Falsification

Fail gates are expressed strictly in terms of kernel invariants (\(\lambda_q, D_{\mathrm{eff}}\)) and instrument-level observables.

This page defines what would decisively falsify each claim, how to pre-register analyses, and how to interpret nulls. Math is hidden by default — use the global “Math” toggle in the header.

What falsification means in DREAM

Invariant-based. Claims are phrased in terms of \(\lambda_q\), \(D_{\mathrm{eff}}\), and directly measured quantities.
No post-hoc rescue. Kernel forms are not adjusted to save a failed claim; only pre-declared ranges may be updated.
Cross-platform checks. Where feasible, confirm with independent instruments (photonics, atoms, imaging, radio) to rule out artifacts.

Pre-registration protocol

Declare invariants: prior ranges for \(\lambda_q\), \(D_{\mathrm{eff}}\), with rationale.
Map instrument scale: specify how baselines, wavelengths, or coherence times map to \(\lambda\).
Fix analysis windows: linear windows in the transform \(y=\ln(-\ln R)\) vs. \(\ln\lambda\); survey bins for cosmology; λ-grids for lensing.
State fail gates: thresholds for exclusion (CIs, Bayes factors, or equivalent).
Publish pipelines: code, injections, and minimal data for re-analysis.

S2 — Retention law Foundation

Claim: contrast and correlation follow the stretched-exponential law and exhibit a transition at \(\lambda_q\).

R(\lambda)=\exp[-(\tfrac{\lambda}{\lambda_q})^{D_{\mathrm{eff}}}],\quad y(\lambda)=\ln(-\ln R)=D_{\mathrm{eff}}\ln\lambda-D_{\mathrm{eff}}\ln\lambda_q.

Decisive null: multiple platforms show no linear window in \(y\) and no shared \(\lambda_q\) within error bars → the retention law fails.

S5 — Strong lensing invariants Early

Claim: under angular smoothing, two retention channels share \(\lambda_q\); CPI ridge skeleton persists as HF texture collapses.

Decisive null: inconsistent \(\widehat{\lambda}_q\) across bands, lack of channel split, or no CPI persistence → claim fails.

S3 — Cosmology & fractal spectrum Partial

Claim: below the cliff, bounded fractality; above, flow to homogeneity with \(D_{\mathrm{eff}}\!\to\!3\).

Decisive null: persistent mismatch between cosmology-inferred and lab-inferred \(D_{\mathrm{eff}}\), or no homogeneity flow → claim fails.

S4 — Vacuum-energy modulation Open

Claim: geometry-dependent shifts at \(\sim10^{-8}\,\mathrm{eV}^4\).

Decisive null: no modulation above \(\sim10^{-10}\,\mathrm{eV}^4\) in ≥3 independent setups → claim fails.

S1 — ALP non-harmonic ladder Relegated / Null

Under retention (S2), any axion-like “dust” lies below the coherence cliff and is unrecoverable in principle. S1 is therefore null, kept only for completeness.

\frac{m_n}{m_1}=n^{1/D_{\mathrm{eff}}}.

Decisive null: moot — the hypothesis is assumed false under S2; any claimed detection must be independently replicated with invariant ratios.

Decision tree

S2 null: framework fails → treat S3–S5 as phenomenology only.
S2 confirmed + S3/S5 support: strongest case; retain \(\widehat{D}_{\mathrm{eff}},\widehat{\lambda}_q\) across domains.
S2 confirmed + S4 null: retention intact; set upper bounds on vacuum-energy modulation.

Reporting checklist

State invariant priors and instrument mapping clearly.
Publish analysis windows and explicit fail gates.
Release code, injections, null channels, and sanity checks.
Treat systematics as nuisance, not as invariant shifts.

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