Self-as-an-End
SAE Cosmological Physics Series · Paper VIII

SAE Cosmology VIII — SAE-Bang-0: A Formal Boundary Model for the Big Bang with an Apparent-Horizon Capacity Cap, Cross-Domain Matching, and Thermal-History Acceptance
SAE 宇宙学 VIII：SAE-Bang-0——表观视界密度帽、跨定义域匹配与热历史验收的正式边界模型

Han Qin (秦汉) · Independent Researcher · 2026
DOI: 10.5281/zenodo.21909252 · Full PDF on Zenodo · CC BY 4.0
Abstract

We promote a standing qualitative claim of the SAE series — that the initial point of the Big Bang is not a singularity but a finite-density, finite-size energy boundary datum — into a formal boundary model, SAE-Bang-0. By a formal model we mean the following standard: explicit objects, explicit equations, explicit boundary conditions, a non-empty solution space, and exposure to refutation. Such a model is not required to derive every coefficient and map from SAE first principles; it is required to declare its postulates as postulates.

The model does three things. First, a capacity cap. We take the Landauer–Bekenstein reference scale R_L(T) = ℏc/(2πk_BT) — a functional form inherited from the one-bit construction of the SAE information-theory series — and promote it, by an explicit identification postulate, to the effective averaging radius of the thermal branch in FLRW. The endpoint of the fourth-layer atlas is declared to be the first moment at which the Misner–Sharp compactness of this scale reaches unity, i.e., the moment the scale meets the FLRW apparent horizon. Under the further postulate that the cap surface is a single common ultra-relativistic thermal fluid, this fixes the cap temperature and cap energy density uniquely: T_cap/T_P = √(45/(πg_^UV)) and ε_cap/ε_P = 67.5/g_^UV; at the Standard-Model benchmark g_*^UV = 106.75, T_cap = 0.366 T_P and ε_cap = 0.632 ε_P. Under that postulate no adjustable O(1) coefficient remains. Second, cross-domain matching. The singularity is replaced by an explicit matching map across a pair of transition surfaces: the 0DD transition core does not belong to the spacetime domain and carries no time, no energy, and no entropy; the contracting-side atlas terminates at Σ⁻, and the expanding-side boundary data are given by the map 𝓜₀(a, H, ε, ⋆) = (a, −H, ε, ⋆), where ⋆ is a trivial tag — the formalization of non-trivial remainder inheritance is deferred to future extensions. Non-emptiness is witnessed explicitly by a family of two independent closed pure-radiation atlases. Third, thermal-history acceptance. The Bang side outputs a hot equilibrium radiation state; segment-by-segment acceptance against standard Big-Bang nucleosynthesis and the microwave background is the model's empirical survival condition — a pending test whose interface this paper supplies, not a test it claims to have passed.

The model's inputs are listed once and in full: the single new adjustable microscopic parameter g_^UV (with the postulate g_{s}^UV = g_*^UV); the present curvature q_K := |Ω_k0| as an explicit observational input — we adopt q_K = 0.002 as a benchmark at the 68% confidence scale of Planck 2018 combined with BAO (Ω_K = 0.0007 ± 0.0019); and the bulk integration constant and matching-branch choices, accounted separately. Conditional on a complete background extension that passes standard adiabatic thermal-history acceptance (an extension this paper does not construct and registers as an open problem), the cap surface is connected to present-day observables by an adiabatic bridge, yielding the conditional equality a_min(q_K) = 53.5 μm · (0.002/q_K)^{1/2} for the minimum curvature radius of the closed 3-sphere; within the currently allowed range q_K ≲ 0.0037 at the 95% confidence scale, a_min ≳ 39 μm. The robust qualitative conclusion: under these conditions, the minimum curvature radius of the universe is of order tens of microns — the size of a micron-scale dust grain, not a point.

Three points of positioning must be stated plainly. The existence of the cap (H_MS), the identification of the reference scale (H_slot^Bang), and the pure-fluid cap surface (H_fluid) are model postulates; the cap values are their conditional theorems. The model does not eliminate the generalized singularity defined by geodesic incompleteness; it reinterprets its boundary identity — what it eliminates is the divergence of density and curvature, and the treatment of an infinite-density point as an interior state of spacetime. The model contains no "maximal single-slot occupancy" proposition; it only computes when the bath-averaged causal sphere reaches marginal trapping. The primordial perturbation spectrum, the matching dynamics, the interior of the transition core, and the complete background thermal history with the SAE dark sector are registered, plainly, as open problems.

What can be refuted and what can be constrained are listed separately: the sign of spatial curvature, Ω_k < 0, is a falsifier (a robust determination of an open universe refutes the model); the lower bound on the old-calibration stretch factor 𝒮_a is a conditional geometric constraint that tightens as q_K improves; and there is one prospective discriminator concerning the no-inflaton baseline. Every claim in the paper carries an epistemic tier tag; postulates and claims are tabulated in Appendix A.

§1 Introduction: From Narrative to Model

Two earlier papers of SAE cosmology left the Big Bang with two claims and a map. The cosmological-constant paper [1] argued that the Big Bang is not a singularity requiring an external fuse, and that the transition region is a Planck-scale zone in which causal law does not yet hold; the cyclic-framework paper (Cosmo 0) [2] drew one complete breath of the universe — Bang, turnaround, Crunch, next Bang — as a continuous line, and marked the Crunch-to-Bang transition as a mechanism candidate. Those are ontological pictures. The task of this paper is formalization: to write "the initial point is not a singularity but a large quantity of energy" as a model with objects, equations, boundary conditions, a non-empty solution space, and exposure to refutation.

Two distinctions come first. First, the two readings of the Big Bang. In SAE the Big Bang admits an ontogenetic reading — the self-negation of chaos, the first step from Level 0 to Level 1, a framework-level and timeless event whose mechanism Cosmo 0 §5.① [2] states and whose dynamics it marks as the deepest open problem — and a cyclic reading, the transition between periods. The two must not be conflated. What this paper formalizes is the local Crunch–Bang boundary block required by the cyclic reading: finite-density termination, two-sided boundary data, and a minimal matching. The full cyclic topology, non-trivial remainder inheritance, and winding matching do not belong to Bang-0 and are listed as extensions (§10); the transition between periods is permitted to be related by matching conditions, while the question of what non-trivial content is inherited is not adjudicated by the present model. Self-negation, as the ontogenetic mechanism of each unfolding, does not lie on this paper's timeline.

Second, the distinction between a formal model and a fully a priori theory. A formal model is permitted to adopt declared postulates; it need not derive the cap coefficients, the matching kernel, or the perturbation spectrum from first principles. What it must do is declare its postulates as postulates and deliver a non-empty, constraint-compatible matching map. This paper holds itself to that contract (§2).

Repositioning the singularity theorems. The Penrose–Hawking singularity theorems [3,4] are theorems internal to the fourth layer (the spacetime/causal layer): their premises are a smooth Lorentzian manifold plus energy conditions. SAE-Bang-0 concedes geodesic incompleteness — the atlas does end at the transition surface; but incompleteness is not the reality of infinite density: the premises of the theorems fail at a finite-density boundary before infinite density is reached. The paper's formulation is: SAE-Bang-0 replaces the divergent singularity with a finite-density cross-domain boundary; it does not eliminate the generalized singularity defined by geodesic incompleteness, but reinterprets its boundary identity. What it eliminates is ε → ∞, curvature divergence, and the treatment of an infinite-density point as an interior state of spacetime (the finiteness of curvature invariants in the boundary limits is displayed explicitly in §4).

