Study 22 — Tasks A–B: Memory-Gated Forward Model

VERIFICATION RECORD — audit companion to The Selective Transient Field from First Principles (Theory Edition V9.7.1). This section publishes the programme's technical record as it exists: the machine-verified release stack and the post-freeze calculation records, rendered from the frozen source files without editorial rewriting. It is written for auditors, not readers; the reader-facing statement of the theory is the Theory Edition. Every document below is hash-pinned; the packages that carry the checkers and data are listed on the record landing page.

Source file STF_Study22_TasksAB_InSession_Result_2026-09-03.md  ·  SHA-256 c299b15e1eb10c56b09186e0cf07e20ea1ed35891e86633384cfcf7525a16fb0  ·  rendered as-is (GitHub-style ```math fences converted to display math).

Date: 2026-09-03. Executed by the site/verification session, not by a GPT instance (Study 22 is on hold for credits; the author asked to continue from where the order left off). Scope: exactly Tasks A and B of STF_Study22_Memory_Gated_Forward_Model_Prompt_2026-09-02.md (rev 1a). Task C (data fit) was NOT started. No observational sample was touched.

1. What was computed

Memory relation as recorded (V8.1 App. C.2 / v9.0 Drude kernel), forward time t, τ = time to merger:

(d/dt + ω_c) y = ω_c Q(t), Z = Q − y, so Z(τ) = Q(τ) − ω_c ∫_τ^∞ e^{−ω_c(τ′−τ)} Q(τ′) dτ′.

Drives (from the order): D1 Q ∝ τ^{−3/4} (readout, large-argument branch); D2 Q ∝ τ^{−3/2} (readout, small-argument branch), which coincides with D3 (compact-port drive S_port ∝ b^{−6} ∝ τ^{−3/2}), so D2 and D3 are one row.

Emission maps: E1 rate ∝ Z; E2 rate ∝ Z²; E3 rate ∝ |dZ/dt|; E4 threshold (rate ∝ Z where Z exceeds 10 % of its window maximum, zero elsewhere).

Windows: W1 [0.1, 54] yr (the V9.6.1 declared window); W2 [0.1, ∞); W3a τ₋ = 0.02 yr; W3b τ₋ = 1.0 yr (both with τ₊ = 54 yr), i.e. τ₋ treated as a nuisance.

Grid: ω_c = 10^{−3} … 10^{+3} yr^{−1} in 13 log steps (τ_c = 1000 … 0.001 yr). Per cell: centroid ⟨τ⟩, centroid·ω_c, and the effective power-law index n_eff (the index of τ^{−n} on the same window with the same mean log-lead).

Blind table frozen before any reading: 416 rows, study22_blind_table.json, table SHA-256 (as embedded) 2ae3cfddbcebb61b…; file SHA-256 b7f0f19d4c717bccaaacc71fa180b1c9627aef4859894fee7ecba15e2c835a73. Scripts: forward_model.py (41d693fc…), analyze_blind_table.py (27dabc97…).

2. Checks

Kernel: the quadrature kernel agrees with the closed forms e^{s}Γ(1−p, s) (p = 3/4) and 2s^{−1/2} − 2√π e^{s} erfc(√s) (p = 3/2) to 1.2 × 10^{−7} relative. Asymptotes verified: Z/Q → 1 for ω_cτ ≪ 1 (0.984 and 0.998 at 10^{−3}); Z/Q → p/(ω_cτ) for ω_cτ ≫ 1 (ratio to asymptote 0.998 at 10^{3}).

Consequence (analytic, confirmed by the table): Z is a broken power law, τ^{−p} inside τ_c and τ^{−p−1} outside, monotonically decreasing in τ with the knee at τ ≈ τ_c. The memory steepens the far tail by exactly one power; it does not suppress emission toward merger, so no drive × map combination has a peak.

