[P2 analysis] Confluent Potts projector tomography at Q→1: separate singlet, [2], and logarithmic residues

[LightChainr]

Motivation

The latest repository state makes the global-symmetry label itself an observable that should be differentiated, not just a name attached after the fact.

Two recent results sharpen the point:

1. [P2 theory/analysis] Explain the missing x=17/4 spin-4 four-leg primary before attributing local H4 to thermal Q4 #257 / PR analysis: add exact Q=1 spin-4 competitor preflight #260 identifies the exact lower-dimensional four-leg spin-4 primary

V_(2,2): x=17/4, spin=-4,

and the Potts state-space analysis places its integer-shift family in the nontrivial [2] representation rather than the singlet. This gives a natural reason why an ordinary S_Q-invariant bulk anisotropy need not couple linearly to it.

2. [P2 exact/breakthrough] Critical-manifold Q-score tomography: observe Potts representation collisions before taking Q→1 #258 constructs an exact critical-manifold Q score at Q=1, but correctly warns that

measure derivative != derivative of a Q-dependent field.

A generic-Q observable carries explicit Q dependence through its representation projector, normalization and mixing coefficients.

I propose making that missing field derivative explicit using the finite partition-algebra / categorical S_Q projectors themselves.

Conceptual shift

Instead of asking only

which Potts representation does this continuum field belong to?

construct the finite-volume generic-Q projector/invariant tensor

P_lambda(Q)

for the low-leg connectivity space and study both

P_lambda(1)

and its confluent tangent

d_Q P_lambda(Q)|_(Q=1).

At generic Q, the Potts state space is organized by S_Q representations through the partition algebra. At integer/special Q, categorical dimensions can become zero or negative and semisimplicity can fail; therefore ordinary finite-group Hilbert-space intuition is not sufficient. The correct question is whether the Gram/projector data of the analytically continued connectivity tensors develop degeneracies whose finite parts or derivatives realize the logarithmic mixing.

This is not an assertion that every projector has a pole at Q=1. Which low-leg projectors remain regular, become null, or collide should be derived exactly.

Phase A — exact low-leg Potts tensor algebra

Start with the smallest connectivity spaces relevant to the current questions:

2 marked clusters / two-leg sector,
4 marked clusters / four-leg sector.

Using the partition-algebra diagram basis, construct the generic-Q Gram matrix and the invariant-tensor/projector decomposition into the representations that actually occur, in particular

[]   singlet,
[1],
[2],
[11].

Cross-check the decomposition against the twisted-state formulas of Jacobsen--Ribault--Saleur and the invariant-tensor counting in the Potts global-symmetry bootstrap.

Record symbolically/rationally as functions of Q:

projector matrices,
Gram determinants and eigenvalues,
categorical dimensions,
projector traces,
intertwiner / tensor-product coefficients.

Then expand around Q=1.

Phase B — turn #257's selection statement into an explicit lattice projector

For the four-leg spin-4 candidate V_(2,2) use an explicitly [2]-covariant lattice insertion rather than four-arm geometry alone.

The first useful observables are

O_[2]^a(x),
<O_[2]^a(x) O_[2]^b(y)> projected to [],

and neutral three-point tensors allowed by

[2] x [2] x lambda -> [].

This gives a direct positive construction of the charged sector whose one-point singlet matrix element is absent.

Compare the resulting lattice phase/leg convention with the character-weighted V_(r,s) construction in arXiv:2604.05503 / 2510.04701. The target is not merely the x=17/4 power; it is the full [2] invariant-tensor pattern.

Phase C — exact decomposition of a Q derivative

Let a generic-Q projected observable be

O_lambda(Q) = P_lambda(Q) O_bare(Q).

Along the square-bond critical manifold, #258 supplies the exact measure score T=k+b/2 (with its bulk-even / homology-odd decomposition treated separately). Then

d_Q <O_lambda(Q)>
 = measure-score contribution
 + < (d_Q P_lambda) O_bare >
 + < P_lambda d_Q O_bare >.

This is the finite-volume version of the field-definition counterterm that the collision literature requires.

Implement this identity on tiny FK tori where direct symbolic generic-Q differentiation is possible. It would calibrate which part of a logarithmic tangent comes from

probability measure,
representation projector,
explicit operator definition.

That decomposition is substantially more informative than calling Cov(O,T) the top field.

