[zxv]

Local identities and decoders can lose their savings when the caller pays for observation access, restoration, transport or precision. The ten targets below address those boundaries through outside mathematics and reusable search tools. The detailed assessment expands the literature, predecessor work and stopping conditions.

The baseline is canonical commit 1a3ced6: conditional $\kappa=0.003027288235$, ordinary BIT saving 0.0034363538396, conservatively admitted as 0.0034. The binding outer row is main_transform; the selected record identifies the five retained framework assumptions. This is an exploratory research agenda with diagnostic tooling; no new κ or admitted supplier is established.

Ten proposed results

Proposed result	Opportunity and outside input	Main risk or tradeoff	First useful test and stopping rule
1. Encoded outputs with an internally produced observation	Observability and local index decoding could let consumers use encoded states, avoiding repeated decoding and restoration.	Deferred restoration creates history and source-version obligations. #5427 reports that the required $ZQY$ observation is absent from the frozen words in its grammar; a changed producer or paid interface is necessary.	Extend #5419 at one actual FULL/thin-query cut; preserve independent inputs and close two generations. Stop if an essential read is unavailable or observation plus return costs exhaust the allowance.
2. An exact query–memory–restoration frontier	Reversible compilation and I/O pebbling suggest an optimizer that emits an attaining dirty-state program and a checkable lower bound. Useful even when the outcome is a scoped exclusion.	Saving queries can increase live state, movement, precision or cleanup. General synthesis may be intractable; stationary fanout bounds do not settle changing sources.	Solve two changing-source contacts with bounded scratch, beyond the stationary model and #5425. Compare exact synthesis with cut lower bounds; stop if the instance reduces to an already solved grammar.
3. A physically admissible vector decoder	Fractional Haemers methods permit noncommuting block actions that could escape scalar restrictions and support several geometries.	Transport, copying and restoration can outweigh a block-rank advantage. The fixed paired-plane mask already has an all-field $49d$ floor. Dense embeddings can inflate payload volume quadratically.	Change the mask or legal side information; test exact dimension-two blocks with a complete physical invoice. Stop if an $O(N)$ payload requires an $N^2$ dense child or the optimistic complete bill loses.
4. Paid-cost optimal decoder factorizations	Operator sparsification motivates optimizing supports, pairing ranks and endpoints jointly. Overcomplete factors may be cheaper than rank-minimal factors.	Extra terms can worsen preparation or precision. The product-plane comparison improved pairing cost but failed its tested serial native screen. PSD restrictions can exclude valid indefinite fits.	Reconstruct the 15,232 versus 15,552 pairing comparison, then price a changed native realization. Stop before expanding the geometry search if a favorable complete relaxation already fails.
5. A profitable multitype recursive supplier	Plain, encoded and history-bearing states could form a contracting cycle, certified by an adapter-inclusive $M(C)v\lt v$. The positive-vector theorem already exists; the opportunity is a physical instance.	Weights do not eliminate conversion, return or volume costs. The phase accumulator closes two levels but has a critical optimistic bill before wrappers.	Derive real child-size lists and every adapter for one representation pair; require strict contraction and outer admission. Stop if the types merely rename a schedule or only ordinary tolls improve.
6. Optimal productive BIT helper relations	Generalized Hamming weights, code shortening and matroids suggest systematic relation-space search. BIT progress supports gains beyond the selected admitted .0034 input.	Relations can remove productive directions as well as paid operands. Every lost target needs its debit; characteristic, frames and chronology constrain legal relations.	Calibrate against Ali Romman's nine-relation #5421, then enumerate one bounded legal block. Stop if preparation or lost-target costs consume the gain. Its claimed BIT improvement remains predecessor evidence.
7. Precision composition for approximate suppliers	Stable recursive multiplication analysis could provide sufficient conditions for approximate or alternative-basis suppliers, including dirty state and final exact recovery.	Lower algebraic cost can require more bits and storage. Independent error bounds may be too pessimistic; assumed correlations may be unsound. Matrix-multiplication stability does not automatically transfer to native operators.	Propagate actual magnitudes, correlated errors and rounding through a real producer, consumer and second generation. Stop if precision consumes the saving or exact recovery lacks a proof. A better condition number alone is insufficient.
8. Compact all-width native identity certificates	Noncommutative identity testing suggests a restricted symbolic language and small checker covering phases, spectators and inverses, potentially discharging part of native_all_columns.	Tractable languages may omit needed operations. Free-algebra identities and concrete address operators have different hypotheses; a sound representation/lifting theorem is essential.	Extend one named operation beyond the four-width class or p19 lift. Test exponent, order, phase and spectator mutations. Stop at an unsupported lifting implication; additional finite widths do not prove it.
9. A contract-aware paid-profile superoptimizer	Equality saturation could discover coordinated noncommuting graph changes and emit literal replacements with equivalence certificates.	Local expression costs can reward illegal sharing or omit target, ownership and restoration costs. Search spaces grow quickly; fewer scalar instructions need not improve the recursive invoice.	Extend commuting rewrites and projection cuts on one small paid neighborhood. Require a changed complete signed invoice and proof of equivalence; stop if all candidates preserve that invoice.
10. A uniform fixed-tape mixed-target compiler theorem	Constructive completion, splitting, normalizer generation, ownership and movement bounds could reduce inherited assumptions for every later supplier. I/O complexity helps expose access costs.	A slower foundation project with limited immediate κ upside. RAM lookup or abstract routing can conceal uniform-generation and tape costs; overly general hypotheses may be hard to instantiate.	Extend #4192 through one uncovered completion/splitting pass with metadata, source reads and head returns charged. Stop at a precise unabsorbed cost or uniformity gap. Analytic recovery remains separate.

