[dcharb78]

Research Package: qq16-interior-dynamic-range-and-adjoint-ledger

• Task ID: joint-query-controller (track = "recursive-interfaces", weight 20, Priority Revision 6 in docs/research/current-priorities.md and docs/research/agent-task-catalog.json, base_commit = "997e6a9ca7b4f95d50e366f61412fd640d72a88a", scope_status = "unchanged")
• Package Path: research/qq16-interior-dynamic-range-and-adjoint-ledger/
• Verification Status: PASS_QQ16_INTERIOR_DYNAMIC_RANGE_AND_ADJOINT_LEDGER — report (author-checked), no new $\kappa$ claimed (no_new_kappa)
• Selected Baseline (certificates/selected-result.json): $\kappa = 0.001672660197076918$ (T37/T16, unchanged exact auditor, zero checker extensions)
• Authors: dcharb78, with AI assistance
• Provenance Boundary: Pinned verification inputs (SOURCES.json, actual parent lineage) are loaded and SHA-256/git-blob verified at runtime (#2697, #2707, #2736, #2932, #2959, #2965; note that #1969, #2540, #2678, #2681, #2697, #2707, #2736, #2932, #2959, and #2965 carry merged_for_archive_only / partially_reviewed status in docs/research/pr-intake-decisions.json); concurrent open PRs (#2995, #3017, #3036, #3068) are not in SOURCES.json and are not read at runtime, serving solely as external comparators / comparison benchmarks.

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1. Summary of Contributions

Prior archived (merged_for_archive_only / partially_reviewed) joint-query-controller packages reported (author-checked):

• the native-F17 conserved-helper producer certificate CONSERVED-QQ16 (#2697, research/joint-query/certificate.json.gz and research/joint-query/evidence/CONSERVED-QQ16.json), which recorded formal triangle-inequality Horner bounds of 55 bits in F (5413950274051752469/256) and 73 bits in B (573411134169296526275133/64);
• the 3,823-event line-normalized 224-column Q_Q16 two-sided decoder (#2707, research/qq16-line-normalized-decoder/RESULT.json);
• the terminal producer-defect transfer law $\delta_R = D(F_0^{-1}\varepsilon_F + \varepsilon_B)$ (#2965, research/qq16-producer-defect-transfer/RESULT.json) with $|F_0^{-1}|{\infty\to\infty} = 49$, $|D|{\infty\to\infty} = 1923$, and normalized error budget 7587/8192 < 1 whenever $|\varepsilon_F|\infty \le H\cdot 2^{-\beta-8}$ and $|\varepsilon_B|\infty \le H\cdot 2^{-\beta-13}$.

However, #2965 explicitly left open the interior fixed-point precision and backward error-propagation ledger across the 21025 (F) and 21401 (B) scalar gates and the 3494 (F) and 3510 (B) native children of CONSERVED-QQ16. Subsequent open PRs (#2995 and #3017 by Wimuks; #3036 and #3068 by Goodhill1) examined only F_0 (either on the 224-column data slice or at individual forward rounding fault sites 24438, 3887, and 4060), leaving the B producer (27019 events, 73-bit Horner bound), the backward adjoint sensitivities of all 480 restored helpers, the common dyadic grid exactness invariant, and the physical recursive bill under ceiling.ceiling(...) unquantified.

This standalone Mode B package (verify.py, Python 3.12 stdlib-only, < 9s runtime) resolves all four questions:

1. Pillar 1 — Exact 712-Column Algebraic Replay & 37-Bit / 52-Bit Dynamic Range Collapse: Replaying all 712 register columns (224 data L[0..223], 480 dirty helpers L[224..703], 8 private scratch C[0..7] mapped to 704..711) across all 37 native cases (F_0 with 26099 events, F_1..F_8 with 26387 events, and B_1_2..B_7_8 with 27019 events) proves that massive algebraic cancellation across the B + restore + cleanup schedule collapses the true interior $\ell_1$ dynamic range by 37 bits in F (55 -> 18 bits, exact peak $\ell_1 = 131072 = 2^{17}$) and by 52 bits in B (73 -> 21 bits, exact peak $\ell_1 = 1154896 < 2^{20.14}$, decomposed into peak data $\ell_1 = 200770$, peak helper $\ell_1 = 954130$, and peak private $\ell_1 = 30$).
2. Pillar 2 — Exact Backward Adjoint Error-Propagation Ledger (F and B, 704 Outputs) & Wrapper-Helper Isolation: Using exact transpose-duality (704 terminal outputs $\times$ 712 input columns verified bit-for-bit between forward and backward passes), we compute the exact worst-case backward $\ell_1$ error-amplification gains from every climb/native_line child (3494 in F with 2950 dirty; 3510 in B with 2928 dirty), add gate (20278 in F, 20654 in B), and scale gate (715 in F and B) into both the 224 data outputs and the 480 restored helpers, and prove Wrapper-Helper Adjoint Isolation: all wrapper native_line children (8 in F, 24 in B) and wrapper scalar gates have exact 0 sensitivity on all 480 restored helpers (helpers_wrapper_child_gain = 0, helpers_wrapper_gate_gain = 0).
3. Pillar 3 — Exact Common Dyadic Grid Invariant (1/16 in F, 1/131072 in B) & 50 + \beta-Bit Precision Floor: Every intermediate rational coefficient across all 712 columns of F and B has a power-of-two denominator bounded by 16 ($2^4$, 4 fractional bits) in F and 131072 ($2^{17}$, 17 fractional bits) in B, with maximum numerator magnitude 65537 ($2^{16}+1$). Executing F on the common dyadic grid $2^{-4}$ and B on $2^{-17}$ makes all 21025 (F) and 21401 (B) scalar gates 100% exact integer operations (eps_gate = 0, non_integer_grid_steps = 0). This completely eliminates the cancellation rounding faults reported in #2995/#3017 (at F_0 event 24438) and #3036/#3068 (at F_0 events 3887 and 4060) and reduces the sufficient fixed-point register width for #2965 target accuracy $H\cdot 2^{-\beta}$ to 50 + \beta bits (21 signed integer + 29 + \beta fractional bits), compared to 75 + \beta bits under rounded-gate execution and 127 + \beta bits under the 73-bit Horner bound (saving 52 bits in B and 37 bits in F).
4. Pillar 4 — Physical Recursive Bill & Packing-Independent ceiling.ceiling(...) Screen: Screening the combined physical recursive bill of CONSERVED-QQ16 (F_0 + B_1_2) plus the #2707 decoder (336 rank-1 lines across the 2-call pair, 168 per call) through research/complex-local-word-ceiling/ceiling.py demonstrates that under the unchanged hosted nine-stage coverage law (centre = 64), the combined profile {1: 2596, 2: 418, 3: 432, 4: 142, 5: 82, 8: 8} has exact packing-independent ceiling a_cap = 0.001633824631 and kappa_cap = 0.001631159602 (dropping to a_cap = 0.001512422742, kappa_cap = 0.001510138774 when 224 rank-9 $H_9$ endpoint adapters—112 per call—are paid to exit F17). Both lie strictly below the selected baseline $\kappa = 0.001672660197076918$. Only under a hypothetical coverage law that eliminates the 64-per-call centre tax (centre = 0) with an F17-native consumer does the surrogate margin reach a_cap = 0.004579628060, kappa_cap = 0.004558750677.

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2. Exact Numerical Tables (RESULT.json)

2.1 Pillar 1: 712-Column Algebraic Replay Across All 37 CONSERVED-QQ16 Cases

Metric	F_0 (s = 0)	F_1..F_8 (s = 1..8)	B_1_2..B_7_8 (28 pairs)
Total expanded events	26099	26387	27019
Scalar gates (add / scale / clear)	21025 (20278 / 715 / 32)	21025 (20278 / 715 / 32)	21401 (20654 / 715 / 32)
Native children (climb core + native_line wrapper)	3494 (3486 + 8)	3494 (3486 + 8)	3510 (3486 + 24)
Children reading dirty helper entries (h_l1 > 0)	2950	2950	2928
Formal path-sum Horner bound (CONSERVED-QQ16.json)	5413950274051752469/256	5413950274051752469/256	573411134169296526275133/64
Formal Horner ceiling width (bits)	55	55	73
Exact interior peak data $\ell_1$ norm (event, reg)	131072 (24611, 93)	131072 (24611, 93)	200770 (26594, 93)
Exact interior peak helper $\ell_1$ norm (event, reg)	273 (19243, 238)	273 (19243, 238)	954130 (26599, 93)
Exact interior peak private $\ell_1$ norm (event, reg)	68 (8162, 704)	68 (8162, 704)	30 (9985, 704)
Exact interior peak total $\ell_1$ norm (event, reg)	131072 (24611, 93)	131072 (24611, 93)	1154896 (26594, 93)
Exact algebraic ceiling width (bits)	18	18	21
Interior dynamic range collapse (bits saved)	37 (55 -> 18)	37 (55 -> 18)	52 (73 -> 21)
Max dyadic denominator ($2^k$) / fractional bits	16 (4 bits)	16 (4 bits)	131072 (17 bits)
Max numerator absolute value	65537	65537	65537
Peak child-entry $\ell_1$ norm (total = data / helper max)	124 (56 / 68)	124 (56 / 68)	166 (158 / 18)
Terminal max data $\ell_1$ norm / helper leak on data	27 / 0	27 / 0	64 / 0
Terminal restored helpers / zeroed private C registers	480 / 8	480 / 8	480 / 8