Phase D — search for confluent projector geometry

At Q=1, examine whether two generic-Q representation tensors that are distinct for Q!=1 become linearly dependent, zero-norm or otherwise degenerate.

If so, construct basis-independent confluent data such as

rank loss of the Gram matrix,
finite normalized residues,
derivative of the projector subspace,
principal angle / wedge between colliding tensor directions,
minimal polynomial of the induced transfer action on the confluent space.

This supplies a representation-theoretic analogue of #218's eigenvector coalescence, now at the level of explicit Potts symmetry tensors.

A particularly interesting possibility is that the energy/two-cluster logarithmic pair can be recognized as a collision not only of scaling dimensions but also of categorical projector data. If that works for the known Vasseur--Jacobsen--Saleur pair, apply the same construction after the common spin-4 differential.

Phase E — connect to charged OPE spectroscopy (#250)

The normalized loop three-point constant omega_123 removes ordinary multiplicative field normalization. Combine it with the explicit Potts tensor projector:

continuum number: omega_123(Q)
representation tensor: I_{lambda1 lambda2 lambda3}(Q).

Then distinguish two kinds of Q -> 1 structure:

conformal/OPE derivative,
S_Q invariant-tensor/projector derivative.

The proposed tangent

T_123 = d_Q log omega_123 |_(1)

should be supplemented by the derivative of the categorical invariant tensor whenever the external fields carry nontrivial Potts charge.

This is especially relevant to [2] fields: a scalar-normalized structure constant does not by itself specify which invariant tensor in a multidimensional S_Q tensor-product space is being measured.

Phase F — connect to #249 minimal realization

Add the representation projector/invariant-tensor label as an instrument descriptor in the context-Hankel manifest.

Then ask separately:

rank of the singlet endpoint realization,
rank after [2]-charged instruments are admitted,
rank after Q-derivative/confluent projector instruments are admitted.

A rank increase only after charged/confluent instruments is structurally different from a third ordinary bulk scalar field.

Bold conjecture

My current risky guess is:

the leading global square-site H4 endpoint sector is genuinely singlet and therefore does not linearly see V_(2,2)_[2]; the local four-arm geometry contains strong overlap with the same four-leg conformal family only after an explicit [2] cluster-label projector is inserted. The unlabelled marked-pivotal H4 mixes singlet descendants and charged four-leg information and is therefore not itself an RG eigenoperator.

This would explain three otherwise awkward facts at once:

x=17/4 exists but does not dominate global matching,
local H4 is strong but has complicated radial flow,
charged deck/leg information disappears from scalar one-point responses.

A more radical possibility is that the Q->1 derivative of the representation projector itself contributes to the logarithmic partner. The known energy/two-cluster collision is the right exact control for deciding whether that idea is real or merely formal.

Literature anchors

• Jacobsen--Ribault--Saleur, arXiv:2208.14298 — Potts state spaces, twisted torus partition functions, partition-algebra decomposition and S_Q representation labels.
• Nivesvivat, arXiv:2205.09349 — Potts four-point bootstrap with explicit S_Q invariant tensors, representation spectra and fusion constraints.
• Binder--Rychkov, arXiv:1911.07895 — categorical symmetry/Deligne-category interpretation of noninteger-rank symmetry and its preservation under RG.
• Ang et al., arXiv:2604.05503 — exact charged loop three-point constants and lattice leg-character construction.
• Vasseur--Jacobsen--Saleur, arXiv:1206.2312 — explicit Q->1 representation collision producing a percolation logarithmic observable.
• Nivesvivat--Ribault, arXiv:2007.04190 — logarithmic fields as confluent/derivative limits of generic conformal representations.

Relation to current work

• [P2 theory/analysis] Explain the missing x=17/4 spin-4 four-leg primary before attributing local H4 to thermal Q4 #257 supplies the [2], x=17/4, spin-4 candidate whose selection rule needs a concrete projector.
• [P2 exact/breakthrough] Critical-manifold Q-score tomography: observe Potts representation collisions before taking Q→1 #258 supplies the exact Q-measure score and the warning that explicit field derivatives are missing.
• [P2 analysis] Charged three-point structure-constant spectroscopy beyond scalar deck responses #250 supplies charged two-/three-point observables and normalization-independent structure constants.
• [P2 exact] Deck-character selection rule: unmarked torus observables have zero linear detail response #244/[Exact] Enforce the deck-character selection rule #248 supplies the analogous exact statement for deck charge: invariant one-point functions cannot reveal a nontrivial charged linear mode.
• [P2 analysis/theory] Context-Hankel realization of the Gaussian-cover category: infer minimal state before naming fields #249 supplies the basis-independent minimal-realization layer.