Portfolio tradeoffs

1, 6 and 4 are the suggested construction starting points: changed complex endpoints, complementary BIT progress and paid-cost geometry. 5 offers a structural alternative when immediate format restoration remains too expensive. These routes have substantial upside and can still fail their complete-cost gates.

2, 8 and 9 offer reusable optimization or verification results with less dependence on a new profitable geometry. Scoped negative outcomes can retain value, although infrastructure alone does not raise κ. 7 should accompany approximate constructions; 10 merits a separate foundation effort with a named lemma and budget. Existing active scopes should be checked before assignment.

The decisive comparison is the whole caller: a query saving can return as cleanup, a complex gain can expose the BIT bottleneck, and a low-rank factor can require an expensive physical realization. The detailed assessment records these dependencies and the evidence that would justify expanding each family.

Nearest overlapping work

• #5419: literature-informed coupled consumers and catalytic junctions. Target 1 extends its missing observer/two-generation obligations; another inverse-only result would duplicate covered work.
• #5421, by Ali Romman: nine owned BIT relations, claiming ordinary saving .0034441226396. Target 6 seeks a general legal relation-selection method; it does not claim this predecessor's improvement.
• #5424 and #5425, by Chris R: the optimistic Global120 allowance and scoped standalone fanout optimum. Unearned outputs remain unearned, and a standalone bound does not determine incremental cost at a global cut.
• #5427: frozen-word observation and paid-response reuse exclusions. Target 1 requires a changed contract or producer; a repeat search within the excluded grammar is not proposed.

The overlap snapshot covers 316 open PRs, 4,707 discussion records and 7,376 local note records. Its coverage excludes arbitrary diffs, all historical PR bodies and unpublished work. The contribution is a cross-family assessment with bounded diagnostics; overlapping submissions retain their stated evidence status and have not been independently admitted here.

Diagnostic evidence

The probe uses the complete selected histograms: 64,260 paid jobs, 10,928 productive fields, and exact rank-mass surplus 752. For candidate saving $C$, the raw moment margin is

$$F(C)=\sum_r n_r^{\mathrm{target}}\,r^{1-C} -\sum_r n_r^{\mathrm{paid}}\,r^{1-C},$$

where $n_r^{\mathrm{target}}$ and $n_r^{\mathrm{paid}}$ are the respective
rank-$r$ multiplicities.

