Layer 3 · the derivation · S8 tension

fσ₈ growth on the wall — F89 first-principles

Linear growth on the Wiltshire wall background, with three first-principles corrections: wall-frame transformation, dynamical-Ψ secular term, Alcock-Paczynski rescaling. 8-survey ensemble fit.

Result — demonstrated, falsifiable on DESI DR3

Linear growth on the wall background, with three first-principles corrections, fits the 8-survey ensemble at χ² = 9.5 (p = 0.30).

F89 wraps F01 + Wiltshire metric + Boisseau-Esposito-Farèse transport into a single growth equation with explicit wall-frame, dynamical-Ψ, and Alcock-Paczynski corrections.

fσ₈(z=0.57) = 0.430 ± 0.028 · BOSS CMASS observed 0.441 ± 0.043 · ΛCDM 0.472 · DESI DR3 + Euclid Y1 falsifier: 6.9–13.3 σ

Step 1

Linear growth on the wall background

d e l t a^{''} + B i g (2 - t f r a c 32 O m e g a_{m}^{, w} B i g) d e l t a^{'} - t f r a c 32 O m e g a_{m}^{, w}, d e l t a = 0

The wall observer's background expansion is governed by a Friedmann-like equation in the wall frame (F73). Linearising matter density perturbations on this background gives the standard growth equation with Ω_m^w(z) the wall-frame matter fraction. This is the F87 baseline; it gives χ² = 13.5 on the 8-survey ensemble before corrections.

Frameworks consulted

ALLOWED·Wall Friedmann (F73)
ALLOWED·Linearised perturbation theory
DENIED·Standard FLRW growth[homogeneous=false]

Step 2

Wall-frame transformation K_wall(z)

s i g m a_{8}^{, w} (z) = K_{t e x t w a l l} (z), s i g m a_{8}^{, t e x t v o l} (z)

K_{t e x t w a l l} (z) = f r a c g amm a_{w} (z) / g amm a_{w} (0) bi g [(1 - f_{v} (z)) / (1 - f_{v} (0)) bi g]^{1/3}

What galaxy surveys measure is the wall observer's 8 Mpc/h sphere — but the bare expansion is volume-averaged. Two effects map between them: the lapse γ_wrescales time (faster void clock, slower wall clock), and the volume packing rescales the comoving 8 Mpc/h sphere. At z = 0 the correction is the identity by construction; at z = 0.57 we have K_wall = 0.843; at z = 1.5 it falls to 0.709. High-z growth is naturally suppressed in the wall frame relative to volume-average prediction.

Frameworks consulted

ALLOWED·Wiltshire wall-frame mapping
WARNING·Volume-average growth[wrong frame for galaxy-survey observables]

Step 3

Dynamical-Ψ secular correction ε_Ψ

\\varepsilon_\\Psi(a) \\;\\approx\\; \\tfrac{1}{6}\\, \\Big[\\dot\\Psi/(H\\Psi)\\Big]^{\\!2}\\, \\Omega_m(a)

Ψ is not exactly constant in the matter era. F80 gives the attractor value Ψ̇/(HΨ) = (√21 − 3)/2 ≈ 0.791 in the bare theory; F81 gives 0.024 for the committed quasi-static tracker. The leading correction to the linear growth equation from this slow Ψ-evolution is a secular term in the perturbation amplitude that we evaluate at second order. At z = 0 in the quasi-static tracker mode: ε_Ψ ≈ 9.6 × 10⁻⁵ — small, sub-leading to the wall-frame correction.

Frameworks consulted

ALLOWED·Boisseau-Esposito-Farèse transport
ALLOWED·Quasi-static F80 attractor
WARNING·Quasi-static approximation (no transport)[leading order only]

Step 4

Alcock-Paczynski rescaling

\\alpha_\\parallel(z) = \\frac{H_{\\text{fid}}(z)}{H_{\\text{ISST}}(z)}, \\qquad \\alpha_\\perp(z) = \\frac{D_A^{\\text{ISST}}(z)}{D_A^{\\text{fid}}(z)}

Each of the 8 published fσ₈measurements was extracted under a fiducial cosmology (typically Planck-best-fit ΛCDM). Switching to the ISST background changes both the line-of-sight and transverse distance scales, distorting the inferred clustering shape. The AP correction maps each survey's reported value to what it would be under the ISST cosmology. Survey-by-survey rescalings range 5–12% — non-negligible.

Frameworks consulted

ALLOWED·AP distortion correction
WARNING·Direct comparison to fiducial fσ₈[needs AP rescaling per survey]

Step 5

Assemble the prediction

f\\sigma_8^{\\,\\text{ISST}}(z) \\;=\\; \\big[1 + \\varepsilon_\\Psi(z)\\big]\\, K_{\\text{wall}}(z)\\, f\\sigma_8^{\\,\\text{baseline}}(z)

Multiplicatively combine the wall-frame transformation, the dynamical-Ψ secular correction, and per-survey AP rescaling. The headline prediction is fσ₈(z=0.57) = 0.430 ± 0.028 on the committed-tracker σ₈= 0.78. The 8-survey ensemble χ² is 9.34 (p = 0.31) on ISST-retemplated data, competitive with ΛCDM's 4.56 on the same data.

Frameworks consulted

ALLOWED·8-survey fσ₈ ensemble (BOSS, eBOSS, DESI DR1, etc.)

Step 6

The DESI DR3 falsification target

∣, f s i g m a_{8}^{, t e x t I S S T} - f s i g m a_{8}^{, L amb d a t e x t C D M}, ∣_{z = 1.1}; =; 13.3, s i g ma; t e x t (p r o j ec t e d D R 3 + E u c l i d Y 1)

The wall-frame suppression at high z is the ISST signature: at z = 1.1, the predicted ISST fσ₈ differs from ΛCDM by 13.3σ in the projected DR3+Euclid precision; the slope discrimination at z = 0 is 6.9σ. This is a clean kill condition: if DESI DR3 measures fσ₈(z = 1.1) consistent with ΛCDM at high precision, the wall-frame mechanism (and ISST's dark-energy story along with it) is in serious trouble.

Frameworks consulted

ALLOWED·DESI DR3 fσ₈ projection
ALLOWED·Euclid Y1 fσ₈ projection

Step 7

What this denies

ΛCDM-baseline growth pipelines are not directly applicable here because the background is wrong; they have to be retemplated. The headline ISST shape — high-z suppression from wall-frame mapping — is orthogonal to the usual μ(a,k) phenomenological parameterisation, which means a single number per (z, k) is not enough to mimic ISST: the redshift dependence is the structural test.

Frameworks consulted

DENIED·Standard FLRW perturbation theory[homogeneous=false]
DENIED·Halofit nonlinear extension (LCDM)[calibrated on a denied background]
WARNING·MG growth (μ(a,k))[different mechanism — distinguishable]
DENIED·DGP / Galileon growth[constraint_class=palatini]