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TEST DESCRIPTION: 4D Geometry Probe — Multi-Slice W-Profile
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Author: Jonathan Shelton
Date: 2026-03-21
Status: PRE-REGISTERED (test description written BEFORE coding)
Protocol: Outcome-agnostic. Observe how the 4D field shape evolves across c.
Project: 4D Research (24-cell focal point)

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1. WHAT WE ARE TESTING
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How does the 4D interference pattern's SHAPE change as the propagation
speed c_4D transitions through the range 1.618 to 1.810?

The current engine saves only the w=0 midplane cross-section. This test
saves the FULL w-profile: intensity at multiple slices along the 4th
spatial dimension. This reveals the 4D geometry of the interference
pattern — whether it is conical, tubular, bulging, or something else.

The transition from 3D-governed behavior (phi regime, c~1.618) through
the anomalous zone (c=1.700-1.732) to the post-transition regime
(c>1.75) should be captured IN FULL, not sampled at endpoints.

RATIONALE: The periodic table's d-block shows that the 4D influence
develops progressively across many elements, not as an abrupt switch.
The transition takes space to develop. Narrowing our measurement window
would miss the development of the transition.

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2. WHAT WE EXPECT TO SEE (pre-registered)
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Based on ZERO assumptions about the 4D geometry shape. We observe:

  A. At phi regime (c~1.618-1.650): The w-profile should be whatever
     the "normal" 3D-influenced behavior looks like. This is our baseline.

  B. Through the transition (c=1.650-1.700): The w-profile should begin
     to change shape. How it changes tells us about the transition.

  C. At the anomalous points (c=1.700, 1.732): Something different from
     both the baseline and the transition. We do not pre-specify what.

  D. Post-transition (c=1.750-1.810): Either a return to baseline-like
     behavior (transition is localized) or a new steady state
     (transition is a one-way development).

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3. WHAT WOULD FALSIFY
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  - If the w-profile is IDENTICAL at all c values, the 4D field shape
    does not depend on propagation speed and the geometry is static
  - If the w-profile changes continuously with no special behavior at
    1.700 or 1.732, the anomalous P/N peaks were a coincidence of the
    midplane slice and the 4D geometry is actually smooth
  - If the w-profile is noisy/random, the simulation is not resolving
    meaningful 4D structure at this grid resolution

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4. WHAT WOULD CONFIRM
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  - If the w-profile CHANGES CHARACTER through the transition in a way
    that correlates with the P/N anomalies seen in the Phase 2 sweep
  - If the anomalous c values show a qualitatively different w-profile
    from the baseline (e.g., concentrated vs spread, shifted vs centered)
  - If the transition develops progressively (matching periodic table
    observation of gradual 4D influence development)

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5. METHODOLOGY
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  ENGINE: Same 4D FDTD engine (engine_4d.py) with ONE modification:
    Save intensity cross-sections at MULTIPLE w-slices, not just w=0.

  W-SLICES: 9 evenly spaced through the 4D domain:
    w_indices = [2, N/8, N/4, 3N/8, N/2, 5N/8, 3N/4, 7N/8, N-3]
    (Avoiding edge PML layers at w=0,1 and w=N-1,N-2)
    This gives slices from near-boundary through center to near-boundary.

  C VALUES: Dense sweep from 1.618 to 1.810, step 0.004 (49 values):
    1.618, 1.622, 1.626, ..., 1.806, 1.810

    This gives ~8 points in the pre-transition zone (1.618-1.650)
    ~13 points through the transition (1.650-1.700)
    ~8 points in the anomalous zone (1.700-1.732)
    ~20 points in the post-transition zone (1.732-1.810)

  RESOLUTION: Phase 2 (N=48). This matches the existing sweep data.
    If Phase 2 shows interesting structure, Phase 3 (N=64) confirmation
    follows for selected c values.

  PERIODS: 50 (same as existing sqrt3 confirmation sweep).

  MEASUREMENTS PER W-SLICE:
    1. Total integrated intensity: sum(|psi|^2) over the 3D slice
    2. Peak count: local maxima above 2-sigma threshold
    3. Radial extent: radius enclosing 90% of integrated intensity
    4. Center of mass offset from geometric center
    5. Peak positions (x,y,z coordinates of top 30 peaks)

  AGGREGATE MEASUREMENTS:
    1. W-profile: integrated intensity as function of w-index
    2. W-profile width: FWHM of the intensity vs w curve
    3. W-asymmetry: ratio of intensity on w>0 side vs w<0 side
    4. W-peak-shift: do peak positions drift as w changes?
    5. Existing metrics: P/N, dual/field ratio, energy, peak count (for
       cross-reference with Phase 2 sweep data)

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6. RUNTIME ESTIMATE
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  Per run: ~8-9 minutes at Phase 2 (based on existing sqrt3 confirmation)
  49 c values: ~7-7.5 hours total
  Plus I/O for saving 9 slices x 49 runs = 441 cross-section files
  Storage: ~441 x 865 KB = ~370 MB

  Well within Hetzner CCX33 capacity (8 cores, 32 GB RAM).

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7. OUTPUT FORMAT
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  Per c value:
    results/geometry_probe/result_c{c:.3f}.json  — all metrics
    results/geometry_probe/slices_c{c:.3f}.npz   — compressed 9-slice stack

  Aggregate:
    results/geometry_probe/probe_summary.json    — all results in one file
    results/geometry_probe/w_profiles.json       — w-profile for each c

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8. CONNECTION TO PRIOR TESTS
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  TLT-4D-001: Established the anomalies at c=1.700 and 1.732 (P/N 2.5x)
  sqrt3 confirmation: Fine sweep confirmed sharp resonances
  THIS TEST: Probes the 4th-dimensional profile across the full transition

  This test does NOT change the engine physics. Only the measurement
  (which w-slices are saved). The FDTD wave equation, sources, PML,
  and C_potential are identical to TLT-4D-001.

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