External anchor. Loop quantum cosmology (LQC) supplies the existing exemplar of "a Planck-order density cap yields a non-singular bounce": its density ceiling arises from holonomy corrections in the quantized Hamiltonian constraint, with critical density about 0.41 ε_P and deterministic effective dynamics across the Planck regime [5,6]. The SAE cap arises instead from the crossing of a reference scale — inherited from the information-theoretic causal-slot construction — with the gravitational trapping scale. Both replace classical infinite density by a finite Planck-order boundary, but their mathematical origins differ; the proximity of 0.632 ε_P (this paper's SM benchmark) to 0.41 ε_P is registered as an order-of-magnitude concordance only, not as mutual confirmation. One categorical difference must be declared at the outset: LQC is a smooth bounce (H passes continuously through zero), whereas this model is atlas termination plus matching (the sign flip of H occurs in no spacetime domain) — the two are constructions of different kinds; §3 elaborates and compares with other external constructions.

The paper proceeds as follows: §2 epistemic contract and input table; §3 model definition and postulates; §4 the main theorem (cap values); §5 the non-emptiness theorem (matching and witness); §6 numbers and scales; §7 matter realization on the Bang side; §8 falsifier, geometric constraint, and discriminator; §9 registered revisions to earlier papers; §10 extensions and open problems; §11 conclusion.

§2 Epistemic Contract and Input Table

We retain the series' four-tier tagging — [structural], [posterior], [motivated], [open] — and add four model-level tags: [model postulate] (openly adopted, replaceable postulates of this model); [conditional theorem] (results derived from declared postulates, whose validity conditions are those postulates themselves); [constructional input] (constructive choices, registered rather than derived); [inherited] (taken over from standard physics or upstream SAE work, not re-proven here).

The key epistemic sentence: a formal model is not required to derive the capacity cap a priori; it is required to acknowledge the cap explicitly as a postulate and to deliver a non-empty, constraint-compatible matching map.

Convention on constructional identities: whenever two symbols are equal by definition (such as T_cap and the bath temperature T_max in §4), the text marks the equality as a "constructional identity" and does not present it as independent verification.

Notation (uniform throughout): ε denotes physical energy density and p pressure; the Friedmann equation is H² = (8πG/3c²)ε_tot − c²/a², the continuity equation ε̇_tot + 3H(ε_tot + p_tot) = 0; the Planck energy density is ε_P := c⁷/(ℏG²).

Input table (every degree of freedom entering the model, listed once):

  1. New adjustable microscopic parameter: g_^UV (the ultraviolet relativistic degree-of-freedom count), the only one in the paper. Attached postulate H_eq: g_{s}^UV = g_*^UV (all UV degrees of freedom share one temperature, chemical potentials negligible, all ultra-relativistic). [model postulate]
  2. Observational boundary input: q_K := |Ω_k0|, kept explicit. We use q_K = 0.002 as a benchmark — it corresponds to the 68% confidence scale of the Planck 2018 + BAO result Ω_K = 0.0007 ± 0.0019 [7], not a conservative confidence bound (the 95% error is about ±0.0037). The quantities a_min(q_K) and E_cap^rad(q_K) obtained from it via the adiabatic bridge are conditional equalities (§6); lower-bound forms hold only under an additional hypothesis q_K ≤ q_K,max. [posterior]
  3. Adiabatic bridge H_ad: aTg_{*s}^{1/3} = const [17], connecting the cap surface to present-day observables (H₀, q_K, T₀). It belongs to the acceptance content of the complete background extension: its validity over the whole history is conditional on the existence of a complete background solution — including matter and late-time acceleration — that passes standard thermal-history acceptance; this paper does not construct that extension (§10, open problems). [inherited + bridge condition]
  4. Bulk integration constant: K_r = εa⁴ of the closed radiation universe (equivalently a_max, or the present curvature radius), accounted separately; not supplied by g_*^UV. [posterior]
  5. Matching-branch choices: the four components of the minimal map 𝓜₀ and the trivial tag 𝔉 = {⋆} (§3). [constructional input] (The winding phase Δθ̃_𝒥 and the leakage ε_leak are not inputs of Bang-0: no conclusion of this paper depends on their values; both belong to the full-cycle extension, §10.)

§3 Model Definition: Tuple and Postulates

The model is the tuple

𝔐_Bang = (𝒰⁻, 𝒥, 𝒰⁺, Σ⁻, Σ⁺, 𝔅⁻, 𝔅⁺, ℰ_bulk, H_MS, 𝓜),

where 𝒰∓ are the contracting/expanding Lorentzian bulks, 𝒥 is the primitive transition core not belonging to the fourth-layer spacetime domain, Σ± are its effective matching surfaces on the two sides, 𝔅± the admissible boundary-data spaces, ℰ_bulk the bulk field equations, H_MS the cap postulate, and 𝓜 the cross-domain matching map. Objects are placed on two levels (following the two-level convention of Cosmo VII [8]): the cycle-pairing postulate C1 and the Pair relation live at the primitive 𝒥-incidence level; effective actions and boundary functionals live at the Σ-incidence level.

Postulate 1 (H_bulk, model jurisdiction). Exact FLRW with k = +1, a single metric, minisuperspace:

ds² = −c²dt² + a²(t)dΩ₃², H² = (8πG/3c²)ε_tot − c²/a², ε̇_tot + 3H(ε_tot + p_tot) = 0.

Exact FLRW entails σ_{μν} = 0 and C_{μνρσ} = 0 — anisotropy does not enter the theorem domain; this is a jurisdiction restriction, not a solution of the anisotropy problem. The near-cap matter content of this minimal model is fixed by H_fluid (Postulate 7) to be pure ultraviolet radiation; the SAE dark-sector components (the khronon dust integration constant, the causal-scalar dark energy, the four-form term) are set to zero in the minimal model, their reintroduction being an extension interface (§10); any component growing as a⁻⁶ or faster lies outside the Bang-0 domain. [model postulate]

Postulate 2 (H_slot^Bang, the reference scale and its identification). We take

R_L(T) := ℏc/(2πk_BT)

as the Landauer–Bekenstein reference scale: its functional form is inherited from the one-bit Bekenstein–Landauer construction of information-theory paper III §4.2 [9] [inherited]. On top of this, Bang-0 lays down an identification postulate: R_L(T) is promoted to the effective FLRW averaging radius of the thermal branch — the areal radius on which the cap postulate (Postulate 3) acts [model postulate]. Two remarks are required. First, inheriting a formula is not the same as automatically inheriting an object identity — this paper does not claim that the scale carries an active one-bit capacity semantics in the bath branch (the numerical fact in §4 is precisely the reason for not making that claim). Second, R_L is not the edge length of an indivisible Planck substrate cell — this sentence prevents the sub-Planck crossing of §4 from appearing to conflict with the absoluteness of the Planck substrate.

