research(rank7-coupled-586-588-carrier-fold): lower helpers 586 & 588 via coupled B+restore carrier folds (κ = 0.0017297760894)

[dcharb78]

Conditional κ = 0.001729776089405120334238 − 10⁻⁴⁶ − 10⁻⁷⁰: Coupled B + restore Dying-Helper Carrier Folds Lowering Helpers 586 & 588 to Rank 7 (width 6) with Literal $T=74$ Allocation (T74_optimal_idle39)

Selected Frontier, Archival Base Status, & Priority Revision 6 Alignment (certificates/selected-result.json, docs/research/pr-intake-decisions.json, docs/research/current-priorities.md):

1. Selected Conditional Bound, Actual Parent Lineage, & External Comparators (base_commit = 997e6a9ca7b4f95d50e366f61412fd640d72a88a, Priority Revision 6, catalogue SHA-256 47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0): The official selected conditional bound on main remains κ = 0.001672660197076918 (PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094, certificates/selected-result.json). This package was developed directly and solely on top of our actual parent lineage—PR Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 (mdumitrean), PR Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015 (jeremykothe2), and PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042 (dcharb78)—which are recorded in docs/research/pr-intake-decisions.json as archived author claims (merged_for_archive_only, partially_reviewed). Concurrent downstream packages on main—PR test(research): independently audit the six restore skips #3050 (petalas), PR Verify six-skip/T740 release: conditional kappa 0.001729737950064279 #3051 (jeremydiviney), PR feat(research): release the T740 six-skip composition #3054 (petalas), PR Contract PR3042's four-ADD restore block to three #3056 (petalas), PR Prove the complete restore-frame optimum of the six-skip words #3057 (petalas), PR Prove an optimal native tariff for the rank-seven restore block #3060 (petalas), PR Propose conditional κ ≈ 0.001729737530919310 via T148 carrier skips (full replay pending) #3064 (dataisfire), PR Prove the complete joint A/B/restore frame optimum #3065 (petalas), and PR Independently verify all patterns in the T740 six-skip packing cap #3103 (Arham-Tahir64) (archived author claims (merged_for_archive_only, partially_reviewed)), alongside open PRs Record the V45 single-lowering margin #3112, Record V45 in the shipped words and cap brackets #3119, Record V45 widths and the inherited call order #3126 (jwr404, V45 invoice/chronology audits), research: restore-only optimum for PR #3042's six-skip words (scoped exact negative, no new κ) #3114 (mdumitrean), research: κ = 0.001729737950064278551242895823 − 10^-52 − 10^-76 #3115 (dataisfire), and Changed helper inventories screened against the packing ceiling (complex-new-helper-inventory) #3125 (navinor, negative ceiling screen of R489/R480 on card complex-new-helper-inventory, outcome=partial, no new $\kappa$)—were not used as inputs to construct this package and are cited solely as external comparators / scoped optimality baselines.
2. Priority Card & Changed Hypothesis (docs/research/current-priorities.md, Priority Revision 6): Under docs/research/current-priorities.md (incumbent-optimization / complex-new-helper-inventory track: "Before reopening a negative result, name the hypothesis you change" and "Independently check one coordinated group and its losing single-change control, then trace the changed costs through the complete invoice"), three recent packages report (author-checked) that the fixed-endpoint premise of PR Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 / PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042 (width_inventory = {4:26, 5:2, 6:16, 7:52, 8:99, 9:213}) is exhausted within its stated scope:
   • PR Prove the complete restore-frame optimum of the six-skip words #3057 & PR Prove the complete joint A/B/restore frame optimum #3065 (petalas, archived author claims merged_for_archive_only, partially_reviewed): Report (author-checked) that on PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042's six-skip words, with scalar operations/order and helper endpoints fixed, all 6,141 A/B/restore frames are already at their global optimum.
   • Open PR research: restore-only optimum for PR #3042's six-skip words (scoped exact negative, no new κ) #3114 (mdumitrean): Reports (author-checked) that with A, B, warps, and helper endpoints ({6:16, 7:52}) fixed, replacing restore by any legal scalar gate sequence and re-solving all frames satisfies cert(W') >= 10,868,103,087 = cert(PR3042)—so PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042's -29,976 restore saving (-12 H1 + 6 H2) exhausts restore-only improvements under fixed A, B, and fixed helper endpoints.
   • PR Verify six-skip/T740 release: conditional kappa 0.001729737950064279 #3051, PR feat(research): release the T740 six-skip composition #3054 & PR Independently verify all patterns in the T740 six-skip packing cap #3103 (archived author claims merged_for_archive_only, partially_reviewed, plus 30-digit refinement open PR research: κ = 0.001729737950064278551242895823 − 10^-52 − 10^-76 #3115): Report (author-checked) that at $T=740$ ($20 \mid T$), the fixed {4:26, 5:2, 6:16, 7:52, 8:99, 9:213} inventory attains its continuous relaxation cap (κ = 0.0017297379500642785512428...).
3. How This Package Breaks the Fixed-Endpoint Ceiling: We change the joint hypothesis that B-phase carrier routing and helper_final terminal subspaces are fixed independently. In PR Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 (construction/STAGE-PRICES.json, lines 773–803), entering helpers 586 and 588 were the only two restored helpers in the entire word whose causal lower bounds $L_{586} = V_{30,76,82}$ and $L_{588} = V_{30,97,127}$ already had dimension 7 ($k = 0$ blocking causal gates), yet Stage-1 frame-only lowering had a negative surrogate margin of -3,345 each (-6,690 joint), and Stage-2 carrier folding at dim 8 alone had a negative surrogate margin of -973 each (-1,946 joint). By coupling a B-phase dying-helper carrier fold on (586, 438) and (588, 440) (B_586_438 and B_588_440) with Stage-1 helper_final lowering on 586 and 588 ({6:16, 7:52} -> {6:18, 7:50}), we unlock a +4,983 per-helper (+9,966 joint) positive cross-term synergy, yielding a +665 per-helper (+1,330 joint) surrogate margin against the rounded guide value BE = 2,091,300 per width-7 $\to$ width-6 reduction (while the exact $\kappa$ root and RELAX-CAP certificate carry the actual claim).
4. Extended Declared-Edit Checker vs. Unchanged Exact Auditor:
   • Extended edit stage (edits/declared_edits.py, edits/edits_review.py, verify.py): The Stage-4 declared-edit applier and independent edit reviewer were extended from single-hunk restore replacements to support multi-hunk B + restore edits with automatic warps index shifting (when B hunks change length) and declared helper_final lowerings to codimension-2 subspaces (V_{30,76,82}, V_{30,97,127}).
   • Unchanged exact auditor (vendor/audit_local.py, vendor/preload_move.py, structure.py): The exact forward/dual linear auditor (vendor/audit_local.py), monotone frame/cleanup engine (vendor/preload_move.py), and two-sided structure certificate checker (structure.py) are unchanged and byte-identical to PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042's (and PR Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 / PR Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's).

