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  DARK MATTER GEOMETRIC CORRELATION TEST
  Project: Prometheus / Time Ledger Theory / Cipher Cross-Scale Validation
  Date: 2026-03-26
  Status: PRE-REGISTERED PREDICTION — analysis pending data download
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HYPOTHESIS (from cipher foundational thesis):
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Galaxy rotation curve "excess" (attributed to dark matter) is NOT caused by
unseen mass. It is caused by GEOMETRIC AMPLIFICATION — the same mechanism
that makes Chromium stiffer than its neighbors (BCC at resonant d-filling)
and that makes copper the best conductor (FCC at d-pos 9).

At galactic scale, the spiral geometry concentrates gravitational effect
through resonance. The {2,3} spiral pattern amplifies gravity the way an
icosahedral cavity amplifies sound — same mechanism, different magnification.

PREDICTION:
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The "dark matter fraction" (rotation curve residual / total gravitational
effect) should CORRELATE with the galaxy's geometric parameters:

  P1: PITCH ANGLE — tighter spirals (lower pitch angle) should show MORE
      geometric amplification (more "dark matter"). Analogous to how
      tighter coordination (more neighbors) amplifies properties more.

  P2: SPIRAL ARM COUNT — galaxies with {2} arms vs {3} arms should show
      different amplification patterns. {2,3} decomposition predicts
      specific ratios, not random scatter.

  P3: MASS DISTRIBUTION SYMMETRY — more symmetric mass distribution should
      produce stronger resonance (analogous to FCC isotropy producing
      the sharpest frequency response).

  P4: MORPHOLOGICAL TYPE — Sa/Sb/Sc Hubble types represent different
      geometric configurations. The residual should cluster by type
      in a way that maps to {2,3} geometric parameters.

ANTI-PREDICTION (what would FALSIFY this):
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If the dark matter fraction is RANDOM with respect to geometric parameters
— no correlation with pitch angle, arm count, or morphology — then the
geometric amplification hypothesis is wrong at galactic scale.

If the dark matter fraction correlates ONLY with total mass (and not
geometry), then standard dark matter theory is correct and geometry
adds nothing.

DATA SOURCE:
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SPARC database (Spitzer Photometry and Accurate Rotation Curves)
  - Lelli, McGaugh, Schombert (2016)
  - 175 galaxies with detailed rotation curves
  - Includes: observed velocity, baryonic velocity (from visible mass),
    morphological type, luminosity, scale length, gas fraction
  - Publicly available

SUPPLEMENTARY DATA (if needed):
  - Galaxy pitch angle catalogs (e.g., Davis et al., Yu & Ho)
  - Spiral arm counts from Galaxy Zoo or NED morphological classifications

ANALYSIS PLAN:
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  STEP 1: Download SPARC data. Extract for each galaxy:
    - V_obs(r): observed rotation velocity at each radius
    - V_bar(r): baryonic (visible mass) predicted velocity
    - Residual(r) = V_obs(r) - V_bar(r) at each radius
    - Integrated residual: total "excess" across the rotation curve
    - Dark matter fraction: residual / total

  STEP 2: Collect geometric parameters for each galaxy:
    - Morphological type (from SPARC)
    - Pitch angle (from supplementary catalog or derivation)
    - Arm count (from morphological classification)
    - Scale length / effective radius (from SPARC)
    - Inclination, distance (from SPARC — needed for corrections)

  STEP 3: Correlation analysis:
    - Dark matter fraction vs pitch angle (P1)
    - Dark matter fraction vs arm count, grouped (P2)
    - Dark matter fraction vs mass distribution symmetry metric (P3)
    - Dark matter fraction clustered by Hubble type (P4)
    - Standard controls: DM fraction vs total mass, vs distance, vs inclination

  STEP 4: Statistical significance:
    - Pearson/Spearman correlation coefficients
    - p-values for each correlation
    - Multiple comparison correction (Bonferroni)
    - Bootstrap confidence intervals

  STEP 5: Compare to {2,3} prediction:
    - If P1 holds: does the pitch angle relationship follow a {2,3} ratio?
    - If P2 holds: is the arm-count effect consistent with {2}/{3} decomposition?
    - If P3 holds: does the symmetry metric map to coordination number?

SUCCESS CRITERIA:
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  MINIMUM: Statistically significant (p < 0.05) correlation between
  dark matter fraction and ANY geometric parameter.

  STRONG: Multiple geometric parameters correlate, and the correlations
  are STRONGER than mass-only correlations.

  EXCEPTIONAL: The geometric correlations follow {2,3} ratios or patterns
  consistent with the cipher's predictions.

OUTPUT-AGNOSTIC. DATA SHOWS WHAT IT SHOWS.
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