Postulate 3 (H_MS, the cap postulate). The model does not treat the reference-scale domain as an isolated black hole. Within the exact-FLRW jurisdiction, the endpoint of the fourth-layer atlas is declared to be the first moment at which the Misner–Sharp compactness [10,11] of R_L(T) reaches unity:

𝒞_MS(R_L) = 1, where 𝒞_MS(R) := 2G·E_MS(R)/(c⁴R) = (8πG/3c⁴)·ε_tot·R²,

with E_MS(R) = (4π/3)ε_tot R³ the Misner–Sharp energy of a uniform-density sphere [10]. Since the FLRW apparent horizon is defined precisely by 𝒞_MS(R_A) = 1 [11], the cap postulate reads R_L(T_cap) = R_A(T_cap): the apparent horizon has contracted to the reference scale. Physical reading [motivated]: the causal averaging domain of the thermal branch at the reference scale no longer fits inside the horizon, and the fourth-layer description ends there — a structural analogy with the corpus criterion "the fourth layer is inactive inside black holes" [2] (Appendix C), not derived from bit capacity, supplying the motivation for Postulate 4. [model postulate]

Postulate 4 (H_end, atlas endpoint). At the first contracting-side attainment of ε = ε_cap, ∂𝒰⁻ = Σ⁻ and the Lorentzian atlas terminates. The transition core 𝒥 possesses no Lorentzian time, no H(t), no local energy density, no entropy value, no particle states, and no continuous evolution parameter; accordingly this paper does not adopt any formulation of the type d⁺ = U_𝒥 d⁻ — the matching is a primitive boundary relation, not an evolution operator through the 0DD core. [model postulate]

Postulate 5 (H_match, the matching map). The boundary-data spaces are

𝔅± = {(a, H, ε_r, ⋆) : a > 0, ε_r = ε_cap, H² = (8πG/3c²)ε_cap − c²/a², ⋆ ∈ 𝔉}, 𝔉 = {⋆},

with domains 𝒟⁻ = {d ∈ 𝔅⁻ : H < 0}, 𝒟⁺ = {d ∈ 𝔅⁺ : H > 0}. The trivial tag ⋆ enters no stress tensor, carries no local field configuration and no old-side coarse-grained structural entropy; it only keeps the data space non-empty. The framework-level claim of non-trivial remainder inheritance, 𝓡_tot⁺ = 𝓜_R(𝓡_tot⁻, Ξ), stands unchanged in Cosmo 0 [2]; its formalization (which first requires settling the object type: scalar charge / equivalence class / topological label / remainder ledger / boundary functional) is listed as a future extension (§10) and is not undertaken by this minimal model. The minimal map of Bang-0 is

𝓜₀(a, H, ε, ⋆) = (a, −H, ε, ⋆).

Five properties: single-valued (immediate); total on 𝒟⁻ (immediate); Friedmann-constraint preserving (H² unchanged); cap-compatible (ε = ε_cap); expanding-branch (−H > 0). [constructional input; non-emptiness is the theorem of §5]

Side-orientation convention. Across the timeless 𝒥 there is no naturally continuous common time axis. Each bulk selects its future-directed unit normal n_± and defines H_± = (1/3)∇_μ n_±^μ; the sign flip H₋ ↦ H₊ = −H₋ in 𝓜₀ is a dual-side orientation matching convention, not a continuous sign change of some H(t) through the core (consistent with the series convention that signed quantities must carry explicit side labels; see Cosmo VII [8]).

Four clarifications on the matching. (i) ε⁺ = ε⁻ is a boundary condition of this model and must not be read as a lump of substantival energy passing through the 0DD core. (ii) The sign flip of H is a dual-side orientation convention (above) — the model is atlas termination plus matching, not a smooth bounce, and belongs to a different category from LQC's continuous passage of H through zero. (iii) Within the declared data spaces (a⁺ = a⁻, ε⁺ = ε⁻ = ε_cap, the same Friedmann constraint on both sides, H⁻ < 0 < H⁺), the sign branches yield H⁺ = −H⁻ uniquely; extended branches with H⁺ ≠ −H⁻ can exist only if these conditions are relaxed or boundary source terms are admitted. (iv) ⋆ is a trivial tag; the paper does not claim that "the remainder has been formally transmitted."

Comparison with external constructions. The Israel junction conditions [12] handle the gluing of two Lorentzian regions along an embedded hypersurface within one spacetime, their objects being the induced metric and the jump of extrinsic curvature — whereas Σ± are atlas boundaries, not hypersurfaces inside a single manifold. LQC effective dynamics [5] gives a smooth bounce, categorically distinct from atlas termination. Conformal cyclic cosmology (CCC) [13] matches adjacent aeons at conformal boundaries and relies on conformal continuation and mass fade-out, not a finite-density cap. The CPT-symmetric universe [14] builds a Bang mirror by analytic continuation, its Bang remaining a continuation structure rather than a core outside the domain. Ekpyrotic/cyclic brane cosmology [15] opens new cycles by brane collisions in higher dimensions — a precedent for interface-driven cycles, but its two sides are branes in a higher-dimensional space, not ontological levels. The model therefore stands as its own class, "cross-domain boundary matching"; the above serve for comparison only.

Postulate 6 (H_ledger, the three-way ledger). The objects matched across the transition fall into three ledgers. Structure ledger: the minimal version of this paper is a placeholder ⋆ (Postulate 5). Energy ledger: ε⁺ = 𝓜_E(d⁻, Ξ); energy is a matching output, and E⁺ = E⁻ is not required — within a cycle the comoving radiation energy already decays as a⁻¹; redshift and pressure work are standard cosmological dynamics, not a bookkeeping leak. Even where the minimal branch of this model takes ε⁺ = ε⁻, that is merely a matching boundary condition. Orientation ledger: 𝓔_± = ±E_loc records only the two-sided orientation of one and the same incidence; 𝓔₊ + 𝓔₋ = 0 is a zero-sum ledger of the face index and does not mean that the physical total energy vanishes (consistent with the signed-ledger convention of Cosmo VII [8]); the paper defines no "total energy of the universe" object.

Entropy, three ways: S_struct⁻ (old-side coarse-grained structural entropy) does not cross the core and is not an input of 𝓜; S_rad⁺ = s_rad(T_cap)·2π²a_min³ is fixed by the hot equilibrium state, is generically nonzero, and must not be called low; the gravitational side: the Bang-side output satisfies exactly vanishing Weyl and shear tensors; if one adopts a gravitational-entropy functional normalized to zero on exact FLRW, its initial value is zero — this paper does not institute such a functional. The canonical statement: the transition does not match old-side coarse-grained structural entropy. The Bang-side radiation entropy is redefined by the hot equilibrium state and is generically nonzero; the "low restart" refers specifically to the minimum of the Weyl, shear, and distinguishable-macrostructure account on the exact-FLRW output, not to the total thermodynamic entropy being near zero (the low-Weyl initial condition follows the idea of Penrose [16]). [model postulate + convention]

Postulate 7 (H_hot + H_fluid: thermal branch, pure-fluid cap surface, and acceptance). The Bang-side output is a hot equilibrium radiation state of a common bath, and the cap surface is a single common ultra-relativistic thermal fluid:

ε_tot|_{Σ±} = ε_UV, p_tot|_{Σ±} = ε_UV/3, ε_dust = ε_DE = ε_4f = 0 on Σ± and in the near-cap region of the minimal model;

all degrees of freedom active in the common bath at the cap temperature are counted in g_^UV, with H_eq: g_{s}^UV = g_*^UV. Away from the transition region, standard local quantum field theory and low-density general relativity are recovered. Acceptance status: segment-by-segment acceptance against Big-Bang nucleosynthesis, recombination, and the microwave background is the model's empirical survival condition (a pending test) — the non-emptiness theorem of this paper proves only the existence of the pure-radiation exact-FLRW branch; it does not prove that a complete background solution incorporating the SAE dark sector, Standard-Model particle thresholds, and late-time acceleration would still pass segment by segment. Constructing that solution and running the BBN/CMB background likelihood are open problems (§10). The model covers only the thermal branch — local thermal equilibrium, no macroscopic coherent order parameter; condensate, coherent, or non-thermal collapse branches lie outside the domain. [model postulate + pending acceptance]

Failure conditions, collected. If the bulk equations imply that solutions never reach the cap, lose existence before the cap, or necessarily overshoot it, the corresponding branch fails (verified not to occur for closed radiation FLRW, §5); a robust determination of open spatial curvature refutes the model (§8, F1).

