[zxv]

Conditional candidate with a complete local replay; awaiting external mathematical review.

The winning uniform templates are finite and specified explicitly by algorithm, but at astronomically large ranks: 2^276540184361 for complex and 2^6552928412 for BIT before covering. Their instruction lists and constants have not been materialized. This is an asymptotic existence result, not a practical algorithm.

Result

$$T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=0.0023.$$

Reference point	κ	Gain from this PR
#1094 · selected frontier	0.001672660197	+37.51%
#4181 · leading conditional claim	0.001730980256 − 10⁻⁴⁶ − 10⁻⁷⁰	+32.87%
This PR	0.0023

#1094 is the maintainer-reviewed selected result, and #4181 is the highest conditional claim we located in public PRs. The increase over #4181 is 5.690 × 10⁻⁴ in the exponent saving. The percentages compare those endpoints, not measured running times.

Recent conditional gains have been 10⁻⁹ to 10⁻⁷ per step, obtained by refilling the same staged words. #4091 proved the pooled ceiling for those words, and #4181 attained it, noting that any further gain needs a changed word or inventory. This PR changes what the recursion counts as useful output. Helpers that finish holding a completed transform of their own arbitrary input keep that work instead of undoing it, so one local unit produces jobs of several ranks. A finite integer compiler then turns that mixed rule back into an ordinary uniform supplier, so the inherited outer theorem can use it.

The key idea: count the work helpers already do

In the uniform-supplier interface every local unit returns its helpers to the identity. Each helper's arbitrary input is borrowed, transformed in passing and then restored. Often, though, a helper ends a controller holding a known, completed transform of its own arbitrary input. Restoring that input pays twice for work that could be delivered as output.

We keep these productive jobs at whatever rank they happen to have and price the controller by its complete mixed balance. With W_k independent productive outputs of rank k and p_r paid recursive calls of rank r, the mixed saving C solves

$$\sum_r p_r\, r^{1-C}=\sum_k W_k\, k^{1-C}.$$

For example, a helper that has completed a width-8 transform of its own input earns nothing under uniform accounting. It must be restored, or forced into a full rank-57 output, before it counts. Under mixed accounting it is simply a rank-8 productive job and contributes 8^{1−C} to the right-hand side. The p3 invoice below books 1,570 such helper jobs of ranks 6–9 beside its 672 main outputs.

That change of accounting changed which controllers were worth building. A pK controller is a path of K call pairs, 2K + 1 local calls in all. An idle window is a stretch of stages in which a bank of 224 fields sits untouched by the main calls, so helper work can be scheduled there. Even the first complete p2 cover beats #4181's supplier endpoint (0.0017310):

Controller (complex)	What changed	Mixed saving C
p2, complete identity cover	Helpers priced at their own ranks	0.0017730
p2, initial-prefix deletion	Early idle windows removed literally	0.0020133
p3, seven calls	Integral boundary factorization; adapters replace idle paths	0.0023051

The p3 controller keeps four idle windows: two of rank 16 on stages 0/1 and 5/6, and two of rank 8 on stages 2 and 4. An exact, integral factorization of its boundary operator removes two more. Exact input/output scalar adapters take the place of those whole idle paths, which did no useful work, and that is what yields rank-25 and rank-57 main outputs. The emitted word records every normalizer, absolute phase, source complement, owned center and helper endpoint.

How the controllers were found. Candidates were discovered with floating-point LP column generation. Each reported construction was then reconstructed as exact rational patterns, and its inventory, chronology, target mass and paid ranks were checked again without the optimizer. Larger menus did not beat p3: paired controllers through p6, 64 p3 single-field masks and 123,904 commuting stage allocations. These are finite search results, not optimality claims.

A favorable balance alone proves nothing. Productive jobs have different ranks, exist at different times and live in different physical frames. The remaining work converts this local tariff into a κ.

From a mixed tariff to κ

1. Compile mixed jobs into one uniform supplier

The inherited outer theorem needs a uniform supplier, in which every output has the same rank. The obvious fix is to rotate the mixed roles so that each array takes every role once, and it fails. Averaging replaces Σ W_k k^τ by a smaller concave expression, and the mixed saving disappears. The compiler instead keeps the hierarchy of sources, complements and exits.

