Finite completion with separate tunnel cells

An exact finite partition may use a different tunnel on each supported cell. This makes finite positive sums of projection traces the right certificates for completing a residual while retaining the selected old blocks. We prove that criterion, then relate the central capacity cost to the ordered overlap of finite-stage representatives. These are two sufficient construction routes; the unrestricted amenability input remains to be proved.

The original finite full-partition statement

Popa's Theorem 4.4.1(1), printed p.222, concerns finite-index inclusions of II₁ factors. The hypothesis is that is an amenable inclusion; part (1) additionally assumes . For every and finite , it asserts finite-dimensional unital , , with

and a finite set , an actual tunnel for each , and projections

The source permits different continuations and different finite lengths. Orthogonality follows from the projection sum: , a sum of positive operators, so . No separability, finite depth, extremality, ergodic core or unique norm trace is imposed in part (1). Those hypotheses belong to other statements and are not inserted here. Taking adjoints of (FP.0) gives .

The source's proof refers to the preceding maximality argument and reduced-algebra heredity. That preceding ergodic-core amalgamation, printed pp.220–221, starts with supports in tail factors. The whole-stage supports of 76.2 instead lie in finite smaller relative commutants. Replacing one membership by the other is not a justified replay of that amalgamation.

Exact human source: Sorin Popa, Classification of amenable subfactors of type II, Acta Mathematica 172 (1994), 163–255, DOI, 4.1.1–4.1.2, printed pp.209–210, and 4.4, printed pp.220–222.

What the existing providers actually supply

The immediate auxiliary inputs are 8.1, inherited trace weights, 53.1, finite-stage expectations, and 57.3, finite tunnel alignment. The existing finite central extraction theorem 85.7 does not itself supply the full partition. The present proofs use finite-factor projection prescription and comparison with their existing programme scope; they claim no blanket transitive prerequisite closure.

The positive trace monoid allows several residual cells

Fix one actual ordinary tunnel of the proper finite-index inclusion and put

Let be the ambient trace of a minimal projection, so . Write exactly as in 78.2, and define the finite positive trace monoid

The empty sum is zero. In the second formula the ranks have no unit-capacity bound. This is a scalar certificate monoid, not a projection-rank set at one common stage.

Lemma FP.1 — exact finite decomposition of a positive certificate. The two formulas in (FP.2) agree. They are independent of the fixed tunnel. For , a finite list of certificates , all at one finite canonical length, can be produced with .

Proof. Move the finitely many projection certificates of a monoid sum to one common level using the actual inclusions. Their rank vectors add to a nonnegative integer vector, giving the second formula. Conversely fix at level , and put

For each coordinate, successive removal of units shows that . Every has integer ranks between zero and , hence gives a projection certificate , by 78.1. Their sum is . If , use the empty list. If , positivity also gives ; omit zero vectors. Finite alignment 57.3 preserves all , hence the monoid.

The canonical projections in this certificate list need not be orthogonal. The actual residual cells will be placed separately. Imposing orthogonality on these canonical representatives would reintroduce an unnecessary common-stage capacity requirement.

Theorem FP.2 — exact multi-cell residual completion. Suppose a finite square from 76.4 at has physical orthogonal supports , residual , and

The following are equivalent:

1. The old selected blocks can be retained exactly and the residual split into finitely many new physical cells, each having an actual whole-tunnel origin as in (FP.1).
2. .

When they hold, the enlarged retains , satisfies both expectation orders, contains in its respective rows, and preserves or improves every approximation estimate to the original physical targets. The old and new cells may use separate continuations. No amenability, depth or trace-uniqueness hypothesis is needed beyond existence of the square being completed.

Proof of necessity. A finite new residual family has . Each belongs to the smaller algebra of some actual finite tunnel, so 78.1 gives . Summing gives . This uses neither a common continuation nor joint canonical orthogonality.

Proof of sufficiency. Use FP.1 to write as a finite sum of positive finite projection traces. In the II₁ factor , choose successive orthogonal projections of these traces. At every step the available trace is the sum of the remaining 's, so finite-factor projection prescription applies. The final complement in has trace zero, hence is zero by faithfulness. For each , exact placement 78.2 gives an actual finite tunnel, with

This construction conjugates its canonical tunnel by a unitary in . Such a unitary fixes pointwise, while every actual contains . Since every commutes with , . Therefore

are finite unital direct sums whose identities form a finite partition of one. For , , and . This proves , , and retention of .

Finite-stage expectation gives . One can obtain this also by taking adjoints of 53.1's restriction . Equivariance under -unitaries and bimodularity over give . The same holds on each old block, so . Pairing with gives , hence

The last equality follows by taking Hilbert-space adjoints. Finally gives . If , no new cell is needed. For , the unit certificate already supplies the needed residual block. No physical old block or target is conjugated.

Thus failure of the single-cell criterion does not by itself obstruct the printed endpoint. The exact unchanged-family obstruction is . It concerns finitely many new residual cells; no countable residual family is being counted as finite.

A one-sided finite-order reduction

Place the canonical representatives of the old support traces in , with ranks . Define the signed residual vector

Let be the actual multiplicity matrix to level , and write . Let , with the inherited trace denoted by . All norm tracial states of , including nonfaithful ones, enter the following test.

Lemma FP.3 — the exact trace-fiber test and a sufficient order test. The following are equivalent:

In particular, it is sufficient that, at some fixed finite level, a trace-zero kernel correction makes every compatible norm trace strictly positive on . For , the sufficient condition is simply for every . No upper bound is needed.

Proof. If a positive integer vector at any finite level represents , move it and to a common later level and take . Trace compatibility gives . Conversely the vector is exactly a positive certificate in FP.2. This proves the equivalence.

For the sufficient test apply the minimum half of 79.2, the finite order test. At the fixed starting level, for ,

The trace space is compact by 79.1 and evaluation is continuous. Strict positivity at every trace therefore gives a strictly positive attained minimum. At a sufficiently late finite stage all the promoted coordinates are nonnegative integers. Their ambient trace is still ; apply FP.1 and FP.2. Unlike a single unit-capacity projection certificate, multiple cells do not require these coordinates to be at most . For zero residual use the empty certificate directly; strict positivity is a sufficient condition, not a characterization at that endpoint or for positive vectors vanishing under some norm traces.

The trace-zero correction can be nonzero even if the originally chosen vector fails order tests. Conversely positive ambient trace alone does not supply a kernel correction or norm-trace positivity. This is the precise arithmetic step still to be derived from actual amenable-inclusion data if one pursues an unchanged-family completion.

Canonical overlap measures the fixed-family cut route

The following independent reduction concerns the fixed representatives and extensions in 80.11–80.12. It is useful when FP.9 is not available; it is not a necessary condition for every multi-cell completion.

Fix canonical projections , . At stage , let be their ranks, , and define as in 80.2. Set

The sum is ordered. All norms and weights are inherited from the actual finite core. The minimum exists since the finite product of finite-dimensional unitary groups is compact and the objective is continuous.

Lemma FP.4 — overlap/capacity equivalence with explicit constants. At every finite stage,

Both quantities decrease under passage to later stages. Consequently, with ,

Proof of the lower bound. In a block of size , put , with total rank sum . Each product has nonnegative trace, and

The inequality is ordinary matrix Cauchy–Schwarz. If , its right side is . If , nonnegativity bounds it below by zero. Multiply by the actual minimal-projection weight and sum over blocks. Since for projections, every rotated family has ordered energy at least . Taking the minimum proves the first inequality.

Proof of the upper bound. The attaining cuts of 80.3 choose with total removed trace , and orthogonal representatives . With the same unitary, write

For each , orthogonality gives . Expanding the ordered overlap once, using , gives

All inequalities are finite positive trace pairings; no commutativity between different tails is assumed. This constructs an actual family satisfying the asserted upper bound. Unitaries available at level remain available at later levels, and inherited trace preserves their energy, so decreases. Monotonicity and the limit of were proved in 80.3 without stationarity. Taking limits proves FP.13; faithfulness of the central trace identifies zero positive-part trace with the central inequality.