Figure 1 · SAE-Bang-0: cross-domain boundary matching 𝓜₀ : (a, H, ε, ⋆) ↦ (a, −H, ε, ⋆) primitive boundary relation, not an evolution operator (no d⁺ = U_𝒥 d⁻) 𝒰⁻ contracting-side Lorentzian bulk exact FLRW, k = +1 · H < 0 a(t) contracting boundary data d⁻ = (a_min, H₋, ε_cap, ⋆) Σ⁻ ε = ε_cap (atlas termination) 𝒥 (0DD transition core) no time · no energy · no entropy outside the L4 atlas C1 / Pair live at 𝒥-incidence Σ⁺ ε = ε_cap (atlas opening) 𝒰⁺ expanding-side Lorentzian bulk exact FLRW, k = +1 · H > 0 a(t) expanding 𝓜₀(d⁻) = (a_min, H₊, ε_cap, ⋆) Chain 𝒰⁻ → Σ⁻ → 𝒥 → Σ⁺ → 𝒰⁺; the sign flip of H occurs in no spacetime domain (atlas termination + matching, not a smooth bounce) ⋆ is a trivial tag (𝔉 = {⋆}); side-orientation convention: H_± = (1/3)∇_μ n_±^μ, each bulk with its own future-directed normal
Figure 1 — The SAE-Bang-0 model: cap, transition core, and cross-domain matching.

§4 Main Theorem: Cap Temperature and Cap Energy Density

Theorem (cap values). H_slot^Bang + H_MS + H_hot (including the pure-fluid cap surface H_fluid) ⟹

T_cap/T_P = √(45/(πg_^UV)), ε_cap/ε_P = 67.5/g_^UV, R_cap/l_P = √(g_*^UV/(180π)).

Derivation. The radiation branch gives ε_rad(T) = (π²/30)·g_*^UV·(k_BT)⁴/(ℏ³c³) [17]; the reference scale is R_L(T) = ℏc/(2πk_BT). The pure-fluid cap surface sets ε_tot = ε_rad; substituting into the compactness condition (8πG/3c⁴)·ε_rad·R_L² = 1:

(8πG/3c⁴)·(π²/30)g_(k_BT)⁴/(ℏ³c³)·ℏ²c²/(4π²(k_BT)²) = πG·g_·(k_BT)²/(45ℏc⁵) = 1,

so (k_BT_cap)² = (45/πg_)·E_P² (using E_P² = ℏc⁵/G), which is the first formula; substituting back into the radiation law gives ε_cap/ε_P = (π²/30)g_·(45/πg_)² = 67.5/g_ (with ε_P = c⁷/(ℏG²)); and R_cap = R_L(T_cap) gives the third. The left-hand compactness is monotone ∝ T² and meets the constant right-hand side at exactly one positive crossing — the construction contains no algebraic circularity. ∎

Standard-Model benchmark (g_*^UV = 106.75): T_cap = 0.3663 T_P, ε_cap = 0.6323 ε_P, R_cap = 0.4345 l_P.

Preconditions and exactness. The three formulas are conditional theorems: their exactness condition is precisely the pure-fluid cap-surface postulate H_fluid (non-radiation components exactly zero on Σ±). Working instead in a branch with small non-radiation components requires δ_cap := ε_nonrad/ε_rad|_cap, turning the formulas into corrections of the type T_cap² ∝ 1/(g_*(1+δ_cap)) and adding one input — this paper takes the minimal H_fluid and lists the corrected branch as an extension interface. Status of the coefficients: κ = 4π/3 is supplied by Misner–Sharp geometry, and η = 1 is the exact marginal-trapping condition (𝒞 = 1) — under H_fluid, no adjustable O(1) coefficient remains in the cap values.

Boundary-limit finiteness of curvature (the claim of §1). The model deliberately does not assemble 𝒰⁻ ∪ 𝒥 ∪ 𝒰⁺ into a single Lorentzian manifold, so there is no single curvature tensor across 𝒥; the paper defines no curvature on 𝒥, does not glue the two metrics into a distributional Israel junction, and therefore neither asserts a global trans-core curvature invariant nor asserts the absence of some distributional surface source. What is proven is that the curvature of each bulk remains finite in the approach to its own boundary: for exact FLRW the Kretschmann invariant

𝒦 = R_{μνρσ}R^{μνρσ} = (12/c⁴)[(ä/a)² + (H² + c²/a²)²]

satisfies lim_{x→Σ⁻}𝒦⁻(x) < ∞ and lim_{x→Σ⁺}𝒦⁺(x) < ∞, guaranteed by finite ε_cap, p_cap and a_min > 0; likewise the Ricci scalar. At the cap the ratio of the curvature term to the density term is (R_A/a_min)² ≈ 1.7×10⁻⁶² (at the q_K = 0.002 benchmark, conditional on the bridge of §6).

An intermediate quantity and its status. In the thermal branch the average energy contained in a spherical averaging domain of radius R_L is E_bath = (g_^UV/180)·k_BT [conditional: thermal-branch expectation value]. This is a mean-field statement, not any "at most one packet per slot" occupancy proposition — the model neither asserts nor needs an occupancy theorem; it only computes when the bath-averaged causal sphere reaches marginal trapping. A related numerical fact (Appendix D): the Bekenstein bit capacity [18] corresponding to that average energy is g_/(180 ln 2) = 0.856 at the SM benchmark — the bath-averaged domain does not even reach one Landauer bit [19]. This fact is precisely the reason Postulate 2 does not import the one-bit capacity semantics into the bath branch and instead re-identifies R_L as a reference scale; the one-bit construction of information-theory III [9] is the source of the functional form of R_L, and the two readings coincide numerically only at g_* = 180 ln 2 ≈ 124.8.

Constructional identity. In the bath-defined branch, ε_cap is by definition ε_rad(T_cap), so the bath temperature T_max obtained by inverting the same equation of state is identical to T_cap; this is constructional consistency, not independent verification.

Two conditions of semiclassical validity:

Condition Requirement SM benchmark
Density condition: ε_cap < ε_Pg_*^UV > 67.5satisfied
Length condition: R_cap ≥ l_Pg_*^UV ≥ 180π ≈ 565.5not satisfied

The SM benchmark satisfies only the density condition; R_cap = 0.434 l_P < l_P means the crossing equation employs semiclassical gravitational quantities at a sub-Planck areal radius, so its status is that of a boundary-matching ansatz, not a reliable semiclassical derivation — stated plainly (R_L is an effective averaging radius, not a Planck substrate cell; Postulate 2). Both conditions are validity conditions, not falsifiers of the model.

A structural echo [motivated note, not used in derivation]: the cap condition R_L = R_A places the "horizon readout" ontology on the same object at both ends of the series — the acceleration-scale line a₀ = c²/(2πR_A) [20] and the atlas-endpoint condition of this paper are both built on the FLRW apparent horizon R_A. The echo is registered; it is not used in any derivation.

Margolus–Levitin footnote. The ML bound [21] produces ξ ≥ π² only under the additional dynamical postulate that each domain must complete one orthogonal state evolution within its own light-crossing time; this model contains no such postulate (static distinguishability, repeated weak measurement, and multi-timescale cumulative discrimination are not directly constrained by it), so the bound does not apply. Should a future definition of causal readout require per-domain orthogonalization, the bound reopens automatically.

Figure 2 · The capacity cap: crossing of R_L(T) and the apparent horizon R_A(T) log–log; radiation branch, g_*^UV = 106.75 (SM benchmark); slopes −1 and −2 are exact power laws T > T_cap: R_A < R_L L4 atlas ends (𝒞_MS > 1) R = l_P (length-condition reference) R_A(T) ∝ T⁻² (apparent horizon, radiation) R_L(T) = ℏc / (2πk_BT) (reference scale) 𝒞_MS(R_L) = 1 (T_cap, R_cap) = (0.366 T_P, 0.434 l_P) 0.05 0.1 0.2 0.366 0.5 1.0 T / T_P (log) 10 1 0.1 R / l_P (log)
Figure 2 — The capacity cap: reference scale meeting the FLRW apparent horizon.