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1. Summary Comparison Table

Package / Record	Intake / Review Status (pr-intake-decisions.json)	κ (headline; candidate rows carry − 10⁻⁴⁶ − 10⁻⁷⁰)	Exact Root / Cap Bracket	Local Profile per Call	Helper Width Inventory	Complete Paid Profile ($T=74$ unless noted)
Selected Result (PR #1094)	Official selected conditional bound (certificates/selected-result.json)	0.001672660197076918	20908252463461475 / 12500000000000000000	Selected k8 baseline	Selected k8 baseline	Selected k8 baseline
PR #2969 (Dugongue)	Archived comparator (allocation/baseline/CLAIM.json)	0.00172403522474810487855	Cap-level at $T=370$	{1:2430, 2:628, 3:282, 4:110, 5:26, 8:8}	{4:26, 5:2, 7:62, 8:100, 9:218} (86 buf)	$T=370$ baseline
PR #2979 (mdumitrean)	Archived author claim (merged_for_archive_only, partially_reviewed)	0.001729033982646513517717	Root 0.0017290339826465... Cap 0.0017290357648755...	{1:2445, 2:706, 3:232, 4:108, 5:18, 8:8}	{4:26, 5:2, 6:16, 7:52, 8:99, 9:213} (91 buf)	{1:1645680, 2:470196, 3:159468, 4:71928, 5:13716, 8:5328, 48:28190}
PR #3015 (jeremykothe2)	Archived author claim (merged_for_archive_only, partially_reviewed)	0.001729150973534224428695	1 skip (63 via 61)	{1:2443, 2:707, 3:232, 4:108, 5:18, 8:8}	{4:26, 5:2, 6:16, 7:52, 8:99, 9:213} (91 buf)	{1:1644348, 2:470862, 3:159468, 4:71928, 5:13716, 8:5328, 48:28190}
PR #3042 (dcharb78, audited in #3050, #3065, #3114)	Archived author claim (merged_for_archive_only, partially_reviewed)	0.001729736166387915986812	Root [0.0017297361663879159868125, ...6] Cap [0.0017297379500642785512428, ...9]	{1:2433, 2:712, 3:232, 4:108, 5:18, 8:8}	{4:26, 5:2, 6:16, 7:52, 8:99, 9:213} (91 buf)	{1:1637688, 2:474192, 3:159468, 4:71928, 5:13716, 8:5328, 48:28190}
PR #3051 / #3054 / #3103 (jeremydiviney / petalas / Arham-Tahir64) & open PR #3115 (dataisfire)	Archived author claims (merged_for_archive_only, partially_reviewed) / open (#3115)	0.001729737950064278551242 (#3054) 0.001729737950064278551242895823 (#3115, − 10⁻⁵² − 10⁻⁷⁶)	Attains {6:16, 7:52} cap at $T=740$ ($20 \mid T$)	{1:2433, 2:712, 3:232, 4:108, 5:18, 8:8}	{4:26, 5:2, 6:16, 7:52, 8:99, 9:213} (91 buf)	$T=740$ cap-attaining profile (6,660 calls)
This PR (research/rank7-coupled-586-588-carrier-fold)	New conditional candidate (author-checked, AI-assisted)	0.001729776089405120334238	Root [0.0017297760894051203342384, 0.0017297760894051203342385] RELAX-CAP [0.0017297774946785970118869, 0.0017297774946785970118870]	{1:2433, 2:724, 3:228, 4:102, 5:20, 8:8}	{4:26, 5:2, 6:18, 7:50, 8:99, 9:213} (91 buf)	{1:1637692, 2:482184, 3:156741, 4:67932, 5:14710, 8:5328, 39:1, 48:28256}

• Exact Gain over PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042 ($T=74$ literal root): +3.99230172e-8 (0.0017297760894051203342384 - 0.0017297361663879159868125).
• Exact Gain over PR Verify six-skip/T740 release: conditional kappa 0.001729737950064279 #3051 / PR feat(research): release the T740 six-skip composition #3054 / PR research: κ = 0.001729737950064278551242895823 − 10^-52 − 10^-76 #3115 ($T=740$ cap-attaining root): +3.81393408e-8 (0.0017297760894051203342384 - 0.0017297379500642785512428).
• Exact Continuous Relaxation Cap Gain (RELAX-CAP over PR feat(research): release the T740 six-skip composition #3054's $T=740$ cap): +3.95446143e-8 (0.0017297774946785970118869 - 0.0017297379500642785512428).
• Exact $T=74$ Integer-Rounding Gap to RELAX-CAP: 1.4053e-9, which is the $20 \mid T$ fractional-basis period effect (idle_per_replica = {1: 234, 3: 1323/20, 5: 75/4, 48: 7637/20} in certificates/RELAX-CAP.json) closed at $T=740$ in PR feat(research): release the T740 six-skip composition #3054 and PR Independently verify all patterns in the T740 six-skip packing cap #3103 (and partially at $T=148$ in PR Propose conditional κ ≈ 0.001729737530919310 via T148 carrier skips (full replay pending) #3064).

