# Version 13 continuation - current reading notice

**Editorial date: 14 September 2026.**

## Binding interpretation
The current Apeiron model postulates one fundamental carrier. Ordinary matter is an organized Apeiron manifestation. Dark matter may be a manifestation or state of the same carrier; it is not a separate fundamental substance in this ontology. Dark energy may be a state or dynamical form of that carrier. These identifications remain hypotheses, without empirical confirmation.

## QCD and Higgs
QCD and Higgs phenomena are investigated as possible organized manifestations of the same carrier. The Standard Model and QCD, including their established equations and results, remain unchanged. A derivation from Apeiron has not been established.

## Three distinct scopes
The v7.13 Float64 classification and the G32 methods status are historical numerical records. In inherited passages, “current”, “Latest”, ORANGE and AP2/AP3 locks refer to the dated record, not to a new assessment on 14 September. The later AP1_GO_SCIENTIFIC_FEASIBILITY_FIRST decision 078 and Audit 079 concern the bounded local O049 witness and its specified response class. They do not establish a complete cosmology, a DM/DE fit or completed independent reproduction.

## Black-hole reconversion
The project status is FEASIBILITY GO for the investigated effective feasibility class. Reconversion is an internal transition of organized Apeiron manifestations toward more fundamental states of the same carrier. It is not inferred from v7.13 or O049. A unique microscopic end state, quantitative reconversion rates and observational evidence remain open; no second substance or external energy sink is required by this interpretation.

## Editorial scope and provenance
This Version 13 reader edition adds a dated reading notice to the supplied document. It does not recalculate, reinterpret as new evidence, or replace any frozen equations, parameters, gates, hashes or numerical results. Original imported files are preserved byte-for-byte in the accompanying historical/frozen source ZIP. Website preparation and this reader edition do not constitute publication, peer review or empirical validation.

The local AP1 source is the unchanged 11 September A079/H1 [Technical Evidence Index](../ap1-go-078-v1.1-2026-09-11-audit079-hygiene/TECHNICAL_EVIDENCE_INDEX_v1.1_2026-09-11_A079_H1.md). [Original imported documents](../historical-frozen-2026-09-14/ORIGINAL_IMPORTED_DOCUMENT_SET_20260914_FROZEN.zip).

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## Current interpretive overlay - 14 September 2026

This section is an **interpretive addendum only**. It changes no frozen AME-1 equation, numerical parameter, gate, source hash or terminal classification.

### Single-carrier principle

Apeiron is treated as one fundamental carrier. Effective sectors used in the operational model must not be read as ontologically separate fundamental substances.

- ordinary matter: organized/emergent Apeiron manifestation;
- dark-matter-like phenomena: possible Apeiron manifestation/state;
- dark-energy-like large-scale behavior: possible Apeiron dynamical/state manifestation;
- Higgs/QCD structures: possible organized low-energy manifestations under separate investigation, with Standard-Model/QCD structure retained;
- black-hole reconversion: possible transition of an organized manifestation toward more fundamental Apeiron states under extreme curvature.

The overlay is a model hypothesis and is not an empirical consequence of the frozen v7.13 calculation.

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# Historical / frozen source record

All dated status labels below belong to that earlier snapshot. Its numerical content is retained.

# Apeiron/AME-1 — Canonical Operational Model Specification

**Version:** 1.0  
**Status:** frozen technical supplement for the first numerical-foundations publication  
**Date:** 2026-08-30  
**Authority:** active execution path in APEIRON_FAST_RUNTIME_LATEST.zip, Library version 53, SHA-256 9b0d9212250a6b475b9a56647d0b34f7d8f87c0912c1ec9c004530a674951985

## 1. Scope and epistemic boundary

This is the canonical operational definition of the homogeneous model that produced the frozen v7.13 classification. It specifies variables, pressure function, background equations, quantum-source closure, discrete fixed-point map, initial data, units and conventions.

