Layer 3 · the derivation · BBN concordance

BBN protection: Ψ frozen during radiation domination

On Branch A with conformal-invariant radiation, the source of Ψ vanishes (T = 0), so G is constant across BBN. The path-ζ analysis (F130b §C.2) corrects an earlier framework error.

Result — Brown convention: trivial pass; Schutz/Hawking: bounded

Ψ is frozen during radiation domination. Standard BBN abundances are inherited unchanged.

On Branch A with conformal-invariant radiation, the trace of the matter stress-energy vanishes, so the Ψ source vanishes and Ψ does not evolve. The effective Newton constant G = 1/Ψ is therefore constant across the BBN epoch.

F130b §C.2 (Brown convention): |ΔG/G|_BBN→today= 0 exactly · Schutz/Hawking: at-most power-law, bounded

Step 1

The Ψ-equation on Branch A

B o x, P s i; =; f r a c 8 p i 3, (1 + f), T

The trace of the modified Einstein equation gives a wave equation for Ψ sourced by the matter trace T = T^μ_μ. This is the canonical Branch-A result; the Branch-B alternative simply has Ψ constant from the outset and reaches the same conclusion trivially.

Frameworks consulted

ALLOWED·F01 trace equation
WARNING·Branch B (Ψ constant)[different branch — also gives trivial pass]

Step 2

Conformal invariance of radiation

T^{m u}_{m u}^{, t e x t r a d}; =; r h o_{r} - 3 p_{r}; =; 0

The electromagnetic field is conformally invariant in 4 dimensions; its stress-energy is traceless, so a radiation-dominated universe has T = 0. This is convention-dependent at higher order: the Brown convention ℒ_m = T/4 gives strict T = 0; the Schutz/Hawking conventions are weakly non-zero but produce only bounded power-law growth, not exponential.

Frameworks consulted

ALLOWED·Standard EM Lagrangian
ALLOWED·Brown matter convention
WARNING·Schutz/Hawking convention[Ψ source non-zero — bounded sub-leading effect]

Step 3

Therefore Ψ is frozen

B o x, P s i = 0;; L o n g r i g h t a r r o w;; P s i (t, v ec x) = P s i_{t e x t B B N} = t e x t co n s t

With zero source and the cosmological boundary conditions of homogeneity and isotropy on the bare radiation-dominated background, the only solution is Ψ = const. The scalar simply cannot evolve while radiation rules the energy budget.

Frameworks consulted

ALLOWED·Static solution to wave equation

Step 4

Newton constant constant across BBN

G (t) = f r a c 1 P s i (t) = t e x t co n s t;; L o n g r i g h t a r r o w;; B i g ∣ f r a c D e l t a G G B i g ∣_{t e x t B B N t o t e x t t o d a y} l e 6 t e x t pp m

The strongest published bound on cosmological variation of G from BBN is a few parts per million across the entire 13.8 Gyr from BBN to today. ISST predicts strictly zero variation on Branch A in the Brown convention; in the most aggressive alternative convention the bound is comfortably met. The (1+f) coupling is irrelevant during radiation domination because the radiation energy density dominates the matter density by a factor of ~10⁴ at the BBN epoch.

Frameworks consulted

ALLOWED·Standard BBN reaction network
ALLOWED·BBN bound on ΔG/G
ALLOWED·Varying-G BBN ruled out

Step 5

Standard BBN abundances inherited

Y_{p} = 0.247, q u a d D / H = 2.55 t im es 1 0^{- 5}, q u a d^{7}! t e x t L i / H = 1.6 t im es 1 0^{- 10}

Because G is unchanged and η ≡ n_b/n_γis set by primordial baryogenesis (which is unchanged in ISST), the standard BBN reaction network produces the same primordial abundances. Helium-4, deuterium, helium-3 all match observation. The lithium discrepancy is inherited unchanged from ΛCDM — likely an astrophysical (stellar depletion) effect, not cosmological. ISST has no prediction here that differs from the standard one.

Frameworks consulted

ALLOWED·Standard BBN code (PArthENoPE)

Step 6

What this denies, and what it doesn't

The frameworks that would predict an observable BBN-era variation in Gare denied at the master-passport level. Any conventional varying-G modified gravity theory has to explain why BBN doesn't see the variation; ISST has the cleaner answer — there's nothing for BBN to see, by construction. Standard primordial baryogenesis remains in place; ISST doesn't modify particle physics, only how matter gravitates.

Frameworks consulted

DENIED·Brans-Dicke varying G[scalar_propagates=false]
DENIED·DEF varying G[scalar_propagates=false]
DENIED·f(R) BBN modification[constraint_class=palatini]
ALLOWED·Primordial baryogenesis (standard)