It builds on the mixed-target lockstep flow of @jacobalansussman's Theorem 1G (#4268). At a fixed top rank N, a Pascal-class bijection matches each paid round's jobs to target roles of a child class at exactly equal integer ranks. Missing demands become top-level sources of rank s, and the deepest classes emit strictly smaller exit jobs of rank e. As in Theorem 1G, each source array splits as s + (N − s), and its complementary N − s axes are completed recursively. Here the already-compiled smaller uniform program pays for them, so every array ends with the full rank-N transform. With L(n) the paid rank-one leaves per productive array, w_s normalized source weights and x_e exit counts normalized by the number of source arrays,

$$L(n)=\sum_s w_s\,L(n-s)+\sum_e x_e\,L(e),\qquad \sum_s w_s=1 .$$

At τ = 1 − C the source and exit potentials are equal. Strict child/target separation (r_max < k_min) makes exit ranks sublinear in n, so the exits become negligible at any slightly larger exponent z₀. Concavity of the complements then gives L(n) ≤ A n^{z₀}. Some explicit finite rank therefore satisfies

$$P_N < B_N\,N^{z}\qquad (z_0 < z < 1),$$

which for rational z = a/b is the integer test P_N^b < B_N^b N^a. One integer multiplier clears every batch denominator, so no fractional piece of a physical array is ever routed. Copied inputs, owned blank storage and the one center reused chronologically inside a unit receive no productive credit. The full derivation is in FINITE-PRECOMPILATION.md §5.

2. Delay the BIT cover until the whole program is compiled

Under the GL cover, a BIT projector P becomes gPg⁻¹. Converting each one immediately to a weighted canonical query creates incompatible role-specific weights, which is where the original interface failed. The fix is to compile the entire finite BIT circuit first and introduce one ambient cover with one common external weight afterward. The enabling algebra is the covariance of complete projector jobs. With C_N the reversal,

$$D_P=\begin{pmatrix}I-P & PC_N\\ C_NP & I-C_NPC_N\end{pmatrix}, \qquad Q_S\,D_Q\,Q_S^{-1}=D_{SQS^{-1}}, \qquad Q_S=\mathrm{diag}(S,\,C_NSC_N).$$

Routing a whole cover fiber by g ↦ gS therefore matches equal-rank completed jobs while the external weight stays independent of the fresh cover coordinate. Only the final rank-one leaves become weighted ordinary children.

The atom exponent also had to change. The retained stopped recurrence has a linear adapter term, and with the old atom exponent 1/1000 that term caps the ordinary BIT saving at 0.001, whatever the supplier achieves. Raising it to 0.003 removes the cap. The compiled program reaches 0.002760, paying the generic cover fallback leaves 0.002750, and the conversion gives

$$(1-0.003)(0.002750)+0.003\,(0.0000384599)=0.0027418653797 .$$

We admit BIT conservatively at 0.0027, so it no longer caps the complex supplier. The cover prime satisfies q > 25,600,000 N^5, and every bad class pays its full fallback bill.

3. Re-derive precision and storage for the new template

A compiled word this large cannot inherit the old supplier's small constants. FINITE-TEMPLATE-CLOSURE.md derives new ones from the template's own finite counts. The semantic precision guard stays linear in depth, C0·d, because children return completed values on one fine grid. Their temporaries are never simultaneously unfinished parent operands. The simultaneous reserve becomes q0(h,p) ≤ C(log₂h + h^0.003 (log₂p)^3 + 1), with reserved bits deferred to later shrinking phases.

4. Pay the complete outer bill

Every retained outer cost has exponent saving strictly above 0.0023:

Outer cost	Saving
Main transform	0.00230197698
Global coordinate routing	0.0023025
Fused Gaussian work	0.00231193094
Group BIT routing	0.002312430925
Boundary matrix work	0.00230297698
Sparse union movement	0.40855297698
Group scans	0.99998
Polynomial products	0.99999
Microbox face corrections	77.00004

The minimum slack is exactly 98849/50000000000 = 0.00000197698. A separate outer auditor recomputes all nine rows and the changed functional inequalities. Boundary matrix work uses Strassen with the exact comparison 7^16 < 2^45.

Complete invoices

The counts below belong to each local p3 mixed unit, and their balance gives the mixed root. BIT figures are before the cover.

Local mixed unit	Complex	BIT
Productive jobs (rank: count)	{6:192, 7:56, 8:93, 9:1229, 25:448, 57:224}	{8:3073, 22:444, 50:222}
Paid calls (rank: count)	{1:17623, 2:5065, 3:1738, 4:714, 5:254}	{1:9261, 2:9394, 3:553, 4:308, 5:714, 6:77, 7:1465}
Independent productive fields	2,242	3,739
Total paid calls	25,394	21,772
Scalar allowance	1,147,304 real primitives	174,908 ADDs (swap = 3 ADDs)
Mixed root C	0.0023050901948…	0.0027698302…

The compiler turns each unit into a uniform supplier with only rank-one paid leaves. Its own counts are finite but enormous, and they are not the unit counts above.

Compiled uniform supplier	Complex	BIT
Explicit uniform rank	2^276540184361	2^6552928412
Uniform saving at that rank	0.002304	0.002760 (before cover)
Saving entering the outer bill	0.002304 (binding)	0.0027 (after cover fallback and conversion)

The exact root enclosures are [0.002305090194876170622919094755, 0.002305090194876170622919094756] (complex) and [0.002769830250, 0.002769830251] (BIT).

Verification

Based on canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244 (priority revision 7). Python standard library only:

python3 -B research/kappa-0023-productive-jobs/verify.py --check-only
python3 -B research/kappa-0023-productive-jobs/verify.py --out /absolute/path/to/new-replay

• A fresh sequential replay outside the checkout passes 23 scientific stages and 59 artifact comparisons. It covers native reconstruction, phases, owners and bills, the finite compiler, the delayed BIT cover, the new constants, the explicit ranks and the independent outer audit. Logs are in research/kappa-0023-productive-jobs/evidence/.
• The review runner fixes a stale-pin ordering defect in the original runner and compares receipts across hosts under declared metadata rules. No mathematical field is waived, and four mutation controls, such as a changed target κ, are rejected.
• The import binds 365 unchanged delivered files plus 16 canonical-main inputs. The critical replay does not rerun every historical negative search.

Where to start reading

• OUTER-TRANSFER-AUDIT.md: proof entry point and the complete outer bill.
• FINITE-PRECOMPILATION.md: the finite integer compiler (step 1).
• DELAYED-COVER-BRIDGE.md: the delayed BIT cover and atom conversion (step 2).
• EXPLICIT-TEMPLATE-BOUNDS.md: the explicit winning ranks.
• FINITE-TEMPLATE-CLOSURE.md: new precision and storage constants (step 3).
• VALIDATION.md: replay runner, comparison rules and import details.

Scope

• Conditional. The claim keeps the inherited analytic decomposition, exact recovery, ordered movement, fixed-tape execution and prime-padding premises.
• Not yet reviewed. The new universal arguments are written proofs backed by finite checks: the compiler, native adapters, common-weight BIT interface and support-local outer transfer. They are neither formally verified nor externally accepted.
• Not materialized. The winning templates' full words and constants have not been generated. The materialized rank-57 complex and rank-50 BIT examples check completion, batching and ownership, but neither reaches the limiting saving.
• The selected frontier and shared verification code are unchanged.

Outstanding research directions

The most direct route past 0.0023 is the H branch. The identity H = I + BC with CB = −2I gives H² = I, which suggests cheap reversible completions over the legal ring. On the #589 BIT word, a minimum three-row invertible completion cuts the local correction to three rank-one calls, and phase-correct complex owner maps pass exact checks. What it lacks is a cheap way to close the controller's endpoints. Three interval corrections remain, costing 672 rank-one requests in complex arithmetic and 666 in BIT, so even the cheapest BIT version falls 21 short. Generic cancellation, register sharing and same-cut compression have been ruled out. A geometry-aware decoder has not been. Details are in LITERATURE-AND-PROBES.md and the architecture overview.

Open next questions

• Review the chain as one argument. Start with the integer source/exit matching and all-column native adapters, then the common-weight BIT interface and the support-local outer transfer.
• Shrink the templates. Tighten the paid-leaf superpotential or find smaller explicit successful ranks without giving up integer batching or complete constants.
• Build a geometry-aware H decoder. Start from the actual query subspaces, the phase-correct complex owners and the three-row BIT completion. A survivor must put preparation, routes, reads, restoration and all endpoints in one invoice.
• Improve the complex tariff. The complex mixed root now binds, so more BIT saving alone cannot advance this point. A changed native word or productive-job contract must improve its complete moment.