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2. Literal edits/EDITS.json Derivation (All 4 Edits & helper_final Lowerings)

Every gate count (len(old) -> len(new)), register ID, rational coefficient, subspace basis, and per-edit profile delta below is extracted programmatically from edits/EDITS.json (identical on base and extra). Across all 4 edits (14 hunks per word) and 2 helper_final lowerings, the touched registers are:

• R1_E42 (14 registers): [18, 22, 34, 40, 45, 54, 61, 63, 92, 103, 108, 226, 228, 230]
• R2_E7 (6 registers): [5, 59, 73, 96, 363, 395]
• B_586_438 (7 registers): [75, 136, 140, 144, 187, 438, 586]
• B_588_440 (7 registers): [84, 162, 172, 196, 221, 440, 588]

Edit ID	Section & Source Addresses	Hunks & Literal Gate Counts (len(old) -> len(new))	Touched Registers	helper_final Lowering (old -> new)	Per-Call Profile $\Delta$	Surrogate Guide $\Delta$ (c1=2,116,671, rounded guide BE=2,091,300; $\kappa$ carries the claim)
R1_E42	restore[162:188]	1 hunk: 26 -> 33	18, 22, 34, 40, 45, 54, 61, 63, 92, 103, 108, 226, 228, 230 (skips E_42 for {22, 40, 54, 63})	None	-8 H1 + 4 H2	-19,984 (+19,984 surrogate margin)
R2_E7	restore[190:203]	1 hunk: 13 -> 15	5, 59, 73, 96, 363, 395 (skips E_7 for {5, 73})	None	-4 H1 + 2 H2	-9,992 (+9,992 surrogate margin)
B_586_438	B[52:53], B[784:785], B[794:795], B[1214:1215], B[1230:1231], restore[85:86]	6 hunks: 6 -> 5 (1 -> 0 at B[52:53], five 1 -> 1)	75, 136, 140, 144, 187, 438, 586	586: E_30 (8) $\to$ V_{30,76,82} (7) ([256,128,70,32,18,12,1], width 7 -> 6)	+6 H2 - 2 H3 - 3 H4 + 1 H5	-2,091,965 vs BE = -2,091,300 $\implies$ +665 surrogate margin
B_588_440	B[57:58], B[789:790], B[799:800], B[1222:1223], B[1238:1239], restore[88:89]	6 hunks: 6 -> 5 (1 -> 0 at B[57:58], five 1 -> 1)	84, 162, 172, 196, 221, 440, 588	588: E_30 (8) $\to$ V_{30,97,127} (7) ([256,128,65,33,18,10,6], width 7 -> 6)	+6 H2 - 2 H3 - 3 H4 + 1 H5	-2,091,965 vs BE = -2,091,300 $\implies$ +665 surrogate margin
Total	B (-2 gates, cleanup +2 at FULL, warps shifted) + restore (+9 gates)	14 hunks (51 -> 58 gates)	34 distinct registers	586, 588 lowered ({6:16, 7:52} -> {6:18, 7:50})	-12 H1 + 18 H2 - 4 H3 - 6 H4 + 2 H5	-4,213,906 vs 2*BE = -4,182,600 $\implies$ +31,306 surrogate margin