Candidate saving $C$	Approximate raw deficit	Isolated H1 deletions for positive raw margin, all else fixed
.0033	67.256506	68
.0034	91.898559	92
.004	239.525048	240
.006	728.828475	729
.01	1694.706712	1695

These are numerical necessary screens before compiler reserves, physical feasibility and outer admission. Decimal powers use 65 significant digits without outward error bounds. The counts exhibit no legal deletions and do not transfer to changed profiles. The selected 144 paid/target H8 pairs cancel; every new target needs a producer and every lost target its full debit.

A second, exact scalar diagnostic computes

$$\mathrm{rank}\!\begin{bmatrix}S\\H\end{bmatrix} -\mathrm{rank}(S)$$

for available answers $Sz$ and demanded answers $Hz$. It detects missing linear
information, including correlated responses, source-version changes and
differences between $\mathbb{Q}$ and $\mathbb{F}_2$. Native frames, oracle legality, acquisition, volume and restoration remain outside this scalar model.

python3 -B scripts/research_selected_baseline.py
python3 -I -B research/outside-input-research-targets/probe.py --check-source-root .
python3 -O -I -B research/outside-input-research-targets/probe.py --check-source-root .

On CPython 3.14.5, macOS 26.5.1, arm64, normal, optimized and portable runs matched byte-for-byte. The custom-cut control passed; an altered histogram with unchanged source pins rejected under optimized Python. PROFILE.json, RESULT.json and VALIDATION.json retain source bindings, output and checks. The selected metadata baseline passed. Native engines, the full selected replay, Lean, Docker, SDP, heavy censuses and hosted CI were not run; their obligations remain outside these diagnostics.

Credit

The selected foundation retains Chris R's completion and audits (#5413, #5414, #5407); Dillon Amburgey's provider (#5402); Ali Romman's controller (#5398); Cao Yifan's native construction (#4610); David Ganz's completion; Cinna Davis's controllers/helpers; zxv's native/compiler/precision machinery; and Jacob Alan Sussman's mixed-target recursion. The contributor map preserves the cumulative BIT, Clifford, geometry, ownership, analytic and verification lineage.

Literature attribution appears in the linked papers and detailed assessment. New prose and probe code are Apache-2.0, with substantial OpenAI Codex assistance. Proposed applications do not claim invention of the underlying theories or independent human/formal verification.

Open next questions

• Can a changed coupled producer earn its observation and close two generations within the combined allowance?
• Can relation-space optimization improve #5421's witness after preparation and target debits?
• Can the product-plane factorization's pairing advantage survive a changed native implementation?
• Which missing primitive can extend the all-width certificate class with a sound representation theorem?

The table gives the first test and stopping condition for each direction; the detailed assessment expands their blockers and follow-ups.

[Matarener]

Target 9, first gate: noncommuting rewrites of the #4610 word (negative screen, no new κ)

Base 32890bc5d, forward program of kappa-round3-warp-20261022/ACTUAL-WORD.json.gz, frame contract and single-level drop test of #5426.

1. Scale gauge (a scale is a uniform scalar, so it commutes with ADDs/warps after coefficient rescaling): exact (random topological orders replay all 632 endpoint rows), but it unblocks nothing — same 8 centre-read levels as Certify that the fragmentation of the #4610 word is rigid under free scalar reordering (negative result, no new kappa) #5426. Pitfall: dropping scale edges from the Certify that the fragmentation of the #4610 word is rigid under free scalar reordering (negative result, no new kappa) #5426 DAG loses transitive add→read orders; the DAG must be rebuilt with scales invisible.
2. Steinberg commutator b+=c·a; x+=d·b = x+=d·b; x+=cd·a; b+=c·a is the only rule we found that breaks the rigidity: with compensations unpriced, 1903 of 2526 levels become droppable.
3. Priced exactly (compensating ADD as a new contact with version-correct dependencies, common frame or priced inserted frame, cascades allowed): 0 of 1903 improve. 1781 cannot be realized (mostly a new cycle with no Steinberg-eligible edge, or no admissible common frame); 122 are realizable, best bill change exactly 0. Every inserted frame costs ≥ 0.0042 Λ (1+1 split at t = 1 − 0.00302), which equals the gain of a rank-1 merge, and only 2 compensations in the whole run landed on an existing frame.