None of these questions is used to justify the 0.0023 claim.

Literature

These sources shaped the construction above, grouped by the part they informed. Each note says what the source suggested and what we actually used; the notes describe our use, not claims by the cited authors. Sources behind unsuccessful branches are included because they explain the turns the search took.

Productive work and mixed accounting

1. Jacob Alan Sussman (2026). “Theorem 1G: mixed-target recursion.” PR #4268 · pinned proof

   The direct algorithmic predecessor. Its integer Pascal-class lockstep flow matches unequal target jobs with smaller paid demands under an exact moment balance, splitting each source as s + (n − s) and paying the remainder recursively. It is the core of our compiler. Our contribution is realizing that mechanism as a finite integer compiler with native adapters, an explicit winning rank and the outer transfer. #4268 claims a DFT improvement and no κ; its Lean results do not verify this compiler or the multiplication claim.

2. Arnold Schönhage (1981). “Partial and Total Matrix Multiplication.” SIAM Journal on Computing 10(3), 434–455. doi:10.1137/0210032

   The classic precedent for assembling useful partial computations into a larger complete one, which is the shape of the productive-job idea. The tensor theorem served only as motivation; the native source/exit schedule and paid complements carry the actual bridge.

3. Péter Vrana (2024). “Partial and weighted matrix multiplication.” arXiv:2408.15728. arXiv

   Reinforced that unequal, weighted useful work deserves its own accounting. It encouraged us to keep mixed ranks rather than round them to a uniform job. Its support-rank theorem is not applied to transform programs.

4. Robert Robere and Jeroen Zuiddam (2021). “Amortized Circuit Complexity, Formal Complexity Measures, and Catalytic Algorithms.” ECCC TR21-035. ECCC

   Motivated one production-and-consumption ledger with an explicit price for every resource. Its repeated-output setting reuses a single input, so it grants no independent-output credit here.

Building the controller over the legal coefficient ring

5. J. J. Koliha (2001). “Block diagonalization.” Mathematica Bohemica 126(1), 237–246. EMIS

   Resolutions of the identity by idempotents pointed us at the operator coefficients of the whole controller rather than its individual calls. That view informed the exact coefficient factorizations tested for p3, including the integral boundary factorization that removes two idle windows.

6. Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel (2023). “Dyadic linear programming and extensions.” arXiv:2309.04601. arXiv

   Made the denominator restriction a first-class constraint. Power-of-two balancing and shear factorizations then avoided an unavailable division by 13. Its general LP algorithm was not implemented.

7. Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel (2021; rev. 2022). “Total dual dyadicness and dyadic generating sets.” arXiv:2111.05749. arXiv

   Separated a rational feasibility witness from a realization over the allowed dyadic ring. This shaped the search's rejection gates.

8. Ingrid Daubechies and Wim Sweldens (1998). “Factoring Wavelet Transforms into Lifting Steps.” Journal of Fourier Analysis and Applications 4(3), 247–269. author manuscript

   Lifting suggested reversible triangular factors for the six-bank transfer in place of generic scalar elimination. Frame contacts, coefficients, preparation and restoration were still checked natively.

9. Christopher M. Brislawn (2024). “The Causal Complementation Algorithm for Lifting Factorization of Perfect Reconstruction Multirate Filter Banks.” arXiv:2408.07970. arXiv

   Suggested factors constrained by when boundary data become available, not unconstrained matrix elimination. We used exact legal shears and verified the actual contact times.

10. Christopher M. Brislawn (2019; final rev. 2024). “Factoring Perfect Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine Approach.” Journal of Computational Algebra 12, 100024. arXiv:1902.09040

    Sharpened the same causal-divisibility question in an earlier handoff. It never licensed an unavailable contact or an uncharged inverse.

11. Hyung-Ju Park (1995). A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems. PhD thesis, UC Berkeley, UCB/ERL M95/39. report

    The factor I − P + zP (§7.4) guided exact projector/delay representations, checked with rational oblique projectors and their full coefficient identities. No orthogonal rotations were imported.

