[unpaidmoon]

Change and mathematical scope

Scoped audit, no new κ. At the pinned #4088 mixed profile and its lower root endpoint, two readings of COMPLEX.md's rank-r transition inside width 73f have opposite moment signs:

• One child of rank r*f multiplies the scalar moment by f^(1-k) and stays strictly negative.
• f copies of each scalar child make the moment strictly positive for every integer f >= 2. At f = 2 the enclosed bill is greater than 56052, and the derivative in f is greater than 68435.

The pinned recipe reports (RECORD-base.json, RECORD-extra.json) remain scalar histograms on ranks {1,2,3,4,5,8}. They do not record a tensor frame. The selected frontier is unchanged.

This does not accept #4100, does not exhibit a native program that emits the rank-(r*f) child, and does not price the left and right monomials on that transition.

Task and overlap

• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.
• Card: uniform-transfer-audit, scope refined to the frame/child coordinate that Audit PR4100’s free-word tensor lift on all four actual programs #4184 leaves open after the free-word cancellation.
• Caller: Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088 histogram in research/staged544-mixed-fourth-entry/RESULT.json (72d7efc181edb21806a878f174847c96dd410ab0 is the retained Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088 head named by the family record; this check uses the copy on the pinned base).
• Audit PR4100’s free-word tensor lift on all four actual programs #4184 audits the algebra only. Independent numerical cross-check of PR4100's exact-root and paid-power ledger (finite layer; no new kappa) #4171 checks ledger powers. Neither compares these two child readings.
• The coverage map in Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 still lists native realization and physical placement of the mixed tensor lift as open. This is that coordinate, restricted to the child-rank alternative.

Package: research/tensor-frame-child-interface/.

Validation

Python 3.12, standard library only. From the repository root:

python3 scripts/research_package.py --package research/tensor-frame-child-interface --check-only
python3.12 scripts/research_package.py --package research/tensor-frame-child-interface

Receipt status: PASS_TENSOR_FRAME_CHILD_GAP.

Controls: f = 1 reproduces the scalar sign; adding 10^9 rank-1 children makes the scalar upper bound positive; both recipe reports are scalar. Logarithms and exponentials use alternating series with explicit tails. The pass/fail comparisons use the exact rational enclosures. The receipt decimals are truncated toward zero.

Not run: native replay, monomial tape accounting, Lean, and any all-size admission.

Credit

The profile, endpoint, and moment shape are Cinna Davis's #4088, with the enclosure lineage named in that package. The C tensor (r f) sentence is Cinna Davis's #4100. ToikY's #4184 certifies the free-word cancellation and leaves this coordinate open; this package does not repeat that algebra.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review is claimed.

Open next questions

What tape-action and child count do the left and right monomials of one actual #4088 warp add when the frame is replaced by its f-fold tensor? The rank-(r*f) child keeps the scalar sign, but an extra paid child or a change in W = 896*360 can erase it. First step: warp 0 of recipe/RECORD-base.json, as explicit F2 row additions, charged once per column for one fixed f >= 2. Blocker: the scalar recipe record has no monomial invoice.

Workflow feedback (optional)

None.

[Matarener]

Scoped negative note on #4470's paid children (no new κ, no PR).

On main ff3cc58e3 I decoded every paid climb of the selected #4470 word (research/joint-helper-lattice, 3,302 climbs per orientation, 2,856 distinct frame edges, 1,086 on main fields) with the repository's own lower_child and the lower_exact decode check (tableau + absolute phase). All 3,302 in both orientations decode as monomial ∘ fourier(r axes) ∘ monomial with physical rank equal to the emitted rank; 391 rank-1 climbs have pure signed-permutation wrappers, the rest carry quadratic phases. Zero children fail the decode, and the core is a tensor product "r fresh axes × 9−r spectators" in every case.

Consequence for the tensor-frame-child question: every paid child of #4470 is already I ⊗ H^{⊗r} with free monomial wrappers, i.e. the single rank-r·f reading that this package shows keeps the scalar sign. There is no further operator structure (Kronecker, Hadamard block, circulant) that could price a child below its rank: the structure class coincides with the whole child class, and by PR679's support argument a monomial wrapper cannot reduce the dense H^{⊗r} to a smaller-support child, while any non-monomial Clifford wrapper in this word is itself a paid Hadamard unit. So a "structured child cheaper than rank" needs a changed tariff for all children (a different σ), not a changed hypothesis for some.

Script and receipts: classify_climbs.py / classify_result.json (≈45 min, standard library plus the clone's Clifford code). Happy to attach them if useful.