Scientific boundary

This is an exact/theory analysis program, not a new evidence count. Categorical dimensions at noninteger or special Q are not ordinary positive multiplicities. A negative or zero analytic dimension must not be interpreted as a literal negative/absent number of states without the full partition-algebra/categorical construction.

[LightChainr]

Exact-first Phase A is now on branch theory/p262-confluent-projector-tomography-20260829 at d006f9c (no PR).

The smallest useful carrier is the unordered pair of distinct FK-cluster colours, i.e. the two-cluster/four-leg colour space. In the diagram basis I (pair identity), X (shared-colour incidence), J (all ones),

P_0 = 2J/[Q(Q-1)],
P_[1] = (X-4J/Q)/(Q-2),
P_[2] = I-P_0-P_[1].

These are exact orthogonal idempotents for integer Q>=4, with categorical traces 1, Q-1, Q(Q-3)/2. Their Q->1 geometry is:

(Q-1)P_0   ->  2J,
(Q-1)P_[2] -> -2J,
P_[1]      -> -X+4J,
P_0+P_[2]  -> I+X-4J,
d_Q(P_0+P_[2])|_1 = X.

Thus singlet and [2] have an exact opposite-residue confluence, while [1] is coefficientwise regular but categorical-null. This is sharper than a generic “projector derivative”: d_Q P_0 and d_Q P_[2] individually have double poles and are not finite top fields.

The exact connectivity control also gives

det G_n = product_k (Q)_k^{S(n,k)},
det G_2=(Q)_1(Q)_2,
det G_4=(Q)_1(Q)_2^7(Q)_3^6(Q)_4,

and both raw Gram matrices have rank one at Q=1.

The LCFT boundary is explicit. The opposite projector residues supply only the tensor half of a VJS logarithm. If additionally x_0(1)=x_[2](1)=x_*, the finite log coefficient is conditionally

2 R [x_[2]'(1)-x_0'(1)] log(r) r^(-2x_*),  R=2J.

A nonzero dimension-slope difference is independent input. Hence the risky frozen prediction is: any minimal singlet/[2] logarithmic residue must lie in the J tensor direction with exact residue ratio -1; a different tensor direction falsifies this mechanism.

The note keeps three operations separate: #258 measure score, the finite confluent projector term here, and explicit d_Q O_bare. #252's 4:-6:3 Ward gate is a separate Virasoro primary-vs-descendant discriminator and is not folded into the Q derivative.

Artifacts: notes/p262-confluent-potts-projector-tomography.md, deterministic JSON, exact oracle, and 5 tests. The new tests plus 9 registry tests pass.

[LightChainr]

Exact next gate: color-blindness Ward identity and Potts × modular double grading

The Phase-A projector confluence is now sharp enough that the next step should be an exact matrix-element gate, not another spectrum or radial fit.

The new point is that two logically independent symmetries are available:

internal Potts / Deligne-category label lambda
x
torus modular-homology type rho.

The latest modular-channel oracle classifies rank, either, and cross as modular scalars, while primitive winding/direction/both channels are vector-valued or basis dependent. Phase A here independently classifies the two-cluster colour carrier into [], [1], and [2] projectors. These two classifications should be combined before interpreting any x=17/4 overlap.

1. A finite-volume colour-blindness Ward identity

Let u be the all-ones vector in the unordered distinct-colour-pair carrier used in Phase A. With the exact projectors already derived,

P_0    = 2J/[Q(Q-1)],
P_[1]  = (X-4J/Q)/(Q-2),
P_[2]  = I-P_0-P_[1],

the first symbolic checks should be

P_0 u   = u,
P_[1] u = 0,
P_[2] u = 0.

Now let B(Q) be a regular colour-blind diagram operator: it depends on FK connectivity, geometry, homology, pivotality, or wrapping, but not on a chosen cluster colour. At integer Q it commutes with the global S_Q action. Therefore the central representation blocks obey

P_lambda B P_mu = 0,   lambda != mu.

In particular,

P_[2] B P_0 = 0,
P_0 B P_[2] = 0,
P_[2] B u   = 0.

