To be peer reviewed

Einstein Probability Dilation (EPD)

Research abstract

Einstein Probability Dilation (EPD) is a measure-theoretic research framework for studying how probability measures transform under positive reweighting (dilation) while preserving normalization and producing controlled changes in expectation values.

EPD treats a probability measure as the primary mathematical object and investigates:

invariant identities induced by reweighting,
composition and iteration of dilations,
fixed points and near-fixed behavior,
whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

EPD is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

Overview

EPD is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., Buffon’s needle). EPD abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:

mathematically well-defined (positivity and normalization),
composable under iteration,
analyzable for invariants and fixed points.

Mathematical framework

Definitions and notation

Let $(\Omega ,\Sigma )$ be a measurable space.

$P$ denotes a probability measure on $(\Omega ,\Sigma )$ .
If $P$ has a density $p$ with respect to a reference measure $\mu$ , then $dP=p\,d\mu$ .
$D:\Omega \to (0,\infty )$ is a measurable dilation field (a positive weight function).
$Z(P,D)$ is the normalization constant: $Z(P,D)=\int _{\Omega }D\,dP$ .
For an observable $f:\Omega \to \mathbb {R}$ integrable under the relevant measure, $\mathbb {E} _{P}[f]=\int _{\Omega }f\,dP$ .

Probability dilation (reweighting)

Given $P$ and $D$ with $0<Z(P,D)<\infty$ , define the EPD transform ${\widetilde {P}}=\mathrm {EPD} (P;D)$ by:

${\widetilde {P}}(A)={\frac {\int _{A}D\,dP}{\int _{\Omega }D\,dP}}\quad {\text{for all }}A\in \Sigma .$

If $dP=p\,d\mu$ , then $d{\widetilde {P}}={\widetilde {p}}\,d\mu$ where

${\widetilde {p}}(x)={\frac {D(x)\,p(x)}{Z}},\quad Z=\int _{\Omega }D(x)\,p(x)\,d\mu .$

Interpretation: the dilation field $D$ shifts probability mass toward regions where $D$ is larger, while renormalization keeps total probability equal to 1.

Example: Lorentz contraction as a positive dilation field

A familiar physical example of a strictly positive factor is the Lorentz factor:

Failed to parse (syntax error): {\displaystyle \gamma(v)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\quad\text{for }|v|\text{<}c.}

Lorentz contraction for a rod of rest length $L_{0}$ moving at speed $v$ is:

$L(v)={\frac {L_{0}}{\gamma (v)}}.$

To connect this idea to EPD (as an illustration only), one may define a positive dilation field based on $\gamma$ , for instance:

$D(x)=\gamma (x)$ (weights states by increasing $\gamma$ ), or
$D(x)={\frac {1}{\gamma (x)}}$ (weights states by decreasing $\gamma$ ),

provided $D(x)$ is strictly positive over the model’s domain and $0<\int _{\Omega }D\,dP<\infty$ .

This example is included purely to illustrate “positive dilation”; it is not presented as a cosmological model.

Buffon sampling, the constant \pi, and a Lorentz-induced dilation factor (toy)

This brief example shows how $\pi$ naturally appears from sampling geometry (Buffon’s needle) and how inserting Lorentz contraction into the sampled length produces a strictly positive, dimensionless candidate dilation field. This is included for mathematical intuition; it is not claimed as a physical derivation.

For the classical Buffon needle problem with needle length $L\leq d$ and line spacing $d$ , the crossing probability is: $P_{\mathrm {cross} }(L;d)={\frac {2L}{\pi d}}.$

If a rest-length $L_{0}$ undergoes Lorentz contraction $L(v)=L_{0}/\gamma (v)$ for Failed to parse (syntax error): {\displaystyle |v|\text{<}c} , then the same form gives: $P_{\mathrm {cross} }(v)={\frac {2L_{0}}{\pi d\,\gamma (v)}}.$

Normalizing to the rest case yields a dimensionless sampling ratio: $D_{\pi }(v):={\frac {P_{\mathrm {cross} }(v)}{P_{\mathrm {cross} }(0)}}={\frac {1}{\gamma (v)}}={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

This $D_{\pi }(v)$ is strictly positive on the domain Failed to parse (syntax error): {\displaystyle |v|\text{<}c} and can be used (illustratively) as a dilation field in the EPD transform: ${\widetilde {P}}(A)={\frac {\int _{A}D_{\pi }(v)\,dP}{\int _{\Omega }D_{\pi }(v)\,dP}}.$

Theorem-style mathematics expansion

Definition (EPD reweighting operator)

Let $P$ be a probability measure on $(\Omega ,\Sigma )$ and $D:\Omega \to (0,\infty )$ measurable with $0<\int _{\Omega }D\,dP<\infty$ . Define $\mathrm {EPD} (P;D)={\widetilde {P}}$ by:

${\widetilde {P}}(A)={\frac {\int _{A}D\,dP}{\int _{\Omega }D\,dP}}.$

Theorem 1 (Normalization and positivity)

${\widetilde {P}}$ is a probability measure. In particular:

${\widetilde {P}}(A)\geq 0$ for all measurable $A$ ,
${\widetilde {P}}(\Omega )=1$ ,
If $P(A)=0$ then ${\widetilde {P}}(A)=0$ (absolute continuity preserved).