2.2 Pillar 2: Backward Adjoint Error-Propagation Ledger (704 Outputs)

Output Sector	Producer Side	Total Child Gain $G_{\text{child}}$	Wrapper Child Gain	Core Child Gain	Max Single-Child Gain	add Gate Gain $G_{\text{add}}$	scale Gate Gain $G_{\text{scale}}$	Combined Scalar Gate Gain $G_{\text{gate}}$
Data X (0..111)	F	132	1	132	2	174091/512	66311/4	2216769809/131072 (16912.6115)
Data Y (112..223)	F	118	1	118	2	528	16392	536179/32 (16755.59375)
All Data (0..223)	F	132	1	132	2	528	66311/4	2216769809/131072 (16912.6115)
Restored Helpers (224..703)	F	59	0	59	—	5920689/32768	131163/4	131599/4 (32899.75)
Data X (0..111)	B	381/2 (190.5)	13	381/2	12	1836679/2	16603/4	3689923/4 (922480.75)
Data Y (112..223)	B	16	1	16	2	257	5/2	258
All Data (0..223)	B	381/2 (190.5)	13	381/2	12	1836679/2	16603/4	3689923/4 (922480.75)
Restored Helpers (224..703)	B	393/2 (196.5)	0	393/2	—	11715/8	61	1515 (1515.0)

2.3 Pillar 4: Physical Recursive Bill Screened by ceiling.ceiling(...)

Profile Screened (m = 73, families = 896, stages = 9, grid = 10^-12)	centre / call	Paid Local Mass / call	Upper $a_{\text{cap}}$ (a_hi)	Upper $\kappa_{\text{cap}} = a_{\text{cap}}/(1+a_{\text{cap}})$	Relation to Selected $\kappa = 0.001672660197076918$
Selected baseline {1: 2412, 2: 418, 3: 432, 4: 142, 5: 82, 8: 8}	64	5586	0.001690054110	0.001687202646	Selected ceiling envelope (+1.45e-5 above selected)
CONSERVED-QQ16 F only (uncharged consumer, centre = 0)	0	5530	0.004730555837	0.004708283041	Uncharged producer-only surrogate margin
CONSERVED-QQ16 B only (uncharged consumer, centre = 0)	0	5546	0.004716424327	0.004694284091	Uncharged producer-only surrogate margin
CONSERVED-QQ16 pair + #2707 decoder (centre = 64 retained)	64	5770	0.001633824631	0.001631159602	Below selected $\kappa$ (-4.15e-5)
CONSERVED-QQ16 pair + #2707 decoder + 224 $H_9$ adapters (centre = 64)	64	6778	0.001512422742	0.001510138774	Below selected $\kappa$ (-1.63e-4)
CONSERVED-QQ16 pair + #2707 decoder (centre = 0, F17-native consumer)	0	5706	0.004579628060	0.004558750677	Requires global centre = 0 coverage law + F17 consumer
CONSERVED-QQ16 pair + #2707 decoder + 224 $H_9$ adapters (centre = 0)	0	6714	0.004239533840	0.004221636071	Requires global centre = 0 coverage law

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3. Six Negative Controls (RESULT.json["negative_controls"])

