δp_min investigation: Phase 1+2 progress

Phase 1 (theoretical) closed; Phase 2 first cut on core3d landed; three audit rows flipped from STUB

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investigation
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Three audit rows flipped from STUB. The δp_min investigation — a Paper III v1.0 prerequisite — is next blocked on dcl_core v0.2.0 exposing a prob_floor parameter.
Author

Jack D. Menendez

Published

May 27, 2026

The δp_min investigation asks, before Paper III’s v1.0 deposit, whether the framework’s minimum probability quantum has a non-zero value that materially affects Paper III’s predicted Roche limit M_\text{min}(d) = \Delta\omega_\text{tongue} \cdot d^3 / R_1. The methodology is a 4-cell grid: two engines (dcl_core.core continuous-amplitude and dcl_core.core3d integer-token) crossed with two δp_min regimes (effectively zero vs explicitly non-zero).

Phase 1 (theoretical pass) and the first column of Phase 2 (numerical probe on core3d) both landed on 2026-05-24. Three audit rows flipped from STUB.

Phase 1: three rows flipped

  • Operational meaning of δp_min pinned: STUB → PASS. The investigation commits to two candidates together: a probability floor in dcl_core.core, and identification with 1/N in dcl_core.core3d, with the per-site-floor-plus-global-renormalisation convention applied per tick (not at extraction) and no separate phase quantum.
  • Cross-engine equivalence, core clamp ↔︎ core3d 1/N: STUB → PART. Structural finding: the two engines are not equivalent at finite N (the integer-token state space is a 1/N-quantised sublattice that core’s continuous-amplitude dynamics does not preserve). The sufficient condition for asymptotic equivalence is the double limit — a \to 0 and N \to \infty — with deviation O(1/N) per tick at matched \epsilon = 1/N. Numerical confirmation of the O(1/N) rate, and pinning the Bresenham-coupled prefactor, are Phase 2 targets.
  • Consistency with Paper I PASS rows under non-zero δp_min: STUB → PART. Analytical upper bounds derived for both Paper I guardrails — hydrogen Bohr recovery requires \epsilon \lesssim 10^{-10}–10^{-12} (order-of-magnitude from the ground-state wavefunction’s box-edge amplitude); Arnold-tongue lock-in requires \epsilon \lesssim \Delta\omega_\text{tongue}^2. Numerical confirmation flips this row PART → PASS.

The two inherited Paper I guardrail rows (hydrogen as joint Arnold-tongue lock-in; gradient detuning breaks joint resonance) remain PASS and must not become FAIL under any δp_min value the investigation finds — if they ever do, that is a Paper I revision, not a silent downgrade here.

Phase 2 first cut on core3d: empirical bound and an engine-protocol surprise

The integer-token right column of the grid is unblocked at the dcl_core v0.1.0 pin, so it ran first. exp_12_dp_min_sweep.py swept the token budget N across nine points from 10^2 to 10^{10} on a 33^3 lattice (single-particle hydrogen in a fixed Coulomb well, 400 ticks per N).

Empirical bound: δp_min ≤ 10^{-7} preserves the engine’s continuum-N baseline at 5% tolerance — substantially less restrictive than the Phase 1 analytical estimate of 10^{-10}–10^{-12}. The integer-quantisation noise is smoothed by the engine’s diffusive dynamics faster than the worst-case wavefunction-tail estimate predicted.

Engine-protocol difference surfaced. core3d’s ground-state r_\text{peak} = 19.63 does not match Paper I’s R_1 = 10.3 at the same 33^3 geometry. Both engines are designed as discretisations of the same Dirac dynamics, but core3d’s hop kernel (uniform average, alternating chirality per tick, periodic boundaries) is a different discretisation than Paper I exp_12’s (directed kinetic hop, both chiralities per tick, open boundaries). They are expected to agree in the continuum limit (a \to 0) but pin different ground-state observables at fixed a. The paper-text for the cross-engine equivalence row was refined to make this explicit; the row stays PART until the left column lands.

No engine errors occurred; A=1 held as a bitwise integer identity at every tick across all 9 sweep points. Total wall-clock for the nine-point sweep: 39 s.

The first cut also surfaced a new finding-target. dcl_core v0.1.0 carries a real fractional bit in its Bresenham-style residual; the v0.1.0 release notes parked the hypothesis that it should be complex — a complex carry would tie the residual to the 51 unaccounted-for generators in Paper II’s per-site automorphism algebra. The δp_min Phase 2 protocols are being shaped to include observables that distinguish coherent (complex) from decoherent (real) quantisation noise, so the investigation can speak to the carry question on the way to its primary deliverable.

What is blocked, and on what

The left column of the 4-cell grid — dcl_core.core with an explicit probability floor — requires a prob_floor: float | None parameter on the continuous-amplitude engine’s amplitude-update primitives. That parameter is the principal upstream coordination output of this investigation. It is additive (default None preserves current behavior) and is the planned content of dcl_core v0.2.0.

Once v0.2.0 is released:

  • This repository’s virtual-env-requirements.txt bumps to the v0.2.0 pin.
  • exp_09_dp_min_floor.py runs on core with prob_floor = 10^-12, 10^-10, 10^-8, …; the resulting \Delta\omega_\text{tongue}(\delta p_\min) measurement closes the Arnold-tongue-width-floor row and provides the direct core-vs-core3d cross-baseline comparison at matched geometry.
  • The 4-cell grid on g_\text{crit} and T_\text{escape}(g) — exp_18_dp_min_grid.py — completes across all four cells, closing the two STUB rows that gate Paper III’s stance decision.

The core3d right column can continue in parallel; exp_18’s right-column points are unblocked at the v0.1.0 pin.

Pointers: