[Gonzalo-Mier]

Change and mathematical scope

This package binds the P112 tetrad-local masked decoder to the selected complex provider's 112 nine-bit ports. The explicit bijection preserves all 12,544 pairings, verifies 6,720 constrained entries, and maps 56 terms of rank 2 with total copy charge 112.

A conditional two-sided rank-32 budget with 166 ideal complex children and two hypothetical unpaid coordinates gives a complex κ ceiling of 0.0137876681081 at a=1/99. The inherited bit-side ceiling is 0.0319843333739, so the complex side limits this relaxation. Both one-unpaid controls fail. The two-unpaid endpoint contracts remain assumptions: no source-bound dirty-input implementation, owner chronology, restoration/consumer invoice or admitted supplier is provided. This is not a physical construction, endpoint-contract change or selected-frontier update; selected κ remains 0.001672660197076918.

Task and overlap

Task complex-rank-shape-decoder, priority revision 6. Scope was refined from pricing a new child-rank shape to binding this decoder to selected ports and screening its conditional ideal budget. Base commit: 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Package: research/selected-p112-tetrad-rank32-route/; exact source pins are in SOURCES.json.

Related work: PR #3344 screens larger children and multiple unpaid coordinates; PR #3567 screens a one-sided tetrad alphabet; PR #3604 composes a selected basis through the centre cut. The cached overlap scan covered 394 open PRs and 2,319 discussions as of 2026-10-18 03:15 UTC; its refresh failed, so it is not a novelty certificate.

Validation

Reproduce with Python 3.12:

python3.12 -I -B research/selected-p112-tetrad-rank32-route/verify.py
python3 scripts/research_package.py --package research/selected-p112-tetrad-rank32-route --check-only
python3.12 scripts/research_package.py --package research/selected-p112-tetrad-rank32-route

The verifier reconstructs the selected word and P112 geometry, checks the port bijection over all 12,544 pairs, verifies the constrained entries and all 56 term ranks, and recomputes both interval-priced profiles. The one-unpaid controls fail.

Credit

The P112 geometry and local identity retain the predecessor package's attribution. The source archive, complex ceiling, centre ledger and bit-pricing engine are pinned in this package. Prepared by Gonzalo Mier with substantial OpenAI Codex assistance; Apache-2.0. No independent human review is claimed.

Open next questions

Implement one eight-port tetrad block against the actual provider interface, then price the full dirty-input path and verify both unpaid-coordinate contracts. The result matters only if the source, chronology and endpoint obligations admit the assumed budget.

[Matarener]

Independent check of the centre charge used in this package (reviewed head ebd6b92c4).

Summary. The copy mass 112 is correct for the one-unpaid line mask, but the budget applies it under the two-unpaid law. Under two unpaid coordinates the decoder target and its tariff change, and the route gives no saving. The coefficient identity itself replays exactly: verify.py passes and reproduces κ ≤ 0.0137876681081.

1. Tariff convention. beta_k = rank_F2 N[X_k,Y_k] prices each term on the singleton port labels. With two unpaid coordinates the ports are paired into Gram-I2 planes, and each port carries its whole plane. philippeVV's ruling (Discussion 1587, applied in research/kport-group-scalar-endpoint-audit) makes the target the group-level strict mask: K_ij = 0 whenever the owning planes are not orthogonal. Pricing is by complete-plane rank. research/paired-plane-rank34-copy-audit/RESULT.json records both conventions for one fit: singleton_label_invoice = 112 and complete_plane_copy_invoice = 224.

2. The tetrad K fits no plane pairing. Using the same 112 selected labels (identical in both packages) and the 56 transported terms, every off-diagonal nonzero K_ij (four per port) would need plane(i) ⟂ plane(j). For each of the 112 ports, the number of Gram-I2 partners j with K_ij = 0 that are also orthogonal to all four K-neighbours is 0. So no perfect matching makes this K a valid two-unpaid decoder. With the PR3335 pairing it violates 372 of the 11,328 forced zeros, and its complete-plane tariff is 492.

3. Budget at the inherited two-unpaid tariff. The same complex-local-word-ceiling engine and the same {1:192, 32:166} frame with centre charge 224 and unpaid=2 give no positive saving for any decoder profile of mass 224, including the idealised {32:7} (moment at a = 1/99 ≈ 1.0117–1.0126). The reason is the deficit 2w − 224 = −224/9 < 0, matching the rank-34 audit and the 2-integral floor.