12. Zinnur Doğanata, P. P. Vaidyanathan and Truong Q. Nguyen (1988). “General synthesis procedures for FIR lossless transfer matrices, for perfect-reconstruction multirate filter bank applications.” IEEE Trans. Acoustics, Speech, and Signal Processing 36(10). doi:10.1109/29.7544 · Caltech copy

    Inspired a transfer-matrix view of the controller with an explicit boundary basis. Its lossless lattices do not license irrational rotations or free frame changes.

13. John Urschel (2023). “Representing the Special Linear Group with Block Unitriangular Matrices.” arXiv:2305.00485. arXiv

    Motivated short block-shear controllers in the earlier redesign. Its field-level factor counts were leads only; dyadic coefficients, contacts and the recursive bill were checked separately.

Arbitrary dirty state, restoration and causal flow

14. Jean-Guillaume Dumas and Bruno Grenet (2024). “In-place accumulation of fast multiplication formulae.” ISSAC 2024; arXiv:2307.12712. arXiv

    Suggested borrowing completed output arrays as dirty scratch and restoring them exactly. Every native contact and reverse operation stayed priced.

15. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff and Florian Speelman (2014). “Computing with a full memory: Catalytic space.” ECCC TR14-053. ECCC

    Supplied the discipline for borrowed scratch: it starts arbitrary and must return exactly. Productive jobs meet a different contract, finishing with the declared transform of their arbitrary input. Keeping the two requirements apart is what lets the construction credit productive work. It also motivated all-column dirty-bank tests rather than zero-initialized examples.

16. Antoine Vinciguerra (2026). “An Operator Approach to Register Programs for Catalytic Computing.” arXiv:2609.18692. arXiv

    Motivated stating restoration as a whole-operator identity and checking whole arbitrary register columns. Its access counts were not imported. Its polynomial-degree and passive-output assumptions differ from our linear jobs, and its matrix lifting and coefficient divisions need not stay within the permitted dyadic coefficients.

17. Marten Folkertsma, Ian Mertz, Sergii Strelchuk and Sathyawageeswar Subramanian (2026). “Fast Cliffords When Your Quantum Memory Is Full.” arXiv:2609.40144. arXiv

    Prompted a dirty-workspace cycling probe, which was rejected. Its restored-workspace premise fails for our accumulating dirty logger.

18. Joseph Connelly and Kenneth Zeger (2016). “Linear Network Coding over Rings, Part II: Vector Codes and Non-Commutative Alphabets.” arXiv:1608.01737. arXiv

    Kept coding over the actual coefficient structure separate from a convenient field-rank count. Acquisition, transport, scatter and erasure therefore stay in the causal interface and the invoice.

19. Mark Braverman and Zhongtian He (2026). “The Multiple Unicast Conjecture is False.” arXiv:2610.10108. arXiv

    Motivated timed, operator-valued carriers with causal local processing. Its characteristic-three code and throughput factors were not transferred.

20. Vadim Kostrykin and Robert Schrader (2001). “The Generalized Star Product and the Factorization of Scattering Matrices on Graphs.” Journal of Mathematical Physics 42, 1563–1598. arXiv:math-ph/0008022

    Suggested composing through retained interface variables in earlier work. It remained an analogy; feedback inverses and recursive costs could not be omitted.

BIT synthesis, state bounds and the H branch

21. Ketan N. Patel, Igor L. Markov and John P. Hayes (2008). “Efficient Synthesis of Linear Reversible Circuits.” Quantum Information and Computation 8(3–4), 282–294. arXiv:quant-ph/0302002

    Motivated explicit binary ADD/SWAP synthesis with shared linear forms. It was used for exact F₂ synthesis and never for characteristic-zero complex arithmetic. The archived BIT note cites the preprint under another title.

22. Andrea Visconti, Chiara V. Schiavo and René Peralta (2018). “Improved upper bounds for the expected circuit complexity of dense systems of linear equations over GF(2).” Information Processing Letters 137. NIST

    Suggested keeping shared dense sums alive until all consumers finish. The resulting deferred reconstruction did not remove the physical frame or the dirty baseline.

23. Alex Parent, Martin Roetteler and Krysta M. Svore (2015). “Reversible circuit compilation with space constraints.” arXiv:1510.00377. arXiv

    Motivated joint lifetime and cleanup accounting. The helper-pair census then rejected all five geometrically eligible coalescings; lifetime overlap alone was not enough for native reuse.