Theorem 2 (Expectation reweighting identity)

If $f$ is integrable under ${\widetilde {P}}$ , then

$\mathbb {E} _{\widetilde {P}}[f]={\frac {\mathbb {E} _{P}[D\,f]}{\mathbb {E} _{P}[D]}}.$

Theorem 3 (Composition / iteration rule)

Let ${\widetilde {P}}=\mathrm {EPD} (P;D_{1})$ and ${\widehat {P}}=\mathrm {EPD} ({\widetilde {P}};D_{2})$ . Then

${\widehat {P}}=\mathrm {EPD} (P;D_{1}D_{2})$

provided $\int _{\Omega }D_{1}D_{2}\,dP$ is finite and nonzero.

Theorem 4 (Fixed-point condition)

$P$ is a fixed point of EPD under dilation $D$ (i.e., $\mathrm {EPD} (P;D)=P$ ) if and only if $D$ is constant $P$ -almost everywhere.

Invariant quantities

EPD naturally highlights quantities that are invariant (or transform in controlled ways) under reweighting, including:

ratios of expectations of the form $\mathbb {E} _{P}[Df]/\mathbb {E} _{P}[D]$ ,
equivalence classes of observables that remain unchanged under specific families of dilations,
fixed-point structure (trivial dilations) and near-fixed behavior (small perturbations).

Core axioms of Einstein Probability Dilation

Axiom 1 — Probability primacy

Probability measures are treated as primary mathematical objects; structure is inferred from invariant properties of measure transformations.

Axiom 2 — Positive dilation

Dilation fields are positive ( $D$ is strictly positive) so that reweighting preserves positivity and supports a consistent normalization constant.

Axiom 3 — Iterative composability

Successive dilations compose multiplicatively ( $D_{1}D_{2}$ ), enabling multi-step (iterative) dynamics on measures.

Falsifiability conditions

A proposed EPD-based model should specify:

the configuration space $\Omega$ ,
a baseline measure $P$ ,
an explicit dilation field $D$ (with parameters and domain),
measurable predictions (statistics, scaling laws, invariants) that can be checked against simulation or data,
a baseline comparison (null model) and evaluation criteria (error measures, goodness-of-fit, or hypothesis tests).

Concrete falsifiability test (clustering example): Choose a baseline mock galaxy catalog (configuration space $\Omega$ ), define an explicit strictly positive dilation field $D(x)=\exp(\lambda \,\phi (x))$ , apply EPD-style reweighting (e.g., importance resampling of the empirical measure), and compute the resulting two-point correlation function $\xi (r)$ using a standard estimator (e.g., normalized Landy–Szalay). Define a quantitative success metric (for example, mean-squared error between modeled and observed $\xi (r)$ over specified separation bins such as $r\leq 60\,\mathrm {Mpc} /h$ ). The model is falsified if no parameter choices within the stated domain can meet the pre-registered threshold under the chosen metric, or if the fit fails out-of-sample on withheld bins.

Numerical simulation and iterative models

Simulation model description

In discrete demonstrations, the “state space” may be a finite set (e.g., bins, configurations, or catalog points), and the probability measure $P$ may be represented as a probability vector (or an empirical measure) over those states.

Two equivalent discrete implementations are common:

weighted evaluation: keep all points and carry weights proportional to $D$ , or
importance resampling: draw a new empirical catalog with sampling probabilities proportional to $D$ .

Demonstration: reweighting mock galaxy catalogs and comparing \xi(r)

This demonstration connects the EPD reweighting idea to a standard observable in large-scale structure: the (monopole) two-point correlation function $\xi (r)$ . The goal here is not a precision cosmology fit, but a clean, falsifiable pipeline:

generate a baseline mock galaxy catalog in a periodic box (approximately unclustered),
define a strictly positive dilation field $D(x)=\exp(\lambda \,\phi (x))$ over galaxy positions,
perform EPD-style importance resampling to obtain a reweighted (clustered) mock catalog,
measure $\xi (r)$ using the properly normalized Landy–Szalay estimator,
tune a small mini-grid over $(\sigma ,\lambda )$ to minimize a simple mismatch score on small scales.