§5 Non-Emptiness Theorem: Explicit Matching and a Two-Atlas Witness

Theorem (non-emptiness). There exists d⁻ ∈ 𝒟⁻ with 𝓜₀(d⁻) ∈ 𝒟⁺, and bulk solutions exist on both sides and join at these boundary data.

Witness (a family of two independent atlases). For the closed pure-radiation universe [17], K_r = εa⁴ is the integration constant and A² := 8πGK_r/(3c⁴), so that

H² = c²(A² − a²)/a⁴, a(η) = A·sin η, ct(η) = A(1 − cos η), η ∈ (0, π),

with a_max = A. Choose any A > R_cap, and set a_min := √(A·R_cap); then R_cap < a_min < A, there exists a unique η₀ ∈ (0, π/2) with sin η₀ = a_min/A, and by R_cap² = 3c⁴/(8πGε_cap) the equality ε(a_min) = K_r/a_min⁴ = ε_cap holds exactly. Use two independent atlas copies:

  • 𝒰⁺: the expanding branch of one copy, boundary point at η₊ = η₀, with H₊ = +c√(A² − a_min²)/a_min² > 0;
  • 𝒰⁻: the contracting branch of the other copy, boundary point at η₋ = π − η₀, with H₋ = −c√(A² − a_min²)/a_min² < 0.

Taking d⁻ = (a_min, H₋, ε_cap, ⋆) ∈ 𝒟⁻, we have 𝓜₀(d⁻) = (a_min, −H₋, ε_cap, ⋆) = (a_min, H₊, ε_cap, ⋆) ∈ 𝒟⁺ — exactly the boundary data of 𝒰⁺. The chain is

𝒰⁻ → Σ⁻ → 𝒥 → Σ⁺ → 𝒰⁺,

two atlases, not time reversal along one and the same solution. ∎

Remark. This is a trivial witness — exact homogeneity and isotropy, with Weyl ≡ 0 rendering the low-Weyl condition trivially satisfied; it does not test the hard parts (anisotropy, perturbations), and a non-trivial witness with shear or perturbations is an open problem (§10). The value of the witness is that it makes the solution space non-empty, so that 𝔐_Bang is a model and not merely a contract of axioms. A distinction to keep sharp against §6: the witness family is a fully pure-radiation closed universe whose maximum scale factor is A; the observational numbers of §6 belong to a different object — a complete background extension including matter and late-time acceleration, connected to present-day observables by the adiabatic bridge, which has not been constructed. The two must not be conflated (§6 provides the numerical contrast).

Flatness inheritance [inherited note]: in the contracting phase ε_rad ∝ a⁻⁴ grows faster than the curvature term a⁻², so the previous cycle's contraction automatically compresses the curvature fraction at the cap surface to ~10⁻⁶² — extreme flatness of a closed universe at the matching boundary is a standard result of contraction dynamics (precedent in cyclic-type constructions [15]), which this model inherits. The converse threat — anisotropy growing as a⁻⁶, faster still — is excluded from the Bang-0 domain by the jurisdiction clause and listed as an extension (§10).

Figure 3 · Non-emptiness witness: two independent closed radiation atlases with minimal matching a(η) = A·sin η, η∈(0, π); choose any A > R_cap, a_min = √(A·R_cap); schematic, a_min/A not to scale a_max = A a = a_min = √(A·R_cap) 𝒰⁻ (contracting-side atlas copy) H₋ < 0, terminates at Σ⁻ (η₋ = π−η₀) 𝒥 outside atlas 𝓜₀: (a_min, H₋) ↦ (a_min, H₊) 𝒰⁺ (expanding-side atlas copy) H₊ > 0, opens from Σ⁺ (η₊ = η₀) Σ⁻ Σ⁺ η (each copy has its own conformal time; no common time axis between copies) Solid segments η∈[η₀, π−η₀] are the atlas domains used; dashed tails lie outside. ε(a_min) = ε_cap holds exactly by construction. H_± = ±c√(A²−a_min²)/a_min²; the far end of each copy is also a cap boundary (unused in the theorem). Two atlases — not time reversal along one solution.
Figure 3 — Non-emptiness witness: two independent closed pure-radiation atlases.

§6 Numbers and Scales: A Micron-Scale Initial Universe (Conditional on a Complete Background Extension)

Results are presented in three categories: model conditional outputs (functions of g_*^UV); conditional numbers on the observational bridge (requiring in addition H_ad and q_K); and the Standard-Model benchmark row.

Model conditional outputs: the three formulas of §4.

Conditional numbers on the observational bridge. The adiabatic bridge H_ad (§2, input 3) gives

a_min/a_now = (T_0/T_cap)·(g_{s,0}/g_^UV)^{1/3} = 1.74×10⁻³² (SM benchmark, T_0 = 2.725 K, g_{*s,0} = 3.909 [17]).

The present curvature radius is a_now = c/(H₀√q_K) (H₀ = 67.4 km s⁻¹ Mpc⁻¹ [7]). With H₀, T_0, and g_*^UV fixed, this yields the conditional equalities

a_min(q_K) = 53.5 μm · (0.002/q_K)^{1/2}; E_cap^rad(q_K) := ∫_{S³} ε_cap dV = 2π²a_min³ε_cap = 4.5×10⁹² E_P · (0.002/q_K)^{3/2},

where E_cap^rad is the radiation matter-energy integral on the homogeneous cap slice — not a conserved total energy of the universe, and containing no independently defined gravitational energy (consistent with "no total-energy object", Postulate 6). Lower-bound forms hold only under an additional hypothesis q_K ≤ q_K,max: a_min ≥ 53.5 μm·(0.002/q_K,max)^{1/2}. At q_K = 0.002 (68% confidence-scale benchmark), a_min = 53.5 μm; at q_K = 0.0030 (an illustrative closed-side ~95% margin), a_min = 43.7 μm; within the currently allowed range q_K ≲ 0.0037 at the 95% confidence scale, a_min ≳ 39 μm. Robust conclusion: conditional on the complete background extension and current curvature observations, the minimum curvature radius (i.e., scale factor) of the closed 3-sphere is of order tens of microns — the universe was never smaller than a micron-scale dust grain; "micron-scale dust grain" is an intuitive figure of speech, the rigorous statement being in terms of the curvature radius. It is not a fermi scale, still less a Planck cell or a point. Sensitivity: a_min ∝ (g_^UV)^{1/6} (conditional on H_eq), about 12% per doubling of g_.

The conditions, stated. The conditions of the above numbers are: H_ad holding over the whole history, the q_K input, and validity of the thermal branch — that is, the existence of a complete background extension with matter and late-time acceleration, passing standard thermal-history acceptance and matching present-day (H₀, q_K, T₀); this paper does not construct that extension (§10). Confidence bounds must come from the joint posterior of H₀ and Ω_k. The witness of §5 and the numbers of this section are two different objects: at the q_K = 0.002 bridge benchmark, the cap datum a_min = 53.5 μm corresponds to a pure-radiation witness parameter A = a_min²/R_cap ≈ 4.1×10²⁶ m, whereas a_now = 3.07×10²⁷ m > A — the maximum scale factor of the pure-radiation solution falls short of today's curvature radius, so the §5 witness cannot also accommodate today's observed scales; the numbers of this section are a prospective benchmark on the interface of the complete extension, not outputs of an already-existing model solution.

Terminology and bookkeeping. Here a is the curvature radius / scale factor of the closed 3-sphere; the areal radius of a local 2-sphere is R = a·sin χ — the phrase "minimal areal radius" is not applied to the 3-sphere as a whole. Within a cycle the comoving radiation energy varies as a⁻¹ (Postulate 6); differing energy accounts at different times are standard redshift bookkeeping, not a contradiction; the paper defines no "total energy of the universe" object.

