” Physical Review D 103, 083533 (2021), arXiv:2007.08991.
- D. Brout et al., “The Pantheon+ Analysis: Cosmological Constraints,” Astrophysical Journal 938, 110 (2022), arXiv:2202.04077.
- E. Bellini and I. Sawicki, “Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity,” JCAP 07, 050 (2014), arXiv:1404.3713.
- LIGO Scientific Collaboration, Virgo Collaboration, Fermi GBM and INTEGRAL, “Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophysical Journal Letters 848, L13 (2017), arXiv:1710.05834.
- E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore, “Modified gravitational-wave propagation and standard sirens,” Physical Review D 98, 023510 (2018), arXiv:1805.08731.

These sources provide context only. They do not modify the internal Apeiron model or elevate AP1–AP3 to completed predictions.