Axioms

Capsule — the six axioms

A1 — Existence (MM). A compact, orientable, smooth 10-dimensional Meta-Manifold with fields exists; “atemporal” in 4D terms.
A2 — Projection (Kernel). A single, fixed, non-invertible kernel realizes 10→4 projection at finite resolution.
A3 — Retention Law (S2). Information/contrast decays with probe scale as a stretched exponential \(R(\lambda)=\exp[-(\lambda/\lambda_q)^{D_{\mathrm{eff}}}]\); this is the foundation.
A4 — Constants are kernel-fixed. \( \{c,\,\hbar,\,G,\,\ldots\} \) are parameters of our kernel slice; they are measured, not derived, inside D.R.E.A.M.
A5 — Topology ↔ classes. Stable MM topological data label particle/gauge classes that project to 4D content.
A6 — Finite resolution ⇒ regimes. A coherence scale \( \lambda_q \) separates quantum-grade retention (below) from classical coarse-grain (above).

Only S2 is elevated to foundation. “S3” (structure focusing via coarea/CPI) provides strong support to S2; “S1” (ALP/dust) is relegated and assumed null below the cliff.

A1 — Existence (MM)

There is a compact, orientable, smooth 10D manifold (MM) with fields. “Atemporal” means 4D time is not fundamental in MM; it appears only after projection.

Why compact & smooth? Compactness ensures finite integrals; smoothness supports differential operators/topological indices used by projection.

Empirical footprint. A1 sets the stage; data touch A2–A6.

A2 — Projection (Kernel)

A single, fixed, non-invertible kernel \(K_{\lambda}\) maps MM fields to 4D observables at finite resolution \( \lambda \) (our slice uses \( \lambda=\lambda_q \)).

O(x)=\int_{\mathrm{MM}} K_{\lambda}(x;X)\,\Phi(X)\,d\mu_{\mathrm{MM}}(X).

Science attaches to invariants of the kernel class: \( \lambda_q \) (coherence), locality radius, regularity/positivity, and effective spectral exponent \( D_{\mathrm{eff}} \).

Guardrails. Normalization, locality in projection, and monotone fidelity ensure a well-posed map \(L^2(\mathrm{MM})\to L^2(\mathbb{R}^{3,1})\).

\textbf{K1:}\ \int K_{\lambda}(x;X)\,d\mu_{\mathrm{MM}}(X)<\infty,\quad \textbf{K2:}\ K_{\lambda}(x;X)\text{ peaks for }X\approx\pi^{-1}(x),\quad \textbf{K3:}\ \partial_{\lambda}\,\alpha(\lambda)\le0.

A3 — Retention Law (S2)

Information fades with probe scale as a stretched exponential:

R(\lambda)=\exp\!\Big[-\big(\tfrac{\lambda}{\lambda_q}\big)^{D_{\mathrm{eff}}}\Big].

Two readouts track the same law with their own \(\lambda_q, D_{\mathrm{eff}}\): high-frequency power \(R_{\mathrm{HF}}(\lambda)\) and structural correlation \(R_{\mathrm{struct}}(\lambda)=\mathrm{corr}^2(e_4|_{\lambda\to0},\,e_4|_{\lambda})\). Fractal-like signatures live only below \(\lambda_q\); homogeneity emerges above.

Operationally. Fit log-linear \( \ln(-\ln R) = D_{\mathrm{eff}}\ln\lambda - D_{\mathrm{eff}}\ln\lambda_q \) across platforms.

A4 — Constants are projection-fixed

The constants \( \{c,\,\hbar,\,G,\,\ldots\} \) are kernel parameters of our slice. We measure them; we do not derive them inside D.R.E.A.M.

Implication. Apparent “drifts” are instrument/scale effects within one projection, not kernel drift.

A5 — Topology ↔ classes

Stable MM topological classes (homotopy/cohomology, characteristic classes) label particle/gauge classes. Individual 4D species and quantized charges are their projections.

Quantization. Integer-valued integrals over cycles give charge quantization robust to finite-resolution averaging.

Q_{\mathrm{U(1)}}=\frac{1}{2\pi}\oint A\in\mathbb{Z},\qquad k=\frac{1}{8\pi^2}\int\operatorname{tr}(F\wedge F)\in\mathbb{Z}.

Selection rules. Class arithmetic constrains allowed processes; anomalies must cancel in the 4D projection.

A6 — Finite resolution ⇒ regimes

A coherence “cliff” near \( \lambda\approx\lambda_q \) separates two regimes: below it interference-grade invariants retain; above it classical coarse-grain dominates.

\text{coherence}(\lambda)\ \propto\ R(\lambda)=\exp\!\Big[-\big(\tfrac{\lambda}{\lambda_q}\big)^{D_{\mathrm{eff}}}\Big].

S3 (support): Coarea focusing and CPI explain why hubs/filaments persist under smoothing, reinforcing S2’s predictions about structure retention and its fading.

Experimental handle. Scan across \( \lambda_q \) (interferometry, photonics, NV/spins, matter-wave) and look for the step-like change in retention readouts.

Assumed vs. tested

Assumed

MM exists and is smooth & compact (A1).
Single fixed kernel at finite resolution (A2).
Constants are kernel-fixed (A4).

De-emphasized. S1 (ALP/dust) is relegated and assumed null below \( \lambda_q \); not used to support the framework.

Tested (invariants)

S2 foundation: retention law \(R(\lambda)=\exp[-(\lambda/\lambda_q)^{D_{\mathrm{eff}}}]\); detect the transition across platforms (A3/A6).
S3 support: coarea focusing + CPI (hubs/filaments persist, then fade as predicted by S2).
Topological quantization/selection rules in 4D (A5).
Ranked-spectrum ratio law \(m_n/m_1=n^{1/D_{\mathrm{eff}}}\) as a cross-domain consequence of \(D_{\mathrm{eff}}\) (supporting test).

Falsification hooks tied to axioms

Against S2 (A3/A6): No retention transition near the posited \( \lambda_q \) across platforms at forecasted sensitivity.
Against A5: Violation of integer quantization or selection rules inconsistent with the class picture.
Against invariant posture: Failure of the ranked-spectrum ratio law \(m_n/m_1=n^{1/D_{\mathrm{eff}}}\) where applicable.

These are clean nulls: we do not rescue the framework by unconstrained kernel tweaks.

Interpretive note (speculative)

Constants feel “fundamental” only because they are kernel-fixed parameters in our slice; alternative local projection conditions might shift some of them. This is interpretive and not required for empirical tests.

Fractal background

Core Axioms