It does **not** assert a unique fundamental covariant action, perturbation theory, ontology or empirical confirmation. For the narrowly scoped numerical-foundations paper, the equations and executable map below are authoritative. An action-level completion would be new physics and cannot be projected backward onto v7.13.

## 2. Units, geometry and time

- Natural units: c = ħ = 1.
- Reduced Planck mass as internal mass unit: M_pl = 1.
- Spatially flat homogeneous FRW background, scale factor a, e-fold variable N = ln(a), expanding branch H > 0.
- A dot denotes differentiation with respect to cosmic time t.
- Dimensionless numerical time:

    τ = H_star t  
    d/dτ = (1/H_star) d/dt

- Noncanonical kinetic scalar:

    X = 0.5 (dot θ)^2

- Scalar fields and mass parameters have mass dimension one. X, pressure and energy density have mass dimension four.

The operational sign convention is fixed by the equations below. A metric-signature convention not used explicitly by the homogeneous implementation is not added here.

## 3. Degrees of freedom and stored state

### 3.1 Homogeneous state

The full eight-column trajectory container is

    y(τ) = (σ, dot σ, θ, dot θ, χ, dot χ, H, N).

In the registered v7.13 map the homogeneous bookkeeping coordinate χ is fixed to zero. The dynamically iterated six-column state is

    u(τ) = (σ, dot σ, θ, dot θ, H, N).

The χ sector enters through the renormalized mode expectation values ρ_q(τ), p_q(τ) and <χ²>_q(τ), not through an independently iterated homogeneous χ.

### 3.2 Discrete vector

The frozen grid has N_τ = 15361 nodes. The solver vector is the flattened and scaled array

    x = vec[
      σ/(1e-2),
      dot σ/(1e-8),
      θ/1,
      dot θ/(1e-8),
      H/(1e-6),
      N/(1e-1)
    ]

with shape (92166,) and Float64 dtype.

| Array | Shape | dtype | Meaning |
|---|---:|---|---|
| t | (15361,) | Float64 | dimensionless τ grid |
| y | (15361, 8) | Float64 | full trajectory container |
| x | (92166,) | Float64 | scaled flattened six-column iterate |
| tau | scalar | Float64 | active terminal value |

During unpacking, first-node σ, dot σ, θ, dot θ and N are reset to the frozen background initial data. First-node H remains in the active state; the image map then imposes the positive Friedmann closure.

### 3.3 Auxiliary and constrained quantities

- H is algebraically reconstructed from the positive Friedmann branch inside the registered image map.
- N obeys dot N = H.
- Quantum modes and Pauli–Villars regulator sectors are auxiliary arrays used to construct ρ_q, p_q and <χ²>_q; they are not independent entries of x.
- Gate diagnostics are not additional dynamical degrees of freedom.

## 4. Frozen parameters

### 4.1 AME parameters

| Symbol | Value | Code name |
|---|---:|---|
| M_pl | 1 | Mpl |
| Λ | 2.3e-3 | Lambda |
| f_σ | 2.5e-2 | f_sigma |
| E_c / V_star | 4.0e-2 | Ec_over_V |
| H_star | 2.0e-6 | Hstar |
| b_star | 0.35 | bstar |
| d b / d(θ/M_pl) | -4.78e-2 | b_slope |
| A_eff | 6.86e-3 | Aeff |
| B_eff | -1.74e-2 | Beff |

    V_star = 3 M_pl² H_star²
    E_c = 0.04 V_star

A_eff and B_eff remain in parameter provenance; the active global pressure completion in Section 5 is the quartic polynomial V_s(θ).