• Control 1 (control_1_horner_vs_exact_algebraic_collapse): Verifies the 37-bit (55 -> 18 in F) and 52-bit (73 -> 21 in B) collapse between formal Horner bounds and exact 712-column algebraic replay (status = "PASS").
• Control 2 (control_2_truncated_restoration_suffix): Truncating the restoration/cleanup tail after the last dirty-helper write onto data wires (event 16850 in F_1, event 18714 in B_1_2) leaves 480 unrestored helpers in F (max helper $\ell_1 = 62$) and 395 unrestored helpers in B (max helper $\ell_1 = 16$), proving the B + restore + cleanup suffix is mandatory (status = "REJECTS_TRUNCATED_RESTORATION").
• Control 3 (control_3_omitted_private_c_clears): Omitting the 32 clear gates on C[0..7] leaves all 8 private registers dirty and corrupts 112 data outputs in F and 8 data outputs in B (status = "REJECTS_OMITTED_PRIVATE_CLEARS").
• Control 4 (control_4_synthetic_wrapper_helper_coupling): Injecting a single synthetic wrapper read from A_copy wire L[112] into helper L[224] raises helpers_wrapper_child_gain from 0 to 1 (status = "REJECTS_WRAPPER_HELPER_COUPLING").
• Control 5 (control_5_single_sign_perturbation_at_peak_helper_event): Flipping the sign at the #3036/#3068 helper-cancellation site (event 3887) and the peak interior $\ell_1$ site (event 24611) in F_1 leaves 136 helpers unrestored, leaks helper dirt onto 4 data outputs, and blows up the terminal data $\ell_1$ norm from 27 to 270668033/4096 (status = "REJECTS_SINGLE_SIGN_PERTURBATION").
• Control 6 (control_6_uncharged_vs_charged_consumer_ceiling): Confirms that under the unchanged centre = 64 coverage law, paying #2707's 336 rank-1 decoder lines forces $\kappa_{\text{cap}} = 0.001631159602 < 0.001672660197076918$ (status = "CONFIRMS_CENTRE_AND_CONSUMER_BOTTLENECK").

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4. Comparison Against Prior & Open PRs on joint-query-controller

Package / PR	Scope Covered	B Replay (27019 events)	712-Col (480 Helpers + 8 C)	Backward Adjoint Ledger	Common Grid (1/16, 1/131072)	ceiling.ceiling(...) Screen
#2697 (CONSERVED-QQ16)	Native producer certificate & Horner bounds (55 / 73 bits)	Horner only (73 bits)	Terminal check	No	No	No
#2707 (qq16-line-normalized-decoder)	3823-event 224-column Q_Q16 decoder (336 rank-1 lines)	N/A	224-col decoder	No	No	No
#2965 (qq16-producer-defect-transfer)	Terminal transfer $\delta_R = D(F_0^{-1}\varepsilon_F + \varepsilon_B)$ (49, 1923)	Terminal only	224-col terminal	No	No	No
#2995 / #3017 (Wimuks)	F_0 224-column data slice & event 24438 rounding fault	No (F_0 only)	No (224 data cols)	Partial (F_0 data)	F_0 data (1/16)	No
#3036 / #3068 (Goodhill1)	F_0 dirty-helper child count (2950) & events 3887/4060	No (F_0 only)	F_0 forward only	No	No	No
This PR	Full 37-case F & B 712-col replay, adjoint ledger, grid & ceiling	Yes (73 -> 21 bits, peak 1154896)	Yes (712 cols across 37 cases)	Yes (F & B, 704 outputs)	Yes (50 + \beta bits, eps_gate = 0)	Yes (0.001631159602 vs 0.004558750677)

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5. Credits, Provenance, and Lineage (#845–#2965 & External Comparators #2995–#3068)

• Package Authors: dcharb78, with AI assistance
• Direct Upstream Predecessors (SOURCES.json, merged_for_archive_only / partially_reviewed):
  • Douglas Colkitt (#2697 research/joint-query/certificate.json.gz & research/joint-query/evidence/CONSERVED-QQ16.json; #2707 research/qq16-line-normalized-decoder/RESULT.json; #2736/#2932/#2959 research/complex-local-word-ceiling/ceiling.py & RESULT.json; #2965 research/qq16-producer-defect-transfer/RESULT.json & verify.py).
  • Prior joint-query-controller and recursive-interfaces ancestor lineage (#845–#2965): #845, #1012, #1094, #1098, #1240, #1659, #1708, #1712, #1748, #1791, #1829, #1918, #1969 (merged_for_archive_only / partially_reviewed), #2048, #2199, #2391, #2540 (Wimuks, merged_for_archive_only / partially_reviewed), #2678 (merged_for_archive_only / partially_reviewed), #2681 (Wimuks, merged_for_archive_only / partially_reviewed), #2697, #2707, #2736, #2771, #2807, #2811, #2873, #2876, #2932, #2959, #2965.
• External Comparators (Concurrent Open PRs Not in SOURCES.json, Not Read at Runtime): Wimuks (#2995, #3017) and Goodhill1 (#3036, #3068), cited strictly as external comparators (reports, author-checked) not in SOURCES.json and not read at runtime.

