Main Theorems

Key results derived from DREAM axioms, formulated as testable relationships between invariants.

Math hidden by default — toggle “Math” in header. Status: Confirmed Partial Speculative.

Capsule — theorem list

T1 — Topological Quantization Confirmed
T2 — Symmetry Projection (Noether under push-forward) Partial
T3 — Coherence Threshold (Retention Law) Confirmed
T4 — Structure Focusing & CPI Partial
T5 — Dimensional Flow to Homogeneity Partial
T6 — Cross-domain Ranked-Spectrum Ratio Law Partial

S2 (Retention Law) is foundation. S1 (“ALP ladder / dust”) is relegated and shown only as speculative.

T1 — Topological Quantization Confirmed

Integer-valued topological invariants on the compact Meta-Manifold label discrete 4D classes; charges and indices remain quantized after projection.

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}.

Operational. Discrete charges and instantons; selection rules; anomaly cancellation.

T2 — Symmetry Projection Partial

If the kernel is equivariant, Noether currents commute with projection: continuous 10D symmetries yield conserved 4D quantities.

K_\lambda(gx;gX)=K_\lambda(x;X)\ \Rightarrow\ \partial_\mu J^\mu=0.

Operational. Test conservation to high precision; search for symmetry-breaking artifacts.

T3 — Coherence Threshold (Retention Law) Confirmed

A coherence scale \(\lambda_q\) separates regimes: below, interference-grade invariants persist; above, coarse-graining dominates.

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

Operational. Two channels with shared \(\lambda_q\): HF power \(R_{\mathrm{HF}}(\lambda)\) and structural correlation \(R_{\mathrm{corr}^2}(\lambda)\).

T4 — Structure Focusing & CPI Partial

Small fiber Jacobians produce hubs/filaments persisting under smoothing; CPI quantifies local microstate budgets and predicts where structure survives longer.

A(x)\sim\langle J_F^{-1}\rangle,\quad M(\lambda,\varepsilon;x)\sim A(x)(\tfrac{\lambda}{\varepsilon})^{D_I},\quad \mathrm{CPI}=\log_{10} M.

Operational. Elevated retention in high-\(A(x)\) regions; slower decay of structure vs HF detail.

T5 — Dimensional Flow to Homogeneity Partial

Under retention, effective dimension flows toward 3 on large scales: fractal signatures are finite-range only; homogeneity emerges above the cliff.

D_{\mathrm{eff}}(\lambda)\nearrow 3\quad \text{as }\lambda\to\infty.

Operational. Counts approach cubic scaling beyond the cliff; cross-scale correlations vanish.

T6 — Cross-domain Ranked-Spectrum Ratio Law Partial

Cumulative retained modes scale as \(N(\Omega)\sim\Omega^{D_{\mathrm{eff}}}\); ranked energies/masses obey ratio laws (base context-specific, ratios invariant).

N(\Omega)\propto\Omega^{D_{\mathrm{eff}}},\quad \rho(\Omega)\propto\Omega^{D_{\mathrm{eff}}-1},\quad \frac{m_n}{m_1}=n^{1/D_{\mathrm{eff}}}.

Operational. Fit \(D_{\mathrm{eff}}\) from retained-mode counts/spectral slopes; check ranked-ratio predictions where ladders apply.

ALP / non-harmonic ladder Speculative

S1 is null under current posture; the ladder relation is shown for completeness only, not as foundation.

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

Interpretive note (speculative)

T3–T5 form the “blur trilogy”: retention sets the law, focusing explains skeleton persistence, and dimensional flow explains the approach to homogeneity.

Fractal background