research: exp_33 black hole cascade — 16/16, ghost heart mechanism

A black hole IS a PAC cascade running to Zeno completion. Four sub-experiments:

- exp_33a (4/4): Cascade-infall isomorphism, BH interval ratio = 1/phi^{3/2}
  to 0.01%, Zeno completion with geometric convergence
- exp_33b (4/4): Cascade temperature = Hawking EXACTLY (ratio 1.0 to 2.2e-16),
  PAC+duality uniquely select phi via g^2 + g = 1
- exp_33c (4/4): Bekenstein-Hawking entropy from cascade counting — area scaling
  exact, 1/4 consistent, PAC-TTN Page curve peaks at N/2 (symmetric to 3.5e-12),
  holographic principle L^2.05 in d=3
- exp_33d (4/4): Ghost heart mechanism — conservation scaffold carries zero
  independent information (SVD rank = N_leaves - 1), universal scaffold accepts
  6 boundary types, Ryu-Takayanagi surface from PAC minimal cut with 368x
  locality advantage, purification S(A) = S(A^c) exact

The holographic principle reframed: not surfaces storing information, but
conservation FORCING actualization to be boundary-determined. The PAC tree
interior is scaffold — like a decellularized ghost heart's extracellular
matrix. Real structure, zero Shannon information.

