# 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.