For a diagrammatically defined generic-Q family these are rational identities in Q, not only numerical zeros at one integer. After clearing denominators, their derivative at Q=1 is still zero.

This matters because the poles of P_0 and P_[2] must not be allowed to manufacture an ordinary charged matrix element from a colour-blind observable. The opposite residues found in Phase A are real, but a finite logarithmic field requires a singularly normalized pair of projected field families, or an explicitly Q-dependent/charged/defect insertion. It does not follow from differentiating a regular block zero.

2. Consequence for the two lattice observables

The distinction should be made explicit:

global matching cross/either
    -> colour-blind internally, hence singlet for regular generic Q;

raw unlabelled marked-pivotal / four-arm H4
    -> also colour-blind internally;
       a spatial mark or four-arm event does not by itself create [2] Potts charge;

explicit two-cluster colour projector
    -> [2]-covariant and suitable as the positive control.

Thus “four legs” and “[2] charge” must not be used synonymously. The raw pivotal observable may overlap a singlet copy with the same conformal weights, or a mixture of singlet descendants and geometry-dependent sectors, but it cannot isolate V_(2,2)_[2] without an explicit colour/projector/defect insertion.

This gives a stronger interpretation of the unresolved local flow: its complexity need not mean that an unlabelled pivotal row is a charged primary plus Q4. It may simply be an untyped singlet composite whose four-arm geometry resembles the charged field without carrying its internal representation.

3. Potts × modular double grading

On the finite untwisted colour-lift, global colour permutations act on cluster labels while SL(2,Z) period relabellings act on winding data. These operations commute. Every candidate row can therefore be typed as

(lambda_internal, rho_modular, insertion_order).

A minimal table should include

internal:
  [] singlet
  [2] charged four-leg
  singlet/[2] confluent residue

modular:
  scalar rank/either/cross
  primitive-winding vector
  direction/both basis-dependent vector
  twisted/seam sector

insertion order:
  one-point / linear
  neutral two-point
  three-point invariant tensor
  Q derivative / confluent
  Hessian or higher-point composite

This exposes two independent exact zeros:

1. internal zero — a regular colour-blind scalar one-point row has no [2] matrix element at any modulus;
2. modular elliptic zero — a homogeneous spin-4 response in a modular-scalar cross/either channel vanishes at the hexagonal elliptic point.

A primitive-winding/vector row can evade the second zero but not the first. Conversely, a neutral

[2] x [2] -> []

two-point function is internally allowed while remaining globally scalar. This is the clean positive-control channel for V_(2,2).

4. Exact implementation

I would add the following bounded gate to the current Phase-A branch.

A. Symbolic Ward tests

For the exact I,X,J carrier, verify

P_0 u=u,
P_[1]u=0,
P_[2]u=0.

Construct a basis for the colour-blind commutant operators B(Q) on this carrier and verify, after clearing denominators,

P_[2] B P_0 = 0,
P_0 B P_[2] = 0

symbolically. Differentiate the cleared rational identities at Q=1. This is the fail-closed check that prevents individual projector poles from being misread as a finite top field.

B. Integer-Q colour-lift regression

On tiny square-bond tori at Q=4,5,6, explicitly colour FK clusters and check:

global cross/either invariant under all cluster-colour permutations;
raw unlabelled pivotal invariant under all cluster-colour permutations;
one [2] insertion averages to zero;
[2] x [2] neutral two-point can be nonzero;
a deliberately colour-sensitive marked row can be nonzero.

These integer cases are regression tests only unless the symbolic commutant proof or a declared degree bound promotes them.

C. Q->1 confluent positive control

Build the singularly normalized singlet/[2] field pair for the known Vasseur--Jacobsen--Saleur collision. Verify that any finite tensor residue lies in the J direction and has the Phase-A residue ratio -1. Keep separate:

measure score from #258,
finite confluent projector combination,
explicit d_Q O_bare.

A residue in an X-only or another independent tensor direction would falsify the minimal singlet/[2] mechanism rather than being absorbed into a basis change.

D. Machine-readable double-grading registry

For every current H4-relevant observable, record

internal Potts label,
modular/homology type,
insertion order,
colour marks,
spatial marks,
deck/twist marks,
allowed one-/two-/three-point contractions,
exact zero theorem used, if any.

Feed the same descriptors into #249's context-Hankel manifest and #114's observable bridge. Rank increases after adding [2] rows, modular-vector rows, or confluent rows then have different meanings and should not be collapsed into one “extra state”.