2.1 R1_E42 (restore[162:188], 26 -> 33 Gates, Skipping {22, 40, 54, 63}, -8 H1 + 4 H2 = -19,984)

• Exact $\mathbb{Q}$-Linear Map (old and new identical across all 14 registers [18, 22, 34, 40, 45, 54, 61, 63, 92, 103, 108, 226, 228, 230]):
  $$X_{18}' = -X_{226}, \quad X_{22}' = -X_{230}, \quad X_{108}' = -X_{22} + 2 X_{34},$$
  $$X_{226}' = -\frac{1}{4} X_{18} - \frac{1}{4} X_{40} + \frac{1}{2} X_{54}, \quad X_{228}' = \frac{1}{4} X_{22} - \frac{1}{2} X_{34} + \frac{1}{4} X_{108},$$
  $$X_{230}' = -\frac{1}{4} X_{22} + \frac{1}{2} X_{34} - \frac{1}{4} X_{40} + \frac{1}{4} X_{45} + \frac{1}{2} X_{54} + \frac{1}{4} X_{61} + \frac{1}{4} X_{63} - \frac{1}{4} X_{92} + \frac{1}{4} X_{103} + \frac{1}{4} X_{228} + \frac{1}{4} X_{230},$$
  and $X_r' = X_r$ for $r \in {34, 40, 45, 54, 61, 63, 92, 103}$.
• Literal Gate Sequence (33 gates in new):
  • Registers 40 and 54 (NEW[0..8] and NEW[20..25]): At $W_B = V_{25,42,51} = [256, 128, 64, 34, 17, 11, 4]$ ($\dim 7$), NEW[0..3] execute 40 += (-2)*54, 40 -= 226, 226 += 40, 40 -= 226. At $E_{42} = [256, 128, 64, 34, 16, 10, 4, 1]$ ($\dim 8$), NEW[4..8] execute 18 += 226, 18 *= -1/4, 18 -= 226, 226 += 18, 18 -= 226, and NEW[20..21] execute 230 -= 18, 230 -= 45. At $\text{FULL} = [256, 128, 64, 32, 16, 8, 4, 2, 1]$ ($\dim 9$), NEW[22..25] execute 40 -= 18, 18 += 40, 40 -= 18, 40 += 2*54. Both 40 and 54 skip $E_{42}$ and climb 7 -> 9 directly (-4 H1 + 2 H2 = -9,992).
  • Register 22 (NEW[9..19]): At $W_A = V_{30,42,52} = [256, 128, 64, 38, 20, 10, 1]$ ($\dim 7$), NEW[9..16] execute 22 += (-2)*34, 22 -= 228, 228 += 22, 22 -= 228, 22 -= 230, 230 += 22, 22 -= 230, 230 += 22 (setting $X_{22}' = -X_{230}$ at $\dim 7$ so 22 skips $E_{42}$ and climbs 7 -> 9 at 'end', -2 H1 + 1 H2 = -4,996), followed by NEW[17..19] at $E_{42}$ ($\dim 8$): 228 += 108, 228 *= 1/4, 108 += (-4)*228.
  • Register 63 via carrier 61 (NEW[26..32], superseding PR Conditional κ = 0.001729150973534224: one declared restore edit (R1) on PR2979's words and literal allocation #3015): NEW[26] executes 61 += 63 at $V_{45} = V_{7,42,45} = [256, 128, 64, 35, 16, 11, 5]$ ($\dim 7$), NEW[27] executes 230 -= 61 at $E_{42}$ ($\dim 8$), and NEW[28] executes 61 -= 63 at $\text{FULL}$ ($\dim 9$, so 63 climbs 7 -> 9 directly, -2 H1 + 1 H2 = -4,996), followed by NEW[29..32] at $E_{42}$ ($\dim 8$): 230 += 92, 230 -= 103, 230 -= 108, 230 *= -1/4.

2.2 R2_E7 (restore[190:203], 13 -> 15 Gates, Skipping {5, 73}, -4 H1 + 2 H2 = -9,992)

• Exact $\mathbb{Q}$-Linear Map (old and new identical across all 6 registers [5, 59, 73, 96, 363, 395]):
  $$X_5' = -X_{395}, \quad X_{59}' = -X_{363}, \quad X_{96}' = X_5 + X_{395}, \quad X_{73}' = X_{73},$$
  $$X_{363}' = -\frac{1}{4} X_{59} + \frac{1}{4} X_{73} + \frac{1}{4} X_{363}, \quad X_{395}' = -\frac{1}{4} X_5 + \frac{1}{4} X_{96} - \frac{1}{4} X_{395}.$$
• Literal Gate Sequence (15 gates in new):
  • Register 73 (NEW[0..6]): NEW[0] executes 363 += 73 at $W_C = V_{7,25,30} = [256, 128, 64, 32, 19, 11, 6]$ ($\dim 7$); NEW[1..5] execute 59 -= 363, 59 *= -1/4, 59 -= 363, 363 += 59, 59 -= 363 at $E_7 = [256, 128, 64, 32, 16, 8, 5, 3]$ ($\dim 8$); and NEW[6] executes 59 += 73 at $\text{FULL}$ ($\dim 9$, so 73 skips $E_7$ and climbs 7 -> 9, -2 H1 + 1 H2 = -4,996).
  • Register 5 (NEW[7..14]): NEW[7] executes 395 += 5 at $W_D = V_{7,51,52} = [256, 128, 64, 37, 21, 8, 3]$ ($\dim 7$); NEW[8..13] execute 96 -= 395, 96 *= 1/4, 96 -= 395, 395 += 96, 96 -= 395, 96 *= -1 at $E_7$ ($\dim 8$); and NEW[14] executes 5 -= 96 at $\text{FULL}$ ($\dim 9$, so 5 skips $E_7$ and climbs 7 -> 9, -2 H1 + 1 H2 = -4,996).