Key definitions

Dilation field: $D(x)=\exp(\lambda \,\phi (x))$ , with Failed to parse (syntax error): {\displaystyle \lambda \text{>} 0} and $\phi (x)\geq 0$ .
Example potential: $\phi (x)=\sum _{k}\exp \!\left(-\|x-s_{k}\|^{2}/(2\sigma ^{2})\right)$ using seed points $s_{k}$ .
Landy–Szalay estimator (normalized counts):

  $\xi (r)={\frac {DD(r)-2DR(r)+RR(r)}{RR(r)}}$ ,
 where  $DD,DR,RR$  are pair counts normalized by the total number of possible pairs in each catalog.

import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# 1) Pair-counting with periodic boundaries
# ==========================================
def pair_counts_weighted(posA, posB, bins, boxsize, wA=None, wB=None,
                         autocorr=False, chunk=512):
    """
    Weighted pair counts in radial bins with periodic boundary conditions.
    - autocorr=True: count unique pairs i<j within posA
    - autocorr=False: count all cross pairs between posA and posB
    Returns raw (un-normalized) weighted counts per bin.
    """
    posA = np.asarray(posA, float)
    posB = np.asarray(posB, float)
    bins = np.asarray(bins, float)
    nb = len(bins) - 1
    L = float(boxsize)

    if wA is None:
        wA = np.ones(len(posA), float)
    if wB is None:
        wB = np.ones(len(posB), float)
    wA = np.asarray(wA, float)
    wB = np.asarray(wB, float)

    counts = np.zeros(nb, float)

    if autocorr:
        N = len(posA)
        for i0 in range(0, N, chunk):
            i1 = min(N, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            # distances from A to ALL posA (periodic)
            d = A[:, None, :] - posA[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wA[None, :]

            # accumulate only i<j
            for ii in range(i1 - i0):
                j_start = i0 + ii + 1
                if j_start >= N:
                    continue
                rr = r[ii, j_start:]
                ww = W[ii, j_start:]
                hist, _ = np.histogram(rr, bins=bins, weights=ww)
                counts += hist
        return counts

    else:
        NA = len(posA)
        for i0 in range(0, NA, chunk):
            i1 = min(NA, i0 + chunk)
            A = posA[i0:i1]
            wAi = wA[i0:i1]

            d = A[:, None, :] - posB[None, :, :]
            d -= L * np.round(d / L)
            r = np.sqrt(np.sum(d * d, axis=2))

            W = wAi[:, None] * wB[None, :]
            hist, _ = np.histogram(r, bins=bins, weights=W)
            counts += hist
        return counts


def landy_szalay_xi(DD_raw, DR_raw, RR_raw, nD, nR):
    """
    Properly normalized Landy–Szalay estimator:
      DD = DD_raw / [nD (nD-1)/2]
      DR = DR_raw / [nD nR]
      RR = RR_raw / [nR (nR-1)/2]
      xi = (DD - 2 DR + RR) / RR
    """
    DD = DD_raw / (nD * (nD - 1) / 2.0)
    DR = DR_raw / (nD * nR)
    RR = RR_raw / (nR * (nR - 1) / 2.0)
    return (DD - 2 * DR + RR) / np.maximum(RR, 1e-30)


# ==========================================
# 2) EPD-style dilation field + resampling
# ==========================================
def gaussian_seed_field(pos, seeds, boxsize, sigma):
    """
    phi(x) = sum_k exp(-||x-seed_k||^2/(2 sigma^2)) with periodic distances.
    Always nonnegative.
    """
    pos = np.asarray(pos, float)
    seeds = np.asarray(seeds, float)
    L = float(boxsize)

    d = pos[:, None, :] - seeds[None, :, :]
    d -= L * np.round(d / L)
    r2 = np.sum(d * d, axis=2)
    phi = np.exp(-0.5 * r2 / (sigma ** 2)).sum(axis=1)
    return phi


def epd_resample_catalog(pos0, phi, lam, rng):
    """
    Importance-resampling version of discrete EPD on an empirical measure:
      weights ∝ D(x)=exp(lam * phi(x))
      resample points with prob ∝ weights
    """
    logw = lam * phi
    logw -= np.max(logw)          # stabilize
    w = np.exp(logw)              # strictly positive
    p = w / w.sum()
    idx = rng.choice(len(pos0), size=len(pos0), replace=True, p=p)
    return pos0[idx]


