[flyingviper]

A source-consuming encoder appears to lower native input34 from a five-dimensional chart to span(73). This package proves that its required scalar map cannot feed the proposed retained suffix using ascending paid transports. No improved kappa or new physical supplier is claimed.

The strongest obstruction is one main coefficient: H[51,51]=1/2, input51 carries direction100, and the greatest suffix cap at output51 is span(45,76), excluding100. A13-edge paid path to fixed main114 independently certifies the exclusion. This rules out every ascending pure-scalar implementation of the nominated whole-encoder map, regardless of cold basis, upfront preparation or interleaved compensation.

A complementary exact identity, alpha = -D^-1[:,78] + (1/2)D^-1[:,431] on the cold coordinates, explains the failure of the original compensation proposal. It covers every invertible cold basis and distributed upfront compensation in both the150-instruction local block (15 cold helpers) and the complete3968-instruction scalar encoder (457 cold registers), retaining all632 arbitrary inputs.

Both suffix orientations and literal/coalesced variants are rebuilt, with every native relation, helper final constraint and adapter pin relaxed. The original encoder with its original suffix passes all3352 forward and3168 inverse support checks. Changed suffix contacts, including undoing the temporary basis immediately after the paired natives, remain outside the theorem, as do nonascending paid transports and native-bearing replacements.

Validation at b16a555e8dbab1ff98416b7899ffc853f77bfaa8:

• Independent agent derivations and package reviews; registered normal and optimized isolated replays pass with16 corruption controls.
• Full632 sparse inverse identities, separate inverse-column round trips and original-source positive control pass.
• Metadata/source pins, exact committed bytes, full source ancestry and whitespace checks pass.
• Official export integrity and optimized Git-free replay with an empty PATH pass. Hosted CI was green on the previous head; the updated head awaits CI.

python3 -B scripts/research_package.py --package research/productive-helper-inverse-support

One scoped package on main base 7c472563474e60185e6ca101a17622133d047e56; no frozen-source, shared-tooling or selected-result edits. It uses #5301's integrated combined source and is distinct from #5324's fixed-height invoice theorem. Credit Cao Yifan's #4610, Cinna Davis's #4692 and #4470/controller lineage, Dugongue, zxv, Jacob Alan Sussman and retained predecessors. Prepared for flyingviper with substantial OpenAI Codex assistance and independent AI agents; no independent human or proof-kernel acceptance is claimed. Apache-2.0.

[jwr404]

Independent review (James Reed, with Claude Code). Head 7b39e20ca5bd9ad0b9cc7e3ee3c145f50fd57785.

Verdict: the obstruction argument is correct, and its decisive identity reproduces under an independent interpreter.

Proof check

• §1, inverse support. This is the subtle direction. A nonzero R⁻¹[i,j] gives a reversed-program dependency path. In forward time that is a chain of forward edges j → x₁ → … → i at decreasing times. Because every register's chart only ascends with time and contacts share a chart, A_i ⊆ chart_i(τ_k) ⊆ … ⊆ chart_j(τ₁) ⊆ B_j. So the claimed containment holds. ✓
• §2 and §4. S⁻¹ = [[1,0],[−D⁻¹c, D⁻¹]], so α (the code's −S⁻¹[:,416]) is D⁻¹c. With R = D Q⁻¹, R⁻¹ = Q D⁻¹. The caps at 78 and 431 exclude direction 73, so §1 forces (QD⁻¹)[i,78] = (QD⁻¹)[i,431] = 0 for any row carrying 73. Since Q's row i is supported on C, multiplying the C-projected identity (1) by it gives β_i = 0, a contradiction, for every invertible cold basis Q. ✓

Independent reproduction of α and identity (1)

I wrote my own sparse interpreter for the 150 scalar events [9961, 10111) of the forward program in productive-helper-fusion-certificate/COMBINED-WORD.json.gz, with ADD dst += c·src and SCALE dst *= c, and none of check.py's matrix code. I then applied the output basis T on 417/419 and the 416/417 exchange myself, formed S = BME⁻¹, and inverted D with my own Fraction Gauss–Jordan.

• Carrier row. Row 416 is entry13 + entry416, and S fixes the carrier. ✓
• α. α is +1/2 on 418, 419, 420, 421, 424, 425 and −1/2 on 430, 431, exactly your local α. ✓
• Identity (1). It holds on all 15 cold coordinates. ✓
• Off the cold set. Identity (1) does not hold on every outside coordinate. That matches your remark that it is "a projected identity on C, not an asserted identity on foreign main coordinates", and the proof only needs the C-projection. Worth keeping that sentence prominent.

The registered replay also passes locally (9 s). I did not independently rebuild the suffix caps (span(25,45,462), span(45,76)) or the 457-register whole-encoder extension. Those rest on your checker's graph propagation. The receipt shows them for both orientations and both suffix variants.