2.3 B_586_438 + helper_final["586"] (6 Hunks, 6 -> 5 Gates, Lowering 586 to $V_{30,76,82}$, +665 Surrogate Margin)

• Mechanism & Exact B vs. cleanup Accounting: Between B[52]: 136 += 1/2*586 and B[784]: 136 -= 1/2*586, 586 only receives B[759]: 586 += 438, so the net effect on 136 is $-\frac{1}{2} X_{438}$. Since 136 later receives B[1215]: 136 += 75 from source register 75 (which has no other readers after B[794] and ends at $\text{FULL}$, $\dim 9$), we delete B[52] (1 -> 0), move 140 -= 1/2*438 to B[784] (1 -> 1) at the 4D join frame $J_{140,438} = [257, 129, 32, 30]$, fold 75 -= 1/2*438 into B[794] (1 -> 1) at the 5D join frame $J_{75,438} = [257, 129, 70, 32, 30]$, lower B[1214]: 144 -= 1/2*438 (1 -> 1) to [256, 128, 70, 32, 30, 1] ($\dim 6$) and B[1230]: 187 -= 1/2*586 (1 -> 1) to [288, 160, 70, 18, 12, 1] ($\dim 6$), and lower restore[85]: 586 -= 438 (1 -> 1) and helper_final["586"] from $E_{30} = [256, 128, 64, 32, 18, 10, 6, 1]$ ($\dim 8$) to $L_{586} = V_{30,76,82} = [256, 128, 70, 32, 18, 12, 1]$ ($\dim 7$, reducing width(586) from $7 \to 6$). Compared with PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042, B loses 1 gate (B[52]), restore is unchanged except for the frame at restore[85], and inverse-reverse cleanup gains 1 gate at $\text{FULL}$ (75 += 1/2*438), which costs no frame steps and leaves the total B + restore + cleanup gate count unchanged.
• Profile Delta & Surrogate Margin: +6 H2 - 2 H3 - 3 H4 + 1 H5 = -2,091,965 units/call vs rounded guide BE = -2,091,300 $\implies$ +665 surrogate margin.

2.4 B_588_440 + helper_final["588"] (6 Hunks, 6 -> 5 Gates, Lowering 588 to $V_{30,97,127}$, +665 Surrogate Margin)

• Mechanism & Exact B vs. cleanup Accounting: Symmetrically, between B[57]: 221 -= 588 and B[789]: 221 += 588, 588 only receives B[763]: 588 += 440, so the net effect on 221 is $+X_{440}$. Since 221 later receives B[1223]: 221 += 84 from source register 84 (which ends at $\text{FULL}$, $\dim 9$), we delete B[57] (1 -> 0), move 172 += 440 to B[789] (1 -> 1) at the 4D join frame $J_{172,440} = [268, 161, 65, 30]$, fold 84 += 440 into B[799] (1 -> 1) at the 5D join frame $J_{84,440} = [262, 161, 65, 20, 10]$, lower B[1222]: 162 += 440 (1 -> 1) to [262, 128, 65, 33, 20, 10] ($\dim 6$) and B[1238]: 196 += 588 (1 -> 1) to [256, 161, 96, 18, 10, 6] ($\dim 6$), and lower restore[88]: 588 -= 440 (1 -> 1) and helper_final["588"] from $E_{30} = [256, 128, 64, 32, 18, 10, 6, 1]$ ($\dim 8$) to $L_{588} = V_{30,97,127} = [256, 128, 65, 33, 18, 10, 6]$ ($\dim 7$, reducing width(588) from $7 \to 6$). Compared with PR Conditional κ = 0.001729736166387916: six restore carrier/pivot skips (R1_E42, R2_E7) on PR2979's words and literal allocation #3042, across B_586_438 and B_588_440 combined, B loses 2 gates (B[52] and B[57]), restore is unchanged except for the frames at restore[85] and restore[88], and cleanup gains 2 gates at $\text{FULL}$ (75 += 1/2*438 and 84 -= 440), costing no frame steps and keeping the total gate count unchanged.
• Profile Delta & Surrogate Margin: +6 H2 - 2 H3 - 3 H4 + 1 H5 = -2,091,965 units/call vs rounded guide BE = -2,091,300 $\implies$ +665 surrogate margin (+1,330 combined with B_586_438).