3. Findings (Task B read-out of the frozen table)

3.1 No combination gives an ω_c-controlled centroid. The local slope a_c = d ln⟨τ⟩ / d ln τ_c never reaches 1 for any convergent case. Maxima: 0.49 (D2/E1, W2), 0.47 (D1/E2, W2), 0.43 (D1/E1, W3a), 0.34 (D1/E1 and D1/E2, W1), ≤ 0.30 for everything else on W1, and ≤ 0.09 for all D2 maps other than E1. The one cell with a_c ≈ 0.9 (D1/E1, W2, τ_c ≥ 10 yr) is spurious: for D1/E1 the tail goes as τ^{−7/4}, whose first moment diverges, so the W2 centroid there is set by the numerical cut-off, which itself scales with 1/ω_c. That case has no centroid without a declared τ₊.

3.2 The lower edge controls the centroid. The sensitivity b = d ln⟨τ⟩ / d ln τ₋ at ω_c ≈ 1 yr^{−1} is 0.36–0.83 for the D1 maps and 0.91–0.96 for the D2 maps; it is ≈ 1.0 for every D2 map at the grid ends. Closed forms match the table: D2/E1 gives ⟨τ⟩ ≈ √(τ_c τ₋) (a_c = b = 1/2, table 0.50/0.50 at ω_c = 10^{−3}); D2/E2 gives ⟨τ⟩ = 2τ₋ (table 0.200 at τ₋ = 0.1); D2/E3 gives 3τ₋ (table 0.29); D2/E4 gives ≈ 2τ₋ (0.216). In words: with a monotone drive the centroid is anchored at the merger-side edge; τ_c enters only through fractional powers, and only when τ₋ ≪ τ_c ≪ τ₊.

3.3 Read-out at the adopted value τ_c = 0.529 yr on W1: centroid 2.46 yr (D1/E1), 0.27 (D1/E2), 0.41 (D1/E3), 0.30 (D1/E4), 0.57 (D2/E1), 0.16 (D2/E2), 0.22 (D2/E3), 0.19 (D2/E4). Effective index n_eff = 1.40 (D1/E1), 2.41, 2.13, 2.06, 1.94 (D2/E1), 3.54, 2.74, 2.81. The D1/E1 index 1.40 sits next to the 11/8 = 1.375 used in the V9.6.1 window profile, but the comparison carries no weight: 11/8 was adopted from the GR chirp exponent, not measured, and n_eff here is dominated by the window (it runs 0.78–1.75 across the grid for that map).

3.4 Consequence for the Study 22 aim (“fit m_s”). The centroid cannot be inverted for m_s in this model class: for the recorded window it depends on τ_c with exponent ≤ 0.34 and on the declared τ₋ = 0.1 yr with exponent ≥ 0.36 (D1) or ≈ 0.9 (D2), and with τ₋ free (W3) the product-type degeneracy ⟨τ⟩ ≈ f(τ_c^{a} τ₋^{1−a}) leaves τ_c unconstrained by the centroid alone. The one place m_s leaves a fingerprint is the knee: the lead-time distribution steepens by exactly one power at τ ≈ τ_c = ħ/(m_s c²), i.e. at 0.53 yr for the adopted mass (under E1 the slope goes from 3/4 to 7/4 for D1 and from 3/2 to 5/2 for D2; E3 and E4 shift both exponents but keep the unit step; E2 doubles them, so its step is two). A knee is a full-shape feature that a centroid cannot see.

4. What Task C would need (not started)

Task C is a shape fit of the lead-time distribution for a knee, with τ₋ marginalised. It requires per-event lead times (the manuscript’s Test 40 distribution), not the summary centroid. That distribution is not deposited: Zenodo 17526550 carries the manuscript PDF only. Any reconstructed sample truncated by a ±5-yr selection window cannot substitute — its range [0.1, 5] yr is too short on the outside to fix the tail and the truncation itself imposes a shape. Until a per-event distribution is available, the Study 22 fit has no admissible input, and the status line stays as after Study 21: m_s adopted, not derived; Study 22 open, blind table frozen.

5. Counters

TARGET_VALUES_USED_IN_DERIVATION = 0 (the value 3.32 yr appears only in §3.3 as a post-freeze read-out comparison, and 0.529 yr only as the grid point read). DATA_ROWS_READ = 0. Blind table frozen at 2026-09-03T08:32:04Z before any read-out.