24. Dmitry Chistikov, Szabolcs Iván, Anna Lubiw and Jeffrey Shallit (2017). “Fractional Coverings, Greedy Coverings, and Rectifier Networks.” STACS 2017, LIPIcs 66, 23. doi:10.4230/LIPIcs.STACS.2017.23

    Motivated shared consumer maps and source-neighborhood covers. Rectifier size does not pay for restoration or native transport.

25. G. David Forney Jr. (2009; rev. 2010). “Minimal realizations of linear systems: The ‘shortest basis’ approach.” arXiv:0910.4336. arXiv

    Framed the question as the state needed across an actual time cut. The nine-center rank checks then ruled out an eight-signal carrier at that cut, leaving other cuts open.

26. Navin Kashyap (2007). “On Minimal Tree Realizations of Linear Codes.” arXiv:0711.1383. arXiv

    State requirements depend on the decomposition and consumer interface. This scoped the same fixed-cut obstruction rather than a universal lower bound.

27. Clément Pernet and Arne Storjohann (2018). “Time and space efficient generators for quasiseparable matrices.” Journal of Symbolic Computation 85, 224–246. arXiv:1701.00396

    Motivated compact exact state across cuts in earlier observer probes, whose measured gaps exposed insufficient state in the tested contract.

28. Itziar Baragaña and Alicia Roca (2019). “Fixed rank perturbations of regular matrix pencils.” arXiv:1907.10657. arXiv

    Distinguished prescribed-rank completion from a loose rank bound. Its field-size hypotheses were not imported into F₂; the BIT completion was proved and tested directly.

29. A. Amparan, I. Baragaña, S. Marcaida and A. Roca (2023; rev. 2024). “Row or column completion of polynomial matrices of given degree.” arXiv:2306.06104. arXiv

    Suggested completing only the designated cheap output rows. Checking invertibility instead of requiring an involution produced the three-row BIT completion. Endpoint debt still blocks a higher claim.

30. Christophe Reutenauer (1994). “A Survey on Noncommutative Rational Series.” FPSAC 1994, 393–402. proceedings

    Motivated testing the coefficient space of all boundary words jointly. The resulting span obstruction rejects the universal two-passage model and leaves geometry-aware coding open.

31. Jean Berstel and Christophe Reutenauer (2010). “Minimization.” Ch. 2 of Noncommutative Rational Series with Applications, Cambridge University Press, 29–46. chapter

    Supplied the minimal-realization viewpoint for the same test. The native obstruction has its own direct proof.

Retained outer inputs

32. Volker Strassen (1969). “Gaussian elimination is not optimal.” Numerische Mathematik 13, 354–356. doi:10.1007/BF02165411

    The retained boundary-matrix algorithm. The exact comparison 7^16 < 2^45 permits the conservative exponent 45/16 in the outer ledger.

33. Timothy S. Trudgian (2014). “Updating the error term in the prime number theorem.” arXiv:1401.2689. arXiv

    Theorem 2 is a cited input to the inherited fine-prime padding argument. The local verifier does not re-prove it.

Attribution

PR	Submitter	Used here
#23	@jacklightChen	Semantic precision and bulk resampling
#555	@eumemic	Punctured BIT supplier predecessor
#589	@DreamingOfClouds	Retained BIT word and transport improvements
#1094	@rohanarun	Selected conditional result
#1673	@Vlado-V	Capped phase-cell fusion outer wrapper
#1758	@jeremykothe2	Gauge-free interiors and radix-distributed union schedule
#1786	@chafreaky	Fine source-prime padding
#3945	@Dugongue	Staged544 owner-fill predecessor
#4076	@rohanarun	Three staged helper entries and T360 fill
#4085	@zxv	Receiver sharing on the staged T360 supplier
#4088	@1337isnot1337	Mixed fourth staged entry
#4091	@RossRichardson	Fixed-inventory pooled ceiling
#4181	@rohanarun	Pooled-ceiling fill; comparison point
#4268	@jacobalansussman	Mixed-target recursion (Theorem 1G), basis of the compiler; claims no κ

Earlier geometry, native checker, owner, compiler and circuit contributions are credited in the frozen packages and source notices.

Prepared with substantial AI assistance, including ChatGPT Pro (Astra 6.0), GPT-6.1 Sol and Opus 5.5. Apache-2.0 for new material; original notices and third-party licenses are retained.