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3. Single-Move Negative Controls & 91-Helper Exhaustion Proofs

3.1 Single-Move Negative Controls on 586 and 588

Control / Configuration	helper_final (586, 588)	B-Phase Gate Topology	Local Profile $\Delta$ (per helper / joint)	Local Climb Cost $\Delta$ (per helper / joint)	Rounded Guide Charge (BE = 2,091,300)	Surrogate Margin (per helper / joint; $\kappa$ carries the claim)
1. Stage-1 Lowering Alone (Fixed Scalar Gates)	$\dim 7$ ($V_{30,76,82}$, $V_{30,97,127}$)	Unfolded (B[52], B[784] and B[57], B[789] kept; frames intersected with $L_h$)	-1 H1 + 7 H2 - 2 H3 - 2 H4 each (-2 H1 + 14 H2 - 4 H3 - 4 H4 joint)	-2,087,955 each (-4,175,910 joint)	+2,091,300 each (+4,182,600 joint)	-3,345 each (-6,690 joint — LOSES)
2. Stage-2 Carrier Fold Alone at dim 8 (Fixed Endpoints)	$\dim 8$ ($E_{30}$, unchanged)	Folded (B[52], B[57] deleted; 140, 172 at $\dim 4$; 75, 84 at $\dim 6 = \text{meet}(B[1214], B[1215])$)	-1 H1 + 2 H2 - 2 H4 + 1 H5 each (-2 H1 + 4 H2 - 4 H4 + 2 H5 joint)	+973 each (+1,946 joint)	0 (width 7 unchanged)	-973 each (-1,946 joint — LOSES)
3. Coupled Carrier Fold + dim 7 Lowering (B_586_438, B_588_440)	$\dim 7$ ($V_{30,76,82}$, $V_{30,97,127}$)	Folded along 5-step subspace towers (140, 172 at $\dim 4$; 75, 84 at $\dim 5$)	+6 H2 - 2 H3 - 3 H4 + 1 H5 each (+12 H2 - 4 H3 - 6 H4 + 2 H5 joint)	-2,091,965 each (-4,183,930 joint)	+2,091,300 each (+4,182,600 joint)	+665 each (+1,330 joint — WINS)

• Why the +4,983 per-helper (+9,966 joint) cross-term synergy occurs: Under Stage-1 lowering alone, keeping B[784]: 136 -= 1/2*586 (and B[789]: 221 += 588) inside $L_{586}$ forces B[784]'s frame from $\dim 6 \to 5$, adding an extra stop on 586 ([1, 3, 5, 6, 7], +6,877) and 136 ([0, 1, 2, 5, 7, 8], +12,108 worse than [0, 1, 2, 7, 8]). Under Stage-2 folding at dim 8 alone, B[1214] stays at $\dim 7$ and restore[85] stays at $\dim 8$, adding an extra stop on 438 (+6,877) while 586 only saves -8,098 ([1, 3, 7, 8]). Coupling both moves drops B[1214] and B[1230] to $\dim 6$ inside $L_{586}$ while bypassing B[784] on 586 and 136, recovering +4,983 per helper over the sum of the two single moves (-3,345 + (-973) + 4,983 = +665 surrogate margin).

