Public-Data Falsification Tests for the Entropy-Response Gravity Framework: Methodology, Tiered Test Suite, and Explicit Postulate-Dependence Caveats

We present a public-data test suite for the Entropy-Response Gravity (ERG) framework, accompanying companion Colab notebook ERG_Framework_Tests.ipynb archived alongside this paper. The suite is organised into two explicit tiers: Tier-1 tests probe derived framework predictions and constitute genuine falsifiers; Tier-2 tests probe downstream postulates whose structural derivations are not yet established (specifically the bipartite modular spectral triple with Pell-Lucas eigenvalues, the resonance scale a_res = a_0*sqrt((4*pi^2-1)/2), and the cosmological feature at z ~ 0.617 where H(z) = sqrt(2)*H_0). Null results from Tier-2 tests constrain the postulates without falsifying the framework's core predictions; we make this distinction operational. The suite uses NuFIT-6.0 oscillation parameters, the SPARC galactic database, ACCEPT cluster data, Pantheon+ supernovae, and DESI BAO measurements, all publicly available. We provide explicit reproduction instructions for each test.

The intent is to expose the framework's empirical commitments transparently, including the speculative commitments, so that observational programmes over the next decade (Euclid DR1, JUNO, DESI-Simons Observatory, CMB-S4 / LiteBIRD, nEXO) can constrain or rule out specific framework components without ambiguity about what is being tested.

Tier-1 tests (six derived predictions, framework falsifiers):
- T1.1 SPARC galaxy rotation curves: standard mu(x) at 6.4 sigma across 173 galaxies
- T1.2 CMB spectral index: n_s = 0.9741 from N_e = 77.23 (ACT DR6 agreement at -0.06 sigma)
- T1.3 Tensor-to-scalar ratio r = 0 exactly (CMB-S4/LiteBIRD test)
- T1.4 Gravitational slip eta = 1.000 (Euclid DR1 October 2026)
- T1.5 Cluster M-T scaling at 12.4 sigma (eFEDS)
- T1.6 Sum of neutrino masses 58.72 +/- 0.30 meV (DESI-SO ~2028)

Tier-2 tests (four postulate-dependent constraints, NOT framework falsifiers):
- T2.1 Silver-ratio mass ratio m_3^2/m_2^2 = (1+sqrt(2))^4 at 0.08 sigma (NuFIT updates)
- T2.2 Cosmological feature at z = 0.617 (DESI BAO future-data test)
- T2.3 Galactic transition scale at r ~ 3 kpc (SPARC stacked residuals)
- T2.4 Cluster temperature feature at g = a_res (ACCEPT)

Plus verified mathematical identity: zeta_{Q(sqrt(2))}(-1) = 1/12 (sympy-rigorous).

The framework operates an explicit discipline rule (10 retractions, 168 caught near-misses, 3 verified derivations, 0 framework-breaking findings). This paper is the operational expression of that rule in publication strategy: rather than publish unverified Tier-2 conjectures as if derived, we publish them as conjectures with explicit caveats and explicit falsification pathways.