Author: lornecodes

foundational/experiments/exp_33_black_hole_cascade/README.md

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+# exp_33: Black Hole Cascade Structure
+
+## Hypothesis
+
+A black hole is a PAC cascade that has run to Zeno completion. The
+bouncing ball's infinite-bounces-in-finite-time (exp_32d) is structurally
+identical to gravitational collapse reaching the singularity in finite
+proper time. This identification is not metaphorical — it is the same
+mathematical structure: a geometric series with ratio 1/phi converging
+to a finite sum.
+
+Three consequences follow:
+1. The event horizon is the boundary between "still cascading" (external
+   observer sees infinite coordinate time) and "cascade complete" (infalling
+   observer reaches singularity in finite proper time).
+2. Hawking radiation is required by PAC conservation — the cascade potential
+   cannot reach zero, so residual potential leaks out at a rate proportional
+   to the cascade gradient at the horizon.
+3. Bekenstein-Hawking entropy counts the number of independent cascade
+   choices tiling the horizon surface.
+
+**Falsification**: If the Schwarzschild geodesic does NOT converge with
+phi-ratio structure, or if the cascade gradient gives wrong T(M) scaling,
+or if cascade counting gives S ~ Volume instead of Area.
+
+## Connection to DFT
+
+- **exp_32d** (bouncing ball): E_n = E_0 * phi^{-n}, finite total time,
+  scale invariance selects e = 1/sqrt(phi). The cascade clock T_n ~ phi^{-n/2}.
+- **exp_32e** (gravity-time duality): g_out = g_in^2 is NECESSARY for
+  conservation + scale invariance + finite closure. At the horizon, this
+  duality is at its extreme.
+- **exp_32f** (cosmological alignment): Dark energy = remaining PAC potential
+  phi^{-n}. The cascade-to-cosmology mapping works at cosmic scales; now
+  we test it at the BH scale.
+- **MAR exp_30**: Schwarzschild metric from cascade density rho_c(r) =
+  rho_crit * r_s/r. The gradient at the horizon drives Hawking radiation.
+- **QG proposal** (milestone6): This experiment begins implementing QG-1
+  (BH interior from cascade), QG-2 (Hawking temp from cascade gradient),
+  and QG-4 (Page curve from PAC conservation).
+- **Herniation hypothesis**: Open question "Is Hawking radiation literally
+  information herniation?" — we address this directly.
+
+## Experiments
+
+### exp_33a — Cascade Zeno Completion (Structural Identification)
+
+Maps the bouncing ball cascade onto Schwarzschild radial infall. The
+energy sequence, convergence structure, and time divergence at the
+horizon are compared quantitatively.
+
+| Test | What it checks |
+|------|---------------|
+| 1. Cascade-infall isomorphism | Energy ratios at phi-ratio radial steps converge to phi |
+| 2. Zeno completion | Proper time finite, geometric series convergence matches bouncing ball |
+| 3. Horizon as cascade boundary | Coordinate time diverges at r_s, proper time finite — gravity-time duality at extreme |
+| 4. Scale invariance | Phi-power radial stepping produces most self-similar cascade |
+
+**Falsification**: Energy ratios converge to non-phi value, or convergence
+structure is non-geometric.
+
+**Key results**: v^2 ratios converge to phi (0.24% at late stage). BH proper-time
+interval ratio matches 1/phi^{3/2} to 0.01% — a natural phi-power emerging from
+the Schwarzschild geodesic. Both cascades Zeno-complete with geometric convergence.
+Horizon cleanly separates finite proper time from divergent coordinate time.
+Phi-power stepping (phi^{1/3}) gives most self-similar cascade.
+
+### exp_33b — Hawking Temperature from Cascade
+
+Derives T_H proportional to 1/M from the cascade density gradient at
+the horizon. The cascade gradient dρ_c/dr|_{r_s} = ρ_crit/r_s gives
+an effective surface gravity, which via the Unruh effect yields the
+Hawking temperature.
+
+| Test | What it checks |
+|------|---------------|
+| 1. T proportional to 1/M | T*M = constant across stellar to supermassive range |
+| 2. Coefficient analysis | Does cascade coefficient match exact Hawking, or introduce phi correction? |
+| 3. PAC necessity | Removing conservation or duality breaks the temperature |
+| 4. Evaporation lifetime | Does reverse cascade give T_evap ~ M^3? |
+
+**Falsification**: Wrong T(M) power law, or coefficient off by more than
+factor 2 from Hawking.
+
+**Key results**: T*M constant to CV = 1.2e-16 (machine precision). Cascade
+temperature coefficient EXACTLY matches Hawking (ratio = 1.0). This is because
+the cascade density profile produces the Schwarzschild metric (MAR exp_30),
+so the surface gravity is identical. The cascade adds interpretation, not
+correction: Hawking radiation = PAC conservation preventing cascade potential
+from reaching zero. Conservation + duality uniquely select g_in = 1/phi
+(algebraic proof: g_in^2 + g_in = 1 has unique solution). M^3 evaporation
+lifetime confirmed, consistent with Stefan-Boltzmann + area scaling.
+
+### exp_33c — Entropy from Cascade Counting
+
+Derives S = A/(4 l_P^2) from counting independent cascade choices on
+the horizon. Tests multiple counting schemes. Simulates PAC tree
+evaporation to produce the Page curve.
+
+| Test | What it checks |
+|------|---------------|
+| 1. Area scaling | S ~ M^2 ~ A (surface), NOT M^3 ~ V (volume) |
+| 2. The 1/4 coefficient | Which cascade counting scheme produces 1/4? |
+| 3. Page curve | PAC tree evaporation: entropy rises to N/2 then falls |
+| 4. Holographic principle | Cascade hierarchy: information capacity = boundary area |
+
+**Falsification**: S scales as volume, or Page curve shows no turnover.
+
+**Key results**: Area scaling S ~ M^2.0 exact across 12 orders of magnitude.
+S/(A/l_P^2) = 0.250000 exactly. Multiple counting schemes consistent with
+1/4 (2D branching argument and 4*ln(phi) cell area), but no unique derivation
+from phi alone — honest gap. Page curve from PAC Tree Tensor Network: turnover
+at k/N = 0.5000 exactly, returns to zero at both ends, symmetric to 3.5e-12.
+The PAC tree IS a holographic tensor network — each conservation bond carries
+H(phi) = 0.665 nats when cut. Shape correlation 0.977 with Page (fatter due
+to hierarchical bonds — physically meaningful). Holographic principle confirmed
+in d=1,2,3 (L^2.05 for d=3, asymptotic fit).
+
+### exp_33d — Holographic Scaffold (The Ghost Heart Mechanism)
+
+The holographic principle reframed: conservation creates an information
+scaffold where the interior is FORCED to actualize from boundary data.
+Like a decellularized ghost heart — dissolve all living cells, the
+extracellular matrix (scaffold) remains, and you can recellularize
+from boundary alone. The PAC tree's interior is scaffold, not content.
+
+| Test | What it checks |
+|------|---------------|
+| 1. Decellularization | Erase interior, reconstruct from boundary — perfect fidelity? |
+| 2. Recellularization | Same scaffold, 6 different boundary conditions — all valid? |
+| 3. Subregion reconstruction | Ryu-Takayanagi surface from PAC minimal cut |
+| 4. Information decomposition | rank(full tree) = N_leaves - 1 (interior adds zero DOF) |
+
+**Falsification**: Reconstruction fidelity < 1.0, interior has independent
+information, or RT surface is not minimal.
+
+**Key results**: Perfect reconstruction (error = 0.0) across D=4 to D=14.
+All 6 boundary types (uniform, PAC, Dirichlet, delta, power-law, thermal)
+produce valid trees — scaffold is universal. RT surface for contiguous
+blocks: O(log N) bonds (area law), vs O(N) for random subsets (368x
+locality advantage). Purification S(A) = S(A^c) exact. SVD confirms
+rank(full tree) = N_leaves - 1 at every depth tested — the interior adds
+exactly zero independent dimensions. I_scaffold = 0.
+
+## Status
+
+| Test | Status | Score | Key Finding |
+|------|--------|-------|-------------|
+| exp_33a | complete | 4/4 | BH IS cascade Zeno completion; interval ratio = 1/phi^{3/2} to 0.01% |
+| exp_33b | complete | 4/4 | Cascade temp = Hawking EXACTLY (ratio 1.0 to 2.2e-16) |
+| exp_33c | complete | 4/4 | Area scaling exact, 1/4 consistent; PAC-TTN Page curve peaks at N/2, symmetric, returns to zero; holographic L^2 in d=3 |
+| exp_33d | complete | 4/4 | Ghost heart mechanism: scaffold carries zero information, RT surface from PAC conservation, 368x locality advantage |
+
+## FDO Links
+
+- `gravity-time-duality`
+- `geometry-precedes-arithmetic`
+- `herniation-hypothesis`
+- `school-entropic-gravity`
+- `pac-comprehensive`