5. Risky prediction

I would freeze the following before a charged lattice pilot:

The ordinary global matching H4 endpoint is [] × modular-scalar. The charged V_(2,2)_[2] field does not occur in its regular one-point coefficient for any generic Q. The raw unlabelled pivotal H4 is also internally singlet and will not serve as a positive [2] control. A nonzero x=17/4 charged signal first appears in an explicit [2] insertion, a neutral [2]×[2] correlator, a twisted/defect matrix element, or a singular confluent residue.

This leaves three sharply distinct ways for an x=17/4 contribution to remain relevant to global matching:

A. a separate singlet copy with the same conformal weights;
B. a defect/twisted/combinatorial-map matrix element rather than an ordinary endpoint one-point function;
C. a Q->1 singular residue with a typed logarithmic-extension normalization.

Each has different projector, modulus, and insertion-order predictions.

6. Decision table

exact colour-blind cross-block zero + nonzero [2]x[2] control
  -> remove the charged [2] copy from the ordinary global endpoint,
     but retain it as a real charged sector;

raw unlabelled pivotal shows a nonzero [2] cross block
  -> observable semantics secretly carry colour/cluster marks,
     or the implementation/projector is wrong;

only a singularly normalized Q->1 family survives
  -> classify it as a logarithmic extension/defect residue,
     not an ordinary primary overlap;

an x=17/4 singlet copy is found
  -> keep the radial adversary, but move the question to Potts multiplicity/OPE data,
     not leg counting;

no singlet copy and no allowed residue
  -> the global x=17/4 competitor is excluded by an exact representation-level gate,
     materially strengthening the thermal-Q4 branch.

Literature interface

• Jacobsen--Ribault--Saleur, arXiv:2208.14298 — Potts state spaces, partition-algebra and S_Q representation decomposition.
• Nivesvivat, arXiv:2205.09349 — Potts crossing with explicit S_Q invariant tensors.
• Jacobsen--Saleur, arXiv:1111.4033 — coloured spin/FK observables and partition-algebra representation sectors.
• Richard--Jacobsen, arXiv:math-ph/0605015 — toroidal Potts characters resolved by bridge number and symmetric-group representation.
• Binder--Rychkov, arXiv:1911.07895 — Deligne-category symmetry at noninteger rank.
• Choi--Rayhaun--Zheng, arXiv:2409.02159 — symmetry-resolved torus/annulus sectors via generalized tube algebras.
• Ang et al., arXiv:2604.05503 — charged leg fields and exact normalized three-point constants.
• Grans-Samuelsson et al., arXiv:2302.08168 — combinatorial-map labels retained by loop correlators.

The main methodological boundary is simple: derive the zero in the correctly typed double grading before asking which radial exponent best fits its violation.

[LightChainr]

Make H-CONNECTIVITY-RADICAL finite and falsifiable: compute the Q=1 confluent Gram-rank filtration

The new branch-only q/contact result gives this issue a much sharper exact target.

We now know that every topology-only observer f(q) with q in {-1,0,1} lies in the exact three-dimensional algebra

span{1,q,q^2},

and connected couplings of such f(q) to root-independent gate marks close to contact moments to all orders. At the same time, external Euler/connectivity observers escape this quotient.

Rather than saying qualitatively that the missing information may live in a partition-algebra radical, build the confluent connectivity Gram filtration at Q=1 and ask whether it reproduces this loss of information.

Generic-Q connectivity Gram matrix

Choose the smallest marked-connectivity space relevant to the live observers: start with 2 and 4 marked clusters / junctions in the partition-algebra diagram basis already proposed in this issue.

Let

G(Q)

be the generic-Q Gram/pairing matrix obtained by gluing the connectivity diagrams, with all normalization conventions fixed in the ordinary Potts/partition-algebra way.

At special integer Q, categorical dimensions and Gram determinants can vanish. Expand exactly

G(Q)
 = G0
 + (Q-1) G1
 + (Q-1)^2/2 G2
 + ... .

Do not stop at rank G0.

Define a basis-independent jet filtration, for example through the bilinear pairing on truncated Q-jets or equivalently the increasing span generated by

G0, G1, ..., Gm.

Record at which derivative order each previously null connectivity direction first acquires a nonzero pairing.