Corollary FP.5 — a concrete finite selection certificate. Let a square in FP.4 approximate a finite with error less than , and let . Align its old cells individually by 57.3 and extend them as in 80.11. If some finite canonical stage permits rotations with ordered overlap

then an exact full finite whole-tunnel partition approximating the same targets within follows.

Proof. FP.14 gives . The physical cuts and residual certificate in 80.4 give a finite full square and error at most

The old targets remain fixed. Canonical rotations do not introduce a new physical conjugation error: at the chosen stage normalizes both and . Replacing by and its alignment by therefore gives the same physical support and same full supported pair at that stage. FP.12 shows equivalently that vanishing fixed-family central cost is exactly the possibility of making these canonical representatives almost orthogonal by such finite-stage rotations. The rotations need not preserve an arbitrary earlier ordinary prefix, and that is not part of this corollary.

A finite diagnostic that separates the two routes

Take the abstract finite algebra , with minimal weights . Its projection trace set is

Three disjoint physical projections of trace in an ambient II₁ factor have residual trace , while . The residual can be split into two cells of trace ; each is unitarily conjugate in the ambient factor to the first minimal projection. The corresponding canonical triple consists of three copies of that minimal projection. It has

Thus a positive fixed-family capacity cost need not forbid unchanged multi-cell scalar completion. This is an exact finite-algebra illustration of FP.1, FP.2 and the sharp constant in FP.12, not an asserted Jones relative-commutant system or a counterexample to the source theorem. The new theorems above apply to actual Jones data only when their stated actual finite-stage inputs are present.

Reproducible checks used Python's standard library, exact fractions and a fixed seed. They verified the eight traces, the two-cell sum, the sharp ordered factor , and 1680 integer cut/rotation constructions for , . The checks supplement the proofs; they do not infer a Jones realization.

Reproduce the exact fraction and 1680 cut checks with the editable figure source.

Figure FP.1. Route A uses the positive trace-fiber certificate (FP.9), decomposes its ranks by (FP.3), and places every residual cell through (FP.5)–(FP.7). Route B uses the ordered overlap/capacity bound (FP.12) and the strict tolerance (FP.16), then the physical-cut proof 80.4. Both produce exact finite support sum one and both expectation orders. The orange panel records the unresolved input from unrestricted amenability. Box positions, widths and colors are schematic; they measure no trace, rank or geometry. Editable figure and exact-check source. Human-source endpoint: Sorin Popa, Theorem 4.4.1(1), printed p.222; all sufficient-route proofs are supplied here.

Exact remaining obligation

FP.2 improves the current residual route in an essential way: the printed endpoint requires a finite positive sum of finite traces, not necessarily a single finite projection of the residual trace. FP.9 is the exact finite-support realization obstruction for a completion that keeps the chosen old blocks unchanged. FP.12–FP.16 give a different quantitative sufficient route that permits small cuts.

The original unrestricted theorem would follow from a construction, for each target family and tolerance, of a 76.4-type selected family whose residual lies in , or of a family satisfying the small-cut certificate, or from another complete argument producing FP.1 directly. The arguments here do not prove that arbitrary amenability supplies any of these alternatives. A negative coordinate of one signed representative, a positive central cost for one fixed family, or an abstract trace diagnostic is not an actual counterexample to Popa 4.4.1(1). No change in its original hypotheses or finite full-support conclusion is made.

Common ordinary stage, retained higher prefix, nested approximations, generating tunnels, the second central local form, unrestricted common-support BF and represented/opposite models remain separate original mathematical obligations. The above-four operator projection expectation identity is also a separate input; no implication from general amenability to that identity is used.

Exercises with complete solutions

Exercise FP.1 — one residual cell or finitely many? In the abstract algebra , give the three minimal projections the inherited weights . List its projection trace set. In an ambient II₁ factor choose three disjoint physical supports of trace . Compute the remaining trace, prove that it has no single projection certificate in , and produce a finite multi-cell certificate. Compute the signed residual vector and an ambient-trace-zero kernel correction giving a nonnegative vector. Explain precisely what would still be required to apply the construction to an actual Jones tunnel.