Path (SOURCES.json)	Usage	git_blob	sha256
certificates/selected-result.json	provenance	b1fea0c94436b236cc2b7d41a7be032811b94c81	bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129
docs/research/agent-task-catalog.json	provenance	67f085acd86acd7d4e03922d9cf4b26bca66fed5	47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0
research/complex-local-word-ceiling/RESULT.json	input	77f55043a679dd722330f1e2499863e36376d9f9	d91d6c46509875abf1f61de67c2609656d3dc685451299f83caa6f922493692d
research/complex-local-word-ceiling/ceiling.py	input	6c0f54527d879396f689523bfde19928a9b6157c	86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc
research/joint-query/certificate.json.gz	input	c6e876d8b48d768214deee265ade428f8983b566	a5ed85fac5ebf3c091482a113ab0d351eb5a7892fe2841a35e16017ec1ff49f7
research/joint-query/evidence/CONSERVED-QQ16.json	input	7a155a27dcf7f1ccd48e40d57a2c5d2724016e9b	b91eb8599c403db55c889bb225a1cd93f46fdb83eb339f101d18843d52b1c920
research/qq16-line-normalized-decoder/RESULT.json	input	ba4167dbf133cff51513bb832802c290edcdffea	58df4ad68e2ff335ee58fa4336a449d0f73f3366d840574e5e30b50f507c358d
research/qq16-producer-defect-transfer/RESULT.json	input	abe7e44c1d75edaef4c4407aa57a0f94faa949e6	d3992b63f7e576a52be846497d37f71a03e89d9481f82d1013743db05dc79cba
research/qq16-producer-defect-transfer/verify.py	input	8dd78e9e36acb63451eae88c3c558f7f9afc348a	1458d3a188a8ec0124e6edd1fdc7d9238c41d7a547fa2c00229763fc698e46c8

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6. Verification Command

python3 -B research/qq16-interior-dynamic-range-and-adjoint-ledger/verify.py

Output (< 9s, Python 3.12 stdlib only):

PASS_QQ16_INTERIOR_DYNAMIC_RANGE_AND_ADJOINT_LEDGER F_collapse_bits 37 B_collapse_bits 52 F_peak_l1 131072 B_peak_l1 1154896 F_child_gain 132 B_child_gain 381/2 centre64_kappa_cap 0.001631159602 no_new_kappa

[rcfc-jreed]

Independent scoped audit of #3181 QQ16 interior dynamic-range / adjoint ledger at 80fc72d1: RESULT matches (30/30)

Code-disjoint checks; does not import package verify.py (local fork with patched HERE/ROOT against nine SOURCES pins at base 997e6a9c…). Package bytes from head 80fc72d14546088e9fce3c06dc12bc67270da0af via GitHub contents API. CLAIM.

No packaging defect in scope. RESULT.json byte-matches fork recomputation (PASS_QQ16_INTERIOR_DYNAMIC_RANGE_AND_ADJOINT_LEDGER).

What passes:

1. All nine SOURCES digests match (CONSERVED-QQ16 + joint-query certificate + producer/decoder/ceiling pins + selected-result/catalog provenance).
2. Pillar receipts: F collapse 55→18 bits (37-bit saving), B collapse 73→21 bits (52-bit saving); peak interior L1 131072 (F) / 1154896 (B); wrapper-helper child gain 0.
3. Backward adjoint data child gains: F 132, B 381/2; centre64 QQ16+decoder κ-cap 0.001631159602 below selected baseline 0.001672660197076918; no_new_kappa.
4. Negative controls 1/4/6 as committed (Horner vs exact PASS; wrapper coupling rejected; centre/consumer bottleneck confirmed).

Not covered: Lean formalization; full written-proof review of upstream #2965/#2995 lineage; new κ; merge/close.

This is a finite pin/seal/math replay, not a formal proof.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).