### 4.2 Portal and regulator parameters

| Quantity | Value |
|---|---:|
| phase scale f_phase | 0.8549468502440697 |
| displacement v | 0.03272387838931097 |
| coupling λ | 1.6740193604000272e-5 |
| bare portal mass m_χ,0 | 9.817951142008693e-5 |
| physical-mode support at initial slice | 0 ≤ k ≤ 0.6 Λ |
| physical quadrature nodes | 128 |
| renormalization scale | μ_R = 1.5 Λ |
| UV order-0 switch | 16 Λ |
| curvature-tail limit | 64 Λ |
| tail nodes per octave | 20 |

## 5. Effective AME pressure and derivatives

Define

    z = σ/f_σ
    b(X, θ) = b_star + b_slope (θ/M_pl) - 5X/Λ⁴
    F(z,b) = z⁴/4 - z²/2 + bz
    V_s(θ) = Σ(i=0..4) c_i θ^i

with

    (c_0,c_1,c_2,c_3,c_4) =
    ( 1.1971212004620583e-11,
      9.87022611239063e-14,
     -1.0336424891583054e-13,
      1.189490499106138e-10,
      1.2236365189926054e-10 )

The homogeneous pressure function is

    P(X,σ,θ) =
      X
      + [5/(3Λ⁴)] (z - 2/√3)² X²
      - E_c F(z,b)
      - V_s(θ).

The executable derivatives are

    P_X =
      1
      + [10/(3Λ⁴)] (z - 2/√3)² X
      + 5 E_c z/Λ⁴

    P_XX = [10/(3Λ⁴)] (z - 2/√3)²

    P_σ =
      { [10/(3Λ⁴)] (z - 2/√3) X²
        - E_c (z³ - z + b) } / f_σ

    P_θ = -E_c z b_slope/M_pl - V_s'(θ)

    P_Xσ =
      { [20/(3Λ⁴)] (z - 2/√3) X
        + 5 E_c/Λ⁴ } / f_σ.

AME stress:

    ρ_AME = 0.5 (dot σ)² + 2X P_X - P
    p_AME = 0.5 (dot σ)² + P.

Hyperbolicity diagnostics:

    P_X > 0
    P_X + 2X P_XX > 0.

## 6. Portal and quantum source

Field-dependent portal mass:

    m_χ²(σ,θ) =
      m_χ,0² + 2λ(v+σ)² cos(2θ/f_phase)

    ∂m_χ²/∂σ = 4λ(v+σ) cos(2θ/f_phase)

    ∂m_χ²/∂θ =
      -[4λ(v+σ)²/f_phase] sin(2θ/f_phase).

Four Pauli–Villars sectors:

    C_j = (1,-3,3,-1)
    J_j = (0,1,2,3)
    M_j² = m_χ² + J_j μ_R².

Modes are advanced by a three-substep Yoshida composition of the implicit-midpoint map for u = a^(3/2) χ. Initial mode data are

    u_k(0) = 1/sqrt(2ω_k(0))
    dot u_k(0) = -i ω_k(0) u_k(0).

The active source combines resolved modes, the analytic order-0 UV tail and auxiliary adiabatic curvature orders 2 and 4. Finite renormalization is fixed on the initial flat-space reference slice as recorded by the active runtime:

- ρ_q(0) = 0;
- <χ²>_q(0) = 0;
- initial quantum pressure is predicted by the fixed prescription rather than tuned as a gate;
- the Einstein–Hilbert response is fixed by the zero-curvature response;
- the finite R² coefficient is zero.

The routine returns ρ_q(τ), p_q(τ) and <χ²>_q(τ). Its source identity is

    dot ρ_q + 3H(ρ_q+p_q)
      - 0.5 (dot m_χ²) <χ²>_q = 0,

evaluated as a normalized numerical diagnostic.

## 7. Registered homogeneous equations

With

    ρ_tot = ρ_AME + ρ_q
    p_tot = p_AME + p_q,

the registered image map integrates

    dσ/dt = v_σ

    dv_σ/dt =
      -3H v_σ + P_σ
      - 0.5 (∂m_χ²/∂σ) <χ²>_q

    dθ/dt = v_θ

    dv_θ/dt =
      { P_θ
        - 0.5 (∂m_χ²/∂θ) <χ²>_q
        - 3H P_X v_θ
        - P_Xσ v_σ v_θ }
      / (P_X + 2X P_XX)

    H = +sqrt[ρ_tot/(3 M_pl²)]

    dN/dt = H.