3.2 Exhaustive 91-Helper Causal Lower-Bound Scan & Structural Obstruction Proofs

Across all 91 restored helpers (26 at end_dim = 4, 30 at end_dim = 7, 35 at end_dim = 8), 586 and 588 are the only helpers with $k = 0$ blocking causal gates. Every remaining candidate with $k \le 2$ blocking gates is locked by one of three exact structural obstructions:

1. Double Cone Lock (584 and 590, $\dim 8 \to 7$ at $E_{51}$, $k = 1$ gate): Complete free-algebra elimination of 584 and 590 (0 gates in A+B+restore) passes all 632 forward/dual inputs in audit_local.py, but fails preload_move.frame_report (28–74 DAG lo <= hi violations across all chronological placements): right consumers 115/130 receive post-warp cold helpers 273/289 (B_WARP[541, 545]) hard-pinned to $W_{\text{right}} = E_{51} \cap E_{67} \cap E_{112}$ ($\dim 7$), while left consumers 168/214 receive entering helpers 554/566 pinned by 273 += 554 / 289 += 566 to $U_{6,\text{left}} \subseteq W_{\text{left}} = E_{51} \cap E_{82} \cap E_{97}$ ($\dim 6$), with $\text{join}(U_{6,\text{left}}, W_{\text{right}}) = E_{51}$ ($\dim 8$).
2. Restored-Dirt Isolation Lock (531, 543 and 554, 555, 566, 567, $\dim 8 \to 7$, $k = 1$ B gate): Removing B[916]: 305 += 531 and B[918]: 321 += 543 drops 531, 543 to $L_{586}, L_{588}$ ($\dim 7$) with 0 DAG violations, but B[660]: 531 += 351 isolates the clean sum $X_{351} + X_1$ while 351 retains $-X_{531} - \frac{1}{2}(X_{587} + X_{436})$. Folding 305 += 531 onto 351 pollutes warped cold helper 305 (B_WARP[611]) and output targets 36, 37, 46, 47 (bad_read = [531, 587], bad_targets = [36, 37, 46, 47]).
3. Grassmann 6 + 6 - 4 = 8 and 5 + 5 - 3 = 7 Fork-and-Uncompute Lock (500, 501, 512, 513 at $\dim 8$ and 572..575, 580..583 at $\dim 7$, $k = 1$ gate): Each 3-gate cold helper $h$ forks partner $p$ to two consumers $c_1, c_2$ at $\dim(\text{lo}[c_1]) = \dim(\text{lo}[c_2]) = 6$ with $\dim(\text{lo}[c_1] \cap \text{lo}[c_2]) = 4$ (forcing $\dim(\text{join}(\text{lo}[c_1], \text{lo}[c_2])) = 6 + 6 - 4 = 8$ at restore: h -= p) or $\dim = 5$ with intersection $\dim = 3$ (forcing $\dim(\text{join}) = 5 + 5 - 3 = 7$), while early splitting c_1 += h + c_1 += p violates $c_1$'s B_WARP boundary (hi_dim = 2, 26 DAG violations per helper).

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4. Verification

• /opt/homebrew/bin/python3.12 -B scripts/research_package.py --package research/rank7-coupled-586-588-carrier-fold $\to$ PASS_REGISTERED_RESEARCH_PACKAGE research/rank7-coupled-586-588-carrier-fold outcome=completed
  • verify_manifest.py: PASS package manifest: 155 files
  • verify.py: PASS_RANK7_JOINT_ENDFRAMES_THREE_STAGE_REPLAY controls 100 (stage1: 20, stage2: 52, stage3: 14, stage4: 14)
  • edits/edits_review.py: PASS_INDEPENDENT_EDIT_REVIEW {'base': 21, 'extra': 21}
  • check_allocation.py: PASS_RANK7_LITERAL_OWNERS_AND_EXACT_MOMENT (T74_optimal_idle39, 271,728 roles, 66,813 physical families, 5 owner controls)
  • check_cap.py: PASS_LITERAL_ALLOCATION_BELOW_FIXED_INVENTORY_CAP_WITH_RECORDED_GAP (gap = 1.4053e-9)
  • reproduce.py: PASS_RANK7_EDITS_COMPLETE_REPLAY (ADMISSION-RESULT.json: PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE, RELEASE-RECEIPT.json: PASS_RELEASE).