Author notes — full detail, auditor-facing

SIM-003 is the long-running attempt to *derive* the framework's foundational r = 0.5 decoherence ceiling from cone geometry rather than impose it. Earlier versions:

• v3 introduced cascade + information-theoretic terms. Matched

the linear coefficient of the quadratic capacity equation; missed the quadratic coefficient. Used p=2 norm and 3D spherical propagation — the latter failed.

• v4 moved to cone-shaped propagation with pulsed overflow.

Fixed v3's 3D failure. Reached 18% angular coefficient-of-variation on the unfolding-angle prediction — better but still wide.

• v6 (skipped v5) added pockets + voids dual-curve representation.

Stalled at global_r = 0.419 — close to 0.5 but persistently below, suggesting an architectural rather than parameter issue.

• v6c (current) adds wave re-injection: when a pulse reflects off

the cone boundary, the reflected energy is re-injected into the cascade as a fresh input, with phase-aware addition. Hypothesis: the 0.419 stall in v6 was because reflected energy was being *absorbed* by the boundary; re-injecting it should drive global_r toward 0.5 cleanly.

Architecture (v6c).

• Two-curve dual representation:
• "Pockets" curve = energy concentrated at cone-apex regions

(small-r, high-density)

• "Voids" curve = energy in cone-bulk regions (large-r, low-density)
• Emergent global_r = mean(r_local) across the full cone volume,

where r_local is computed from the local pocket/void ratio.

• Wave re-injection: reflected pulses are reintroduced at the source

plane with phase-aware superposition.

• Convergence criterion: global_r stable to within 1% over 1000

cascade steps.

Currently running on Hetzner (5.78.189.153). Single-box CPU run, ~96 hours estimated. As of 2026-05-12: ~36 hours in, global_r currently tracking ~0.487 (vs v6's stalled 0.419). The trend is moving toward 0.5, but it's premature to call convergence — the last 8 hours have seen global_r drift from 0.480 to 0.487, which is the right direction but not yet stable.

Pre-registered outcomes.

• Success: global_r converges to 0.500 ± 0.003 over the last 1000

steps. This would *derive* the r=0.5 ceiling from cone geometry with no parameter tuning.

• Partial success: global_r converges to 0.49–0.51 but with wider

variance. The framework's ceiling is approximately right but the derivation isn't tight enough to call canonical.

• Failure: global_r stalls below 0.49 or oscillates without

converging. Indicates the wave-reinjection hypothesis is insufficient and a deeper architectural change is needed (likely moving to 6D cascade — see SIM-003 6D extension memo).

What this is and is not.

• IS: an attempt to derive r=0.5 (a load-bearing framework parameter)

from first principles via cone-geometry cascade simulation.

• IS NOT: a fit. The r=0.5 ceiling was set in the framework BEFORE

SIM-003 began. If the simulation converges to 0.5, that's a prediction-matches-observation result. If it stalls below, the framework either has the right ceiling but the wrong derivation path, or both.

Why this matters. r=0.5 is the most foundational parameter in TLT. It governs the dimensional cascade, defines when overflow occurs, and propagates into the energy boundaries, the framerate formula, and the spiral coordinate. A clean derivation would close one of the framework's biggest open derivation chains.

Float64 precision concern. Earlier SIM-003 runs hit float64 precision limits when r_local approached 0.5 (cancellation errors in the pocket/voids ratio). v6c added compensated summation in the relevant accumulators. The fix appears to be working — no precision artifacts observed in the first 36 hours.