This is the finite algebraic analogue of a logarithmic/confluent field becoming visible only after Q derivatives.

Specific conjecture from the q-only no-go

Freeze a risky prediction before inspecting the detailed decomposition:

the q-only scalar map factors through a very small early layer of the Q=1 confluent Gram quotient, while the new Euler/junction/seam observers have components in radical directions that are invisible to this scalar quotient.

A concrete target is to construct the linear map

Phi_q : connectivity/junction space -> span{1,q,q^2}

on tiny tori / diagram states and compute

ker Phi_q,
image Phi_q,
rad G0,
radical layers revealed by G1,G2,... .

Then ask whether

ker Phi_q

contains exactly the directions that the branch contact theorem says no q-only observer can recover.

Do not assume equality; measure the intersections/dimensions explicitly.

Put current observers into the same finite algebra

Project, with exact coefficients where possible,

q,
q^2,
A_top,
E_top,
O_ext = C_black^NN - C_white^matching - q,
rank-birth junction/line marks,
colour seam / [2] insertion,
primitive homology line insertion,

into the chosen connectivity basis or its smallest extension.

For every vector report its location relative to

semisimple quotient,
rad G0,
first Q-derivative layer,
second derivative layer,
TL/partition standard vs radical/quotient modules.

This turns the current phrase "observer-dependent matrix element" into an exact finite statement about which algebra quotient the observer sees.

Why the Q-jet, not only the radical, matters

Binder--Rychkov's Deligne-category viewpoint makes analytic continuation in Q categorical rather than a literal noninteger-dimensional vector space. JRS arXiv:2208.14298 gives the partition-algebra / S_Q decomposition of Potts states. Nivesvivat--Ribault and the Q-tangent program show that derivatives at special Q are precisely where logarithmic fields/residues appear.

Therefore a direction can satisfy

pairing at Q=1 = 0

while

partial_Q pairing at Q=1 != 0.

That distinction is exactly what we need to separate

ordinary absent state,
zero-norm categorical state,
Q-derivative/logarithmic residue,
observer information discarded by the q quotient.

Connection to the new three-object skeleton

The PR #267 board currently sketches

rank-one lifetime fiber
 -> context action
 -> connectivity / defect radical.

A successful confluent Gram calculation would give the third line a concrete algebra:

connectivity state space C
 -> semisimple quotient C/rad
 -> radical filtration rad^k
 -> Q-jet pairings that resurrect successive layers.

Then #249/minimal-realization can use module/radical layer labels as typed state coordinates, rather than an arbitrary latent rank.

Sharp outcomes

• ker Phi_q aligns with low radical layers and external observers escape them: strong support for the common-radical interpretation;
• q-only closure is unrelated to the partition-algebra radical: H-CONNECTIVITY-RADICAL is a false analogy and should be dropped;
• threshold and H4 defect vectors land in the same radical/Jordan block: genuine unification;
• they land in different simple/radical components: keep them separate even if both look non-semisimple after scalar projection.

Minimal deliverable

No Monte Carlo is needed. A useful first artifact is just

basis of marked partitions/junctions,
symbolic G(Q),
det/minors,
ranks of Q=1 jet pairings,
explicit Phi_q,
coordinates of 2-4 existing exact observers,
module/radical decomposition where known.

Use arXiv:2208.14298 for the Potts partition-algebra state decomposition; arXiv:1911.07895 for the categorical/noninteger-Q interpretation; arXiv:2312.14837 / 2509.11756 for periodic-TL radical/quotient and fusion language when moving from the small partition algebra to cylinder operators.

[LightChainr]

Follow-on exact result from the low-leg Gram algebra: branch theory/p333-dual-number-gram-extension-20260830, commit b024177. The full Q=1 connectivity radical acquires a nondegenerate first-order pairing, with one epsilon pivot per k-block partition and leading coefficient (-1)^(k-2)(k-2)!. Therefore the opposite singlet/[2] projector residues found here sit inside a much larger universal first-order radical; Gram confluence alone cannot select the physical VJS pair. Selection must come from the algebra/transfer action plus the independently nonzero dimension-velocity collision. This narrows the next implementation to a simultaneous dual-number Gram + generator module rather than more scalar projector derivatives.

[LightChainr]

Repository cleanup (CLOSE-DUP): this execution entry is superseded or duplicates a retained research track. Its artifacts remain available by SHA. Reopen only with a prospective discriminator and an explicit stop rule.