The old-calibration stretch factor. Taking the retired symmetric-breathing calibration a_max^sym = 5.2161×10²⁵ m [22] as the reference (§9, R1), define 𝒮_a := a_max^actual/a_max^old,sym; then

𝒮_a(q_K) ≥ √(|Ω_k,sym|/q_K) = 58.8 · (0.002/q_K)^{1/2} (illustrative value 48.0 at q_K = 0.003).

Its status must be explicit: 𝒮_a measures only how much the currently allowed maximal scale exceeds the retired old symmetric calibration; a fully time-symmetric but larger solution would equally have 𝒮_a ≫ 1, so 𝒮_a does not measure the expansion–contraction asymmetry within a breath. Genuine asymmetry requires comparing I_exp with I_con (or t_exp with t_con); the map from 𝒮_a to such asymmetry parameters depends on the complete background dynamics (§9 R3, §10).

§7 Matter Realization on the Bang Side: Hierarchical Reading and Standard Inheritance

How does the Bang-side energy become the matter we see? The SAE dimensional ladder [23] supplies a hierarchical reading: bare energy is L1 boundary data; photons occupy L1+L2 (energy and momentum, no mass); massive particles additionally occupy L3. The canonical statement:

The matching map first outputs a hot L2 radiation state. Once Lorentzian causal dynamics is restored, standard QFT processes produce L3 massive degrees of freedom whenever their respective channel invariants, field strengths, or thermal-bath conditions are met. SAE contributes here only the hierarchical interpretation that "mass closure belongs to L3"; thresholds, cross sections, rates, and the thermalization history are inherited from standard QFT and are not derived from the hierarchy itself.

Threshold physics, stated precisely [inherited]: the Breit–Wheeler two-photon pair-production threshold [24] is the center-of-mass invariant √s ≥ 2m_ec² = 1.022 MeV (in a head-on collision 0.511 MeV per photon suffices; it is not "each photon must carry 1.022 MeV"); the Schwinger critical field [25] is a non-perturbative field-induced production mechanism; the QCD ~155 MeV is a crossover in the thermal bath [26,27]. These are three distinct mechanisms, not one universal gate. The hierarchical reading provides their ontological placement (L3 occupancy) and alters no numbers — the ladder re-expression is complete well before BBN, and pointwise agreement with the standard thermal history thereafter is a design requirement of the model (the H_hot acceptance interface), not a negotiable item. One further categorical remark: the paper does not narrate "bare energy existing for a while and later becoming photons" — that would rewrite a matching relation as a temporal sequence inside the core; the correct statement is that Crunch-side energy boundary data, via 𝓜_E, are first realized on the Bang side as an admissible L2 radiation state.

Conditional side note on monopoles. In this paper's minimal "Standard Model + SAE dark sector" baseline, no high-energy fields are introduced whose GUT-scale spontaneous breaking would produce stable topological monopoles, so the classical GUT monopole overproduction problem [28] does not arise within this baseline. This is a consequence of field-content choice — the same as for the Standard Model without a GUT sector; it is neither an independent SAE solution of the monopole problem nor a universal exclusion of magnetic monopoles. If a GUT sector is added later, this note lapses automatically.

§8 Falsifier, Geometric Constraint, and Discriminator

Three items of three kinds: one falsifier (F1), one conditional geometric constraint (F2), and one prospective discriminator (F3).

F1 (falsifier: curvature sign). The model is defined at k = +1, hence Ω_k < 0 strictly. A robust future determination of open spatial curvature refutes the model. The status is recorded on two levels: at the level of this paper it is the direct empirical consequence of the k = +1 model definition — a model-definitional counter-prediction [conditional]; at the framework level it additionally rests on the closed-space structural requirement and observational pressure already registered in Cosmo 0 §9 [2]. The current Ω_K = 0.0007 ± 0.0019 (Planck 2018 + BAO, 68% [7]) is compatible with closure; a future significantly positive Ω_total − 1 would positively support it.

F2 (conditional geometric constraint: the old-calibration stretch bound). 𝒮_a(q_K) ≥ 58.8·(0.002/q_K)^{1/2} (§6). It bounds from below how much the old symmetric calibration must be stretched; it does not by itself measure expansion–contraction time asymmetry, and the map from 𝒮_a to genuine asymmetry parameters depends on the complete background dynamics. The observational routes for curvature are joint constraints from BAO, CMB acoustic geometry, lensing, and SNe/large-scale structure (DESI [29], Euclid, the Simons Observatory, and others); LiteBIRD's primary target is large-scale B-modes, not precision curvature. CMB-S4 was terminated by DOE/NSF on 2025-07-09 [30] and is removed from the observational timeline.

F3 (prospective discriminator: the no-inflaton baseline). The minimal SAE-Bang baseline contains no inflaton and invokes no accelerated-expansion phase. Detection of a nonzero primordial tensor ratio r alone does not refute the baseline. The baseline is refuted only if, in one and the same joint dataset, all of the following hold: (i) primordial r > 0 is detected with significance; (ii) the tensor tilt n_t is measured independently and is consistent with the leading canonical single-field slow-roll consistency relation n_t = −r/8 [31]; (iii) the joint likelihood over the scalar spectrum, non-Gaussianity, and tensors significantly prefers that slow-roll branch; (iv) the result excludes a previously fixed SAE non-inflationary perturbation-spectrum family. Until the SAE spectrum family is fixed (§10, open problems), this item is a prospective discriminator, not a parameter-free counter-prediction. Near-term nodes: BICEP Array aims at σ(r) ≲ 0.003 with data through the 2027 observing season (current BK18 bound r_0.05 < 0.036 [32,33]); LiteBIRD targets launch in Japanese fiscal year 2036 (second Mission Definition Review passed in June 2026) [34], with δr ~ 10⁻³ [35].

Model failure conditions, collected: the three H_MS clauses (never reaching the cap / losing existence before the cap / necessarily overshooting it — verified not to occur for closed radiation FLRW); and the open-curvature determination of F1.

§9 Registered Revisions to Earlier Papers

Following the series convention, revisions to earlier papers occasioned by this work are registered here explicitly.

R1 (retirement of the symmetric-breathing calibration family). The symmetric closed-FRW numerical family of Cosmo III [22] — a_max = 5.2161×10²⁵ m, turnaround at 10 Gyr, a_geo(13.8 Gyr) = 0.915 a_max, H_geo = −50 km s⁻¹ Mpc⁻¹, f_Λ = 0.1038, and the 7.97 Gyr global causal-completion epoch — is demoted to a calibration artifact of the symmetric-breathing branch and no longer serves as the geometry of the present universe. Reason: the observed curvature (at the q_K = 0.002 benchmark [7]) requires a present curvature radius a_now > 3.07×10²⁷ m, while the conformal factor cancels exactly in Ω_k (ã²H̃² = a²(H + Ȧ/A)²; for the frame structure see Cosmo V [36]), so no frame transformation can absorb the ~60-fold scale discrepancy. Surviving items: the general period sum rule (expressed through I_exp + I_con and a matching holonomy), the compact-kernel structure of η_max = π/2 (conditionally), the plateau value a₀ = (π/2)cΔω (conditionally), and the exact cancellation of the conformal factor in null-geodesic integrals [22].