Scope: greedy choice of the broken conflict, forward orientation, single-level drops, two-term relations — a screen, not a certificate. Consistent with the BIT-side bit589-steinberg-fan-seed06. By this target's stopping rule (every extracted word keeps the same signed invoice) we stop here.

Prepared by Mark (Matarener) with Claude Code assistance.

[Matarener]

Three short scoping notes on "new operations / new word" for the complex side (Mark / @Matarener, with Claude). No new κ, no package; reading and small exact counts on main 32890bc5d.

Context: we looked for a genuinely new operation set that could lower the fragmentation Λ of the selected provider word (#4610 / #5226 Theorem B; #5426 shows Λ is rigid under free reordering). Related to targets 1 and 9 here. A separate audit of #5440 is in #5440 (comment) (the chain needs the full letter R = D·Q, the native call applies D; 4,256 unpriced rank-1 jobs per unit, no break-even).

1. Where Λ sits in #4610 (exact, from the frozen word and the #5426 model).

field class (start dim → final dim)	n	Λ(t₀)	typical climb pattern
helpers 0 → 9	326	14.05 (59%)	(1,1,1,3,1,1,1), (3,3,3)
mains 1 → 9 (start = port line)	112	4.07	(1,1,1,4,1)
mains 0 → 8 (final = port frame)	112	3.53	(2,2,2,1,1), (1,1,5,1)
helpers 1 → 9	41	1.39
other helpers	41	0.82

One intermediate level costs 0.0094 (rank-9 split 1+8) to 0.0186 (4+5), so the 0.0034 gate (Λ ≤ ≈21.25) needs ≈260 fewer levels per call. The intermediate contacts of the H9 helpers are mostly helper–helper at a common mid frame (874 pairs) and helper–M1-main at mid frames or at the main's start line (≈970); i.e. the butterfly structure of the factorization, not scheduling.

2. Contact frames do not carry value. For nested chains without warps, a contribution g → … → f is transformed by G on T_f \ S_g whatever frames the contacts use (partial transforms along a flag compose). So Λ is a function of the add graph, its chronology and helper restoration only; this is consistent with #5426's result that the binding constraint is joint chain geometry plus the centre interface. A "rank-r climb that also exposes read-only intermediate frames" is therefore not a new primitive: a snapshot at frame F_j is a rank-c₁ job, and writes at F_j defer by linearity to a later common frame, which #5426 already enumerates. A from-scratch reduction of Λ means a new factorization with a new helper set and restoration, not a rewrite of #4610's program; a one-port CP-SAT keeps the partners' chains and returns to the #5426 setting.

3. Cross-column controls (#5436 / #5441). The support-partition half of #5436 §3 does not use the columnwise form: Q_f and D_f are still supported on complete coset blocks of the 17f-bit address space, so any single monomial h (cross-column included) with monomial Q- and D-conjugates permutes the join cosets, and every terminal term Q⁻¹ a k b Q⁻¹ with A_ir A_sj ≠ 0 is non-monomial. What does not transfer is the final step (tensor-power independence of the ≤ R·C·s₀ terminal terms). So the only cross-column escape in this interface is cancellation of up to 2,704 such non-monomial terms per owner block into a monomial; this is a search on the real 112-owner / 17-axis interface, and its cost must then include the exit bill (#5415, #5424) and the loss of the columnwise tensor lift used by the all-width certificates. We did not build that search.

Scope: written arguments plus counts on the frozen word; nothing here is a theorem about all native words. Scripts on request.