# ==========================================
# 3) Load real xi0(r) for reference (BOSS)
# ==========================================
# NOTE: Set this path to wherever you saved the file locally.
# The example below matches the earlier DR11 multipoles file name.
path = "samushia_2013_CMASSDR11_xi_multipoles.dat"
vals = np.loadtxt(path)
xi_obs = vals[:16]  # monopole xi0

# BOSS bins: edges 24..152 step 8 => centers 28..148
bins = np.arange(24, 152 + 8, 8)
r_centers = (bins[:-1] + bins[1:]) / 2


# ==========================================
# 4) Build baseline mock + randoms (periodic box)
# ==========================================
rng = np.random.default_rng(7)

boxsize = 200.0
N_data0 = 1400
N_rand = 4500

pos0 = rng.uniform(0, boxsize, size=(N_data0, 3))
posR = rng.uniform(0, boxsize, size=(N_rand, 3))

# Precompute RR once (random-random doesn't change during grid search)
RR = pair_counts_weighted(posR, posR, bins, boxsize, autocorr=True, chunk=512)


# ==========================================
# 5) Define seed locations (kept fixed during tuning)
# ==========================================
K = 12
seeds = rng.uniform(0, boxsize, size=(K, 3))


# ==========================================
# 6) Objective: simple "success" on small-r bins
# ==========================================
def mse_on_bins(xi_model, xi_target, r_centers, rmax=60.0):
    """
    Mean squared error over bins with r <= rmax (simple success criterion).
    """
    m = r_centers <= rmax
    return float(np.mean((xi_model[m] - xi_target[m]) ** 2))


# ==========================================
# 7) Mini-grid search over (lambda, sigma)
# ==========================================
sigmas = [6.0, 8.0, 10.0, 12.0, 15.0]   # seed width (correlation length scale)
lams   = [0.5, 1.0, 2.0, 3.0, 4.0, 5.0] # dilation strength

best = None

# Baseline xi (should be ~0)
DD0 = pair_counts_weighted(pos0, pos0, bins, boxsize, autocorr=True, chunk=512)
DR0 = pair_counts_weighted(pos0, posR, bins, boxsize, autocorr=False, chunk=512)
xi_base = landy_szalay_xi(DD0, DR0, RR, nD=len(pos0), nR=len(posR))

for sigma in sigmas:
    phi = gaussian_seed_field(pos0, seeds, boxsize, sigma)

    for lam in lams:
        pos1 = epd_resample_catalog(pos0, phi, lam, rng)

        DD1 = pair_counts_weighted(pos1, pos1, bins, boxsize, autocorr=True, chunk=512)
        DR1 = pair_counts_weighted(pos1, posR, bins, boxsize, autocorr=False, chunk=512)
        xi_epd = landy_szalay_xi(DD1, DR1, RR, nD=len(pos1), nR=len(posR))

        score = mse_on_bins(xi_epd, xi_obs, r_centers, rmax=60.0)

        if best is None or score < best[0]:
            best = (score, sigma, lam, xi_epd)

best_score, best_sigma, best_lam, xi_best = best

# ==========================================
# 8) Plot baseline vs best EPD vs observed
# ==========================================
plt.figure()
plt.plot(r_centers, xi_base, marker='o', label="Mock baseline (uniform Poisson)")
plt.plot(r_centers, xi_best, marker='o',
         label=f"Best EPD-resampled (sigma={best_sigma}, lambda={best_lam})")
plt.plot(r_centers, xi_obs, marker='o', linestyle='--',
         label="Observed BOSS DR11 ξ₀(r) (reference)")
plt.axhline(0, linewidth=1)
plt.xlabel("r [Mpc/h]")
plt.ylabel("ξ(r) (monopole estimate)")
plt.title("EPD reweighting of mock catalogs: mini-grid tuning (toy demonstration)")
plt.legend()
plt.show()

print("=== Mini-grid search result ===")
print(f"Best score (MSE on r<=60): {best_score:.6e}")
print(f"Best sigma: {best_sigma}  Best lambda: {best_lam}")

How to run: download the BOSS multipoles text file to your computer and set path in the script to that local filename. Then run the script in Python (NumPy + Matplotlib required). The baseline curve should lie near $\xi \approx 0$ , while the EPD-resampled mock typically shows positive small-scale clustering.

Scope and limitations

EPD is a mathematical framework for measure transformations. It does not claim:

a replacement theory for GR/QM,
empirical confirmation without explicit predictions and tests.

Future directions

develop canonical families of dilation fields and invariants,
clarify “structure-from-measure” diagnostics,
publish reproducible simulation notebooks and parameter sweeps,
compare multiple dilation families under shared evaluation criteria.

References

Probability / measure theory

Computing / simulation

Copyright and licensing

This Wikiversity page is authored by Howard Richardson. Please attribute appropriately when reusing or adapting this material.

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