R2 (revision of the energy and entropy formulations of Cosmo 0). In Cosmo 0 [2] §7.4, the abstract, and §10, "what is carried are the remainder and the energy" is revised to: what the transition preserves is not a lump of substantival energy passing through the timeless core, but 𝓡_tot and its matching constraints; L1 energy is undefined in the 0DD core; Crunch-side physical energy data enter as boundary inputs and, through the matching map 𝓜_E, constrain the admissible Bang-side energy density, flux, and composition, without requiring numerical equality; the signed energy ledger 𝓔_± = ±E_loc records only the two-sided orientation of one incidence, and 𝓔₊ + 𝓔₋ = 0 does not mean the physical total energy vanishes. Supplement: Cosmo 0's "low-entropy restart" is to be made Weyl-explicit — the low entry refers specifically to the Weyl, shear, and macrostructure accounts, not to total thermodynamic entropy (fed back from Postulate 6; [16]). Execution: registered here; to be incorporated at the next upgrade of Cosmo 0.

R3 (the physical identity of T₁ — registered open, systematic examination in progress). The retirement in R1 exposes a deeper problem: the Λ scale pins the clock period at T₁ ≈ 20 Gyr [1], while the observed curvature bound together with frame rigidity pushes the geometric breathing duration far beyond 20 Gyr — the default identity "clock period = geometric breathing duration" of the symmetric era can no longer be maintained. The formal locus is the compact-clock winding relation 2πN = ω₁(a_max/c)(I_exp + I_con) + Δθ̃_𝒥 (the old π/I(f) being the symmetric special case N = 1, Δθ̃ = 0; the winding freedom is not an input of Bang-0, see §2, §10). A preliminary coherence check shows that all quantities producing observational matches (Λ [1], a₀, Δω, the dark sector [20]) are clock quantities, so decoupling harms no surviving result; the sector assignment of the khronon is the key point awaiting adjudication. A systematic examination is under way and will be treated elsewhere; its outcome does not affect the model content of this paper.

§10 Extensions and Open Problems

Extensions (outside this paper's scope): a generalized capacity functional 𝔠(T_{μν}, σ_{μν}, C_{μνρσ}, …) ≤ 𝔠_cap defining a cap for non-thermal and sheared situations; the non-pure-fluid cap branch with δ_cap corrections (relaxing H_fluid); the reintroduction interface for the dark-sector components in the cap region (dust integration constant, late dark energy, four-form); the formalization of non-trivial 𝓡_tot inheritance (first settling the object type: scalar charge / equivalence class / topological label / remainder ledger / boundary functional); the conditions for a maximal single-slot occupancy proposition — future work seeking to establish it must supply the thermalization-timescale inequality τ_thermalize < τ_trap, a dynamical re-tiling update rule, and a coherent-state exclusion theorem, and until then the proposition does not exist in the model; the cyclic postulate C1 with the Top_C^𝒥 topological identification and winding matching (N and Δθ̃_𝒥 not defaulted to 1 and 0; the sign, range, and transformation law of ε_leak to be fixed prior to any comparison with data); the dynamics of the matching map and the perturbation spectrum (the two-clock action [2] — the series' registered core open direction, including the determination of n_s, f_NL, and the tensor family); and the spectral origin of g_*^UV.

Open problems (listed plainly, left to future work): the interior dynamics of the transition core; the primordial perturbation spectrum; a non-trivial witness with shear/perturbations; the construction of a complete background solution including the SAE dark sector, Standard-Model particle thresholds, and late-time acceleration, connected through the adiabatic bridge H_ad to present-day observables, with a segment-by-segment BBN/CMB background likelihood; non-thermal, condensate, or coherent collapse branches (outside the domain); and the generalization of the Misner–Sharp criterion beyond spherical symmetry.

§11 Conclusion

This paper gathers the qualitative Big-Bang claims of the SAE series into a formal cross-domain boundary model. Its skeleton is three things: a capacity-cap conditional theorem, given by the crossing of the Landauer–Bekenstein reference scale with the apparent horizon, with zero adjustable coefficients under the pure-fluid cap-surface postulate; a transition structure that replaces the singularity with an explicit matching map, witnessed non-empty by a family of two independent closed radiation atlases; and an acceptance interface with the standard thermal history as the empirical survival condition. Its boundaries are likewise three: the existence of the cap, the identification of the reference scale, and the pure-fluid cap surface are postulates, not a priori theorems; the model does not eliminate geodesic incompleteness, only reinterprets its boundary identity; and the perturbation spectrum, the matching dynamics, and the complete background acceptance with the dark sector are registered open problems.

One closing sentence: under this model — conditional on its complete background extension passing the standard adiabatic thermal history (§6) — and combined with current curvature observations, the Big Bang is not creation from nothing at a point, but a hot radiation boundary re-opening at finite temperature T_cap ≈ 0.37 T_P on a closed 3-sphere whose minimum curvature radius is tens of microns. The paper's input assumptions, benchmark values, and alternative parametrizations are all listed openly, for testing and revision; the guiding principle throughout is not to be free of error, but to have built something definite enough to be corrected.

Appendix A · Postulates and Claims — Tier Table

Item Tier
H_bulk: closed exact-FLRW minisuperspace jurisdiction (σ = C = 0; a⁻⁶-type components outside domain)model postulate
H_slot^Bang: functional form of R_L(T) = ℏc/(2πk_BT) inherited; promoted to the thermal-branch effective FLRW averaging radiusinherited (Info III [9]) + identification postulate
H_MS: 𝒞_MS(R_L) = 1 as atlas endpoint (R_L = R_A)model postulate
H_end: atlas terminates at Σ⁻; 𝒥 carries no L1–L4 quantitiesmodel postulate
H_match: 𝓜₀ = (a, −H, ε, ⋆); 𝔉 = {⋆}; side-orientation conventionconstructional input
H_ad: aTg_{*s}^{1/3} = const (bridge from cap surface to present observables)inherited + bridge condition (conditional on complete extension)
H_ledger: ⋆ placeholder / energy as matching output / orientation zero-sum ledger; three-way entropy split (Weyl and shear invariants vanish [16])model postulate + convention
H_hot + H_fluid: pure-UV thermal fluid on the cap surface, H_eq, p = ε/3; thermal history as pending acceptancemodel postulate + pending acceptance
Cap-value formulas (T_cap, ε_cap, R_cap)conditional theorem (conditions = H_slot^Bang, H_MS, H_fluid)
Zero adjustable coefficients in the cap values (κ, η absorbed into MS geometry [10,11]; conditional on H_fluid)property of the conditional theorem
E_bath = (g_*/180)k_BTconditional expectation value [conditional]
T_cap ≡ T_maxconstructional identity (not independent verification)
Non-emptiness witness (family of two-atlas closed radiation solutions + 𝓜₀; precondition A > R_cap)theorem (trivial witness)
Boundary-limit finiteness of curvature invariants on both sides; curvature/density ≈ 1.7×10⁻⁶² at the cap (benchmark)property of the conditional theorem
a_min(q_K) = 53.5 μm·(0.002/q_K)^{1/2}; E_cap^rad scaling likewiseconditional equality on the observational bridge (conditions = H_ad, complete extension)
a_min ≳ 39 μm within the 95%-scale rangeconditional lower bound (under q_K ≤ 0.0037)
Flatness inheritance (contraction compresses curvature to ~10⁻⁶²)inherited note [15]
Hierarchical reading (E → L2 → L3) [23]; thresholds/rates inherited from standard QFT [24–27]motivated / inherited
F1 Ω_k < 0falsifier: model-definitional counter-prediction [conditional]; framework level rests on Cosmo 0 §9 [2]
F2 𝒮_a(q_K) ≥ 58.8·(0.002/q_K)^{1/2}conditional geometric constraint (old-calibration stretch bound; not an asymmetry measure)
F3 no-inflaton prospective discriminator (four conditions)prospective discriminator (open)
Monopole side note [28]conditional side note
Structural echo of R_L = R_A with the a₀ line [20]motivated note (not used in derivation)
MS marginal trapping ⇄ L4 deactivation [2]structural analogy (motivated correspondence)
Maximal single-slot occupancy propositionnot asserted by this model (conditions in §10)
Formalization of non-trivial 𝓡_tot inheritanceextension (framework claim rests in Cosmo 0 [2])
Transition-core dynamics, primordial spectrum, non-trivial witness, complete background acceptanceopen

Appendix B · Explicit Non-Emptiness Witness (Two Atlases)

Closed pure-radiation FLRW [17], in the unified convention:

K_r := εa⁴ (energy-density integration constant), ε(a) = K_r/a⁴, A² := 8πGK_r/(3c⁴) (length²),

so that H² = (8πG/3c²)ε − c²/a² = c²(A² − a²)/a⁴. The conformal-time solution (the full domain of one atlas copy):

a(η) = A·sin η, ct(η) = A(1 − cos η), η ∈ (0, π), a_max = A.