Solution. The identity has trace . A projection in has a rank vector in , so its eight traces are

The physical support sum has trace , leaving . This is absent from the eight-element set: a single bounded rank vector cannot represent it. Nevertheless . The positive certificate vector splits into by (FP.3). These two canonical representatives coincide; canonical orthogonality is not required. In the ambient factor split the physical into two orthogonal projections of trace . Their sum is by faithfulness. Each is unitarily conjugate in that factor to the first minimal projection.

The three old canonical supports have rank sum , while the identity ranks are . Thus and . Put . Then

The first coordinate of exceeds the unit capacity one. Its two bounded pieces supply the finite multi-cell certificate; demanding would lose that route. The second and third characters of evaluate at zero, so the strict all-trace positivity test is not necessary for this exact positive certificate. The first character evaluates it at two, explaining why an upper all-trace bound below one is not part of the multi-cell criterion.

These are finite-algebra and factor-comparison calculations. They do not identify with an actual smaller Jones relative commutant, prove amenability of an inclusion, or produce its full larger row. To invoke Theorem FP.2 one must supply the actual finite whole-tunnel pairs and their inherited weights, the old selected square, and the actual finite-alignment/exact-placement inputs. With those hypotheses, each new physical cell can use its own actual continuation and the theorem gives both expectation identities and preserves the old target errors. No Jones realization or counterexample is inferred from this numerical example.

Exercise FP.2 — check the two sufficient certificates. Let an actual selected square and its canonical finite-stage data have all the hypotheses of FP.2–FP.5.

(a) At some level suppose is an integer vector with , , and for every compatible norm tracial state. Explain how the one-sided order test yields finitely many residual cells. Must every be below one?

(b) Independently, suppose a finite stage has , , and a proposed rotated family has ordered overlap . Suppose the target norms are at most one and the original target errors are strictly below . Verify the overlap/cut inequalities and the strict certificate for final tolerance . Give an explicit final error bound and describe which physical supports may change.

(c) Explain why neither part proves that general amenability supplies its assumed certificate.

Solution. For (a), the minimum limit (FP.10) is at least . Choose a finite later level with . Every coordinate of is then positive; it remains an integer vector and its actual ambient trace remains . If some coordinates exceed their block sizes, apply the explicit finite decomposition (FP.3). Each piece has bounded valid ranks, and the sum of its traces is exactly . Theorem FP.2 places the corresponding finitely many orthogonal physical residual cells separately. No upper all-trace bound is required: the certificate is a finite sum, not a single unit-capacity projection. Compactness and the stated uniform positivity justify the finite choice; positive ambient trace alone would not justify it.

For (b), the minimum is at most the proposed . The lower bound applies to every proposed rotated family, while the upper bound constructs a possibly different family. The supplied numbers are consistent with

The last inequality here is strict, illustrating why the proof says that the constructed family satisfies the upper bound rather than always attaining equality. With , the required threshold is

Indeed , and , so the strict threshold holds. By , the extra target error is at most

Together with the original strict error below , the final error is strictly below . The sharper bound using the given capacity is , but the overlap certificate already suffices. Optimal physical cuts remove total trace from the old supports. Their remainder is enlarged by that exact amount, and the finite canonical complement supplies its trace certificate. A cut old support can change or disappear; zero cells are omitted. Every resulting cell has a full actual whole-tunnel pair and their sum is exactly one. Both expectation orders hold. The targets remain fixed; no earlier ordinary-prefix retention is asserted.

For (c), part (a) assumes a particular actual kernel correction and positivity on every compatible norm trace. Part (b) assumes a particular actual finite selection with verified small overlap and capacity. The proofs convert those data into finite exact partitions; they do not derive either dataset from unrestricted amenability. Supplying such data for every target family and tolerance, or constructing the original partition by another complete argument, is the residual input of (FP.0)–(FP.1).

Original programme proof text: GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026; CC0 1.0. Human mathematical context and exact source credit are retained above.