4. One-unpaid control. Where 112 is the right price, the existing centre costs 64. With the tetrad profile {2:56}, centre 112 and unpaid=1, the moment at 1/99 is 1.0104 > 1, as p112-tetrad-local-decoder already notes (M(0) = 585/584).

Because #3604 inherits the centre charge 112, its two-unpaid ceilings depend on the same step. I have not separately assessed its kernel-invariant coverage mechanism.

The scripts are standard-library Python against this PR's tree, and I can share them. Prepared with substantial AI assistance (Claude).

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Generated by Claude Code

[Matarener]

Independent check of the centre charge used in this package (reviewed head ebd6b92c4).

Summary. The copy mass 112 is correct for the one-unpaid line mask, but the budget applies it under the two-unpaid law. Under two unpaid coordinates the decoder target and its tariff change, and the route gives no saving. The coefficient identity itself replays exactly: verify.py passes and reproduces κ ≤ 0.0137876681081.

1. Tariff convention. beta_k = rank_F2 N[X_k,Y_k] prices each term on the singleton port labels. With two unpaid coordinates the ports are paired into Gram-I2 planes, and each port carries its whole plane. philippeVV's ruling (Discussion 1587, applied in research/kport-group-scalar-endpoint-audit) makes the target the group-level strict mask: K_ij = 0 whenever the owning planes are not orthogonal. Pricing is by complete-plane rank. research/paired-plane-rank34-copy-audit/RESULT.json records both conventions for one fit: singleton_label_invoice = 112 and complete_plane_copy_invoice = 224.

2. The tetrad K fits no plane pairing. Using the same 112 selected labels (identical in both packages) and the 56 transported terms, every off-diagonal nonzero K_ij (four per port) would need plane(i) ⟂ plane(j). For each of the 112 ports, the number of Gram-I2 partners j with K_ij = 0 that are also orthogonal to all four K-neighbours is 0. So no perfect matching makes this K a valid two-unpaid decoder. With the PR3335 pairing it violates 372 of the 11,328 forced zeros, and its complete-plane tariff is 492.

3. Budget at the inherited two-unpaid tariff. The same complex-local-word-ceiling engine and the same {1:192, 32:166} frame with centre charge 224 and unpaid=2 give no positive saving for any decoder profile of mass 224, including the idealised {32:7} (moment at a = 1/99 ≈ 1.0117–1.0126). The reason is the deficit 2w − 224 = −224/9 < 0, matching the rank-34 audit and the 2-integral floor.

4. One-unpaid control. Where 112 is the right price, the existing centre costs 64. With the tetrad profile {2:56}, centre 112 and unpaid=1, the moment at 1/99 is 1.0104 > 1, as p112-tetrad-local-decoder already notes (M(0) = 585/584).

Because #3604 inherits the centre charge 112, its two-unpaid ceilings depend on the same step. I have not separately assessed its kernel-invariant coverage mechanism.

The scripts are standard-library Python against this PR's tree, and I can share them. Prepared with substantial AI assistance (Claude).

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Generated by Claude Code

[rcfc-jreed]

Independent scoped audit of #3566 P112 tetrad decoder ↔ selected ports at ebd6b92c: binding + rank-32 budget agree (34/34)

Code-disjoint checks; does not import package verify.py (local fork against pinned SOURCES at base 20ecf289…). Package bytes from head ebd6b92c43239d76b1a59e29f574563494388f6a via GitHub contents API. CLAIM.

No packaging defect in scope. RESULT.json matches fork recomputation.

What passes:

1. All thirteen SOURCES digests match blobs at the package base pin (including selected-replay.tar.gz and predecessor geometry/ceiling/levers).
2. Explicit 112-port bijection: 12 544 parity pairings; map sha d66c6db9…; 56 decoder terms with copy profile {2:56} / mass 112; 14 tetrad blocks / 28 four-frames; 6720 masked identity entries.
3. Conditional two-unpaid rank-32 budgets: complex κ cap 1747553197/126747553197 ≈ 0.0137876681081 (caps the joint ideal); bit κ cap ≈ 0.0319843333739; both one-unpaid controls fail at a=1/99; κ = a/(1+a) identities hold exactly.
4. Selected κ record unchanged at 0.001672660197076918; package correctly marks itself as not a supplier/selected improvement.
5. Status PASS_SELECTED_P112_TETRAD_RANK32_BUDGET.

Not covered: source-bound dirty-input implementation; owner chronology / restoration invoice; admitted physical supplier; new κ; formal-kernel proof.

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

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