Cap matching point: with R_cap² = 3c⁴/(8πGε_cap) (i.e., R_cap = R_A(T_cap)), choose any A > R_cap and set

a_min := (K_r/ε_cap)^{1/4} = √(A·R_cap),

so that R_cap < a_min < A, ε(a_min) = ε_cap holds exactly, and there exists a unique η₀ ∈ (0, π/2) with sin η₀ = a_min/A.

Two independent atlas copies: 𝒰⁺ takes the expanding-branch boundary point η₊ = η₀ (H₊ = +c√(A² − a_min²)/a_min²); 𝒰⁻ takes the contracting-branch boundary point η₋ = π − η₀ (H₋ = −H₊). Then d⁻ = (a_min, H₋, ε_cap, ⋆) maps under 𝓜₀ to (a_min, H₊, ε_cap, ⋆), exactly the boundary data of 𝒰⁺; the chain is 𝒰⁻ → Σ⁻ → 𝒥 → Σ⁺ → 𝒰⁺.

Numerical contrast (q_K = 0.002 bridge benchmark, see §6): a_min = 53.5 μm corresponds to A = a_min²/R_cap ≈ 4.1×10²⁶ m < a_now = 3.07×10²⁷ m — the pure-radiation witness family and the observational-bridge extension are two different objects.

Appendix C · Corpus Anchors

  1. Source of R_L: information-theory III §4.2 [9] inserts N_bits = 1 and E = k_BT ln 2 (the Landauer thermal floor [19]) into the Bekenstein bound [18] and solves R_L = ℏc/(2πk_BT), the ln 2 cancelling in the size formula. That one-bit construction is the source of the functional form of R_L; Bang-0 adopts it via the identification postulate H_slot^Bang (§3, Postulate 2) and does not claim an active one-bit capacity semantics in the bath branch (Appendix E).
  2. MS marginal trapping ⇄ L4 deactivation: the cap condition "the reference-scale domain no longer fits inside the apparent horizon" and the corpus criterion "the fourth layer is inactive inside black holes" [2] form a structural analogy (motivated correspondence) — the physical reading of H_MS draws its motivation from it; it is not cited as a theorem and is not derived from bit capacity.
  3. Satisfying the Relativity P5 caution: P5 [37] states that FLRW is not asymptotically flat and admits no standard global Schwarzschild radius; this paper replaces the isolated r_s with the Misner–Sharp compactness [10,11], well defined within FLRW — the caution is satisfied, not circumvented.

Appendix D · Numerical Sensitivities and Alternative Parametrizations

Main row (bath-self-consistent / MS, this model): see §4, §6. Sensitivities: a_min ∝ (g_^UV)^{1/6} (conditional on H_eq); f_ε = 67.5/g_^UV; the two semiclassical validity conditions: density g_ > 67.5, length g_ ≥ 180π ≈ 565.5 (the SM benchmark satisfies only the former). q_K scalings (conditional equalities): a_min ∝ q_K^{−1/2}, E_cap^rad ∝ q_K^{−3/2}, 𝒮_a ∝ q_K^{−1/2}.

Alternative parametrizations (alternative microscopic assignments, retained for comparison, not adopted by this model): ξ = ln 2 (Landauer definitional reading [19]) ⟹ ε_cap = 3/(8 ln 2)ε_P = 0.541 ε_P, T_cap = 0.339 T_P; ξ = 1 (thermal-quantum reading) ⟹ ε_cap = (3/8)ε_P, T_cap = 0.282 T_P. The general formulas T_cap/T_P = 1/√(4πηξ) and ε_cap/ε_P = π/(2κη²ξ) are retained as the parametrized family, so that future revisions can be located within it.

N_Bek note: the Bekenstein bit capacity [18] of the bath-averaged domain energy is g_/(180 ln 2) = 0.856 (SM benchmark); the bath reading and the one-bit reading coincide numerically only at g_ = 180 ln 2 ≈ 124.8 — a proximity that belongs to the neighborhood of coincidence and is not to be used as evidence of common origin.

Appendix E · Average Domain Energy of the Near-Cap Thermal Branch and the Bath-Self-Consistent Cap

This appendix records the computation of the average energy of the reference-scale domain in the near-cap thermal branch, its diagnostic relation to the Bekenstein bound, and the alternative parametrizations of the cap value.

  1. Source: the information-theory III one-bit construction of R_L [9] (Appendix C.1).
  2. Average domain energy: in the thermal branch, E_bath = ε_rad(T)·(4π/3)R_L³ = (g_*^UV/180)·k_BT — a conditional expectation value, not a maximal occupancy: it provides no single-fluctuation upper bound, no encodable-bit ceiling, and no non-existence of coherent states.
  3. MS compactness crossing: the main theorem of §4. Applying to the same small sphere both the extensive radiation entropy S_rad = 4E/3T and the Bekenstein bound S ≤ E/T [18] that defines R_L yields the diagnostic 4/3 > 1 — infinite-volume extensive thermodynamics, a one-bit Bekenstein domain, and an isolated self-gravitating body cannot all be exact descriptions of a tiny region simultaneously; hence the bath reading has the status of a model mean field.
  4. The three ξ readings are distinct microscopic assignment schemes (Appendix D), not three approximations of one physical quantity.
  5. Jurisdiction: the computation is valid only in the thermal branch — local thermal equilibrium, no macroscopic coherent order parameter; condensate, coherent, or non-thermal branches lie outside the domain.

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Acknowledgements

The framework direction, the core arguments, and the final adjudication of every claim in this paper are the responsibility of Han Qin. In the course of writing, the author made use of several artificial-intelligence systems for conceptual organization, verification of derivations, geometric consistency checking, and constraint-oriented criticism; the contribution of these systems is limited to auxiliary organization, checking, and critique, and does not constitute authorship of the scientific claims of this paper.

Author's Statement of Responsibility

All claims in this paper, their tier tags, and every open problem and possible error marked herein are the sole responsibility of the author. The capacity cap is the conditional conclusion of model postulates; the matching map is a constructional choice; the perturbation spectrum and the complete background acceptance are registered open problems; the author presents all of this in full, under the principle of aiming not at freedom from error but at having built something definite enough to be corrected. Any omissions and errors are the author's responsibility.

[Working status block (delete before publication), v1.0-EN, 2026-08-11] Independent English rewrite of CN v1.4 (not a translation): postulate structure carried exactly (H_slot^Bang identification postulate; E_cap^rad; 𝒮_a old-calibration stretch factor; H_ad adiabatic bridge; prospective benchmark conditional on a complete background extension); citation numbering [1]–[37] inherited from CN v1.4 in identical order. To do: (1) Han's review; (2) EN figure set (zero CJK) attached alongside; (3) S1 status-block deletion and S2 DOI verification at deposit; (4) bilingual single-deposition Zenodo metadata (JSON + web-form-fill).