foundational/experiments/exp_33_black_hole_cascade/meta.yaml

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+schema: "2.0"
+status: active
+proficiency: research
+description: >
+  Maps the PAC cascade onto black hole physics. A black hole IS a cascade
+  running to Zeno completion: infinite levels in finite proper time, with
+  the event horizon as the boundary between "still cascading" (external)
+  and "cascade complete" (infalling). Derives Hawking temperature from
+  cascade density gradient and Bekenstein-Hawking entropy from cascade
+  state counting. Begins implementing QG-1/QG-2/QG-4 from the quantum
+  gravity proposal.
+files:
+  - README.md
+child_directories:
+  - scripts
+  - results
+  - journals
+key_results:
+  - "exp_33a: 4/4, BH IS cascade Zeno completion (v^2 ratios -> phi 0.24%, BH interval = 1/phi^{3/2} to 0.01%)"
+  - "exp_33b: 4/4, cascade temperature EXACTLY matches Hawking (ratio 1.0 to 2.2e-16), PAC+duality uniquely select phi"
+  - "exp_33c: 4/4, area scaling exact (S ~ M^2), 1/4 coefficient consistent; PAC-TTN Page curve peaks at N/2 (symmetric to 3.5e-12, returns to zero); holographic L^2.05 in d=3"
+  - "exp_33d: 4/4, ghost heart mechanism — scaffold carries zero information (error 0.0), universal scaffold (6 boundary types), RT surface O(log N) with 368x locality advantage, rank(full)=N_leaves-1 exact"
+dependencies:
+  internal:
+    - exp_32_geometric_primacy
+    - minimum_actualization_resolution
+    - milestone6
+  external:
+    - numpy>=1.24
+    - scipy>=1.10