The Raychaudhuri relation

    dH/dt = -(ρ_tot+p_tot)/(2 M_pl²)

is evaluated independently rather than used as the evolution equation in this image map.

## 8. Initial data

    θ(0) = -0.690
    ε_0 = 2.35e-5
    N(0) = 0

    dot θ(0) =
      -sqrt(2 ε_0) M_pl H_star / sqrt(1.3)

    dot σ(0) = 0.

σ(0) = f_σ z_+, where z_+ is the positive local minimum satisfying

    z³ - z + b(θ(0)) = 0
    3z² - 1 > 0.

The initial Hubble value is the positive Friedmann root. Homogeneous portal bookkeeping data are χ(0) = dot χ(0) = 0. No spatial boundary conditions occur because the registered model is homogeneous.

## 9. Canonical discrete fixed-point map

For active endpoint τ_end,

    Δτ = τ_end / 15360.

The canonical image map G(x;τ_end):

1. creates the uniform 15361-node τ grid and frozen background seed;
2. unpacks x into the six active trajectory columns using Section 3.2;
3. imposes frozen first-node conditions;
4. calculates renormalized quantum-source arrays with the fixed portal and PV prescription;
5. advances (σ, dot σ, θ, dot θ, N) by classical fourth-order Runge–Kutta while imposing the positive Friedmann root for H;
6. lifts the result to the eight-column container and repacks the six active columns.

Residual and fixed-point diagnostic:

    R(x;τ) = x - G(x;τ)
    δ_FP = ||R||_infinity.

This is complete for the frozen homogeneous discrete map. It does not license continuum, perturbative, global-stability or observational claims.

## 10. Active-source provenance

| Source file | SHA-256 | Role |
|---|---|---|
| model.py | db4c0fa9cf4f2013fef299f558ca7b6b509d17c1acbd76f84ece6bd2dd0c5a65 | parameters and pressure primitives |
| frw_noncanonical_background_v6_6.py | f96b13df83e6d6dce8f9bec073e2aab24c23660073cee994d25a327f2b61f7f1 | homogeneous equations and initial data |
| frw_pv_fixed_mpl_renorm_v6_17.py | aecd4dc3202604228d1a46c1a4cb439b874e129314e55378e1fbb5b999df482d | stress tensor, renormalization and diagnostics |
| frw_pv_covariant_uv_complete_v6_22.py | 9c54b4095845687b9824dab66da91031436b67a5decd01055ef487b14b4c19b7 | UV-tail closure |
| frw_pv_uv_yoshida_v6_31.py | 8fa1b93a2f24377c751f3d9fdffe700394cdb2e00842fc4dc5f35d19ae3943e1 | registered mode evolution |
| rb19_anderson_uv_v6_24.py | 8323bbf6c6f3e23e1d466addf5cceb94af2517934fddd50ce8fcef4ae04fcc2c | state packing |
| rb23_friedmann_closure_uv_v6_28.py | 873d3eca436d28b4261623957917087b0dd2efdf3b95509b9fb97d1fe4f5d7bc | Friedmann map and Raychaudhuri audit |
| rb_recovery_fixedtau_v6_64.py | 5082e44cf9c60ca9ef1eaa0ba1ccda7d968d19f61ea1d3d2e9c7581be82ee72b | fixed-point and solver base |
| rb_recovery_adaptive_v7_13.py | 8854c5847f9da3e67994701a20e8f4f6e9372c0a65da3f2f3ea0fd7017f81205 | v7.13 wrapper and schedule |

Filename version labels record dependency origin. They appear only because these files form the active hash-frozen execution path of the Latest runtime. No alternative historical model is reconstructed.

## 11. Licensed publication statement

Maximum supported wording:

> Apeiron/AME-1 v7.13 is a frozen homogeneous FRW numerical model defined operationally by the equations, parameters, state map, quantum closure, discretization and hash-identified active runtime sources in this supplement.

Inadmissible stronger wording:

> The supplement proves a unique fundamental or covariant theory of Apeiron.


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